Abstract
The study of specific physiological processes from the perspective of network physiology has gained recent attention. Modeling the global information integration among the separated functionalized modules in structural and functional brain networks is a central problem. In this article, the preferentially cutting–rewiring operation (PCRO) is introduced to approximatively describe the above physiological process, which consists of the cutting procedure and the rewiring procedure with specific preferential constraints. By applying the PCRO on the classical Erdös–Rényi random network (ERRN), three types of isolated nodes are generated, based on which the common leaves (CLs) are formed between the two hubs. This makes the initially homogeneous ERRN experience drastic changes and become heterogeneous. Importantly, a statistical analysis method is proposed to theoretically analyze the statistical properties of an ERRN with a PCRO. Specifically, the probability distributions of these three types of isolated nodes are derived, based on which the probability distribution of the CLs can be obtained easily. Furthermore, the validity and universality of our statistical analysis method have been confirmed in numerical experiments. Our contributions may shed light on a new perspective in the interdisciplinary field of complexity science and biological science and would be of great and general interest to network physiology.
1 Introduction
The collective behaviors that emerged on different kinds of complex systems have become the central topics under investigation since the seminal “small-world” and “scale-free” network models were successively proposed by Strogatz and Barabási (; ). Several typical types of spatiotemporal dynamical behaviors, such as synchronous phenomena (; ; Zhang et al., 2015; ; ; ), self-sustained oscillations (; ; ; ; ; ), and chimera and chimeralike states (; ; ; ; ; ; Zhang H. et al., 2024), have been reported. For example, exposed the relay and complete synchronization in heterogeneous multiplex networks of chaotic maps. discussed the emergence of self-sustained patterns in small-world excitable media. Kachhara and Ambika revealed the frequency chimera state induced by differing dynamical timescales (). and Zhang H. et al. (2024), respectively, uncovered the chimeralike oscillation modes on excitable scale-free networks and the alternate attractor chimeralike states on rings of chaotic Lorenz-type oscillators.
One of the most classical network models, the Erdös–Rényi random network (ERRN), was proposed by P. Erdös and A. Rényi (; ) and is utilized to explore these issues. Many interesting phenomena were found, and great achievements were realized by this paradigmatic network model (; Xu and Liu, 2008; ; ; ; ; ; ; ; ; ). For example, Gong et al. discussed the synchronization of Erdös–Rényi networks (). Tattini et al. investigated the coherent periodic activity on excitatory Erdös–Rényi neural networks and exposed the key role of network connectivity (). discovered the scaling of percolation transitions on Erdös–Rényi networks under centrality-based attacks. Kartal et al. studied the complex dynamics of the COVID-19 mathematical model on the Erdös–Rényi network (). Qian et al. first reported the emergence of the self-sustained oscillations on excitable Erdös–Rényi random networks and exposed the determinants (; ) and then revealed the effects of time delay and connection probability on the corresponding oscillations and synchronization transitions ().
In addition to the collective behaviors that can self-organize to emerge on ERRNs consisting of different types of local units, the statistical properties of the ERRN are also an important issue. For example, Erdös and Rényi were the first to study the distribution of the maximum and minimum degree in a random graph (), and the full degree distribution was derived later by . Chung and Lu discussed the diameter of sparse random graphs (). investigated the return probabilities and hitting times of random walks on sparse Erdös–Rényi graphs. explored the random sequential renormalization and agglomerative percolation on Erdös–Rényi networks. first studied the distribution of diameters for Erdös–Rényi random graphs and then discussed the distribution of shortest path lengths of subcritical Erdös–Rényi networks (). However, whether the statistical properties of an ERRN with specific operation can be theoretically derived is still unknown.
Nowadays, investigating the physiological processes from the perspective of network physiology is an important topic in the interdisciplinary field of complexity science and biological science. Several excellent contributions were achieved in this field (; ; ; ; ; ; ; Zhang Z. et al., 2024; ). For example, summarized the new field of network physiology, that is, building the human physiolome. discussed the perspectives on understanding aberrant brain networks in epilepsy. reviewed the adaptive networks in functional modeling of physiological systems.
As we know, there exists a cost-efficiency trade-off between the physical cost of the network and the information integration among the whole system in organizing structural and functional brain networks. To save wiring costs, brain networks tend to build module structures to implement localized functions. To achieve global information integration among the separated functionalized modules, long-range synapses can be created via synaptic plasticity on these local structures. More importantly, the newly reshaped long-range synapses will preferentially connect to specific hub regions to fulfill special physiological functions among the whole brain systems. So we would ask whether the appropriate functional model can be proposed to describe this physiological process from the perspective of network physiology? We think this is an important issue of great and general interest to network physiology.
In this article, the preferentially cutting–rewiring operation (PCRO) is proposed to approximatively describe the above physiological process. We have found that, by applying the PCRO on the ERRN with certain preferential constraints, the topological feature of the given network will change dramatically. Then, the theoretical statistical properties of the operated ERRN are studied. The remainder of the article is organized as follows. Section 2 introduces the cutting and rewiring procedures of the PCRO. In Section 3, we apply the PCRO to the classical ERRN. The statistical analysis method proposed in Section 4 supports studying the theoretical statistical properties of an ERRN with a PCRO. The validity and the universality of our statistical analysis method are, respectively, confirmed in Sections 5, 6. Finally, we give the conclusion in the last section.
2 The cutting and rewiring procedures of a PCRO
In this part, we first introduce the PCRO, which is adopted to regulate the structure of a given network. The PCRO proposed in the present article contains two different stages: the cutting procedure and the rewiring procedure with specific preferential constraints, which are denoted by , (the preferential constraint indicators used in the cutting procedure) and (the constraint indicator in the rewiring procedure), respectively.The schematic diagram of the PCRO is presented in Figure 1, and the specific preferential constraints are selected as , , and . Here, presents the actual value of the average degree in the network before the cutting procedure, and is the maximum degree in the corresponding network after the cutting procedure. Figure 1A displays the initial structure without a PCRO, which consists of 16 nodes. The seven green cells and 15 denote the nodes possessing degrees within , which satisfy the preferential constraint in the cutting procedure. Consequently, these seven green cells are considered candidates for performing the following cutting operation. To do this, we introduce the PCRO probability condition to determine whether each selected candidate will be operated or not in the cutting operation. Specifically, for each green candidate, we execute the cutting operation with probability . If the PCRO probability condition is satisfied (i.e., ), all the connections of this candidate will be discarded; otherwise (i.e., ), they will be reserved completely. The yellow cell denotes the isolated node that initially existed in the network, which is named the naturally isolated node (NIN). The other gray cells indicate the remainder of ordinary nodes without any operation in the cutting procedure.
FIGURE 1
By applying the cutting procedure with a specific PCRO probability , the structure of the initial network will change dramatically, and the corresponding result is illustrated in Figure 1B. In the current case, the initially green candidates , and 15 satisfy the PCRO probability condition, and their corresponding connections are all discarded. These operated candidates are called actively deleted isolated nodes (ADINs) and are colored pink. Meanwhile, the green candidates and 9 do not meet the PCRO probability condition; all their links are reserved, and they turn into gray ordinary nodes. Furthermore, after the cutting procedure, the originally ordinary gray cell becomes an isolated node due to its links connecting to the initially green candidates and 14, which are deleted in the cutting procedure. This type of newly generated isolated node is defined as a passively deleted isolated node (PDIN) and is colored orange. Moreover, the red cells and 6 present the two nodes satisfying the preferential constraint in the rewiring procedure, that is, the nodes possessing degree after the cutting procedure, which are selected as the target nodes to be rewired in the following rewiring procedure.
As the cutting procedure is completed, the second stage of the PCRO starts, that is, the rewiring procedure. In this stage, the above three types of isolated nodes, that is, the yellow NIN, the pink ADIN, and the orange PDIN, will be rewired to the two red target nodes. In the following discussion, these rewired nodes are called the common leaves (CLs) and are colored blue. Figure 1C shows the network structure after the rewiring operation, in which many blue CLs are formed between the two red target nodes. This makes the two target nodes possess relatively large degrees, which are consequently called hubs. Based on the illustrations shown in Figure 1, we can conclude that the PCRO is an effective method of regulating the structure of the given network, by which many CLs can be formed between the two hubs. The initially homogeneous network will become heterogeneous by applying the PCRO.
3 The ERRNs with a PCRO
The classical ERRN is utilized to test the effects of the PCRO. A classical ERRN can be constructed only based on a simple rule, that is, the connections between every pair of nodes in the ERRN are linked with a specific connection probability. In the present article, the initially homogeneous ERRN without a PCRO is composed of nodes and is constructed with connection probability . Consequently, the total number of connections in the ERRN is expected to be . Here, we should mention that by manipulating the connection probability , one can produce a number of ERRNs with different properties. There are many network realizations for a given connection probability . Furthermore, these two network structure parameters, that is, and , will be used in the following if there are no special instructions.
Figures 2A–D display the heterogeneous ERRNs constructed for four different parameters of the PCRO, that is, four different PCRO probabilities [Figure 2A], [Figure 2B], [Figure 2C], and [Figure 2D]. These figures clearly show three types of nodes: red hubs, blue CLs, and gray ordinary cells. Importantly, they can visualize the increasing heterogeneity of the network with increasing PCRO probability. The four network structures shown in Figures 2A–D are obtained from the initial homogeneous ERRN by a PCRO with the same preferential constraints utilized in Figure 1 (i.e., , , and ). This further confirms that the PCRO method proposed here has the effect of regulating the structure of the given network, such that the initially homogeneous network structure becomes heterogeneous in a controlled way.
FIGURE 2
4 The statistical analysis method
Now, we would ask whether the statistical properties of the ERRN with a PCRO can be theoretically derived, especially the probability distribution of CL formed in the cutting and rewiring procedures. A statistical analysis method is proposed to explore this issue. As shown in Figure 1B, three types of isolated nodes exist in an ERRN with a PCRO: the yellow NIN, the pink ADIN, and the orange PDIN. The rewiring procedure of these isolated nodes with a certain preferential constraint changes them into the CLs between the two hubs. The probability distributions of NIN, ADIN, and PDIN (denoted by , , and , respectively) are the vital tasks we aim to analyze first, based on which the theoretical analysis of the probability distribution of CL (indicated by ) can be realized. Furthermore, as these three probability distributions of the isolated nodes are related to the node satisfying the preferential constraint in the cutting procedure (NSPCCP) [colored in green in Figure 1A], we first give the deduction of the analytical expression of the probability distribution of the NSPCCP (denoted by ).
4.1 The probability distribution of NSPCCPs
For an arbitrary ER random network with a specific system size and connection probability , we assume that the probability for each actual value of an edge in the network to be . So the edge distribution in the given ER random network satisfies the following binomial distribution:By calculating the statistical average for all possible , the expectation value of average edge in the network can be obtained asThe actual value of the average degree for each edge in the network obeysSimilarly, for the same ER random network with , the probability distribution for the degree of an arbitrary node in the given ER random network can also be described by the following binomial distribution:By calculating the statistical average for all possible , we can gain the expectation value of average degree in the network asThis indicates that for a given ER random network with nodes, is determined by the connection probability .
We further introduce an approximation that, for a given ER random network with nodes, there exists an equivalent connection probability that can lead to the expectation value of average degree approximatively equals to the actual value of average degree ; that is,With this approximation, one can conveniently obtain the relationship between the actual value of an edge in the network and asIn this case, can be considered the equivalent connection probability of an ER random network to acquire the actual value of edge we wanted. Moreover, for each actual value of edge , there exists a corresponding equivalent connection probability .
Based on the above approximation, we can apply the equivalent connection probability to approximatively obtain the conditional degree distribution of an ER random network with nodes and edges. In this case, the conditional probability for an arbitrary node in the network being the NSPCCP obeysNow, we assume that there exist NSPCCPs in the given ER random network. Therefore, the conditional probability distribution of for a given should satisfy the following binomial distribution:By calculating the statistical average for all possible , the distribution of NSPCCPs in an ER random network can be derived as
4.2 The probability distribution of ADINs
The probability distribution of ADINs can be conveniently obtained based on the . The ADINs are those nodes satisfying the preferential constraint in the cutting procedure, and they belong to the category of NSPCCP. They are selected by the given PCRO probability , and all links of these ADINs are deleted in the cutting procedure. Here, we assume that an ER random network with a given contains NSPCCPs, among which there exist ADINs. For a given , the conditional probability of should satisfy the binomial distributionBy calculating the statistical average for all possible , the probability distribution of ADINs in an ER random network can be derived as
4.3 The probability distribution of PDINs
We can further solve the probability distribution of PDINs with the aid of . Here, we first explore the conditional probability distribution of PDINs in an ER random network with a given . Consider an arbitrary node with degree in the network. We focus on an edge connected to node with degree . The conditional degree distribution of node (denoted by ) obeys the following binomial distributionHere, is the equivalent connection probability of obtaining an ER random network with a given . Because of the homogeneity of the ER network, the conditional degree distributions of the other remainder neighbors of the -th node (except neighbor ) should be the same and are described by Equation 2. If all of the -th node’s neighbors are selected by a given PCRO probability and all links of these selected neighbors are deleted in the cutting procedure, the -th node would become a PDIN. To realize this situation, the following three conditions should be satisfied simultaneously.
Condition I:
The -th node does not belong to the ADIN and NIN categories. The conditional probability in realizing condition I includes two cases. (i) When the degree of the -th node , the -th node is not selected in the PCRO with probability ; (ii) When the degree of -th node or , .
By combining (i) and (ii), can be written by the following piecewise function:
Condition II:
The degrees of the neighbor nodes of the -th node satisfy the preferential constraint in the cutting procedure, that is, . According to Equation 2, the conditional probability II should satisfy
Condition III:
All the neighbor nodes of the -th node are selected by the given , and all the links of these neighbors are deleted in the cutting procedure. The conditional probability III can be calculated as
By multiplying the above three conditional probabilities, we can obtain the conditional probability for the -th node being the PDIN in the case of specific As the -th node is an arbitrary element in the given ERRN, its degree distribution should also satisfy the conditional binomial distribution of Equation 2. By calculating the statistical average for all possible of the -th node, the above conditional probability for a given can be obtained asBy further calculating the statistical average for all possible and , the probability for an arbitrary node in the ERRN being the PDIN can be derived as
Based on the above , the probability distribution of PDINs in an ER random network can be easily obtained according to the binomial distribution. Here, we assume that PDINs exist in the network. The probability distribution of PDINs then follows
4.4 The probability distribution of NINs
Here, we discuss the last type of isolated node, that is, the NINs colored in yellow in Figure 1B. For an arbitrary ER random network with a given , an NIN can be considered the node with degree . According to the degree distribution in the given ER random network of Equation 1, the probability that an arbitrary node in the network is a NIN can be obtained asHere, we assume that NINs exist in the ER random network. So the probability distribution of NINs obeys the following binomial distribution:
4.5 The probability distribution of CLs
Thus far, the probability distributions of NINs, ADINs, and PDINs have been derived analytically. Furthermore, these three types of isolated nodes are rewired in the rewiring procedure with a certain preferential constraint and become the CLs between the two hubs. As , , and are uncorrelated with each other, the probability distribution of CLs can be gained based on the superposition principle as
5 The validity and applicability of the statistical analysis method
In this section, we try to apply the statistical analysis method on an ERRN with a PCRO to verify the correctness of the above conclusions. The preferential constraints , , and , and the network structure parameters and , which are used as the example in Figure 2, are still utilized in this part.
We first test the probability distributions of ADINs, PDINs, NINs, and CLs in the operated ERRNs with a specific PCRO probability. The corresponding results for are respectively displayed in Figures 3A–D, where the numerical simulations (blue circles) and theoretical predictions (red curves) coincide very well. In simulations, samples are performed for each set of parameters, and this standard will also be implemented in the following tests. The consistency of the results exposed in Figure 3 confirms the correctness of the statistical analysis method proposed in the present article.
FIGURE 3
Now, we further verify the statistical analysis method for other PCRO probabilities. Here, we only utilize the probability distribution of CLs as the example, which is derived based on the probability distributions of ADINs, PDINs, and NINs. Figures 4A–C respectively, reveal the numerical results (blue circles) and theoretical predictions (red curves) of the probability distribution of CLs in the operated ERRNs for different PCRO probabilities [Figure 4A], [Figure 4B] and [Figure 4C]. It is displayed explicitly in Figure 4 that, even for different PCRO probabilities, the theoretical predictions can still match well with the experimental data. This strongly confirms the validity and applicability of our statistical analysis method.
FIGURE 4
Based on the above results, we can now apply the statistical analysis method to forecast the average number of CLs in the operated ERRN, which are formed in the cutting and rewiring procedures of the PCRO and largely determined by the PCRO probability condition. Figure 5 presents the numerical results (blue circles) and theoretical predictions (red curves) of the average number of CLs in the operated ERRN on the PCRO probability . The numerical and theoretical numbers of CLs both increase significantly as the PCRO probability increases, confirming not only the effectiveness of the statistical analysis method proposed here but also the effects of the PCRO in regulating the structure of the given network. At this point, the validity and applicability of the statistical analysis method have been verified in the ERRNs with a PCRO.
FIGURE 5
6 The universality of the statistical analysis method
It is necessary to inspect the universality of our statistical analysis method. The average number of CLs in the operated ERRNs with a PCRO is still utilized as an example to explore this issue. Based on the preferential constraints and the network structure parameters utilized in Figure 2, we first test our method with other preferential constraints in the PCRO. The corresponding results are respectively displayed in Figure 6A (, , and , that is, only the preferential constraint in the cutting procedure is changed); Figure 6B (, , and , that is, only the preferential constraint in the rewiring procedure is changed); and Figure 6C (, , , that is, both the preferential constraints in the cutting and rewiring procedures are changed). Here, and present the actual values of the average degree in the network before and after the cutting procedure. and are the maximum degrees in the corresponding network. The theoretical predictions (red curves) revealed with other preferential constraints coincide well with the numerical results (blue circles). This confirms that the statistical analysis method proposed here is irrelevant to the preferential constraints in the PCRO.
FIGURE 6
The statistical analysis method is applicable to general ERRN structures. The preferential constraints are the same as those in Figure 2, and the PCRO probability is employed for the following discussion. The theoretical predictions (red curves) and the numerical results (blue circles) obtained for other different connection probabilities and system sizes are displayed in Figures 7A, B, respectively. Note good alignment between theoretical predictions and numerical results. This further verifies the universality of our statistical analysis method. Here, we should also mention that the idea of the theoretical deduction can be extended and applied to other paradigmatic network models, such as homogeneous random networks, small-world networks, and even scale-free networks. However, the corresponding formulas and conclusions may not be the same.
FIGURE 7
7 Mimicking epileptic-seizure-related synchronization phenomena in brain systems
Modeling specific physiological processes and physiological functions from the perspective of network physiology should be discussed. In this part, we use the PCRO method proposed in this article to mimic the epileptic-seizure-related synchronization phenomena in pathological brain systems. This issue was first studied from the perspective of network physiology by and . They revealed that, in addition to the empirical brain network, the small-world networks with intermediate rewiring probability can also reproduce the epileptic-seizure-related synchronization phenomena that closely resemble the ones seen during epileptic seizures in humans (see Figure 7 in and Figure 3 in ). In this case, the corresponding network structure properties are found to be (the average clustering coefficient) and (the mean shortest path length), by which the balance of regularity and randomness of the given network is revealed. The authors claimed that the network topology with a certain balance of regularity and randomness is the key factor in determining the self-initiation and self-termination of episodes of seizure-like synchronization.
Based on the discussions presented in the above sections, we can conclude that the proposed PCRO method also has the effect of regulating the network structure. By applying the PCRO to the paradigmatic network models, the corresponding structures will undergo a transition from homogeneous to heterogeneous. So we would ask whether a similar balance of regularity and randomness can also be induced by the PCRO, by which the same epileptic-seizure-related synchronization phenomena can be mimicked by our scenario.
An ERRN consisting of FitzHugh–Nagumo (FHN) neurons with the rotational coupling scheme is utilized to address this issue. The system size and the parameter setting of the FHN network (, , and ) are all same as the ones adopted by and except for the coupling strength . The connection probability of the initial ERRN is chosen as . Figure 8A first displays the dependence of the average clustering coefficient (red line) and the mean shortest path length (black line) of the operated FHN network on the PCRO probability . The preferential constraints are selected as , , and . It is shown that, as the PCRO probability increases, the average clustering coefficient decreases gradually, while the mean shortest path length increases. Importantly, as approaches 1.0, the approximate network structure properties of and are those for the emergence of epileptic-seizure-related synchronization on the small-world network with intermediate rewiring probability can be obtained. This means that, in these parameter regions, our PCRO method can also induce a certain balance of regularity and randomness on the operated FHN network, based on which the corresponding epileptic-seizure-related synchronization phenomena are expected to be observed.
FIGURE 8
The global Kuramoto order parameter (black curve) of the operated FHN network obtained at a PCRO probability of is revealed in Figure 8B. The horizontal red line marks the threshold of . The definition of the epileptic-seizure-related synchronization, which was first introduced by Schöll et al., is also utilized here; that is, the global Kuramoto order parameter should satisfy for more than 8 s. The two vertical pink dashed lines, respectively, indicate the instants of the self-initiation and the self-termination of episodes of seizure-like synchronization, between which an epileptic-seizure-related synchronization can be identified on the FHN network with a PCRO. Figure 8C shows the corresponding space-time plot of the dynamical phases by which the PCRO method that induced the epileptic-seizure-related synchronization is further verified. These results can confirm the possible application of the PCRO method proposed in this article in mimicking specific physiological phenomena in real cases.
8 Conclusion
In conclusion, a preferentially cutting–rewiring operation is proposed in the present article to regulate the structure of the given network. It consists of two distinct stages: the cutting procedure and the rewiring procedure with specific preferential constraints. By applying the PCRO on the classical ERRN with specific constraints and a certain PCRO probability, the initially homogeneous structure changes drastically. Three types of isolated nodes are generated: the NINs, the ADINs, and the PDINs, based on which the CLs are formed between the two hubs in the operated network. Furthermore, as the PCRO probability increases, the number of CLs increases significantly, which makes the initially homogeneous ERRN become heterogeneous. This confirms that the PCRO introduced in this article has effects on regulating the network structure.
The statistical properties of the ERRN with a PCRO are theoretically studied using a statistical analysis method. We have analytically derived the statistical expressions of the probability distributions of NINs, ADINs, and PDINs, based on which the probability distribution of CLs is acquired easily. More importantly, the theoretical predictions obtained from these analytical formulas have been confirmed in numerical simulations and coincide with the experimental data very well. Furthermore, these analytical expressions are applied to forecast the average number of CLs in the operated ERRN. The coincidence of the numerical and theoretical results confirms the validity and applicability of the statistical analysis method proposed here. Finally, the universality of the statistical analysis method has also been verified. Our method is general and can be applied to ERRNs with arbitrary preferential constraints and topologies.
Modeling specific physiological processes and physiological functions from the perspective of network physiology is an important and central issue under investigation in the interdisciplinary field of complexity science and biological science. The PCRO method proposed in this article, which consists of the cutting procedure and the rewiring procedure, may give us a clue in understanding the physiological process of the global information integration among the localized functional modules in structural and functional brain networks to implement specific physiological functions. The reasons are as follows. In the PCRO cutting procedure, we discard the links of the nodes satisfying the preferential constraint with a certain PCRO probability condition, and isolated nodes are produced in the original network. As these isolated nodes originally satisfied the given preferential attribute, they can be roughly considered the localized functional modules in anatomical space, which are selected and will be integrated into achieving global information communication across the whole brain system for specific physiological functions. In the PCRO rewiring procedure, we reconnect these isolated nodes to the hubs satisfying the corresponding preferential constraint, which can be roughly regarded as the specialized regulatory centers (i.e., the hub regions) in brain networks to perform global physiological functions among the whole brain system. Furthermore, in a recent contribution (), the PCRO-induced oscillation mode transition from the originally single-mode oscillations to the newly multi-mode oscillations has been confirmed to emerge among the preferentially operated nodes (i.e., the integrated local modules), which we think is beneficial for understanding the complicated global multimodal physiological functions in integrated structural and functional brain networks. More importantly, the probability distributions of the three different types of isolated nodes formed in the PCRO are derived according to the statistical analysis method proposed in this article, based on which the probability distribution of the preferentially operated common leaves is acquired explicitly. We think the statistical analysis method and the precise theoretical formulas exposed in this article can shed light on a deep comprehension of these amazing physiological phenomena in highly complex and heterogeneous brain networks. We do hope our results will be of great interest to network physiology.
Statements
Data availability statement
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.
Author contributions
YQ: conceptualization, data curation, formal analysis, funding acquisition, investigation, methodology, project administration, resources, software, supervision, validation, visualization, writing–original draft, and writing–review and editing. JC: formal analysis, investigation, software, and writing–original draft. JH: data curation, formal analysis, investigation, methodology, resources, and writing–review and editing. SZ: resources, software, and writing–review and editing. WC: data curation, formal analysis, and writing–review and editing. ZL: conceptualization, data curation, formal analysis, funding acquisition, investigation, methodology, project administration, resources, software, supervision, validation, visualization, writing–original draft, and writing–review and editing. XC: conceptualization, data curation, formal analysis, funding acquisition, investigation, methodology, project administration, resources, software, supervision, validation, visualization, writing–original draft, and writing–review and editing. ZZ: conceptualization, data curation, formal analysis, funding acquisition, investigation, methodology, project administration, resources, software, supervision, validation, visualization, writing–original draft, and writing–review and editing.
Funding
The author(s) declare financial support was received for the research, authorship, and/or publication of this article. This work is supported by the National Natural Science Foundation of China (Grant Nos. 12375033, 12375031, and 62402010), the Natural Science Basic Research Plan in Shaanxi Province of China (Grant Nos. 2022JZ-03, 2022GD-TSLD-27, and 2024SF-YBXM-134), the Shaanxi Fundamental Science Research Project for Mathematics and Physics (Grant Nos. 22JSY021 and 23JSQ051), the Baoji University of Arts and Sciences Innovative Research Project of Postgraduates (Grant No. YJSCX24YB42), the Youth Innovation Team of Shaanxi Universities, and the Key Science and Technology Innovation Team of Shaanxi Province (Grant No. 2024RSCXTD66).
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.
References
1
AlmeiraN.BilloniO. V.PerottiJ. I. (2020). Scaling of percolation transitions on Erdös-Rényi networks under centrality-based attacks. Phys. Rev. E101, 012306. 10.1103/PhysRevE.101.012306
2
AndrzejakR. G.RuzzeneG.MalvestioI. (2017). Generalized synchronization between chimera states. Chaos27, 053114. 10.1063/1.4983841
3
BarabásiA.-L.AlbertR. (1999). Emergence of scaling in random networks. Science286, 509–512. 10.1126/science.286.5439.509
4
BizhaniG.GrassbergerP.PaczuskiM. (2011). Random sequential renormalization and agglomerative percolation in networks: application to Erdös-Rényi and scale-free graphs. Phys. Rev. E84, 066111. 10.1103/PhysRevE.84.066111
5
BollobásB. (1981). Degree sequences of random graphs. Discrete Math.33, 1–19. 10.1016/0012-365x(81)90253-3
6
ChungF.LuL. (2001). The diameter of sparse random graphs. Adv. Appl. Math.26, 257–279. 10.1006/aama.2001.0720
7
ErdösP.RényiA. (1959). On random graphs i. Publ. Math. Debr.6, 290–297. 10.5486/pmd.1959.6.3-4.12
8
ErdösP.RényiA. (1960). On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci.5, 17.
9
FerrariU.LucibelloC.MoroneF.ParisiG.Ricci-TersenghiF.RizzoT. (2013). Finite-size corrections to disordered systems on Erdös-Rényi random graphs. Phys. Rev. B88, 184201. 10.1103/physrevb.88.184201
10
FretterC.LesneA.HilgetagC. C.HüttM. T. (2017). Topological determinants of self-sustained activity in a simple model of excitable dynamics on graphs. Sci. Rep.7, 42340. 10.1038/srep42340
11
GersterM.BernerR.SawickiJ.ZakharovaA.S̆kochA.HlinkaJ.et al (2020). FitzHugh-Nagumo oscillators on complex networks mimic epileptic-seizure-related synchronization phenomena. Chaos30, 123130. 10.1063/5.0021420
12
GhoshS.SarG. K.MajhiS.GhoshD. (2023). Antiphase synchronization in a population of swarmalators. Phys. Rev. E108, 034217. 10.1103/physreve.108.034217
13
GongB.YangL.YangK. (2005). Synchronization on erdös-rényi networks. Phys. Rev. E72, 037101. 10.1103/PhysRevE.72.037101
14
GoodfellowM.AndrzejakR. G.MasollerC.LehnertzK. (2022). What models and tools can contribute to a better understanding of brain activity?Front. Netw. Physiology2, 907995. 10.3389/fnetp.2022.907995
15
HagerstromA. M.MurphyT. E.RoyR.HövelP.OmelchenkoI.SchöllE. (2012). Experimental observation of chimeras in coupled-map lattices. Nat. Phys.8, 658–661. 10.1038/nphys2372
16
HartmannA. K.MézardM. (2018). Distribution of diameters for Erdös-Rényi random graphs. Phys. Rev. E97, 032128. 10.1103/PhysRevE.97.032128
17
IseleT.SchöllE. (2015). Effect of small-world topology on wave propagation on networks of excitable elements. New J. Phys.17, 023058. 10.1088/1367-2630/17/2/023058
18
IvanovP. C. (2021). The new field of network physiology: building the human physiolome. Front. Netw. Physiology1, 711778. 10.3389/fnetp.2021.711778
19
KachharaS.AmbikaG. (2021). Frequency chimera state induced by differing dynamical timescales. Phys. Rev. E104, 064214. 10.1103/PhysRevE.104.064214
20
KartalN.KartalS. (2023). Complex dynamics of COVID-19 mathematical model on Erdös-Rényi network. Int. J. Biomathematics5, 2250110. 10.1142/s1793524522501108
21
KatzavE.BihamO.HartmannA. K. (2018). Distribution of shortest path lengths in subcritical Erdös-Rényi networks. Phys. Rev. E98, 012301. 10.1103/PhysRevE.98.012301
22
LeiZ.PuS.ZhangH.YaoC.QianY.ZhengZ. (2022). Bistability-induced chimeras in one-dimensional paced excitable rings with nonlocal couplings. Europhys. Lett.139, 62001. 10.1209/0295-5075/ac8e92
23
LeiZ.ZhangC.WangY.WeiZ.QianY.ZhengZ. (2023). Chimeralike oscillation modes in excitable scale-free networks. Phys. Rev. Res.5, 013006. 10.1103/physrevresearch.5.013006
24
MartinO. C.S̆ulcP. (2010). Return probabilities and hitting times of random walks on sparse Erdös-Rényi graphs. Phys. Rev. E81, 031111. 10.1103/physreve.81.031111
25
MasoumiR.OloomiF.SajjadiS.ShiraziA. H.JafariG. R. (2022). Modified heider balance on erdös-rényi networks. Phys. Rev. E106, 034309. 10.1103/PhysRevE.106.034309
26
OmelchenkoI.Omel’chenkoO. E.HövelP.SchöllE. (2013). When nonlocal coupling between oscillators becomes stronger: patched synchrony or multichimera states. Phys. Rev. Lett.110, 224101. 10.1103/PhysRevLett.110.224101
27
QianY. (2014). Emergence of self-sustained oscillations in excitable Erdös-Rényi random networks. Phys. Rev. E90, 032807. 10.1103/PhysRevE.90.032807
28
QianY.ChenZ.YangR.GaoH.LeiZ.ZhengZ. (2024a). Remote response modes on the paced excitable C. elegans network. Phys. Rev. Res.6, 033014. 10.1103/physrevresearch.6.033014
29
QianY.CuiX.ZhengZ. (2017). Minimum Winfree loop determines self-sustained oscillations in excitable Erdös-Rényi random networks. Sci. Rep.7, 5746. 10.1038/s41598-017-06066-6
30
QianY.HanJ.YangR.ChenW.LeiZ.ZhengZ. (2024b). Emergence of chimeralike oscillation modes in excitable complex networks with preferentially cutting-rewiring operation. Chaos34, 063143. 10.1063/5.0167251
31
QianY.HuangX.HuG.LiaoX. (2010a). Structure and control of self-sustained target waves in excitable small-world networks. Phys. Rev. E81, 036101. 10.1103/PhysRevE.81.036101
32
QianY.LiaoX.HuangX.MiY.ZhangL.HuG. (2010b). Diverse self-sustained oscillatory patterns and their mechanisms in excitable small-world networks. Phys. Rev. E82, 026107. 10.1103/PhysRevE.82.026107
33
QianY.ZhangZ. (2017). Effects of time delay and connection probability on self-sustained oscillations and synchronization transitions in excitable Erdös-Rényi random networks. Commun. Nonlinear Sci. Numer. Simul.47, 127–138. 10.1016/j.cnsns.2016.11.012
34
RosenblumM. (2024). Feedback control of collective dynamics in an oscillator population with time-dependent connectivity. Front. Netw. Physiology4, 1358146. 10.3389/fnetp.2024.1358146
35
RoxinA.RieckeH.SollaS. A. (2004). Self-sustained activity in a small-world network of excitable neurons. Phys. Rev. Lett.92, 198101. 10.1103/PhysRevLett.92.198101
36
RybalovaE.StrelkovaG.SchöllE.AnishchenkoV. (2020). Relay and complete synchronization in heterogeneous multiplex networks of chaotic maps. Chaos30, 061104. 10.1063/5.0008902
37
SchmeltzerC.SorianoJ.SokolovI. M.RüdigerS. (2014). Percolation of spatially constrained Erdös-Rényi networks with degree correlations. Phys. Rev. E89, 012116. 10.1103/PhysRevE.89.012116
38
SchöllE. (2021). Partial synchronization patterns in brain networks. Europhys. Lett.136, 18001. 10.1209/0295-5075/ac3b97
39
SchöllE.SawickiJ.BernerR.IvanovP. C. (2022). Editorial: adaptive networks in functional modeling of physiological systems. Front. Netw. Physiology2, 996784. 10.3389/fnetp.2022.996784
40
SemenovaN.ZakharovaA.AnishchenkoV.SchöllE. (2016). Coherence-resonance chimeras in a network of excitable elements. Phys. Rev. Lett.117, 014102. 10.1103/PhysRevLett.117.014102
41
ShiJ.KiriharaK.TadaM.FujiokaM.UsuiK.KoshiyamaD.et al (2022). Criticality in the healthy brain. Front. Netw. Physiology1, 755685. 10.3389/fnetp.2021.755685
42
SinhaN.JoshiR. B.SandhuM. R. S.NetoffT. I.ZaveriH. P.LehnertzK. (2022). Perspectives on understanding aberrant brain networks in epilepsy. Front. Netw. Physiology2, 868092. 10.3389/fnetp.2022.868092
43
SinhaS.SaramäkiJ.KaskiK. (2007). Emergence of self-sustained patterns in small-world excitable media. Phys. Rev. E76, 015101(R). 10.1103/PhysRevE.76.015101
44
TattiniL.OlmiS.TorciniA. (2012). Coherent periodic activity in excitatory Erdös-Rényi neural networks: the role of network connectivity. Chaos22, 023133. 10.1063/1.4723839
45
VenkadeshS.ShaikhA.ShakeriH.BarretoE.Van HornJ. D. (2024). Biophysical modulation and robustness of itinerant complexity in neuronal networks. Front. Netw. Physiology4, 1302499. 10.3389/fnetp.2024.1302499
46
WalterS.NunnenkampA.BruderC. (2014). Quantum synchronization of a driven self-sustained oscillator. Phys. Rev. Lett.112, 094102. 10.1103/PhysRevLett.112.094102
47
WattsD. J.StrogatzS. H. (1998). Collective dynamics of ‘small-world’ networks. Nature393, 440–442. 10.1038/30918
48
WuY.WangN.LiL.XiaoJ. (2012). Anti-phase synchronization of two coupled mechanical metronomes. Chaos22, 023146. 10.1063/1.4729456
49
XuX.LiuF. (2008). Continuous-time quantum walks on Erdös-Rényi networks. Phys. Lett. A372, 6727–6732. 10.1016/j.physleta.2008.09.042
50
ZhangH.ChenZ.LiuF.LeiZ.ZhengZ.QianY. (2024a). Alternate attractor chimeralike states on rings of chaotic Lorenz-type oscillators. New J. Phys.26, 023016. 10.1088/1367-2630/ad2411
51
ZhangX.BoccalettiS.GuanS.LiuZ. (2015). Explosive synchronization in adaptive and multilayer networks. Phys. Rev. Lett.114, 038701. 10.1103/PhysRevLett.114.038701
52
ZhangZ.WangX.LiH.ChenY.QuZ.MiY.et al (2024b). Uncovering hidden nodes and hidden links in complex dynamic networks. Sci. China-Physics, Mech. and Astronomy67, 240511. 10.1007/s11433-023-2303-7
Summary
Keywords
network physiology, biological science, brain networks, complex systems, network models
Citation
Qian Y, Cao J, Han J, Zhang S, Chen W, Lei Z, Cui X and Zheng Z (2024) A statistical analysis method for probability distributions in Erdös–Rényi random networks with preferential cutting–rewiring operation. Front. Netw. Physiol. 4:1390319. doi: 10.3389/fnetp.2024.1390319
Received
23 February 2024
Accepted
27 September 2024
Published
17 October 2024
Volume
4 - 2024
Edited by
Eckehard Schöll, Technical University of Berlin, Germany
Updates
Copyright
© 2024 Qian, Cao, Han, Zhang, Chen, Lei, Cui and Zheng.
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: Yu Qian, qianyu0272@163.com; Zhao Lei, leizhao1010@163.com; Xiaohua Cui, xhcui@bnu.edu.cn; Zhigang Zheng, zgzheng@hqu.edu.cn
Disclaimer
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.