# Diffusion and Synchronization Dynamics Reveal the Multi-Scale Patterns of Spatial Segregation

- Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Tarragona, Spain

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.

## 1 Introduction

The expansion of urbanization and progressive increase of the population in cities has intensified the concern over the many dimensions of segregation—i.e., school, economic or ethnics—that have a tangible impact in the health, education and equal opportunities of citizens [1–8]. 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 [9–14]. 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 [15–21], yet understanding the role played by each of the scales and their interplay still remains a challenge.

Dynamical processes in general, and in particular diffusion [22–30] and synchronization [31–35] dynamics, have been widely studied in complex networks on account of their relation with the spread of diseases and information [36, 37] and real-world phenomena in social or economic systems [38–40]. Interestingly, they provide insights on the topological scales and structure of networks and reveal the existence of functional meso-scale structures [27, 30, 32, 34, 41].

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.

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 Supplementary Material Section 2 and Supplementary Figures S8–S10). 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.

## 2 Results

### 2.1 Diffusion Dynamics and Income Segregation

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 *k* ∈ *K*, we consider a graph *G*(*V*, *E*) with adjacency matrix *A* = {*a*_{ij}} in which the spatial units are represented as a set of nodes *V* connected by a set of edges *E*. The adjacency matrix *A* we have considered takes *a*_{ij} = 1 when spatial units *i* and *j* are adjacent and *a*_{ij} = 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 *i* that belong to socioeconomic category (or class) *k*, written as

where *i* that belong to category *k*. As extreme cases, *k* living in *i*, and *i* belong to category *k*. Of course, the normalization condition

is fulfilled for all nodes *i*.

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 *k* 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 [42, 43].

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 [44–47] 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.

In fact, we focus on one of the best-case scenarios where the values of *k* to regions where they are less abundant.

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 *k* residing in each census tract (see Table 1).

Once we have the set of initial node states

where

is the degree of node *i*. 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 *k*.

The diffusion dynamic lasts until the stationary state, *∀i* is reached, and we denote the spanned time as *τ*_{diff}(*k*). Since the time to reach the stationary state can be infinitely large, we have considered that it is reached when the variance of *τ*_{diff}(*k*) are related to a more homogeneous distribution of the population within a category *k*, and the other way around when it is higher. In the extreme case in which all units have the same initial value of *τ*_{diff}(*k*) 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 *x*^{k} or the topology of the graph by running the same diffusion dynamics on the same graph but where the values of *x*^{k} have been reshuffled, thus defining the average null-model diffusion time

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.

We analyze the normalized diffusion times *k* 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 Figure 1A we display *x*^{k} as well as the Moran’s I (see Supplementary Material Section 1 and Supplementary Figure S1). Still, there are strong quantitative differences, with Cleveland and Detroit displaying higher values for most of the categories, in contrast to Boston and Denver.

**FIGURE 1**. Diffusion dynamics as a measure for income segregation. **(A)** Synchronization time for each of the 16 income categories in Boston, Cleveland, Denver and Detroit. **(B)** Median value of **(C)** Ranking for the median value of

Since

By applying diffusion dynamics we implicitly assume that *x*^{k} 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 *i*, we first construct the normalized time-series for each spatial unit across those years as

and then cluster, for each category *k*, the temporal profiles of all the nodes. For the clustering, we have made use of the k-means algorithm [52, 53], 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 Figure 2, where a non-monotonic behavior is observed in most of the cases, with oscillatory behaviors through time of varying amplitude.

**FIGURE 2**. 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 *k*, *t* for the lower **(A)** and higher **(B)** income categories.

### 2.2 Synchronization Dynamics and Income Segregation

According to the oscillations in the temporal evolution of *i* as an individual Kuramoto oscillator, with an initial phase *i* that belongs to a category *k* within the range [0, *π*] as

The interaction between spatial units is given by the Kuramoto model

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 *ω*_{i} = 1, *∀i*. In order to account separately the segregation of each category *k*, our approach assumes that there is no interaction between categories and, thus, *k*.

We use the standard order parameter |*z*^{k}| to assess the global level of synchronization for a category *k* in a city, where

and *N* is the total number of spatial units or Kuramoto oscillators [35]. We consider that a city has reached the synchronized state when |*z*^{k}| > 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 *τ*_{sync}(*k*) 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

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.

In Figure 3A 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 *x*^{k} does not determine directly the normalized synchronization times in Supplementary Figure S1.

**FIGURE 3**. Synchronization time as a measure for income segregation. **(A)** Synchronization time for each of the 16 income categories in Boston, Cleveland, Denver and Detroit. **(B)** Ranking for the median value of **(C)** Average value of *i* for the two main clusters detected. **(D)** Location and cluster assignment for each of the analyzed cities.

Likewise with diffusion, we calculate the median and variance of *k* = 1 and *k* = 16). The ranking is shown in Figure 3B 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 Figure 3A is consistent with our observations, with Boston and Denver on the bottom of the ranking and Detroit and Cleveland on the top of it.

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 *k*, 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

Although

Beyond the global quantification of segregation, we can also evaluate the local level of segregation of a concrete census tract *i* at a given time step *t* by computing

where *i* at time *t* and Φ^{k}(*t*) is the average phase of all the oscillators in a city in a given time *t* [32]. When *i* has synchronized, from which we can obtain *i* has reached the global synchronized state at a time

Figure 4 displays the normalized synchronization times for each of the census tracts in Denver and Detroit, focusing on three very distinct income categories: low income, Figures 4A,E; middle income, Figures 4B,F; and high income, Figures 4C,G. 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. Figures 4D,H 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.

**FIGURE 4**. Local synchronization time as a measure for income segregation. Normalized synchronization time for each census tract in Denver **(A–C)** and Detroit **(E–G)** for three different income categories: **(A,E)** class 1 (low income); **(B,F)** class 8 (middle income); **(C,G)** class 16 (high income). We provide as a reference the median income of each census tract in **(D)** Denver and **(H)** Detroit.

As we detail in Supplementary Figure S5, the spatial patterns of segregation product of the synchronization dynamics are significantly different to those obtained from first-neighbor quantities such as the Moran’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 *k* 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 *k* might not show a large local spatial correlation if their neighbors have similar values but could, instead, produce high values of

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 Figure 5 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.

**FIGURE 5**. Synchronization time and median income. Normalized synchronization time as a function of the median income averaged over bins of $5,000 for class 1 **(A)**, class 8 **(B)** and class 16 **(C)**.

## 3 Discussion

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 [15–21]. 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.

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 [56–59].

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.

## Data Availability Statement

Publicly available datasets were analyzed in this study. This data can be found here: https://www.census.gov/programs-surveys/acs.

## Author Contributions

AB performed the research. AB, SG, and AA designed the research and wrote the manuscript.

## Funding

AB acknowledges financial support from the Ministerio de Ciencia e Innovación under the Juan de la Cierva program (FJC2019-038958-I). We acknowledge support by Ministerio de Economía 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).

## Conflict of Interest

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.

## Publisher’s Note

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.

## Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2022.833426/full#supplementary-material

## References

1. Kennedy BP, Kawachi I, Glass R, Prothrow-Stith D. Income Distribution, Socioeconomic Status, and Self Rated Health in the united states: Multilevel Analysis. *BMJ* (1998) 317:917–21. doi:10.1136/bmj.317.7163.917

2. Elliott JR. Social Isolation and Labor Market Insulation: Network and Neighborhood Effects on Less-Educated Urban Workers. *Sociological Q* (1999) 40:199–216. doi:10.1111/j.1533-8525.1999.tb00545.x

3. Collins WJ, Margo RA. Residential Segregation and Socioeconomic Outcomes. *Econ Lett* (2000) 69:239–43. doi:10.1016/s0165-1765(00)00300-1

4. Ross NA, Nobrega K, Dunn J. Income Segregation, Income Inequality and Mortality in north American Metropolitan Areas. *GeoJournal* (2001) 53:117–24. doi:10.1023/a:1015720518936

5. Mayer SE. How Economic Segregation Affects Children's Educational Attainment. *Social Forces* (2002) 81:153–76. doi:10.1353/sof.2002.0053

6. Acevedo-Garcia D, Lochner KA. *Residential Segregation and Health*. Neighborhoods and Health (2003). p. 265–87. doi:10.1093/acprof:oso/9780195138382.003.0012Residential Segregation and Health

7. Wheeler CH. *Urban Decentralization and Income Inequality: Is Sprawl Associated with Rising Income Segregation across Neighborhoods? FRB of St. Louis* (2006). Working Paper No. 2006–037A.

8. Owens A. Income Segregation between School Districts and Inequality in Students' Achievement. *Sociol Educ* (2018) 91:1–27. doi:10.1177/0038040717741180

10. Dawkins CJ. Measuring the Spatial Pattern of Residential Segregation. *Urban Stud* (2004) 41:833–51. doi:10.1080/0042098042000194133

11. Brown LA, Chung S-Y. Spatial Segregation, Segregation Indices and the Geographical Perspective. *Popul Space Place* (2006) 12:125–43. doi:10.1002/psp.403

12. Dawkins C. The Spatial Pattern of Black-white Segregation in Us Metropolitan Areas: an Exploratory Analysis. *Urban Stud* (2006) 43:1943–69. doi:10.1080/00420980600897792

13. Wong DWS, Shaw S-L. Measuring Segregation: An Activity Space Approach. *J Geogr Syst* (2011) 13:127–45. doi:10.1007/s10109-010-0112-x

14. Rey SJ, Smith RJ. A Spatial Decomposition of the Gini Coefficient. *Lett Spat Resour Sci* (2013) 6:55–70. doi:10.1007/s12076-012-0086-z

15. Farber S, Páez A, Morency C. Activity Spaces and the Measurement of Clustering and Exposure: A Case Study of Linguistic Groups in Montreal. *Environ Plan A* (2012) 44:315–32. doi:10.1068/a44203

16. Louf R, Barthelemy M. Patterns of Residential Segregation. *PLOS ONE* (2016) 11:e0157476. doi:10.1371/journal.pone.0157476

17. Chodrow PS. Structure and Information in Spatial Segregation. *Proc Natl Acad Sci U.S.A* (2017) 114:11591–6. doi:10.1073/pnas.1708201114

18. Olteanu M, Randon-Furling J, Clark WAV. Segregation through the Multiscalar Lens. *Proc Natl Acad Sci U.S.A* (2019) 116:12250–4. doi:10.1073/pnas.1900192116

19. Sousa S, Nicosia V. *Quantifying Ethnic Segregation in Cities through Random Walks* (2020). *arXiv* (2020) arXiv:2010.10462.

20. Bassolas A, Nicosia V. First-passage Times to Quantify and Compare Structural Correlations and Heterogeneity in Complex Systems. *Commun Phys* (2021) 4:1–14. doi:10.1038/s42005-021-00580-w

21. Bassolas A, Sousa S, Nicosia V. Diffusion Segregation and the Disproportionate Incidence of Covid-19 in African American Communities. *J R Soc Interf* (2021) 18:20200961. doi:10.1098/rsif.2020.0961

22. Gómez S, Díaz-Guilera A, Gómez-Gardeñes J, Pérez-Vicente CJ, Moreno Y, Arenas A. Diffusion Dynamics on Multiplex Networks. *Phys Rev Lett* (2013) 110:028701. doi:10.1103/PhysRevLett.110.028701

23. Solé-Ribalta A, De Domenico M, Kouvaris NE, Díaz-Guilera A, Gómez S, Arenas A. Spectral Properties of the Laplacian of Multiplex Networks. *Phys Rev E Stat Nonlin Soft Matter Phys* (2013) 88:032807. doi:10.1103/PhysRevE.88.032807

24. De Domenico M, Solé-Ribalta A, Cozzo E, Kivelä M, Moreno Y, Porter MA, et al. Mathematical Formulation of Multilayer Networks. *Phys Rev X* (2013) 3:041022. doi:10.1103/physrevx.3.041022

25. Li Y, Chen W, Wang Y, Zhang ZL. Influence Diffusion Dynamics and Influence Maximization in Social Networks with Friend and Foe Relationships. *Proc sixth ACM Int Conf Web search Data mining* (2013) 657–66. doi:10.1145/2433396.2433478

26. Delvenne JC, Lambiotte R, Rocha LE. Diffusion on Networked Systems Is a Question of Time or Structure. *Nat Commun* (2015) 6:7366–10. doi:10.1038/ncomms8366

27. De Domenico M. Diffusion Geometry Unravels the Emergence of Functional Clusters in Collective Phenomena. *Phys Rev Lett* (2017) 118:168301. doi:10.1103/physrevlett.118.168301

28. Masuda N, Porter MA, Lambiotte R. Random Walks and Diffusion on Networks. *Phys Rep* (2017) 716-717:1–58. doi:10.1016/j.physrep.2017.07.007

29. Cencetti G, Battiston F. Diffusive Behavior of Multiplex Networks. *New J Phys* (2019) 21:035006. doi:10.1088/1367-2630/ab060c

30. Bertagnolli G, De Domenico M. Diffusion Geometry of Multiplex and Interdependent Systems. *Phys Rev E* (2021) 103:042301. doi:10.1103/PhysRevE.103.042301

31. Arenas A, Díaz-Guilera A, Pérez-Vicente CJ. Synchronization Processes in Complex Networks. *Physica D: Nonlinear Phenomena* (2006) 224:27–34. doi:10.1016/j.physd.2006.09.029

32. Arenas A, Díaz-Guilera A, Pérez-Vicente CJ. Synchronization Reveals Topological Scales in Complex Networks. *Phys Rev Lett* (2006) 96:114102. doi:10.1103/physrevlett.96.114102

33. Gómez-Gardeñes J, Moreno Y, Arenas A. Paths to Synchronization on Complex Networks. *Phys Rev Lett* (2007) 98:034101. doi:10.1103/PhysRevLett.98.034101

34. Gómez-Gardeñes J, Moreno Y, Arenas A. Synchronizability Determined by Coupling Strengths and Topology on Complex Networks. *Phys Rev E Stat Nonlin Soft Matter Phys* (2007) 75:066106. doi:10.1103/PhysRevE.75.066106

35. Arenas A, Díaz-Guilera A, Kurths J, Moreno Y, Zhou C. Synchronization in Complex Networks. *Phys Rep* (2008) 469:93–153. doi:10.1016/j.physrep.2008.09.002

36. Gómez-Gardeñes J, Soriano-Paños D, Arenas A. Critical Regimes Driven by Recurrent Mobility Patterns of Reaction-Diffusion Processes in Networks. *Nat Phys* (2018) 14:391–5. doi:10.1038/s41567-017-0022-7

37. Zhang Z-K, Liu C, Zhan X-X, Lu X, Zhang C-X, Zhang Y-C. Dynamics of Information Diffusion and its Applications on Complex Networks. *Phys Rep* (2016) 651:1–34. doi:10.1016/j.physrep.2016.07.002

38. Pluchino A, Latora V, Rapisarda A. Changing Opinions in a Changing World: A New Perspective in Sociophysics. *Int J Mod Phys C* (2005) 16:515–31. doi:10.1142/s0129183105007261

39. Calderón C, Chong A, Stein E. Trade Intensity and Business Cycle Synchronization: Are Developing Countries Any Different? *J Int Econ* (2007) 71:2–21. doi:10.1016/j.jinteco.2006.06.001

40. Erola P, Díaz-Guilera A, Gómez S, Arenas A. *Modeling International Crisis Synchronization in the World Trade Web*, 7. Networks and Heterogeneous Media (2012). 385–97. doi:10.3934/nhm.2012.7.385Modeling International Crisis Synchronization in the World Trade Web*Nhm*

41. Motter AE, Zhou C, Kurths J. Network Synchronization, Diffusion, and the Paradox of Heterogeneity. *Phys Rev E Stat Nonlin Soft Matter Phys* (2005) 71:016116. doi:10.1103/PhysRevE.71.016116

42. Schelling TC. Dynamic Models of Segregation†. *J Math Sociol* (1971) 1:143–86. doi:10.1080/0022250x.1971.9989794

44. Zhang J. A Dynamic Model of Residential Segregation. *J Math Sociol* (2004) 28:147–70. doi:10.1080/00222500490480202

45. Clark WAV. Changing Residential Preferences across Income, Education, and Age. *Urban Aff Rev* (2009) 44:334–55. doi:10.1177/1078087408321497

46. Zhang J. Tipping and Residential Segregation: a Unified Schelling Model*. *J Reg Sci* (2011) 51:167–93. doi:10.1111/j.1467-9787.2010.00671.x

47. DeLuca S, Garboden PME, Rosenblatt P. Segregating Shelter. *Annals Am Acad Polit Soc Sci* (2013) 647:268–99. doi:10.1177/0002716213479310

49. Adelman RM. Neighborhood Opportunities, Race, and Class: The Black Middle Class and Residential Segregation. *City & Community* (2004) 3:43–63. doi:10.1111/j.1535-6841.2004.00066.x

50. Thomas M, Moye R. Race, Class, and Gender and the Impact of Racial Segregation on Black-white Income Inequality. *Sociol Race Ethn* (2015) 1:490–502. doi:10.1177/2332649215581665

51. Florida R, Mellander C. *Segregated City: The Geography of Economic Segregation in America’s Metros*. Martin Prosperity Institute (2015).

52. Hartigan JA, Wong MA. Algorithm as 136: A K-Means Clustering Algorithm. *Appl Stat* (1979) 28:100–8. doi:10.2307/2346830

53. Likas A, Vlassis N, J. Verbeek J. The Global K-Means Clustering Algorithm. *Pattern Recognition* (2003) 36:451–61. doi:10.1016/s0031-3203(02)00060-2

54. Moran PAP. The Interpretation of Statistical Maps. *J R Stat Soc Ser B (Methodological)* (1948) 10:243–51. doi:10.1111/j.2517-6161.1948.tb00012.x

55. Fotheringham AS, Wong DWS. The Modifiable Areal Unit Problem in Multivariate Statistical Analysis. *Environ Plan A* (1991) 23:1025–44. doi:10.1068/a231025

56. Xu Y, Belyi A, Santi P, Ratti C. Quantifying Segregation in an Integrated Urban Physical-Social Space. *J R Soc Interf* (2019) 16:20190536. doi:10.1098/rsif.2019.0536

57. Tóth G, Wachs J, Di Clemente R, Á J, Ságvári B, Kertész J, et al. Inequality Is Rising where Social Network Segregation Interacts with Urban Topology. *Nat Commun* (2021) 12:1–9. doi:10.1038/s41467-021-21465-0

58. Bokányi E, Juhász S, Karsai M, Lengyel B. Universal Patterns of Long-Distance Commuting and Social Assortativity in Cities. *Scientific Rep* (2021) 11:1–10. doi:10.1038/s41598-021-00416-1

59. Moro E, Calacci D, Dong X, Pentland A. Mobility Patterns Are Associated with Experienced Income Segregation in Large Us Cities. *Nat Commun* (2021) 12:4633–10. doi:10.1038/s41467-021-24899-8

Keywords: urban segregation, spatial heterogeneity, synchronization, phase-oscillators, diffusion

Citation: Bassolas A, Gómez S and Arenas A (2022) Diffusion and Synchronization Dynamics Reveal the Multi-Scale Patterns of Spatial Segregation. *Front. Phys.* 10:833426. doi: 10.3389/fphy.2022.833426

Received: 11 December 2021; Accepted: 04 April 2022;

Published: 25 April 2022.

Edited by:

Haroldo V. Ribeiro, State University of Maringá, BrazilReviewed by:

Zoltan Neda, Babeș-Bolyai University, RomaniaMatjaž Perc, University of Maribor, Slovenia

Copyright © 2022 Bassolas, Gómez and Arenas. 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.

*Correspondence: Aleix Bassolas, aleix.bassolas@gmail.com