Spontaneous Formation of Dynamical Groups in an Adaptive Networked System

In this work, we investigate a model of an adaptive networked dynamical system, where the coupling strengths among phase oscillators coevolve with the phase states. It is shown that in this model the oscillators can spontaneously differentiate into two dynamical groups after a long time evolution. Within each group, the oscillators have similar phases, while oscillators in different groups have approximately opposite phases. The network gradually converts from the initial random structure with a uniform distribution of connection strengths into a modular structure which is characterized by strong intra connections and weak inter connections. Furthermore, the connection strengths follow a power law distribution, which is a natural consequence of the coevolution of the network and the dynamics. Interestingly, it is found that if the inter connections are weaker than a certain threshold, the two dynamical groups will almost decouple and evolve independently. These results are helpful in further understanding the empirical observations in many social and biological networks.

Modularity frequently occurs in many social and biological networked systems [1], which is generally believed to correspond to certain functional groups [2]. Usually, in modular networks, the intra connections are stronger than the inter connections [3,4,5,6]. However, they both play important roles in maintaining the network structure and functions. P. Csermely pointed out that the strong links can define the system, while the weak links are crucial to the stabilization of complex system [7]. Such examples can be found in many situations, such as the connectivity of social networks [3], group survival [8], social efficiency [9], firm efficiency [10], and ecosystem stability [11]. Furthermore, in many networks, such as the natural food webs [12], mobile networks [3], author collaboration networks [13], metabolic netwoks [14] and neural networks [15], it is found that most of interactions are weak, and only a few interactions are strong, which usually leads to a power law distribution of the connection strengths [3,13,14,15].
In the past decade, there are extensive works exploring networked complex systems. Mainly, these works focus on either the topological structures of the networks [16], or the dynamics on the networks [17]. Nevertheless, in various realistic systems, especially the biological and social systems, in principle the network topology and dynamics are strongly dependent on each other. Thus any network structures and dynamical patterns that emerged are actually the results of the coevolution of the network dynamics and topology [18]. For example, the change of the synaptic coupling strength between neurons depends on the relative timing of the presynaptic and postsynaptic spikes in neural networks [19], and in the mobile communication networks [3], the connection strengths are determined by the dynamical behaviors of the mobile agents.
Recently attentions have been paid to the adaptive coevolutionary networks [18]. These include the adaptive rewiring links [20,21], and the adaptive altering connection strength [22,23,24,25] based on the states of local dynamics. However, the previous studies still focus mainly on the topological properties of the networks, while neglecting the dynamical evolution and characteristics, which are actually one very important aspect of networked dynamical systems. We noticed that in many social and biological networked systems, with the evolution of the network topology, dynamically the system may form different functional groups corresponding to different dynamical states. One such example is the mammalian brain, in which the connections are plastic [19]. It is known that the mammalian brain is composed of a number of functional groups, within which the nodes can be regarded as sharing similar dynamical states. However, so far, how the dynamical groups are generated during the coevolution of network structure and dynamics has not been investigated from the point of view of complex networks.
Motivated by this idea, in the present work, we set up a toy adaptive network model consisting of phase oscillators. Due to the simplicity of the dynamics, phase oscillators have been frequently used to describe many simplified real dynamical systems, such as biological networks, chemical oscillators and so on [26]. In our model, the coupling functions adopt the higher order Fourier modes, and the connection strengths are coupled with the local dynamical states following the plasticity function. Particularly, we investigate what kinds of the dynamical states and network structures can be formed as the result of the coevolution of both network dynamics and topology. Mainly, our study presents three new results. (i) The dynamical groups can be spontaneously formed in our model, i.e., in-phase and anti-phase synchronized states simultaneously exist in our system. In the previous work [27], though the desynchronized states and the synchronized states coexist and are both stable, the network only tends to be one of the two states, depending on its initial mean coupling. While in our model, the oscillators within (between) groups tend to in-phase (anti-phase) synchronization. (ii) The connection strengths in the network can self-organize into a power law distribution from the initial random distribution. In addition, communities, which correspond to the dynamical groups in our model, can also be spontaneously formed. The community structure and the power law distribution of the connection strengths are common in many empirical networked systems. (iii) The resource constraint can significantly affect the formation of the dynamical groups. If the total connection strength is a finite constant, the network tends to split into two dynamical groups: within each group the oscillators are in-phase synchronized, while the oscillators in different groups are antiphase synchronized. However, if there is no resource constraint, the two groups finally merge into one.
In our model, the dynamical equations for the networked phase oscillators read: Here, m, k = 1, 2, . . . , N are the oscillator (node) indices, and γ is the uniform coupling strength. θ m and ω m are the instantaneous phase and intrinsic frequency of the mth oscillator, respectively. W = {w mk }(w mk = w km ) is the weighted coupling matrix, where w mk > 0 is the coupling strength if nodes m and k are directly connected, and w mk = 0 otherwise. In order to generate possible dynamical groups in our model, we tentatively choose the coupling function Γ(φ) as the higher order of Fourier modes, i.e. Γ(φ) = sin(hφ) (h = 2, 3, 4, · · ·) [28], where the parameter h can control the number of groups. Without losing generality, we set h = 2 throughout this paper.
In the coevolutionary networked system, how the network topology couples with the dynamics is crucial to both the dynamical pattern and topological structure that result. In our model, we propose a coupling rule for the connection strength w mk based on following hypothesis: w mk is a finite real number, and the connections will be strengthened (weakened) if the phase differences are smaller (bigger) than some threshold α. Actually, this can be regarded as an extension of the spike-timing dependent plasticity (STDP) rule [19]. In fact, in many realistic networked systems, individuals with similar states usually tend to form the group which has relatively stronger intra connections inside. For instance, in human society, individuals with similar attributes are easily organized into the same communities [4,29,30]. Meanwhile, similarity will breed connection [30], indicating that the relations among individuals with similar attributes may be constantly strengthened, while those among individuals with dissimilar attributes may be gradually weakened. Based on the above consideration, the change of the connection strength is assumed to satisfy the following equation: where ∆θ mk = |θ k − θ m | (0 ≤ ∆θ mk ≤ π) is the phase difference between oscillators m and k. w mk in the right hand side of the equation ensures that the rate of change rate of the link weight is proportional to itself, and w mk ≥ 0 always. ǫ is a constant which can be chosen to make the time scale of the network topology evolution much longer than that of the local dynamics of the oscillators. The function Θ(φ, α) determines how the coupling strength evolves according to the phase difference between oscillators.
In this study, we set Θ(φ, α) = e −2|φ−π/2| . The function Λ(φ), which is similar to the sign function, controls either the strengthening or weakening of the connections based on the phase differences. For simplicity, we assume the form Λ(φ) = Γ(φ). The form Θ(φ, α)Λ(φ) is similar to the STDP rule, which has been widely used in neural network studies [19,27]. The difference is that the STDP rule [19,27] depends on the relative timing ∆t of presynaptic and postsynaptic spikes and the critical window τ , while the plasticity function in our model depends on the phase difference ∆θ and the connection strength itself. In addition, the exponential function Θ(φ, α) = e −2|φ−π/2| is modulated by the sine function Λ(φ) = sin(2φ), which makes the plasticity function not a monotone function on the same side of the threshold value, e.g., ∆θ < π/2. With the above assumptions, the model is fully described bẏ In this study, the natural frequencies and initial conditions of the oscillators are chosen randomly from the range [0, 1] and [−π, π], respectively. It is known that in many practical adaptive networks, the "resource", which can be represented by the summation of all connection strength in the network, is usually limited. Consequently, all connections will compete for this resource. Therefore, in our model we define the "resource" as M = L w , where L is the number of total connections and w is the average connection strength. In our simulation, we use the normalization w = 1 during the evolution in order to make the "resource" M = L, i.e., the total "resource" to be allocated is a constant during evolution. The collective behavior of the dynamical system can be conveniently described by two order parameters R and F . The order parameter R, which characterizes whether the global coherence occurs or not, is defined as where s m is the strength of node m, i.e. s m = k w mk . This type of order parameter has been widely used to characterize the phase synchronization in complex network [31]. From the definition in papers [31], it seems natural to use Eq. (4) as the order parameter in weighted networks. The order parameter F , which measures the fraction of all link weights synchronized in networks [32], is defined as In adaptive oscillator networks where the connections are coupled with the dynamical states, the order parameters R and F can be jointly used to characterize whether the local coherence within subnetwork takes place. For example, if R ≈ 0 and F ≫ R after a long time evolution from random initial phases on random networks, it indicates that the local synchronization (rather than the global synchronization) emerges within subnetworks, i.e., the dynamical groups have been generated in the system. First, we consider a simplified situation: a two-oscillator system. In this case, the dynamics can be rewritten in terms of two variables, ∆θ = (θ 1 − θ 2 ) and w, as From the above equations, we can see that if |∆ω| = |ω 1 − ω 2 | = 0, the system will have stable equilibrium states ∆θ * = 0 or π, and the final connection strength w * is a finite constant. These two states correspond to the in-phase synchronization and the antiphase synchronization of the two oscillators, respectively. If |∆ω| = 0, strictly speaking the two-oscillator system does not have any equilibrium states. This implies that the coupling strength will always be varying during the evolution. Nevertheless, if the rate of change of the connection strength is much slower than the phase dynamics, we can approximately regard w as a constant. In this case, we can obtain the stable equilibrium states of ∆θ provided that |∆ω| ≤ 2γw, i.e., In our numerical simulations, the above analysis has been verified. Next, we consider the case of a many-oscillator system. Without losing generality, the initial network topology is chosen as a random structure, and the initial connection strengths are chosen uniformly from the range (0, 2]. To monitor the evolution, we record the instantaneous phases of all the oscillators ({θ m (t)}). Interestingly, it is found that after the transient period, the oscillators can spontaneously separate into two dynamical groups. Within each group, all oscillators have similar phases. Meanwhile, the two dynamical groups as a whole tend to approximate anti-phase synchronization as shown in Fig. 1(a). Through extensive numerical simulations, we found that the sizes of the two groups depend on the initial conditions. In general, they are almost equal to each other when the initial phases are chosen uniformly. Of course, if all the oscillators are identical, the coevolution can still generate two dynamical groups as nonidentical system. In this case, the phase states within each group are strictly identical. The collective behavior of the dynamical system with multiple dynamical groups can also be described by the following parameter [24], If the order parameter R ′ converges to 1 and the order parameter R converges to 0, this also implies that dynamical groups have formed. The difference between F and R ′ is that F can characterize the properties of the dynamical states and the topology of weighted networks simultaneously, while R ′ can mainly characterize the properties of the dynamical states. In order to explain the formation of different dynamical groups in our model Eq. (1), we rewrite it in a more convenient form by defining the local order parameter according to Eq. (8) Here r ′ m with 0 < r ′ m < 1 measures the local coherence among the neighbors of oscillator m. ψ m is the average phase, and s m is the strength of node m, i.e. s m = k w mk . With this definition, Eq. (1) becomeṡ When γ → 0, Eq.(10) yields θ m ≈ ωt + θ m (0), that is, the oscillators evolve according to their own natural frequencies. The oscillators are neither in-phase nor anti-phase synchronized, i.e. r ′ m → 0 as t → ∞. On the other hand, in the limit of strong coupling, the oscillators tend to anti-phase synchronized, r ′ m → 1 and 2ψ m − 2θ m ≈ 2q m π(q m = 0, ±1), i.e. 2ψ m − 2θ m − 2q m π ≈ 0. Consequently, Eq. (10) can be rewritten aṡ where ψ ′ m = ψ m − q m π. Thus, the phase difference ∆θ mn = θ m − θ n between m and n becomes d∆θ From d∆θmn dt = 0, we can obtain the equilibrium value ∆θ mn , i.e., where ∆ω mn = ω m − ω n , and sm−sn sn+sm (ψ ′ m − θ m + ψ ′ n − θ n ) is the high-order infinitesimal, which could be neglected. When oscillators m and n tend to in-phase (anti-phase) synchronization, ψ ′ m − ψ ′ n ≈ qπ (q = 0, ±1), so the equilibrium values of the phase difference ∆θ * mn are As shown in Fig. 1 (b), our numerical simulations of the phase differences are consistent with the analytical results. Physically, the spontaneous formation of two different dynamical groups in our model can be attributed to the adaptive evolution rule described by Eq. (2). Based on this equation, the connection strength among oscillators with initially close phases will be enhanced. Meanwhile, if two oscillators initially have large phase difference (e.g., ∆θ > π/2), the connection strength between them will be weakened during evolution. As a combined effect of these two "forces", the networked oscillators self-organize into two dynamical groups after a long time evolution. Within the same group, the oscillators have similar states, while oscillators in different groups have approximate anti-phases. Interestingly, in many social and biological systems, we often find two groups are formed with opposite states. For instance, in human society, individuals with homogeneous character, e.g., the same generation or living in the same neighborhood, are disposed to associate [4], and conflicting (accordant) characters could weaken (strengthen) the social contacts. In food webs, if the living habits of predator and prey are similar (different), the predator-prey relationships are strong (weak) [6]. Our model thus can shed light on the origin of the formation of such dynamical groups.
With the formation of dynamical groups, how the network structure evolves is another important question. In this work, we do not consider the rewiring of network connections. Instead, we fix the network topology and focus on how the network connections compete for the limited "resource", i.e., the reallocation of the connection strength. At every time steps, we normalize w = 1, i.e. w * mn = M wmn j>i w ij , in order to make the "resource" M = L. In Fig. 2, we illustrate using one typical example the properties of the network structure. As shown in Fig. 2(a), the oscillators in the network self-organize into two dynamical groups with different phase states, i.e., oscillators within the same group have similar but nonidentical states, while oscillators in different groups have approximate anti-phases. The formation of the dynamical groups can be characterized by the two order parameters R and F . As shown in Fig. 2 (b), F keeps increasing during the evolution, but R always maintains very small values. This suggests that local dynamical patterns (rather than a global one) gradually form in the system. To characterize the emerging network structure, we show the average strength of the inter connections w inter and the intra connections w intra as a function of time in Fig. 2 (c). It is evident that the average strength of the inter connections w inter decreases, while the intra connection strength w intra keeps increasing with time. These results indicate that with the appearance of the dynamical groups, the distribution of the connection strengths in the network also changes. From the initial random distribution, the connection strengths within the groups are gradually strengthened, while the connection strengths between the two groups are weakened simultaneously. In this way, after a long time evolution, the topological structure of the networked system has the following characteristics, as shown in Figs. 2 (d)-(e). First, the network evolves into a modular structure with the formation of dynamical groups. Secondly, it is found that the network consists of many weak connections and a few strong connections. Thirdly, to be specific, we have verified that the distribution of the connection strengths follows a power law, as compared to the initial random distribution. It should be pointed out that this power-law distribution of the link weights in the present model is a natural consequence of the coevolution of the network topology and dynamics. These results are consistent with the empirical observations of social systems [3,13], biological systems [12,14] and neural network [15,25]. For instance, in neural networks [15,25], the synaptic strengths of experimental data follow a power law distribution. As shown by Fig. 2, with the evolution of the networked dynamics, the oscillators begin to separate into two groups with different states. In Fig. 2(c), it is shown that during the evolution process the average intra connection strength is gradually enhanced while the average inter connection strength is weakened always. Here, the question is: what would the two groups behave when the connections between them become weak enough? In Fig. 3, we further explore this situation. Interestingly, it is found that when the inter connections between the two groups are too weak, e.g., w inter < 0.1  The distribution of the active connection strengths, which follows a power law. This result is the average of 20 runs with different initial conditions. All network parameters are the same as that in Fig. 2 [33], the two dynamical groups effectively decouple and evolve independently according to their own frequencies. As shown in Fig. 3(a), the frequencies of the two groups are almost equal to each other, and during the evolution, their phases will slowly approach the same value and then begin to separate. This occurs repeatedly, which lead to the regular oscillation of the global order parameter R as shown in Fig. 3(b). Meanwhile, when the phase differences between the two groups become small enough, according to Eq. (3), the inter connection strength will be enhanced. However, this trend will not last for long since the phase differences of the two groups will begin to be significant soon. As shown in Fig. 3(c), although both the average inter connection strength and the average intra connection strength each oscillate with a small range, the trends are an overall decrease for the average inter connection strength and increase for the average intra connection strength. This implies that these two dynamical groups will become more and more independent after a long time evolution. Moreover, even in the collective oscillatory regime, the distribution of the link weights still follow a power law (as shown in Fig. 2(d)).
In realistic networked systems, if the connections are extremely weak, it may be impossible to measure them. As a consequence, any observed real network should consist of connections whose strengths are strong enough to be measured. In our model, we found that there exists a large number of weak links and many of these have no opportunity to be enhanced again. Therefore, from the practical point of view they may not be observable at all after a long time evolution. To distinguish them, we can define the active connections as follows: if w mk exceeds a threshold value, the connection between oscillator m and k is regarded as "active"; otherwise it is "inactive". The threshold can be reasonably taken as the average of the inter connections, i.e., w inter . Using this criterion, we obtain observable networks after a long time evolution based on our model. Numerically, we let the networked system evolve from many different initial conditions. After t = 5000, we start taking snapshots of w mk . After discarding the "inactive" connections, we obtain the observable networks only consisting of the "active" connections. It is found that in these observable networks, the modular property becomes even more distinct. As shown in Fig. 4, the oscillators can be reasonably partitioned into two communities, and the distribution of connection strengths still approximately satisfies the power law relation. These results suggest that the widely observed community structure and the power-law distribution of link weights in complex networks could emerge simultaneously from the coevolution of the network topology and dynamics.
In the above studies, we have limited the total connection strength as a constant in the network. This consideration makes sense in certain practical circumstances. For example, the bandwidth of a local area network in a university is always limited. However, in other networks, e.g., the social acquaintance network, there is no need to limit the total connection strength during the network evolution. In this case, how would the dynamics and the network structure coevolve? In the following, we investigate one such example. It is found that the initial stage of the network evolution is quite similar to the case when the total network connection strength is limited. As shown in Fig.  5(a), the global order parameter R remains small while the local order parameter F keeps increasing. This indicates that the two dynamical groups have been generated. In Figs. 5(b) and (c), it is shown that the average intra connection strength continues to increase, while the average inter connection strength keeps decreasing. This is just the reason which leads to the formation of the dynamical groups. When the inter connection strength among the groups is small enough, the two groups almost decouple and they behave just like two independent oscillators. However, with the further increase of time, contrary to the previous situations, the inter connection strength starts to gradually increase as shown in Fig. 5(c). Due to this strengthening of the inter connections, the two dynamical groups eventually will merge into one and all oscillators will achieve in-phase synchronization. Therefore, our results suggest that during the evolution of a network, the limitation of the total connection strength is in favor of the formation of stable dynamical groups.
In summary, we have investigated a coevolutionary networked model. In this model, the node dynamics are described by phase oscillators, and the connections among oscillators are coupled with the dynamical states. By adopting a simple evolution rule, it is shown that the evolution of the networked system naturally leads to two dynamical groups with different phase states. Simultaneously, with the formation of the dynamical pattern, the network also converts from the initial random structure with a uniform distribution of connection strengths to the final modular network with a power-law distribution of the connection strengths. Interestingly, it is found that if the total connection strength is limited as a constant, the two dynamical groups will almost decouple eventually when the inter connection is too weak. On the contrary, if the total connection strength does not have an imposed limit, the two dynamical groups will finally merge into one with all the oscillators achieving in-phase synchronization. In our numerical simulations, the above results have been qualitatively verified on networks with sizes up to N = 1000. Although the model studied is simple, it essentially captures the interplay between the network topology and dynamics. Thus it can exhibit reasonable results which are useful for us to better understand the behaviors of many real networked dynamical systems, such as the evolution of social networks [4], and the evolution of food webs [6].
In our model, the connection strengths are assumed to respond immediately to the change of phase difference. Nevertheless, time-delay inevitably exists in realistic networked systems. For example, electric signals can only propagate along neural axons