Complex Network Modelling with Power-law Activating Patterns and Its Evolutionary Dynamics

TL;DR

This work addresses how power-law inter-event times in a closed network affect cooperation and fixation dynamics. It develops a two-state activation framework with power-law dwell times, derives a closed-form stationary distribution for activated size via Markov-renewal theory, and analyzes a two-player Prisoner’s Dilemma on the activated subgraph using weak selection and birth-death-like updates. Key findings include a binomial stationary distribution for activated sizes, a coalescent-based fixation-probability analysis, and a cooperative-condition bound that depends on network size, degree, and activation parameters; simulations on synthetic and four real networks corroborate these results and reveal differences in resilience and cooperation under temporal heterogeneity. The framework offers a principled lens to study social physics in time-evolving networks and suggests that temporal activation patterns can facilitate cooperation under social dilemmas, with implications for designing robust, cooperative dynamics in real systems.

Abstract

Complex network theory provides a unifying framework for the study of structured dynamic systems. The current literature emphasizes a widely reported phenomenon of intermittent interaction among network vertices. In this paper, we introduce a complex network model that considers the stochastic switching of individuals between activated and quiescent states at power-law rates and the corresponding evolutionary dynamics. By using the Markov chain and renewal theory, we discover a homogeneous stationary distribution of activated sizes in the network with power-law activating patterns and infer some statistical characteristics. To better understand the effect of power-law activating patterns, we study the two-person-two-strategy evolutionary game dynamics, demonstrate the absorbability of strategies, and obtain the critical cooperation conditions for prisoner's dilemmas in homogeneous networks without mutation. The evolutionary dynamics in real networks are also discussed. Our results provide a new perspective to analyze and understand social physics in time-evolving network systems.

Figures (10)

• Figure 1: An example of the proposed network model. Green and grey vertices are in activated and quiescent states respectively. (a) The state transition of a single vertex with power-law activating patterns. An activated vertex becomes quiescent after a power-law period with the parameter $\mu$, while a quiescent vertex turns activated after another power-law period with the parameter $\lambda$. (b) The evolving of network structure with the power-law activating patterns of vertices. We consider a network with six vertices and nine edges. We show the state transition time stamps of nodes $a$ and $d$ and the network snapshots at $t_1$ and $t_2$. Green periods indicate the activated duration. If both ends of an edge are activated, this edge is then activated. (Color online)
• Figure 2: An example of the strategy update. (a) The strategy update process of a $C$ player in the green square considers its neighbors. In a prisoner's dilemma game, the center vertex with the strategy $D$ possesses the highest payoff and fitness in the community because it connects to the largest number of cooperators. Therefore, the center vertex has the highest probability of spreading its strategy to the individual in the green square. (b) The time axis shows the strategy update timestamp of three individuals during activation. They update strategies by independent Poisson processes, thus only one vertex can be chosen to update at one certain moment. (Color online)
• Figure 3: Size distributions of the activated subgraph. The activated subgraph size shows binomial distribution as Proposition \ref{['proposition: 1']} suggests. (a) $\lambda=2.60$. (b) $\lambda=3.50$. (c) $\lambda=6.40$. We set $\mu=[2.60, 3.50, 6.40]$ and network size $N=1000$ for cross simulation. Initially, each vertex has the equal probability to be activated or quiescent. We collect the sizes of activated subgraphs if $t>50$ and calculate the frequencies until $t=600$. Results for $\mu=2.60$, $3.50$, and $6.40$ are shown in red, green, and blue respectively. Black plots indicate the theoretical distribution as Proposition \ref{['proposition: 1']}. (Color online)
• Figure 4: Degree distributions of the activated subgraph. The power-law activation patterns maintain the homogeneity of WSN but break the heterogeneity of BAN. (a) WSN, $\mu=2.60$. (b) WSN, $\mu=3.70$. (c) BAN, $\mu=2.60$. (d) BAN, $\mu=3.70$. We fix the parameters $N=2000$, $\lambda=3.50$ and set $\mu=[2.60, 3.70]$, $k=[8, 16, 24]$ for cross experiments. We collect the degree distributions if $t>20$ every $10$ simulation time until $t=200$. Results for $k=8$, $16$, and $24$ are shown in red, green, and blue respectively. Degree distributions of WSNs and BANs are shown in double-linear axes and double-logarithmic axes respectively. (Color online)
• Figure 5: Mean degree as the function of $\lambda$. Mean degrees of activated subgraphs have positive correlations to $\lambda$. (a) WSN, $N=500$, $k=4$. (b) BAN, $N=500$, $k=4$. (c) Mouse visual cortex network, $N=193$. (d) Infect Dublin, infectious contact network, $N=410$. We set $\lambda\in[2.60,6.40]$ and $\mu\in\{2.60, 3.50, 6.40\}$ for cross simulations. We collect the mean degrees of activated subgraphs in the time interval $t\in[50,150]$ and take the average for each data point. The results for $\mu=2.60, 3.50, 6.40$ are shown in red, green, and blue curves respectively. (Color online)

Theorems & Definitions (15)

• Definition 1
• Definition 2
• Definition 3
• Lemma 1
• Lemma 2
• Proposition 1
• proof
• Corollary 1
• Corollary 2
• Proposition 2