<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE article PUBLIC "-//NLM//DTD Journal Publishing DTD v2.3 20070202//EN" "journalpublishing.dtd">
<article article-type="research-article" dtd-version="2.3" xml:lang="EN" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">
<front>
<journal-meta>
<journal-id journal-id-type="publisher-id">Front. Phys.</journal-id>
<journal-title>Frontiers in Physics</journal-title>
<abbrev-journal-title abbrev-type="pubmed">Front. Phys.</abbrev-journal-title>
<issn pub-type="epub">2296-424X</issn>
<publisher>
<publisher-name>Frontiers Media S.A.</publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id pub-id-type="publisher-id">833426</article-id>
<article-id pub-id-type="doi">10.3389/fphy.2022.833426</article-id>
<article-categories>
<subj-group subj-group-type="heading">
<subject>Physics</subject>
<subj-group>
<subject>Original Research</subject>
</subj-group>
</subj-group>
</article-categories>
<title-group>
<article-title>Diffusion and Synchronization Dynamics Reveal the Multi-Scale Patterns of Spatial Segregation</article-title>
<alt-title alt-title-type="left-running-head">Bassolas et al.</alt-title>
<alt-title alt-title-type="right-running-head">Dynamical Patterns of Multi-Scale Segregation</alt-title>
</title-group>
<contrib-group>
<contrib contrib-type="author" corresp="yes">
<name>
<surname>Bassolas</surname>
<given-names>Aleix</given-names>
</name>
<xref ref-type="corresp" rid="c001">&#x2a;</xref>
<uri xlink:href="https://loop.frontiersin.org/people/1599707/overview"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname>G&#xf3;mez</surname>
<given-names>Sergio</given-names>
</name>
<uri xlink:href="https://loop.frontiersin.org/people/97084/overview"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Arenas</surname>
<given-names>Alex</given-names>
</name>
<uri xlink:href="https://loop.frontiersin.org/people/206044/overview"/>
</contrib>
</contrib-group>
<aff>
<institution>Departament d&#x2019;Enginyeria Inform&#xe0;tica i Matem&#xe0;tiques</institution>, <institution>Universitat Rovira i Virgili</institution>, <addr-line>Tarragona</addr-line>, <country>Spain</country>
</aff>
<author-notes>
<fn fn-type="edited-by">
<p>
<bold>Edited by:</bold> <ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/72779/overview">Haroldo V. Ribeiro</ext-link>, State University of Maring&#xe1;, Brazil</p>
</fn>
<fn fn-type="edited-by">
<p>
<bold>Reviewed by:</bold> <ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/96326/overview">Zoltan Neda</ext-link>, Babe&#x219;-Bolyai University, Romania</p>
<p>
<ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/73038/overview">Matja&#x17e; Perc</ext-link>, University of Maribor, Slovenia</p>
</fn>
<corresp id="c001">&#x2a;Correspondence: Aleix Bassolas, <email>aleix.bassolas@gmail.com</email>
</corresp>
<fn fn-type="other">
<p>This article was submitted to Social Physics, a section of the journal Frontiers in Physics</p>
</fn>
</author-notes>
<pub-date pub-type="epub">
<day>25</day>
<month>04</month>
<year>2022</year>
</pub-date>
<pub-date pub-type="collection">
<year>2022</year>
</pub-date>
<volume>10</volume>
<elocation-id>833426</elocation-id>
<history>
<date date-type="received">
<day>11</day>
<month>12</month>
<year>2021</year>
</date>
<date date-type="accepted">
<day>04</day>
<month>04</month>
<year>2022</year>
</date>
</history>
<permissions>
<copyright-statement>Copyright &#xa9; 2022 Bassolas, G&#xf3;mez and Arenas.</copyright-statement>
<copyright-year>2022</copyright-year>
<copyright-holder>Bassolas, G&#xf3;mez and Arenas</copyright-holder>
<license xlink:href="http://creativecommons.org/licenses/by/4.0/">
<p>This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.</p>
</license>
</permissions>
<abstract>
<p>Urban systems are characterized by populations with heterogeneous characteristics, and whose spatial distribution is crucial to understand inequalities in life expectancy or education level. Traditional studies on spatial segregation indicators focus often on first-neighbour correlations but fail to capture complex multi-scale patterns. In this work, we aim at characterizing the spatial distribution heterogeneity of socioeconomic features through diffusion and synchronization dynamics. In particular, we use the time needed to reach the synchronization as a proxy for the spatial heterogeneity of a socioeconomic feature, as for example, the income. Our analysis for 16 income categories in cities from the United States reveals that the spatial distribution of the most deprived and affluent citizens leads to higher diffusion and synchronization times. By measuring the time needed for a neighborhood to reach the global phase we are able to detect those that suffer from a steeper segregation. Overall, the present manuscript exemplifies how diffusion and synchronization dynamics can be used to assess the heterogeneity in the presence of node information.</p>
</abstract>
<kwd-group>
<kwd>urban segregation</kwd>
<kwd>spatial heterogeneity</kwd>
<kwd>synchronization</kwd>
<kwd>phase-oscillators</kwd>
<kwd>diffusion</kwd>
</kwd-group>
</article-meta>
</front>
<body>
<sec id="s1">
<title>1 Introduction</title>
<p>The expansion of urbanization and progressive increase of the population in cities has intensified the concern over the many dimensions of segregation&#x2014;i.e., school, economic or ethnics&#x2014;that have a tangible impact in the health, education and equal opportunities of citizens [<xref ref-type="bibr" rid="B1">1</xref>&#x2013;<xref ref-type="bibr" rid="B8">8</xref>]. In fact, quantifying the extent of segregation and the identification of economically and socially isolated neighborhoods has been a topic of wide interest that first led to the development of global metrics, and which were later extended to spatial metrics [<xref ref-type="bibr" rid="B9">9</xref>&#x2013;<xref ref-type="bibr" rid="B14">14</xref>]. Most of the initial spatial measures were limited to first neighbour indices, which facilitated the development of multi-scalar indices that provide a more nuanced picture of segregation [<xref ref-type="bibr" rid="B15">15</xref>&#x2013;<xref ref-type="bibr" rid="B21">21</xref>], yet understanding the role played by each of the scales and their interplay still remains a challenge.</p>
<p>Dynamical processes in general, and in particular diffusion [<xref ref-type="bibr" rid="B22">22</xref>&#x2013;<xref ref-type="bibr" rid="B30">30</xref>] and synchronization [<xref ref-type="bibr" rid="B31">31</xref>&#x2013;<xref ref-type="bibr" rid="B35">35</xref>] dynamics, have been widely studied in complex networks on account of their relation with the spread of diseases and information [<xref ref-type="bibr" rid="B36">36</xref>, <xref ref-type="bibr" rid="B37">37</xref>] and real-world phenomena in social or economic systems [<xref ref-type="bibr" rid="B38">38</xref>&#x2013;<xref ref-type="bibr" rid="B40">40</xref>]. Interestingly, they provide insights on the topological scales and structure of networks and reveal the existence of functional meso-scale structures [<xref ref-type="bibr" rid="B27">27</xref>, <xref ref-type="bibr" rid="B30">30</xref>, <xref ref-type="bibr" rid="B32">32</xref>, <xref ref-type="bibr" rid="B34">34</xref>, <xref ref-type="bibr" rid="B41">41</xref>].</p>
<p>Here we use previous knowledge on diffusion and synchronization dynamics to assess the multi-scale patterns of residential segregation. By moving the focus from the network topology and organization to the node states, we are able to measure how well distributed a population with a certain characteristic is using the time needed to reach the absorbing state. Our framework requires thus the implementation of a population dynamic to drive the system towards the homogeneous state, in our case diffusion and synchronization dynamics. None of them constitute here an attempt to model or predict the changes in the spatial distribution of a population characteristic but are highly stylized simplifications of their evolution that allow us to measure the time needed to attain the homogeneous state, which we consider to be the non-segregated scenario. Dynamical approaches are thus introduced here not because they provide a realistic approximation to the evolution of population dynamics but because they offer a significant advantage to measure multi-scale correlations as they do not require to take distance explicitly into account. Moreover, the assumption that cities converge towards uniformity is rather unrealistic without a heavy external driver, and is only a means to construct our measures.</p>
<p>As case studies we provide an analysis on the distribution of citizens of a certain income category in cities from the United States, and the distribution of a set of socioeconomic indicators in the city of Paris throughout an average day (see <xref ref-type="sec" rid="s9">Supplementary Material Section 2</xref> and <xref ref-type="sec" rid="s9">Supplementary Figures S8&#x2013;S10</xref>). The analysis on the spatial organization of income categories reveals that the most deprived and affluent sectors display higher diffusion and synchronization times linked to a higher heterogeneity, and allow us to split the cities in two groups depending on the difference on the level of segregation. Finally, we evaluate the level of synchronization at the neighborhood level which allow us to spot the more sensitive places in a city.</p>
</sec>
<sec id="s2">
<title>2 Results</title>
<sec id="s2-1">
<title>2.1 Diffusion Dynamics and Income Segregation</title>
<p>Citizens exhibit a huge diversity of characteristics usually captured by socioeconomic indicators such as education level, income or ethnicity, and they are often heterogeneously distributed in space: those individuals with similar characteristics tend to live close between them. To assess the heterogeneity of a population with a characteristic <italic>k</italic> &#x2208; <italic>K</italic>, we consider a graph <italic>G</italic>(<italic>V</italic>, <italic>E</italic>) with adjacency matrix <italic>A</italic> &#x3d; {<italic>a</italic>
<sub>
<italic>ij</italic>
</sub>} in which the spatial units are represented as a set of nodes <italic>V</italic> connected by a set of edges <italic>E</italic>. The adjacency matrix <italic>A</italic> we have considered takes <italic>a</italic>
<sub>
<italic>ij</italic>
</sub> &#x3d; 1 when spatial units <italic>i</italic> and <italic>j</italic> are adjacent and <italic>a</italic>
<sub>
<italic>ij</italic>
</sub> &#x3d; 0 otherwise, which is the traditional connectivity matrix used to capture residential segregation. Still, other types of (weighted) matrices could be considered to assess, for example, the impact of mobility in segregation. The state of a node <inline-formula id="inf1">
<mml:math id="m1">
<mml:msubsup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
</mml:math>
</inline-formula> is given by the fraction of citizens living in node <italic>i</italic> that belong to socioeconomic category (or class) <italic>k</italic>, written as<disp-formula id="e1">
<mml:math id="m2">
<mml:msubsup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msubsup>
<mml:mrow>
<mml:mi>n</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
</mml:mrow>
<mml:mrow>
<mml:munder>
<mml:mrow>
<mml:mo>&#x2211;</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mo>&#x2032;</mml:mo>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:munder>
<mml:msubsup>
<mml:mrow>
<mml:mi>n</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mo>&#x2032;</mml:mo>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:msubsup>
</mml:mrow>
</mml:mfrac>
<mml:mo>,</mml:mo>
</mml:math>
<label>(1)</label>
</disp-formula>where <inline-formula id="inf2">
<mml:math id="m3">
<mml:msubsup>
<mml:mrow>
<mml:mi>n</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
</mml:math>
</inline-formula> is the total number of citizens in unit <italic>i</italic> that belong to category <italic>k</italic>. As extreme cases, <inline-formula id="inf3">
<mml:math id="m4">
<mml:msubsup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0</mml:mn>
</mml:math>
</inline-formula> when there are no citizens of category <italic>k</italic> living in <italic>i</italic>, and <inline-formula id="inf4">
<mml:math id="m5">
<mml:msubsup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:math>
</inline-formula> when all the citizens in node <italic>i</italic> belong to category <italic>k</italic>. Of course, the normalization condition<disp-formula id="e2">
<mml:math id="m6">
<mml:munder>
<mml:mrow>
<mml:mo>&#x2211;</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
<mml:mo>&#x2208;</mml:mo>
<mml:mi>K</mml:mi>
</mml:mrow>
</mml:munder>
<mml:msubsup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:math>
<label>(2)</label>
</disp-formula>is fulfilled for all nodes <italic>i</italic>.</p>
<p>To measure the multi-scalar patterns of segregation, our assumption is that cities suffering from stronger residential segregation are further from the stationary state where the citizens of category <italic>k</italic> are homogeneously distributed in space. Although cities are in continuous change and most likely far from equilibrium, similar approaches such as the long-standing Schelling and the Alonso-Muth-Mills models have been able to draw relevant conclusions from the equilibrium state [<xref ref-type="bibr" rid="B42">42</xref>, <xref ref-type="bibr" rid="B43">43</xref>].</p>
<p>By adopting diffusion dynamics we do not refuse the high complexity of population dynamics influenced by a wide variety of demographic, economic, political, and behavioral factors [<xref ref-type="bibr" rid="B44">44</xref>&#x2013;<xref ref-type="bibr" rid="B47">47</xref>] but avoid introducing further parameters and factors that could hinder our aim of characterizing the segregation of a particular population category. Bear in mind that our final goal is by no means to assess real-world migration processes but to construct a multi-scalar measure of segregation that does not explicitly include the distance and the use of more complex and realistic approaches that would complicate the interpretation of the results. Diffusion constitutes one of the most basic approximations to how information, or any other characteristic, is transmitted through a system. Although far from the real behavior, it provides one of the simplest scenarios where the flow of population follows a gradient.</p>
<p>In fact, we focus on one of the best-case scenarios where the values of <inline-formula id="inf5">
<mml:math id="m7">
<mml:msubsup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
</mml:math>
</inline-formula> converge towards equilibrium following a gradient, which could be interpreted as the change of residence of citizens of category <italic>k</italic> to regions where they are less abundant.</p>
<p>We focus on the economic segregation in the metropolitan areas of the United States with more than 1 million inhabitants and analyze a dataset containing the number of households within an income interval <italic>k</italic> residing in each census tract (see <xref ref-type="table" rid="T1">Table 1</xref>).</p>
<table-wrap id="T1" position="float">
<label>TABLE 1</label>
<caption>
<p>Income range (in US dollars) corresponding to each category (or class).</p>
</caption>
<table>
<tbody valign="top">
<tr>
<td align="left">
<bold>Class</bold>
</td>
<td align="center">
<bold>Income ($)</bold>
</td>
</tr>
<tr>
<td align="left">1</td>
<td align="center">Less than 10,000</td>
</tr>
<tr>
<td align="left">2</td>
<td align="center">10,000&#x2013;14,999</td>
</tr>
<tr>
<td align="left">3</td>
<td align="center">15,000&#x2013;19,999</td>
</tr>
<tr>
<td align="left">4</td>
<td align="center">20,000&#x2013;24,999</td>
</tr>
<tr>
<td align="left">5</td>
<td align="center">25,000&#x2013;29,999</td>
</tr>
<tr>
<td align="left">6</td>
<td align="center">30,000&#x2013;34,999</td>
</tr>
<tr>
<td align="left">7</td>
<td align="center">35,000&#x2013;39,999</td>
</tr>
<tr>
<td align="left">8</td>
<td align="center">40,000&#x2013;44,999</td>
</tr>
<tr>
<td align="left">9</td>
<td align="center">45,000&#x2013;49,999</td>
</tr>
<tr>
<td align="left">10</td>
<td align="center">50,000&#x2013;59,999</td>
</tr>
<tr>
<td align="left">11</td>
<td align="center">60,000&#x2013;74,999</td>
</tr>
<tr>
<td align="left">12</td>
<td align="center">75,000&#x2013;99,999</td>
</tr>
<tr>
<td align="left">13</td>
<td align="center">100,000&#x2013;124,999</td>
</tr>
<tr>
<td align="left">14</td>
<td align="center">125,000&#x2013;149,999</td>
</tr>
<tr>
<td align="left">15</td>
<td align="center">150,000&#x2013;199,999</td>
</tr>
<tr>
<td align="left">16</td>
<td align="center">200,000 or more</td>
</tr>
</tbody>
</table>
</table-wrap>
<p>Once we have the set of initial node states <inline-formula id="inf6">
<mml:math id="m8">
<mml:msubsup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
</mml:math>
</inline-formula>, their evolution through time is determined by the diffusion dynamics<disp-formula id="e3">
<mml:math id="m9">
<mml:mfrac>
<mml:mrow>
<mml:mi>d</mml:mi>
<mml:msubsup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
</mml:mrow>
<mml:mrow>
<mml:mi>d</mml:mi>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:mfrac>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>s</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
<mml:munderover accentunder="false" accent="false">
<mml:mrow>
<mml:mo>&#x2211;</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
</mml:munderover>
<mml:msub>
<mml:mrow>
<mml:mi>a</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mfenced open="(" close=")">
<mml:mrow>
<mml:msubsup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>j</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:mo>&#x2212;</mml:mo>
<mml:msubsup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
</mml:mrow>
</mml:mfenced>
<mml:mo>,</mml:mo>
</mml:math>
<label>(3)</label>
</disp-formula>where<disp-formula id="e4">
<mml:math id="m10">
<mml:msub>
<mml:mrow>
<mml:mi>s</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:munderover accentunder="false" accent="false">
<mml:mrow>
<mml:mo>&#x2211;</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
</mml:munderover>
<mml:msub>
<mml:mrow>
<mml:mi>a</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
</mml:math>
<label>(4)</label>
</disp-formula>is the degree of node <italic>i</italic>. For simplification purposes, we have opted to use a normalized diffusion dynamic, with diffusion strength equal to 1. Note that we have independent diffusion processes for each category <italic>k</italic>.</p>
<p>The diffusion dynamic lasts until the stationary state, <inline-formula id="inf7">
<mml:math id="m11">
<mml:msubsup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:mo>&#x3d;</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msup>
</mml:math>
</inline-formula>, <italic>&#x2200;i</italic> is reached, and we denote the spanned time as <italic>&#x3c4;</italic>
<sub>diff</sub>(<italic>k</italic>). Since the time to reach the stationary state can be infinitely large, we have considered that it is reached when the variance of <inline-formula id="inf8">
<mml:math id="m12">
<mml:msubsup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
</mml:math>
</inline-formula>, in time, becomes lower than 0.0001. We hypothesize that lower values of <italic>&#x3c4;</italic>
<sub>diff</sub>(<italic>k</italic>) are related to a more homogeneous distribution of the population within a category <italic>k</italic>, and the other way around when it is higher. In the extreme case in which all units have the same initial value of <inline-formula id="inf9">
<mml:math id="m13">
<mml:msubsup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
</mml:math>
</inline-formula>, the diffusion time <italic>&#x3c4;</italic>
<sub>diff</sub>(<italic>k</italic>) would attain its minimum value. As we aim to compare cities with different characteristics, we control for confounding factors such as the particular distribution of <italic>x</italic>
<sup>
<italic>k</italic>
</sup> or the topology of the graph by running the same diffusion dynamics on the same graph but where the values of <italic>x</italic>
<sup>
<italic>k</italic>
</sup> have been reshuffled, thus defining the average null-model diffusion time <inline-formula id="inf10">
<mml:math id="m14">
<mml:msubsup>
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mtext>diff</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mtext>null</mml:mtext>
</mml:mrow>
</mml:msubsup>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> calculated over 500 reshuffling realizations. The relative diffusion time we will use throughout this manuscript can then be written as<disp-formula id="e5">
<mml:math id="m15">
<mml:msub>
<mml:mrow>
<mml:mover accent="true">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mo>&#x303;</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mtext>diff</mml:mtext>
</mml:mrow>
</mml:msub>
<mml:mfenced open="(" close=")">
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:mfenced>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mtext>diff</mml:mtext>
</mml:mrow>
</mml:msub>
<mml:mfenced open="(" close=")">
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mrow>
<mml:msubsup>
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mtext>diff</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mtext>null</mml:mtext>
</mml:mrow>
</mml:msubsup>
<mml:mfenced open="(" close=")">
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mfrac>
<mml:mo>.</mml:mo>
</mml:math>
<label>(5)</label>
</disp-formula>A relative diffusion time equal to one means that it is compatible with the null model, i.e., there are no remarkable spatial dependencies, while a greater value suggests that spatial heterogeneities delay the arrival to the stationary state.</p>
<p>We analyze the normalized diffusion times <inline-formula id="inf11">
<mml:math id="m16">
<mml:msub>
<mml:mrow>
<mml:mover accent="true">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mo>&#x303;</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mtext>diff</mml:mtext>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> by running simulations for all US cities above 1 million of inhabitants and each of the 16 income categories <italic>k</italic> as a proxy for how heterogeneously distributed is the population; we have excluded New York City, whose adjacency network does not provide an accurate picture of residential segregation due to the particular geography of Manhattan. In <xref ref-type="fig" rid="F1">Figure 1A</xref> we display <inline-formula id="inf12">
<mml:math id="m17">
<mml:msub>
<mml:mrow>
<mml:mover accent="true">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mo>&#x303;</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mtext>diff</mml:mtext>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> in Boston, Cleveland, Detroit and Denver observing a common qualitative behavior: smaller values for middle-income categories, and higher ones for the categories in the extremes of the income distribution. Our results suggest that the wealthier and most deprived citizens suffer from stronger segregation and display a more clustered spatial distribution. More interestingly, category 9 seems to be the more homogeneously distributed across space, in agreement with the results observed in [<xref ref-type="bibr" rid="B20">20</xref>] and with the mean and standard deviation of <italic>x</italic>
<sup>
<italic>k</italic>
</sup> as well as the Moran&#x2019;s I (see <xref ref-type="sec" rid="s9">Supplementary Material Section 1</xref> and <xref ref-type="sec" rid="s9">Supplementary Figure S1</xref>). Still, there are strong quantitative differences, with Cleveland and Detroit displaying higher values for most of the categories, in contrast to Boston and Denver.</p>
<fig id="F1" position="float">
<label>FIGURE 1</label>
<caption>
<p>Diffusion dynamics as a measure for income segregation. <bold>(A)</bold> Synchronization time for each of the 16 income categories in Boston, Cleveland, Denver and Detroit. <bold>(B)</bold> Median value of <inline-formula id="inf13">
<mml:math id="m18">
<mml:msub>
<mml:mrow>
<mml:mover accent="true">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mo>&#x303;</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mtext>diff</mml:mtext>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> across income categories as a function of its variance. <bold>(C)</bold> Ranking for the median value of <inline-formula id="inf14">
<mml:math id="m19">
<mml:msub>
<mml:mrow>
<mml:mover accent="true">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mo>&#x303;</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mtext>diff</mml:mtext>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> for the studied set of US cities.</p>
</caption>
<graphic xlink:href="fphy-10-833426-g001.tif"/>
</fig>
<p>Since <inline-formula id="inf15">
<mml:math id="m20">
<mml:msub>
<mml:mrow>
<mml:mover accent="true">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mo>&#x303;</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mtext>diff</mml:mtext>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> takes a set of 16 values for each city, we calculate their median and variance values over all categories to ease the comparison between the set of cities studied. While the median value provides information on the segregation across all economic categories, its variance reports the variability among them. <xref ref-type="fig" rid="F1">Figure 1B</xref> shows this median value of <inline-formula id="inf16">
<mml:math id="m21">
<mml:msub>
<mml:mrow>
<mml:mover accent="true">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mo>&#x303;</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mtext>diff</mml:mtext>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf17">
<mml:math id="m22">
<mml:mtext>med</mml:mtext>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mover accent="true">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mo>&#x303;</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mtext>diff</mml:mtext>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> as a function of its variance, <inline-formula id="inf18">
<mml:math id="m23">
<mml:mtext>var</mml:mtext>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mover accent="true">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mo>&#x303;</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mtext>diff</mml:mtext>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula>. The prior cities appear ordered as Detroit, Cleveland, Boston and Denver, although the variance is very similar for Cleveland and Boston, likely due to the high values observed for low-income categories in Boston. Finally, we provide in <xref ref-type="fig" rid="F1">Figure 1C</xref> the ranking of the selected US cities according to <inline-formula id="inf19">
<mml:math id="m24">
<mml:mtext>med</mml:mtext>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mover accent="true">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mo>&#x303;</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mtext>diff</mml:mtext>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula>, as a measure of the overall segregation in cities. On top of it, we find cities such as Milwaukee or Detroit, which have been reported to suffer from economic and ethnic segregation [<xref ref-type="bibr" rid="B49">49</xref>&#x2013;<xref ref-type="bibr" rid="B51">51</xref>].</p>
<p>By applying diffusion dynamics we implicitly assume that <italic>x</italic>
<sup>
<italic>k</italic>
</sup> evolves homogeneously towards consensus, which more than a realistic scenario, it is a means to calculate the time needed to reach consensus and obtain a measure of segregation. To further inspect the actual change of <inline-formula id="inf20">
<mml:math id="m25">
<mml:msubsup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
</mml:math>
</inline-formula> between 2011 and 2019 in each of the spatial units <italic>i</italic>, we first construct the normalized time-series for each spatial unit across those years as<disp-formula id="e6">
<mml:math id="m26">
<mml:mi>P</mml:mi>
<mml:mfenced open="(" close=")">
<mml:mrow>
<mml:msubsup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:mo>,</mml:mo>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:mfenced>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msubsup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:mfenced open="(" close=")">
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mrow>
<mml:munder>
<mml:mrow>
<mml:mo>&#x2211;</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mo>&#x2032;</mml:mo>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:munder>
<mml:msubsup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:mfenced open="(" close=")">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mo>&#x2032;</mml:mo>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mfrac>
<mml:mo>,</mml:mo>
</mml:math>
<label>(6)</label>
</disp-formula>and then cluster, for each category <italic>k</italic>, the temporal profiles of all the nodes. For the clustering, we have made use of the k-means algorithm [<xref ref-type="bibr" rid="B52">52</xref>, <xref ref-type="bibr" rid="B53">53</xref>], grouping together those units with a similar temporal evolution, and setting the number of clusters to 3. The resulting time-series of the corresponding centroids for the highest and lowest income categories are depicted in <xref ref-type="fig" rid="F2">Figure 2</xref>, where a non-monotonic behavior is observed in most of the cases, with oscillatory behaviors through time of varying amplitude.</p>
<fig id="F2" position="float">
<label>FIGURE 2</label>
<caption>
<p>Average temporal evolution of the abundance of households within the lowest and highest income. Temporal evolution of centroids after performing a k-means clustering on the normalized abundance of households with category <italic>k</italic>, <inline-formula id="inf21">
<mml:math id="m27">
<mml:mi>P</mml:mi>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:msubsup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:mo>,</mml:mo>
<mml:mi>t</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula>, as a function of time <italic>t</italic> for the lower <bold>(A)</bold> and higher <bold>(B)</bold> income categories.</p>
</caption>
<graphic xlink:href="fphy-10-833426-g002.tif"/>
</fig>
</sec>
<sec id="s2-2">
<title>2.2 Synchronization Dynamics and Income Segregation</title>
<p>According to the oscillations in the temporal evolution of <inline-formula id="inf22">
<mml:math id="m28">
<mml:msubsup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
</mml:math>
</inline-formula> (<xref ref-type="fig" rid="F2">Figure 2</xref>), diffusion dynamics appear to be a rather simplistic approach to assess the time needed to converge. Even thought we do not aim to mimic the real evolution of <inline-formula id="inf23">
<mml:math id="m29">
<mml:msubsup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
</mml:math>
</inline-formula>, we seek for a dynamic that at least can resemble its real behaviour in a qualitative way. Thus, despite still constituting a stylized approximation, a dynamical process with an oscillatory behavior, like a system of coupled Kuramoto oscillators, appears to be a better way to assess the spatial heterogeneity of socioeconomic indicators across cities. To analyze segregation in terms of synchronization dynamics, we treat each of the spatial units <italic>i</italic> as an individual Kuramoto oscillator, with an initial phase <inline-formula id="inf24">
<mml:math id="m30">
<mml:msubsup>
<mml:mrow>
<mml:mi>&#x3b8;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mn>0</mml:mn>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> that is set by distributing the fraction of population in node <italic>i</italic> that belongs to a category <italic>k</italic> within the range [0, <italic>&#x3c0;</italic>] as<disp-formula id="e7">
<mml:math id="m31">
<mml:msubsup>
<mml:mrow>
<mml:mi>&#x3b8;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:mfenced open="(" close=")">
<mml:mrow>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:mfenced>
<mml:mo>&#x3d;</mml:mo>
<mml:msubsup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:mi>&#x3c0;</mml:mi>
<mml:mo>.</mml:mo>
</mml:math>
<label>(7)</label>
</disp-formula>The interaction between spatial units is given by the Kuramoto model<disp-formula id="e8">
<mml:math id="m32">
<mml:msubsup>
<mml:mrow>
<mml:mover accent="true">
<mml:mrow>
<mml:mi>&#x3b8;</mml:mi>
</mml:mrow>
<mml:mo>&#x307;</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:mfenced open="(" close=")">
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:mfenced>
<mml:mo>&#x3d;</mml:mo>
<mml:msubsup>
<mml:mrow>
<mml:mi>&#x3c9;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:mo>&#x2b;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>s</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
<mml:munderover accentunder="false" accent="false">
<mml:mrow>
<mml:mo>&#x2211;</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
</mml:munderover>
<mml:msub>
<mml:mrow>
<mml:mi>a</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x2061;</mml:mo>
<mml:mi>sin</mml:mi>
<mml:mfenced open="(" close=")">
<mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:msubsup>
<mml:mrow>
<mml:mi>&#x3b8;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>j</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:mfenced open="(" close=")">
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:mfenced>
<mml:mo>&#x2212;</mml:mo>
<mml:msubsup>
<mml:mrow>
<mml:mi>&#x3b8;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:mfenced open="(" close=")">
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:mfenced>
<mml:mo>,</mml:mo>
</mml:math>
<label>(8)</label>
</disp-formula>where we have modified the traditional interaction term between oscillators by dividing the angle difference by two, allowing for the interaction between regions displaying extreme values of <inline-formula id="inf25">
<mml:math id="m33">
<mml:msubsup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
</mml:math>
</inline-formula>. Additionally, to facilitate the global synchronization of the system, we set all the individual natural frequencies of the oscillators to the same value, i.e., <italic>&#x3c9;</italic>
<sub>
<italic>i</italic>
</sub> &#x3d; 1, <italic>&#x2200;i</italic>. In order to account separately the segregation of each category <italic>k</italic>, our approach assumes that there is no interaction between categories and, thus, <inline-formula id="inf26">
<mml:math id="m34">
<mml:msubsup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
</mml:math>
</inline-formula> synchronize independently of <italic>k</italic>.</p>
<p>We use the standard order parameter &#x7c;<italic>z</italic>
<sup>
<italic>k</italic>
</sup>&#x7c; to assess the global level of synchronization for a category <italic>k</italic> in a city, where<disp-formula id="e9">
<mml:math id="m35">
<mml:msup>
<mml:mrow>
<mml:mi>z</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msup>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
</mml:mfrac>
<mml:munderover accentunder="false" accent="false">
<mml:mrow>
<mml:mo>&#x2211;</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
</mml:munderover>
<mml:msup>
<mml:mrow>
<mml:mi>e</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:msubsup>
<mml:mrow>
<mml:mi>&#x3b8;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>j</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
</mml:mrow>
</mml:msup>
<mml:mo>,</mml:mo>
</mml:math>
<label>(9)</label>
</disp-formula>and <italic>N</italic> is the total number of spatial units or Kuramoto oscillators [<xref ref-type="bibr" rid="B35">35</xref>]. We consider that a city has reached the synchronized state when &#x7c;<italic>z</italic>
<sup>
<italic>k</italic>
</sup>&#x7c; &#x3e; 0.999. As in the case of diffusion, we assess how the distribution of initial phases determines the synchronization of the system, a city in our case, by measuring the time <italic>&#x3c4;</italic>
<sub>sync</sub>(<italic>k</italic>) required to reach the synchronized state. The more heterogeneously distributed the initial phases are, the higher the time the system requires to synchronize. To distinguish between the effect produced by the spatial distribution <inline-formula id="inf27">
<mml:math id="m36">
<mml:msubsup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
</mml:math>
</inline-formula> from its overall distribution as well as the topology of the graph, we also measure the average time the system needs to synchronize when the same phases are redistributed at random, <inline-formula id="inf28">
<mml:math id="m37">
<mml:msubsup>
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mtext>sync</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mtext>null</mml:mtext>
</mml:mrow>
</mml:msubsup>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula>. The normalized synchronization time of the system is then given by the ratio<disp-formula id="e10">
<mml:math id="m38">
<mml:msub>
<mml:mrow>
<mml:mover accent="true">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mo>&#x303;</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mtext>sync</mml:mtext>
</mml:mrow>
</mml:msub>
<mml:mfenced open="(" close=")">
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:mfenced>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mtext>sync</mml:mtext>
</mml:mrow>
</mml:msub>
<mml:mfenced open="(" close=")">
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mrow>
<mml:msubsup>
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mtext>sync</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mtext>null</mml:mtext>
</mml:mrow>
</mml:msubsup>
<mml:mfenced open="(" close=")">
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mfrac>
<mml:mo>.</mml:mo>
</mml:math>
<label>(10)</label>
</disp-formula>Like for diffusion, a synchronization time close to one means that the spatial distribution of phases is compatible with the null model, and a larger value indicates that spatial heterogeneities delay the appearance of a synchronized state.</p>
<p>In <xref ref-type="fig" rid="F3">Figure 3A</xref> we inspect the normalized synchronization time in Boston, Cleveland, Detroit and Denver when spatial units interact through Kuramoto-like dynamics. All four of them share similar features, with central classes displaying smaller synchronization times compared to the most disadvantaged and wealthier ones. An expected result since those individuals in the extremes of the income distribution tend to be more isolated and clustered together compared to middle-income citizens. Despite sharing qualitative features, the cities shown display sharp quantitative differences. Almost all categories appear to be significantly more isolated in Detroit and Cleveland compared to Denver and Boston, where <inline-formula id="inf29">
<mml:math id="m39">
<mml:msub>
<mml:mrow>
<mml:mover accent="true">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mo>&#x303;</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mtext>sync</mml:mtext>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> looks much flat. Overall, the synchronization results are compatible with the diffusion ones, likely because both dynamical processes share common features. We have further checked that the mean <italic>x</italic>
<sup>
<italic>k</italic>
</sup> does not determine directly the normalized synchronization times in <xref ref-type="sec" rid="s9">Supplementary Figure S1</xref>.</p>
<fig id="F3" position="float">
<label>FIGURE 3</label>
<caption>
<p>Synchronization time as a measure for income segregation. <bold>(A)</bold> Synchronization time for each of the 16 income categories in Boston, Cleveland, Denver and Detroit. <bold>(B)</bold> Ranking for the median value of <inline-formula id="inf30">
<mml:math id="m40">
<mml:msub>
<mml:mrow>
<mml:mover accent="true">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mo>&#x303;</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mtext>sync</mml:mtext>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> for the studied set of US cities. <bold>(C)</bold> Average value of <inline-formula id="inf31">
<mml:math id="m41">
<mml:mi>P</mml:mi>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mover accent="true">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mo>&#x303;</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mtext>sync</mml:mtext>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> as a function of each income category <italic>i</italic> for the two main clusters detected. <bold>(D)</bold> Location and cluster assignment for each of the analyzed cities.</p>
</caption>
<graphic xlink:href="fphy-10-833426-g003.tif"/>
</fig>
<p>Likewise with diffusion, we calculate the median and variance of <inline-formula id="inf32">
<mml:math id="m42">
<mml:msub>
<mml:mrow>
<mml:mover accent="true">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mo>&#x303;</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mtext>sync</mml:mtext>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> over all categories to be able to compare between analyzed cities (see <xref ref-type="sec" rid="s9">Supplementary Figure S2</xref> for the individual rankings of <inline-formula id="inf33">
<mml:math id="m43">
<mml:msub>
<mml:mrow>
<mml:mover accent="true">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mo>&#x303;</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mtext>sync</mml:mtext>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> for the categories <italic>k</italic> &#x3d; 1 and <italic>k</italic> &#x3d; 16). The ranking is shown in <xref ref-type="fig" rid="F3">Figure 3B</xref> and has cities such as Detroit, Cleveland, Milwaukee or Memphis close to the top, which are well-known for being among the most economically segregated cities in the United States. The location in the ranking of the cities in <xref ref-type="fig" rid="F3">Figure 3A</xref> is consistent with our observations, with Boston and Denver on the bottom of the ranking and Detroit and Cleveland on the top of it.</p>
<p>Our index is given by the median value of the normalized synchronization times, yet depending on the dimension of segregation we aim to capture, we can also construct an index based on a population-weighted average. Whereas the median gives equal weight to each economic category focusing on the segregation suffered by residents of category <italic>k</italic>, the weighted average provides an overall picture of segregation taking the population of each category into account. We show the ranking obtained for the weighted average index and its relation with <inline-formula id="inf34">
<mml:math id="m44">
<mml:mtext>med</mml:mtext>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mover accent="true">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mo>&#x303;</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mtext>sync</mml:mtext>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> in <xref ref-type="sec" rid="s9">Supplementary Figures S6, S7</xref>. Additionally, we show in <xref ref-type="sec" rid="s9">Supplementary Figures S3, S4</xref> how <inline-formula id="inf35">
<mml:math id="m45">
<mml:msub>
<mml:mrow>
<mml:mover accent="true">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mo>&#x303;</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mtext>sync</mml:mtext>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> significantly correlates with the traditional Moran&#x2019;s I [<xref ref-type="bibr" rid="B54">54</xref>] as well as a multi-scale quantity based on class mean first passage times developed in [<xref ref-type="bibr" rid="B20">20</xref>, <xref ref-type="bibr" rid="B21">21</xref>], reinforcing the idea that synchronization (and diffusion) dynamics indeed capture the patterns of residential segregation. Despite the dynamics we have used are stylized versions of the real behavior of the quantity <inline-formula id="inf36">
<mml:math id="m46">
<mml:msubsup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
</mml:math>
</inline-formula> and do not capture the full complexity of its temporal evolution, it is able to capture segregation with values comparable to other segregation indicators.</p>
<p>Although <inline-formula id="inf37">
<mml:math id="m47">
<mml:msub>
<mml:mrow>
<mml:mover accent="true">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mo>&#x303;</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mtext>sync</mml:mtext>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> is larger for extreme categories in most of the cities, some of them like Denver display smaller variations than others such as Detroit and, therefore, it might be of interest to group cities according to the change in synchronization times. By running a k-means algorithm on the normalized value of <inline-formula id="inf38">
<mml:math id="m48">
<mml:mi>P</mml:mi>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mover accent="true">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mo>&#x303;</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mtext>sync</mml:mtext>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> so that <inline-formula id="inf39">
<mml:math id="m49">
<mml:msub>
<mml:mrow>
<mml:mo movablelimits="false" form="prefix">&#x2211;</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mi>P</mml:mi>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mover accent="true">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mo>&#x303;</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mtext>sync</mml:mtext>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:math>
</inline-formula>, we can split the cities of study between those with higher and smaller differences in <inline-formula id="inf40">
<mml:math id="m50">
<mml:msub>
<mml:mrow>
<mml:mover accent="true">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mo>&#x303;</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mtext>sync</mml:mtext>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula>, see <xref ref-type="fig" rid="F3">Figure 3C</xref>. In <xref ref-type="fig" rid="F3">Figure 3D</xref>, we display the cluster assigned to each metropolitan area, where no strong spatial pattern is observed. Still, the cities in the Midwest, which are known for being economically segregated, fall into the red cluster, together with other cities such as Baltimore or Los Angeles. If, instead, we focus on the blue cluster, we have cities such as Sacramento or Washington D.C. Among the cities discussed in <xref ref-type="fig" rid="F3">Figure 3A</xref>, Denver falls into the group with more homogeneous segregation (in blue) and the rest into the one with more unequal segregation patterns (in red).</p>
<p>Beyond the global quantification of segregation, we can also evaluate the local level of segregation of a concrete census tract <italic>i</italic> at a given time step <italic>t</italic> by computing<disp-formula id="e11">
<mml:math id="m51">
<mml:msubsup>
<mml:mrow>
<mml:mi>&#x3c1;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:mfenced open="(" close=")">
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:mfenced>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>cos</mml:mi>
<mml:mfenced open="(" close=")">
<mml:mrow>
<mml:msubsup>
<mml:mrow>
<mml:mi>&#x3b8;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:mfenced open="(" close=")">
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:mfenced>
<mml:mo>&#x2212;</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mi mathvariant="normal">&#x3a6;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msup>
<mml:mfenced open="(" close=")">
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mfenced>
<mml:mo>,</mml:mo>
</mml:math>
<label>(11)</label>
</disp-formula>where <inline-formula id="inf41">
<mml:math id="m52">
<mml:msubsup>
<mml:mrow>
<mml:mi>&#x3b8;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> is the phase of unit <italic>i</italic> at time <italic>t</italic> and &#x3a6;<sup>
<italic>k</italic>
</sup>(<italic>t</italic>) is the average phase of all the oscillators in a city in a given time <italic>t</italic> [<xref ref-type="bibr" rid="B32">32</xref>]. When <inline-formula id="inf42">
<mml:math id="m53">
<mml:msubsup>
<mml:mrow>
<mml:mi>&#x3c1;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mo>&#x3e;</mml:mo>
<mml:mn>0.999</mml:mn>
</mml:math>
</inline-formula> we consider that oscillator <italic>i</italic> has synchronized, from which we can obtain <inline-formula id="inf43">
<mml:math id="m54">
<mml:msubsup>
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mtext>loc</mml:mtext>
</mml:mrow>
</mml:msubsup>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula>. However, given that <inline-formula id="inf44">
<mml:math id="m55">
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c1;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:msubsup>
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mtext>loc</mml:mtext>
</mml:mrow>
</mml:msubsup>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> can oscillate through time, we only consider that a unit <italic>i</italic> has reached the global synchronized state at a time <inline-formula id="inf45">
<mml:math id="m56">
<mml:msubsup>
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mtext>loc</mml:mtext>
</mml:mrow>
</mml:msubsup>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> when <inline-formula id="inf46">
<mml:math id="m57">
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c1;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x3e;</mml:mo>
<mml:msubsup>
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mtext>loc</mml:mtext>
</mml:mrow>
</mml:msubsup>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> does not go below 0.999 anymore, otherwise our methodology could fail to capture long-range correlations. In order to provide a metric for each spatial unit, simulations last until each of the spatial units have fullfiled the synchronization criteria. Normalizing <inline-formula id="inf47">
<mml:math id="m58">
<mml:msubsup>
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mtext>loc</mml:mtext>
</mml:mrow>
</mml:msubsup>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> by its null model counterpart, it yields <inline-formula id="inf48">
<mml:math id="m59">
<mml:msubsup>
<mml:mrow>
<mml:mover accent="true">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mo>&#x303;</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mtext>loc</mml:mtext>
</mml:mrow>
</mml:msubsup>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula>, a measure of the local synchronization time.</p>
<p>
<xref ref-type="fig" rid="F4">Figure 4</xref> displays the normalized synchronization times for each of the census tracts in Denver and Detroit, focusing on three very distinct income categories: low income, <xref ref-type="fig" rid="F4">Figures 4A,E</xref>; middle income, <xref ref-type="fig" rid="F4">Figures 4B,F</xref>; and high income, <xref ref-type="fig" rid="F4">Figures 4C,G</xref>. To ease the comparison between income categories, the range of values is common for all the maps, evincing the strong differences between Detroit and Denver, especially for the low and high-income categories. <xref ref-type="fig" rid="F4">Figures 4D,H</xref> also report the income per census tract. The shape of the segregation in Detroit can be outlined by the lower-income downtown and the richer suburbs, being the most segregated parts, and a less-segregated region in-between. In the case of Denver, we only slightly see high values for the low-income category in the North of the city and the high-income category in the South.</p>
<fig id="F4" position="float">
<label>FIGURE 4</label>
<caption>
<p>Local synchronization time as a measure for income segregation. Normalized synchronization time for each census tract in Denver <bold>(A&#x2013;C)</bold> and Detroit <bold>(E&#x2013;G)</bold> for three different income categories: <bold>(A,E)</bold> class 1 (low income); <bold>(B,F)</bold> class 8 (middle income); <bold>(C,G)</bold> class 16 (high income). We provide as a reference the median income of each census tract in <bold>(D)</bold> Denver and <bold>(H)</bold> Detroit.</p>
</caption>
<graphic xlink:href="fphy-10-833426-g004.tif"/>
</fig>
<p>As we detail in <xref ref-type="sec" rid="s9">Supplementary Figure S5</xref>, the spatial patterns of segregation product of the synchronization dynamics are significantly different to those obtained from first-neighbor quantities such as the Moran&#x2019;s I. Instead of focusing on those regions whose proportion of citizens is high (or low) compared to its neighbors, our methodology highlights those with a ratio of population within a category <italic>k</italic> distinct than the average, either because it is high or low, and spatially isolated from those regions with average values. In other words, a region with a high proportion of residents of category <italic>k</italic> might not show a large local spatial correlation if their neighbors have similar values but could, instead, produce high values of <inline-formula id="inf49">
<mml:math id="m60">
<mml:msubsup>
<mml:mrow>
<mml:mover accent="true">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
<mml:mo>&#x303;</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mtext>loc</mml:mtext>
</mml:mrow>
</mml:msubsup>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> if it is isolated from those regions displaying a proportion of citizens closer to the city average. As the majority of spatial measures, our approach can also suffer the so-called modifiable areal unit problem [<xref ref-type="bibr" rid="B55">55</xref>] in a similar fashion. However, given that our methodology captures mid and long-range correlations instead of local differences, it might be less affected by such small local changes.</p>
<p>Finally, we inspect if the synchronization time of a region displays any type of connection with its actual income. To do so, we plot in <xref ref-type="fig" rid="F5">Figure 5</xref> the normalized local synchronization time as a function of the median income averaged over all the census tracts within bins of $5,000 in four US cities. Again we see that segregation is much stronger in Detroit followed by Cleveland and Boston. High-income regions are more segregated in Boston compared to Cleveland. In general terms, the census tracts with a median income between $50,000 and $80,000 seem to be the less segregated ones as they synchronize faster for both low and high-income categories. These results are in agreement with the cluster assignment of the previous cities, with Detroit, Cleveland and Boston in the red cluster where low and high-income categories need more time to synchronize, and Denver in the blue cluster where only the high-income categories need more time to synchronize.</p>
<fig id="F5" position="float">
<label>FIGURE 5</label>
<caption>
<p>Synchronization time and median income. Normalized synchronization time as a function of the median income averaged over bins of $5,000 for class 1 <bold>(A)</bold>, class 8 <bold>(B)</bold> and class 16 <bold>(C)</bold>.</p>
</caption>
<graphic xlink:href="fphy-10-833426-g005.tif"/>
</fig>
</sec>
</sec>
<sec id="s3">
<title>3 Discussion</title>
<p>Traditional spatial segregation indicators that focus on local scale of segregation fail in most cases to capture the presence of long-range correlations, thus highlighting the need of multi-scale indices [<xref ref-type="bibr" rid="B15">15</xref>&#x2013;<xref ref-type="bibr" rid="B21">21</xref>]. Our framework does not consider any specific scale, but uses a dynamical approach that captures the patterns of segregation across the multiple scales. We have revealed how categories in the extreme of the income distribution are more heterogeneously distributed in space compared to middle classes, displaying larger diffusion and synchronization times. This approach has also allowed us to group together those cities that display common features of segregation. In this context, it is important to note that our work does not attempt to model the evolution of income segregation nor can be used as a forecasting tool, but takes modeling assumptions to assess the level of segregation that a distribution of population exhibits.</p>
<p>Despite the main manuscript focuses on the economic segregation, our methodology can be used to assess the heterogeneity in the spatial distribution of any characteristic. Moreover, it can go beyond the spatial component of segregation by including in the analysis other types of graphs, e.g., the daily mobility network of citizens. In this way, we could assess how citizens of diverse socioeconomic environments interact through mobility [<xref ref-type="bibr" rid="B56">56</xref>&#x2013;<xref ref-type="bibr" rid="B59">59</xref>].</p>
<p>Summarizing, we show how diffusion and synchronization dynamics can be used in some systems to assess the heterogeneity in the distribution of node features. While the present work focuses on the initial phases of oscillators and their synchronization time, node metadata could also be understood as an internal frequency and provide further insights on feature correlation across topological scales.</p>
</sec>
</body>
<back>
<sec id="s4">
<title>Data Availability Statement</title>
<p>Publicly available datasets were analyzed in this study. This data can be found here: <ext-link ext-link-type="uri" xlink:href="https://www.census.gov/programs-surveys/acs">https://www.census.gov/programs-surveys/acs</ext-link>.</p>
</sec>
<sec id="s5">
<title>Author Contributions</title>
<p>AB performed the research. AB, SG, and AA designed the research and wrote the manuscript.</p>
</sec>
<sec id="s6">
<title>Funding</title>
<p>AB acknowledges financial support from the Ministerio de Ciencia e Innovaci&#xf3;n under the Juan de la Cierva program (FJC2019-038958-I). We acknowledge support by Ministerio de Econom&#xed;a y Competitividad (PGC2018-094754-BC21, FIS2017-90782-REDT and RED2018-102518-T), Generalitat de Catalunya (2017SGR-896 and 2020PANDE00098), and Universitat Rovira i Virgili (2019PFR-URV-B2-41). AA acknowledges also ICREA Academia and the James S. McDonnell Foundation (220020325).</p>
</sec>
<sec sec-type="COI-statement" id="s7">
<title>Conflict of Interest</title>
<p>The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.</p>
</sec>
<sec sec-type="disclaimer" id="s8">
<title>Publisher&#x2019;s Note</title>
<p>All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.</p>
</sec>
<sec id="s9">
<title>Supplementary Material</title>
<p>The Supplementary Material for this article can be found online at: <ext-link ext-link-type="uri" xlink:href="https://www.frontiersin.org/articles/10.3389/fphy.2022.833426/full#supplementary-material">https://www.frontiersin.org/articles/10.3389/fphy.2022.833426/full&#x23;supplementary-material</ext-link>
</p>
<supplementary-material xlink:href="DataSheet1.PDF" id="SM1" mimetype="application/PDF" xmlns:xlink="http://www.w3.org/1999/xlink"/>
</sec>
<ref-list>
<title>References</title>
<ref id="B1">
<label>1.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Kennedy</surname>
<given-names>BP</given-names>
</name>
<name>
<surname>Kawachi</surname>
<given-names>I</given-names>
</name>
<name>
<surname>Glass</surname>
<given-names>R</given-names>
</name>
<name>
<surname>Prothrow-Stith</surname>
<given-names>D</given-names>
</name>
</person-group>. <article-title>Income Distribution, Socioeconomic Status, and Self Rated Health in the united states: Multilevel Analysis</article-title>. <source>BMJ</source> (<year>1998</year>) <volume>317</volume>:<fpage>917</fpage>&#x2013;<lpage>21</lpage>. <pub-id pub-id-type="doi">10.1136/bmj.317.7163.917</pub-id> </citation>
</ref>
<ref id="B2">
<label>2.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Elliott</surname>
<given-names>JR</given-names>
</name>
</person-group>. <article-title>Social Isolation and Labor Market Insulation: Network and Neighborhood Effects on Less-Educated Urban Workers</article-title>. <source>Sociological Q</source> (<year>1999</year>) <volume>40</volume>:<fpage>199</fpage>&#x2013;<lpage>216</lpage>. <pub-id pub-id-type="doi">10.1111/j.1533-8525.1999.tb00545.x</pub-id> </citation>
</ref>
<ref id="B3">
<label>3.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Collins</surname>
<given-names>WJ</given-names>
</name>
<name>
<surname>Margo</surname>
<given-names>RA</given-names>
</name>
</person-group>. <article-title>Residential Segregation and Socioeconomic Outcomes</article-title>. <source>Econ Lett</source> (<year>2000</year>) <volume>69</volume>:<fpage>239</fpage>&#x2013;<lpage>43</lpage>. <pub-id pub-id-type="doi">10.1016/s0165-1765(00)00300-1</pub-id> </citation>
</ref>
<ref id="B4">
<label>4.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Ross</surname>
<given-names>NA</given-names>
</name>
<name>
<surname>Nobrega</surname>
<given-names>K</given-names>
</name>
<name>
<surname>Dunn</surname>
<given-names>J</given-names>
</name>
</person-group>. <article-title>Income Segregation, Income Inequality and Mortality in north American Metropolitan Areas</article-title>. <source>GeoJournal</source> (<year>2001</year>) <volume>53</volume>:<fpage>117</fpage>&#x2013;<lpage>24</lpage>. <pub-id pub-id-type="doi">10.1023/a:1015720518936</pub-id> </citation>
</ref>
<ref id="B5">
<label>5.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Mayer</surname>
<given-names>SE</given-names>
</name>
</person-group>. <article-title>How Economic Segregation Affects Children&#x27;s Educational Attainment</article-title>. <source>Social Forces</source> (<year>2002</year>) <volume>81</volume>:<fpage>153</fpage>&#x2013;<lpage>76</lpage>. <pub-id pub-id-type="doi">10.1353/sof.2002.0053</pub-id> </citation>
</ref>
<ref id="B6">
<label>6.</label>
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Acevedo-Garcia</surname>
<given-names>D</given-names>
</name>
<name>
<surname>Lochner</surname>
<given-names>KA</given-names>
</name>
</person-group>. <source>Residential Segregation and Health</source>. <publisher-name>Neighborhoods and Health</publisher-name> (<year>2003</year>). p. <fpage>265</fpage>&#x2013;<lpage>87</lpage>. <pub-id pub-id-type="doi">10.1093/acprof:oso/9780195138382.003.0012</pub-id>
<article-title>Residential Segregation and Health</article-title> </citation>
</ref>
<ref id="B7">
<label>7.</label>
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Wheeler</surname>
<given-names>CH</given-names>
</name>
</person-group>. <source>Urban Decentralization and Income Inequality: Is Sprawl Associated with Rising Income Segregation across Neighborhoods? <italic>FRB of St. Louis</italic>
</source> (<year>2006</year>). <comment>Working Paper No. 2006&#x2013;037A</comment>. </citation>
</ref>
<ref id="B8">
<label>8.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Owens</surname>
<given-names>A</given-names>
</name>
</person-group>. <article-title>Income Segregation between School Districts and Inequality in Students&#x27; Achievement</article-title>. <source>Sociol Educ</source> (<year>2018</year>) <volume>91</volume>:<fpage>1</fpage>&#x2013;<lpage>27</lpage>. <pub-id pub-id-type="doi">10.1177/0038040717741180</pub-id> </citation>
</ref>
<ref id="B9">
<label>9.</label>
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Cliff</surname>
<given-names>AD</given-names>
</name>
<name>
<surname>Ord</surname>
<given-names>JK</given-names>
</name>
</person-group>. <source>Spatial Processes: Models &#x26; Applications</source>. <publisher-name>Taylor &#x26; Francis</publisher-name> (<year>1981</year>). </citation>
</ref>
<ref id="B10">
<label>10.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Dawkins</surname>
<given-names>CJ</given-names>
</name>
</person-group>. <article-title>Measuring the Spatial Pattern of Residential Segregation</article-title>. <source>Urban Stud</source> (<year>2004</year>) <volume>41</volume>:<fpage>833</fpage>&#x2013;<lpage>51</lpage>. <pub-id pub-id-type="doi">10.1080/0042098042000194133</pub-id> </citation>
</ref>
<ref id="B11">
<label>11.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Brown</surname>
<given-names>LA</given-names>
</name>
<name>
<surname>Chung</surname>
<given-names>S-Y</given-names>
</name>
</person-group>. <article-title>Spatial Segregation, Segregation Indices and the Geographical Perspective</article-title>. <source>Popul Space Place</source> (<year>2006</year>) <volume>12</volume>:<fpage>125</fpage>&#x2013;<lpage>43</lpage>. <pub-id pub-id-type="doi">10.1002/psp.403</pub-id> </citation>
</ref>
<ref id="B12">
<label>12.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Dawkins</surname>
<given-names>C</given-names>
</name>
</person-group>. <article-title>The Spatial Pattern of Black-white Segregation in Us Metropolitan Areas: an Exploratory Analysis</article-title>. <source>Urban Stud</source> (<year>2006</year>) <volume>43</volume>:<fpage>1943</fpage>&#x2013;<lpage>69</lpage>. <pub-id pub-id-type="doi">10.1080/00420980600897792</pub-id> </citation>
</ref>
<ref id="B13">
<label>13.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Wong</surname>
<given-names>DWS</given-names>
</name>
<name>
<surname>Shaw</surname>
<given-names>S-L</given-names>
</name>
</person-group>. <article-title>Measuring Segregation: An Activity Space Approach</article-title>. <source>J Geogr Syst</source> (<year>2011</year>) <volume>13</volume>:<fpage>127</fpage>&#x2013;<lpage>45</lpage>. <pub-id pub-id-type="doi">10.1007/s10109-010-0112-x</pub-id> </citation>
</ref>
<ref id="B14">
<label>14.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Rey</surname>
<given-names>SJ</given-names>
</name>
<name>
<surname>Smith</surname>
<given-names>RJ</given-names>
</name>
</person-group>. <article-title>A Spatial Decomposition of the Gini Coefficient</article-title>. <source>Lett Spat Resour Sci</source> (<year>2013</year>) <volume>6</volume>:<fpage>55</fpage>&#x2013;<lpage>70</lpage>. <pub-id pub-id-type="doi">10.1007/s12076-012-0086-z</pub-id> </citation>
</ref>
<ref id="B15">
<label>15.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Farber</surname>
<given-names>S</given-names>
</name>
<name>
<surname>P&#xe1;ez</surname>
<given-names>A</given-names>
</name>
<name>
<surname>Morency</surname>
<given-names>C</given-names>
</name>
</person-group>. <article-title>Activity Spaces and the Measurement of Clustering and Exposure: A Case Study of Linguistic Groups in Montreal</article-title>. <source>Environ Plan A</source> (<year>2012</year>) <volume>44</volume>:<fpage>315</fpage>&#x2013;<lpage>32</lpage>. <pub-id pub-id-type="doi">10.1068/a44203</pub-id> </citation>
</ref>
<ref id="B16">
<label>16.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Louf</surname>
<given-names>R</given-names>
</name>
<name>
<surname>Barthelemy</surname>
<given-names>M</given-names>
</name>
</person-group>. <article-title>Patterns of Residential Segregation</article-title>. <source>PLOS ONE</source> (<year>2016</year>) <volume>11</volume>:<fpage>e0157476</fpage>. <pub-id pub-id-type="doi">10.1371/journal.pone.0157476</pub-id> </citation>
</ref>
<ref id="B17">
<label>17.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Chodrow</surname>
<given-names>PS</given-names>
</name>
</person-group>. <article-title>Structure and Information in Spatial Segregation</article-title>. <source>Proc Natl Acad Sci U.S.A</source> (<year>2017</year>) <volume>114</volume>:<fpage>11591</fpage>&#x2013;<lpage>6</lpage>. <pub-id pub-id-type="doi">10.1073/pnas.1708201114</pub-id> </citation>
</ref>
<ref id="B18">
<label>18.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Olteanu</surname>
<given-names>M</given-names>
</name>
<name>
<surname>Randon-Furling</surname>
<given-names>J</given-names>
</name>
<name>
<surname>Clark</surname>
<given-names>WAV</given-names>
</name>
</person-group>. <article-title>Segregation through the Multiscalar Lens</article-title>. <source>Proc Natl Acad Sci U.S.A</source> (<year>2019</year>) <volume>116</volume>:<fpage>12250</fpage>&#x2013;<lpage>4</lpage>. <pub-id pub-id-type="doi">10.1073/pnas.1900192116</pub-id> </citation>
</ref>
<ref id="B19">
<label>19.</label>
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Sousa</surname>
<given-names>S</given-names>
</name>
<name>
<surname>Nicosia</surname>
<given-names>V</given-names>
</name>
</person-group>. <source>Quantifying Ethnic Segregation in Cities through Random Walks</source> (<year>2020</year>). <comment>
<italic>arXiv</italic> (2020) arXiv:2010.10462</comment>. </citation>
</ref>
<ref id="B20">
<label>20.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Bassolas</surname>
<given-names>A</given-names>
</name>
<name>
<surname>Nicosia</surname>
<given-names>V</given-names>
</name>
</person-group>. <article-title>First-passage Times to Quantify and Compare Structural Correlations and Heterogeneity in Complex Systems</article-title>. <source>Commun Phys</source> (<year>2021</year>) <volume>4</volume>:<fpage>1</fpage>&#x2013;<lpage>14</lpage>. <pub-id pub-id-type="doi">10.1038/s42005-021-00580-w</pub-id> </citation>
</ref>
<ref id="B21">
<label>21.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Bassolas</surname>
<given-names>A</given-names>
</name>
<name>
<surname>Sousa</surname>
<given-names>S</given-names>
</name>
<name>
<surname>Nicosia</surname>
<given-names>V</given-names>
</name>
</person-group>. <article-title>Diffusion Segregation and the Disproportionate Incidence of Covid-19 in African American Communities</article-title>. <source>J R Soc Interf</source> (<year>2021</year>) <volume>18</volume>:<fpage>20200961</fpage>. <pub-id pub-id-type="doi">10.1098/rsif.2020.0961</pub-id> </citation>
</ref>
<ref id="B22">
<label>22.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>G&#xf3;mez</surname>
<given-names>S</given-names>
</name>
<name>
<surname>D&#xed;az-Guilera</surname>
<given-names>A</given-names>
</name>
<name>
<surname>G&#xf3;mez-Garde&#xf1;es</surname>
<given-names>J</given-names>
</name>
<name>
<surname>P&#xe9;rez-Vicente</surname>
<given-names>CJ</given-names>
</name>
<name>
<surname>Moreno</surname>
<given-names>Y</given-names>
</name>
<name>
<surname>Arenas</surname>
<given-names>A</given-names>
</name>
</person-group>. <article-title>Diffusion Dynamics on Multiplex Networks</article-title>. <source>Phys Rev Lett</source> (<year>2013</year>) <volume>110</volume>:<fpage>028701</fpage>. <pub-id pub-id-type="doi">10.1103/PhysRevLett.110.028701</pub-id> </citation>
</ref>
<ref id="B23">
<label>23.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Sol&#xe9;-Ribalta</surname>
<given-names>A</given-names>
</name>
<name>
<surname>De Domenico</surname>
<given-names>M</given-names>
</name>
<name>
<surname>Kouvaris</surname>
<given-names>NE</given-names>
</name>
<name>
<surname>D&#xed;az-Guilera</surname>
<given-names>A</given-names>
</name>
<name>
<surname>G&#xf3;mez</surname>
<given-names>S</given-names>
</name>
<name>
<surname>Arenas</surname>
<given-names>A</given-names>
</name>
</person-group>. <article-title>Spectral Properties of the Laplacian of Multiplex Networks</article-title>. <source>Phys Rev E Stat Nonlin Soft Matter Phys</source> (<year>2013</year>) <volume>88</volume>:<fpage>032807</fpage>. <pub-id pub-id-type="doi">10.1103/PhysRevE.88.032807</pub-id> </citation>
</ref>
<ref id="B24">
<label>24.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>De Domenico</surname>
<given-names>M</given-names>
</name>
<name>
<surname>Sol&#xe9;-Ribalta</surname>
<given-names>A</given-names>
</name>
<name>
<surname>Cozzo</surname>
<given-names>E</given-names>
</name>
<name>
<surname>Kivel&#xe4;</surname>
<given-names>M</given-names>
</name>
<name>
<surname>Moreno</surname>
<given-names>Y</given-names>
</name>
<name>
<surname>Porter</surname>
<given-names>MA</given-names>
</name>
<etal/>
</person-group> <article-title>Mathematical Formulation of Multilayer Networks</article-title>. <source>Phys Rev X</source> (<year>2013</year>) <volume>3</volume>:<fpage>041022</fpage>. <pub-id pub-id-type="doi">10.1103/physrevx.3.041022</pub-id> </citation>
</ref>
<ref id="B25">
<label>25.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Li</surname>
<given-names>Y</given-names>
</name>
<name>
<surname>Chen</surname>
<given-names>W</given-names>
</name>
<name>
<surname>Wang</surname>
<given-names>Y</given-names>
</name>
<name>
<surname>Zhang</surname>
<given-names>ZL</given-names>
</name>
</person-group>. <article-title>Influence Diffusion Dynamics and Influence Maximization in Social Networks with Friend and Foe Relationships</article-title>. <source>Proc sixth ACM Int Conf Web search Data mining</source> (<year>2013</year>) <fpage>657</fpage>&#x2013;<lpage>66</lpage>. <pub-id pub-id-type="doi">10.1145/2433396.2433478</pub-id> </citation>
</ref>
<ref id="B26">
<label>26.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Delvenne</surname>
<given-names>JC</given-names>
</name>
<name>
<surname>Lambiotte</surname>
<given-names>R</given-names>
</name>
<name>
<surname>Rocha</surname>
<given-names>LE</given-names>
</name>
</person-group>. <article-title>Diffusion on Networked Systems Is a Question of Time or Structure</article-title>. <source>Nat Commun</source> (<year>2015</year>) <volume>6</volume>:<fpage>7366</fpage>&#x2013;<lpage>10</lpage>. <pub-id pub-id-type="doi">10.1038/ncomms8366</pub-id> </citation>
</ref>
<ref id="B27">
<label>27.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>De Domenico</surname>
<given-names>M</given-names>
</name>
</person-group>. <article-title>Diffusion Geometry Unravels the Emergence of Functional Clusters in Collective Phenomena</article-title>. <source>Phys Rev Lett</source> (<year>2017</year>) <volume>118</volume>:<fpage>168301</fpage>. <pub-id pub-id-type="doi">10.1103/physrevlett.118.168301</pub-id> </citation>
</ref>
<ref id="B28">
<label>28.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Masuda</surname>
<given-names>N</given-names>
</name>
<name>
<surname>Porter</surname>
<given-names>MA</given-names>
</name>
<name>
<surname>Lambiotte</surname>
<given-names>R</given-names>
</name>
</person-group>. <article-title>Random Walks and Diffusion on Networks</article-title>. <source>Phys Rep</source> (<year>2017</year>) <volume>716-717</volume>:<fpage>1</fpage>&#x2013;<lpage>58</lpage>. <pub-id pub-id-type="doi">10.1016/j.physrep.2017.07.007</pub-id> </citation>
</ref>
<ref id="B29">
<label>29.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Cencetti</surname>
<given-names>G</given-names>
</name>
<name>
<surname>Battiston</surname>
<given-names>F</given-names>
</name>
</person-group>. <article-title>Diffusive Behavior of Multiplex Networks</article-title>. <source>New J Phys</source> (<year>2019</year>) <volume>21</volume>:<fpage>035006</fpage>. <pub-id pub-id-type="doi">10.1088/1367-2630/ab060c</pub-id> </citation>
</ref>
<ref id="B30">
<label>30.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Bertagnolli</surname>
<given-names>G</given-names>
</name>
<name>
<surname>De Domenico</surname>
<given-names>M</given-names>
</name>
</person-group>. <article-title>Diffusion Geometry of Multiplex and Interdependent Systems</article-title>. <source>Phys Rev E</source> (<year>2021</year>) <volume>103</volume>:<fpage>042301</fpage>. <pub-id pub-id-type="doi">10.1103/PhysRevE.103.042301</pub-id> </citation>
</ref>
<ref id="B31">
<label>31.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Arenas</surname>
<given-names>A</given-names>
</name>
<name>
<surname>D&#xed;az-Guilera</surname>
<given-names>A</given-names>
</name>
<name>
<surname>P&#xe9;rez-Vicente</surname>
<given-names>CJ</given-names>
</name>
</person-group>. <article-title>Synchronization Processes in Complex Networks</article-title>. <source>Physica D: Nonlinear Phenomena</source> (<year>2006</year>) <volume>224</volume>:<fpage>27</fpage>&#x2013;<lpage>34</lpage>. <pub-id pub-id-type="doi">10.1016/j.physd.2006.09.029</pub-id> </citation>
</ref>
<ref id="B32">
<label>32.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Arenas</surname>
<given-names>A</given-names>
</name>
<name>
<surname>D&#xed;az-Guilera</surname>
<given-names>A</given-names>
</name>
<name>
<surname>P&#xe9;rez-Vicente</surname>
<given-names>CJ</given-names>
</name>
</person-group>. <article-title>Synchronization Reveals Topological Scales in Complex Networks</article-title>. <source>Phys Rev Lett</source> (<year>2006</year>) <volume>96</volume>:<fpage>114102</fpage>. <pub-id pub-id-type="doi">10.1103/physrevlett.96.114102</pub-id> </citation>
</ref>
<ref id="B33">
<label>33.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>G&#xf3;mez-Garde&#xf1;es</surname>
<given-names>J</given-names>
</name>
<name>
<surname>Moreno</surname>
<given-names>Y</given-names>
</name>
<name>
<surname>Arenas</surname>
<given-names>A</given-names>
</name>
</person-group>. <article-title>Paths to Synchronization on Complex Networks</article-title>. <source>Phys Rev Lett</source> (<year>2007</year>) <volume>98</volume>:<fpage>034101</fpage>. <pub-id pub-id-type="doi">10.1103/PhysRevLett.98.034101</pub-id> </citation>
</ref>
<ref id="B34">
<label>34.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>G&#xf3;mez-Garde&#xf1;es</surname>
<given-names>J</given-names>
</name>
<name>
<surname>Moreno</surname>
<given-names>Y</given-names>
</name>
<name>
<surname>Arenas</surname>
<given-names>A</given-names>
</name>
</person-group>. <article-title>Synchronizability Determined by Coupling Strengths and Topology on Complex Networks</article-title>. <source>Phys Rev E Stat Nonlin Soft Matter Phys</source> (<year>2007</year>) <volume>75</volume>:<fpage>066106</fpage>. <pub-id pub-id-type="doi">10.1103/PhysRevE.75.066106</pub-id> </citation>
</ref>
<ref id="B35">
<label>35.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Arenas</surname>
<given-names>A</given-names>
</name>
<name>
<surname>D&#xed;az-Guilera</surname>
<given-names>A</given-names>
</name>
<name>
<surname>Kurths</surname>
<given-names>J</given-names>
</name>
<name>
<surname>Moreno</surname>
<given-names>Y</given-names>
</name>
<name>
<surname>Zhou</surname>
<given-names>C</given-names>
</name>
</person-group>. <article-title>Synchronization in Complex Networks</article-title>. <source>Phys Rep</source> (<year>2008</year>) <volume>469</volume>:<fpage>93</fpage>&#x2013;<lpage>153</lpage>. <pub-id pub-id-type="doi">10.1016/j.physrep.2008.09.002</pub-id> </citation>
</ref>
<ref id="B36">
<label>36.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>G&#xf3;mez-Garde&#xf1;es</surname>
<given-names>J</given-names>
</name>
<name>
<surname>Soriano-Pa&#xf1;os</surname>
<given-names>D</given-names>
</name>
<name>
<surname>Arenas</surname>
<given-names>A</given-names>
</name>
</person-group>. <article-title>Critical Regimes Driven by Recurrent Mobility Patterns of Reaction-Diffusion Processes in Networks</article-title>. <source>Nat Phys</source> (<year>2018</year>) <volume>14</volume>:<fpage>391</fpage>&#x2013;<lpage>5</lpage>. <pub-id pub-id-type="doi">10.1038/s41567-017-0022-7</pub-id> </citation>
</ref>
<ref id="B37">
<label>37.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Zhang</surname>
<given-names>Z-K</given-names>
</name>
<name>
<surname>Liu</surname>
<given-names>C</given-names>
</name>
<name>
<surname>Zhan</surname>
<given-names>X-X</given-names>
</name>
<name>
<surname>Lu</surname>
<given-names>X</given-names>
</name>
<name>
<surname>Zhang</surname>
<given-names>C-X</given-names>
</name>
<name>
<surname>Zhang</surname>
<given-names>Y-C</given-names>
</name>
</person-group>. <article-title>Dynamics of Information Diffusion and its Applications on Complex Networks</article-title>. <source>Phys Rep</source> (<year>2016</year>) <volume>651</volume>:<fpage>1</fpage>&#x2013;<lpage>34</lpage>. <pub-id pub-id-type="doi">10.1016/j.physrep.2016.07.002</pub-id> </citation>
</ref>
<ref id="B38">
<label>38.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Pluchino</surname>
<given-names>A</given-names>
</name>
<name>
<surname>Latora</surname>
<given-names>V</given-names>
</name>
<name>
<surname>Rapisarda</surname>
<given-names>A</given-names>
</name>
</person-group>. <article-title>Changing Opinions in a Changing World: A New Perspective in Sociophysics</article-title>. <source>Int J Mod Phys C</source> (<year>2005</year>) <volume>16</volume>:<fpage>515</fpage>&#x2013;<lpage>31</lpage>. <pub-id pub-id-type="doi">10.1142/s0129183105007261</pub-id> </citation>
</ref>
<ref id="B39">
<label>39.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Calder&#xf3;n</surname>
<given-names>C</given-names>
</name>
<name>
<surname>Chong</surname>
<given-names>A</given-names>
</name>
<name>
<surname>Stein</surname>
<given-names>E</given-names>
</name>
</person-group>. <article-title>Trade Intensity and Business Cycle Synchronization: Are Developing Countries Any Different?</article-title> <source>J Int Econ</source> (<year>2007</year>) <volume>71</volume>:<fpage>2</fpage>&#x2013;<lpage>21</lpage>. <pub-id pub-id-type="doi">10.1016/j.jinteco.2006.06.001</pub-id> </citation>
</ref>
<ref id="B40">
<label>40.</label>
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Erola</surname>
<given-names>P</given-names>
</name>
<name>
<surname>D&#xed;az-Guilera</surname>
<given-names>A</given-names>
</name>
<name>
<surname>G&#xf3;mez</surname>
<given-names>S</given-names>
</name>
<name>
<surname>Arenas</surname>
<given-names>A</given-names>
</name>
</person-group>. <source>Modeling International Crisis Synchronization in the World Trade Web</source>, <volume>7</volume>. <publisher-name>Networks and Heterogeneous Media</publisher-name> (<year>2012</year>). <fpage>385</fpage>&#x2013;<lpage>97</lpage>. <pub-id pub-id-type="doi">10.3934/nhm.2012.7.385</pub-id>
<article-title>Modeling International Crisis Synchronization in the World Trade Web</article-title>
<source>Nhm</source> </citation>
</ref>
<ref id="B41">
<label>41.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Motter</surname>
<given-names>AE</given-names>
</name>
<name>
<surname>Zhou</surname>
<given-names>C</given-names>
</name>
<name>
<surname>Kurths</surname>
<given-names>J</given-names>
</name>
</person-group>. <article-title>Network Synchronization, Diffusion, and the Paradox of Heterogeneity</article-title>. <source>Phys Rev E Stat Nonlin Soft Matter Phys</source> (<year>2005</year>) <volume>71</volume>:<fpage>016116</fpage>. <pub-id pub-id-type="doi">10.1103/PhysRevE.71.016116</pub-id> </citation>
</ref>
<ref id="B42">
<label>42.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Schelling</surname>
<given-names>TC</given-names>
</name>
</person-group>. <article-title>Dynamic Models of Segregation&#x2020;</article-title>. <source>J Math Sociol</source> (<year>1971</year>) <volume>1</volume>:<fpage>143</fpage>&#x2013;<lpage>86</lpage>. <pub-id pub-id-type="doi">10.1080/0022250x.1971.9989794</pub-id> </citation>
</ref>
<ref id="B43">
<label>43.</label>
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Fujita</surname>
<given-names>M</given-names>
</name>
</person-group>. <source>Urban Economic Theory</source>. <publisher-name>Cambridge University Press</publisher-name> (<year>1989</year>). </citation>
</ref>
<ref id="B44">
<label>44.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Zhang</surname>
<given-names>J</given-names>
</name>
</person-group>. <article-title>A Dynamic Model of Residential Segregation</article-title>. <source>J Math Sociol</source> (<year>2004</year>) <volume>28</volume>:<fpage>147</fpage>&#x2013;<lpage>70</lpage>. <pub-id pub-id-type="doi">10.1080/00222500490480202</pub-id> </citation>
</ref>
<ref id="B45">
<label>45.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Clark</surname>
<given-names>WAV</given-names>
</name>
</person-group>. <article-title>Changing Residential Preferences across Income, Education, and Age</article-title>. <source>Urban Aff Rev</source> (<year>2009</year>) <volume>44</volume>:<fpage>334</fpage>&#x2013;<lpage>55</lpage>. <pub-id pub-id-type="doi">10.1177/1078087408321497</pub-id> </citation>
</ref>
<ref id="B46">
<label>46.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Zhang</surname>
<given-names>J</given-names>
</name>
</person-group>. <article-title>Tipping and Residential Segregation: a Unified Schelling Model&#x2a;</article-title>. <source>J Reg Sci</source> (<year>2011</year>) <volume>51</volume>:<fpage>167</fpage>&#x2013;<lpage>93</lpage>. <pub-id pub-id-type="doi">10.1111/j.1467-9787.2010.00671.x</pub-id> </citation>
</ref>
<ref id="B47">
<label>47.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>DeLuca</surname>
<given-names>S</given-names>
</name>
<name>
<surname>Garboden</surname>
<given-names>PME</given-names>
</name>
<name>
<surname>Rosenblatt</surname>
<given-names>P</given-names>
</name>
</person-group>. <article-title>Segregating Shelter</article-title>. <source>Annals Am Acad Polit Soc Sci</source> (<year>2013</year>) <volume>647</volume>:<fpage>268</fpage>&#x2013;<lpage>99</lpage>. <pub-id pub-id-type="doi">10.1177/0002716213479310</pub-id> </citation>
</ref>
<ref id="B49">
<label>49.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Adelman</surname>
<given-names>RM</given-names>
</name>
</person-group>. <article-title>Neighborhood Opportunities, Race, and Class: The Black Middle Class and Residential Segregation</article-title>. <source>City &#x26; Community</source> (<year>2004</year>) <volume>3</volume>:<fpage>43</fpage>&#x2013;<lpage>63</lpage>. <pub-id pub-id-type="doi">10.1111/j.1535-6841.2004.00066.x</pub-id> </citation>
</ref>
<ref id="B50">
<label>50.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Thomas</surname>
<given-names>M</given-names>
</name>
<name>
<surname>Moye</surname>
<given-names>R</given-names>
</name>
</person-group>. <article-title>Race, Class, and Gender and the Impact of Racial Segregation on Black-white Income Inequality</article-title>. <source>Sociol Race Ethn</source> (<year>2015</year>) <volume>1</volume>:<fpage>490</fpage>&#x2013;<lpage>502</lpage>. <pub-id pub-id-type="doi">10.1177/2332649215581665</pub-id> </citation>
</ref>
<ref id="B51">
<label>51.</label>
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Florida</surname>
<given-names>R</given-names>
</name>
<name>
<surname>Mellander</surname>
<given-names>C</given-names>
</name>
</person-group>. <source>Segregated City: The Geography of Economic Segregation in America&#x2019;s Metros</source>. <publisher-name>Martin Prosperity Institute</publisher-name> (<year>2015</year>). </citation>
</ref>
<ref id="B52">
<label>52.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Hartigan</surname>
<given-names>JA</given-names>
</name>
<name>
<surname>Wong</surname>
<given-names>MA</given-names>
</name>
</person-group>. <article-title>Algorithm as 136: A K-Means Clustering Algorithm</article-title>. <source>Appl Stat</source> (<year>1979</year>) <volume>28</volume>:<fpage>100</fpage>&#x2013;<lpage>8</lpage>. <pub-id pub-id-type="doi">10.2307/2346830</pub-id> </citation>
</ref>
<ref id="B53">
<label>53.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Likas</surname>
<given-names>A</given-names>
</name>
<name>
<surname>Vlassis</surname>
<given-names>N</given-names>
</name>
<name>
<surname>J. Verbeek</surname>
<given-names>J</given-names>
</name>
</person-group>. <article-title>The Global K-Means Clustering Algorithm</article-title>. <source>Pattern Recognition</source> (<year>2003</year>) <volume>36</volume>:<fpage>451</fpage>&#x2013;<lpage>61</lpage>. <pub-id pub-id-type="doi">10.1016/s0031-3203(02)00060-2</pub-id> </citation>
</ref>
<ref id="B54">
<label>54.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Moran</surname>
<given-names>PAP</given-names>
</name>
</person-group>. <article-title>The Interpretation of Statistical Maps</article-title>. <source>J R Stat Soc Ser B (Methodological)</source> (<year>1948</year>) <volume>10</volume>:<fpage>243</fpage>&#x2013;<lpage>51</lpage>. <pub-id pub-id-type="doi">10.1111/j.2517-6161.1948.tb00012.x</pub-id> </citation>
</ref>
<ref id="B55">
<label>55.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Fotheringham</surname>
<given-names>AS</given-names>
</name>
<name>
<surname>Wong</surname>
<given-names>DWS</given-names>
</name>
</person-group>. <article-title>The Modifiable Areal Unit Problem in Multivariate Statistical Analysis</article-title>. <source>Environ Plan A</source> (<year>1991</year>) <volume>23</volume>:<fpage>1025</fpage>&#x2013;<lpage>44</lpage>. <pub-id pub-id-type="doi">10.1068/a231025</pub-id> </citation>
</ref>
<ref id="B56">
<label>56.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Xu</surname>
<given-names>Y</given-names>
</name>
<name>
<surname>Belyi</surname>
<given-names>A</given-names>
</name>
<name>
<surname>Santi</surname>
<given-names>P</given-names>
</name>
<name>
<surname>Ratti</surname>
<given-names>C</given-names>
</name>
</person-group>. <article-title>Quantifying Segregation in an Integrated Urban Physical-Social Space</article-title>. <source>J R Soc Interf</source> (<year>2019</year>) <volume>16</volume>:<fpage>20190536</fpage>. <pub-id pub-id-type="doi">10.1098/rsif.2019.0536</pub-id> </citation>
</ref>
<ref id="B57">
<label>57.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>T&#xf3;th</surname>
<given-names>G</given-names>
</name>
<name>
<surname>Wachs</surname>
<given-names>J</given-names>
</name>
<name>
<surname>Di Clemente</surname>
<given-names>R</given-names>
</name>
<name>
<surname>&#xc1;</surname>
<given-names>J</given-names>
</name>
<name>
<surname>S&#xe1;gv&#xe1;ri</surname>
<given-names>B</given-names>
</name>
<name>
<surname>Kert&#xe9;sz</surname>
<given-names>J</given-names>
</name>
<etal/>
</person-group> <article-title>Inequality Is Rising where Social Network Segregation Interacts with Urban Topology</article-title>. <source>Nat Commun</source> (<year>2021</year>) <volume>12</volume>:<fpage>1</fpage>&#x2013;<lpage>9</lpage>. <pub-id pub-id-type="doi">10.1038/s41467-021-21465-0</pub-id> </citation>
</ref>
<ref id="B58">
<label>58.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Bok&#xe1;nyi</surname>
<given-names>E</given-names>
</name>
<name>
<surname>Juh&#xe1;sz</surname>
<given-names>S</given-names>
</name>
<name>
<surname>Karsai</surname>
<given-names>M</given-names>
</name>
<name>
<surname>Lengyel</surname>
<given-names>B</given-names>
</name>
</person-group>. <article-title>Universal Patterns of Long-Distance Commuting and Social Assortativity in Cities</article-title>. <source>Scientific Rep</source> (<year>2021</year>) <volume>11</volume>:<fpage>1</fpage>&#x2013;<lpage>10</lpage>. <pub-id pub-id-type="doi">10.1038/s41598-021-00416-1</pub-id> </citation>
</ref>
<ref id="B59">
<label>59.</label>
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Moro</surname>
<given-names>E</given-names>
</name>
<name>
<surname>Calacci</surname>
<given-names>D</given-names>
</name>
<name>
<surname>Dong</surname>
<given-names>X</given-names>
</name>
<name>
<surname>Pentland</surname>
<given-names>A</given-names>
</name>
</person-group>. <article-title>Mobility Patterns Are Associated with Experienced Income Segregation in Large Us Cities</article-title>. <source>Nat Commun</source> (<year>2021</year>) <volume>12</volume>:<fpage>4633</fpage>&#x2013;<lpage>10</lpage>. <pub-id pub-id-type="doi">10.1038/s41467-021-24899-8</pub-id> </citation>
</ref>
<ref id="B60">
<label>60.</label>
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Manson</surname>
<given-names>S</given-names>
</name>
<name>
<surname>Schroeder</surname>
<given-names>J</given-names>
</name>
<name>
<surname>Riper</surname>
<given-names>DV</given-names>
</name>
<name>
<surname>Ruggles</surname>
<given-names>S</given-names>
</name>
</person-group>. <source>IPUMS National Historical Geographic Information System: Version 14.0 [Database]</source>. <publisher-loc>Minneapolis, MN</publisher-loc>: <publisher-name>IPUMS</publisher-name> (<year>2019</year>). <comment>[Dataset]</comment>. <pub-id pub-id-type="doi">10.18128/D050.V14.0</pub-id> </citation>
</ref>
</ref-list>
</back>
</article>