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 (Watts and Strogatz, 1998; Barabási and Albert, 1999). Several typical types of spatiotemporal dynamical behaviors, such as synchronous phenomena (Wu et al., 2012; Walter et al., 2014; Zhang et al., 2015; Andrzejak et al., 2017; Rybalova et al., 2020; Ghosh et al., 2023), self-sustained oscillations (Roxin et al., 2004; Sinha et al., 2007; Qian et al., 2010a; Qian et al., 2010b; Isele and Schöll, 2015; Fretter et al., 2017), and chimera and chimeralike states (Hagerstrom et al., 2012; Omelchenko et al., 2013; Semenova et al., 2016; Kachhara and Ambika, 2021; Lei et al., 2022; Lei et al., 2023; Zhang H. et al., 2024), have been reported. For example, Rybalova et al. (2020) exposed the relay and complete synchronization in heterogeneous multiplex networks of chaotic maps. Sinha et al. (2007) 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 (Kachhara and Ambika, 2021). Lei et al. (2023) 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 (Erdös and Rényi, 1959; Erdös and Rényi, 1960) and is utilized to explore these issues. Many interesting phenomena were found, and great achievements were realized by this paradigmatic network model (Gong et al., 2005; Xu and Liu, 2008; Tattini et al., 2012; Ferrari et al., 2013; Schmeltzer et al., 2014; Almeira et al., 2020; Masoumi et al., 2022; Kartal and Kartal, 2023; Qian, 2014; Qian et al., 2017; Qian and Zhang, 2017). For example, Gong et al. discussed the synchronization of Erdös–Rényi networks (Gong et al., 2005). Tattini et al. investigated the coherent periodic activity on excitatory Erdös–Rényi neural networks and exposed the key role of network connectivity (Tattini et al., 2012). Almeira et al. (2020) 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 (Kartal and Kartal, 2023). Qian et al. first reported the emergence of the self-sustained oscillations on excitable Erdös–Rényi random networks and exposed the determinants (Qian, 2014; Qian et al., 2017) and then revealed the effects of time delay and connection probability on the corresponding oscillations and synchronization transitions (Qian and Zhang, 2017).

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 (Erdös and Rényi, 1959), and the full degree distribution was derived later by Bollobás (1981). Chung and Lu discussed the diameter of sparse random graphs (Chung and Lu, 2001). Martin and S̆ulc (2010) investigated the return probabilities and hitting times of random walks on sparse Erdös–Rényi graphs. Bizhani et al. (2011) explored the random sequential renormalization and agglomerative percolation on Erdös–Rényi networks. Hartmann and Mézard (2018) 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 (Katzav et al., 2018). 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 (Ivanov, 2021; Shi et al., 2022; Sinha et al., 2022; Goodfellow et al., 2022; Schöll et al., 2022; Venkadesh et al., 2024; Rosenblum, 2024; Zhang Z. et al., 2024; Qian et al., 2024a). For example, Ivanov (2021) summarized the new field of network physiology, that is, building the human physiolome. Sinha et al. (2022) discussed the perspectives on understanding aberrant brain networks in epilepsy. Schöll et al. (2022) 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.

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 k 1 , k 2 (the preferential constraint indicators used in the cutting procedure) and k 3 (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 k 1 = 1 , k 2 = ⟨ k ⟩ A V , and k 3 = k M a x ′ . Here, ⟨ k ⟩ A V presents the actual value of the average degree in the network before the cutting procedure, and k M a x ′ 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 i = 2,3,5,9,12,14 , and 15 denote the nodes possessing degrees within k 1 ≤ k i ≤ k 2 , 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 P P C R O 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 p . If the PCRO probability condition is satisfied (i.e., p ≤ P P C R O ), all the connections of this candidate will be discarded; otherwise (i.e., p > P P C R O ), they will be reserved completely. The yellow cell i = 10 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.

By applying the cutting procedure with a specific PCRO probability P P C R O , 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 i = 2, 5, 12, 14 , 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 i = 3 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 i = 4 becomes an isolated node due to its links connecting to the initially green candidates i = 5 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 i = 1 and 6 present the two nodes satisfying the preferential constraint in the rewiring procedure, that is, the nodes possessing degree k i = k 3 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.

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 N = 100 nodes and is constructed with connection probability P E R = 0.05 . Consequently, the total number of connections in the ERRN is expected to be P E R N ( N − 1 ) / 2 . Here, we should mention that by manipulating the connection probability P E R , one can produce a number of ERRNs with different properties. There are many network realizations for a given connection probability P E R . Furthermore, these two network structure parameters, that is, N = 100 and P E R = 0.05 , 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 P P C R O = 0.25 [Figure 2A], P P C R O = 0.50 [Figure 2B], P P C R O = 0.75 [Figure 2C], and P P C R O = 1.00 [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., k 1 = 1 , k 2 = ⟨ k ⟩ A V , and k 3 = k M a x ′ ). 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.

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 P N I N , P A D I N , and P P D I N , respectively) are the vital tasks we aim to analyze first, based on which the theoretical analysis of the probability distribution of CL (indicated by P C L ) 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 P N S P C C P ).

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 k 1 = 1 , k 2 = ⟨ k ⟩ A V , and k 3 = k M a x ′ , and the network structure parameters N = 100 and P E R = 0.05 , 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 P P C R O = 0.50 are respectively displayed in Figures 3A–D, where the numerical simulations (blue circles) and theoretical predictions (red curves) coincide very well. In simulations, S = 1 0 6 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.

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 P P C R O = 0.25 [Figure 4A], P P C R O = 0.50 [Figure 4B] and P P C R O = 0.75 [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.

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 N ̄ C L in the operated ERRN on the PCRO probability P P C R O . 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.

It is necessary to inspect the universality of our statistical analysis method. The average number of CLs N ̄ C L 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 ( k 1 = ⟨ k ⟩ A V , k 2 = k M a x , and k 3 = k M a x ′ , that is, only the preferential constraint in the cutting procedure is changed); Figure 6B ( k 1 = 1 , k 2 = ⟨ k ⟩ A V , and k 3 = ⟨ k ⟩ A V ′ , that is, only the preferential constraint in the rewiring procedure is changed); and Figure 6C ( k 1 = ⟨ k ⟩ A V , k 2 = k M a x , k 3 = ⟨ k ⟩ A V ′ , that is, both the preferential constraints in the cutting and rewiring procedures are changed). Here, ⟨ k ⟩ A V and ⟨ k ⟩ A V ′ present the actual values of the average degree in the network before and after the cutting procedure. k M a x and k M a x ′ 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.

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 P P C R O = 0.50 is employed for the following discussion. The theoretical predictions (red curves) and the numerical results (blue circles) obtained for other different connection probabilities P E R and system sizes N 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.

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 Gerster et al. (2020) and Schöll (2021). 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 Gerster et al. (2020) and Figure 3 in Schöll (2021)). In this case, the corresponding network structure properties are found to be C = 0.25 (the average clustering coefficient) and L = 3.0 (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 ( N = 90 ) and the parameter setting of the FHN network ( α = 0.5 , ε = 0.05 , and φ = π 2 − 0.1 ) are all same as the ones adopted by Gerster et al. (2020) and Schöll (2021) except for the coupling strength σ = 0.07 . The connection probability of the initial ERRN is chosen as P E R = 0.032 . Figure 8A first displays the dependence of the average clustering coefficient C (red line) and the mean shortest path length L (black line) of the operated FHN network on the PCRO probability P P C R O . The preferential constraints are selected as k 1 = 1 , k 2 = ⟨ k ⟩ A V , and k 3 = k M a x ′ . It is shown that, as the PCRO probability increases, the average clustering coefficient decreases gradually, while the mean shortest path length increases. Importantly, as P P C R O approaches 1.0, the approximate network structure properties of C and L 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.

The global Kuramoto order parameter r (black curve) of the operated FHN network obtained at a PCRO probability of P P C R O = 0.97 is revealed in Figure 8B. The horizontal red line marks the threshold of r = 0.8 . 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 r > 0.8 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.

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 (Qian et al., 2024b), 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.