Bursty Switching Dynamics Promotes the Collapse of Network Topologies

TL;DR

We address how bursty, intermittent temporal links shape the evolution of network structure and dynamics. By modeling edge states as independent renewal processes, the work derives closed-form stationary statistics for the activated subgraph, including a Binomial edge-activation law with parameter $q_0$ and a derived activated-degree distribution, and analyzes random-walk and evolutionary-game consequences. The results show that increasing activation probability can cause topology collapse and fragmentation, slow information spread via random walks, and, paradoxically, promote cooperation in donation games under switching topology. The framework provides a quantitative tool for studying social and technological networks with intermittent interactions and informs design and control of time-varying networks.

Abstract

Time-varying connections are crucial in understanding the structures and dynamics of complex networks. In this paper, we propose a continuous-time switching topology model for temporal networks that is driven by bursty behavior and study the effects on network structure and dynamic processes. Each edge can switch between an active and a dormant state, leading to intermittent activation patterns that are characterized by a renewal process. We analyze the stationarity of the network activation scale and emerging degree distributions by means of the Markov chain theory. We show that switching dynamics can promote the collapse of network topologies by reducing heterogeneities and forming isolated components in the underlying network. Our results indicate that switching topologies can significantly influence random walks in different networks and promote cooperation in donation games. Our research thus provides a simple quantitative framework to study network dynamics with temporal and intermittent interactions across social and technological networks.

Figures (6)

• Figure 1: Illustration of stochastic edge-switching topology in complex network. The underlying network (left panel) is static with 6 vertices and 9 edges. Each edge switches between dormant and activated states independently in a continuous time axis (middle panel). The time spent in activated and dormant states is governed by general distributions $f(t)$ and $g(t)$. At the time $t$, the edge set of activated subgraph $\mathcal{G}_A(t)$ does not contain the dormant edges $(A, B)$ and $(C, D)$.
• Figure 2: Network topology properties. (a) Evolution of activated edge numbers with time. The underlying network is RRG with $N=100$ and $\left\langle k\right\rangle=4$. We set $\lambda\in\{0.50, 1.00, 2.00\}$ and $\alpha=2.60$ with randomly assigned initial condition. (b) Stationary distribution of activated edge number. The network and switching topology settings are the same as (a). Each data point is averaged in the stable state with $t\in[0.5\times10^3,10^3]$. Theoretical results are shown in curves with corresponding colors. (c) Degree distribution of activated subgraph $\mathcal{G}_A(t)$. The underlying networks are BAN (upper panel) and WSN (lower panel) with $N=10^3$ and $k\in\{4,8\}$. The switching parameters are $\lambda=1$ and $\alpha=2.6$. We plot the numerical degree distributions in data points in $t\in[50,100]$ with the interval 2. Curves are the theoretical degree distribution. (d) Density of $\mathcal{G}_A(t)$. The underlying networks are BAN (upper panel) and WSN (lower panel) with $N=2\times10^3$ and $k\in\{4,8,12\}$. The switching parameters are $\lambda\in[0.5,2.0]$ and $\alpha=2.6$. Each data point is the average of over 100 independent realizations in $t\in[100,300]$. Theoretical results are shown in curves. (e) The spectral radius of the average adjacency matrix of $\mathcal{G}_A(t)$. The network and switching topology settings are the same as (d). (f) The relative size of the largest component. The underlying networks are BAN (upper panel) and WSN (lower panel) with $N=10^3$ and $k\in\{2,4,8\}$. The switching parameters are the same as (d) and (e). Each data point is the average of over 10 independent realizations in $t\in[50,150]$. For (c)-(f), the reconnection probability of WSNs is unified as $0.25$.
• Figure 3: The impact of edge switching on the maximum component ratio in real networks. We set $\alpha\in\{2.60,3.70,4.60\}$ and $\lambda\in[0.5, 2.0]$ in six real-world network data sets for cross simulations. Each data point is the average of the ratios of the largest component over $t\in[80, 200]$. (a) The yeast protein network with jeong2001lethality, (b) The inferred network by high-throughput protein-protein interactions cho2014wormnet, (c) The inferred network by small/medium-scale protein-protein interactions cho2014wormnet, (d) The bipartite network contains persons who appeared in at least one crime case nr-aaai15, (e) The network of fiber tracts in brains bigbrain, (f) The retweet and mentions network from the UN conference held in Copenhagen ahmed2010time.
• Figure 4: Edge cover ratio for random walks. (a) An example of random walks with network switching topology. Our focal random walk steps by Poisson process and is now in vertex $A$. In the left panel, edges $(A, B)$ and $(A, D)$ are activated, and the focal walker can step into $B$ or $D$ with equal probability. In the right panel, all edges around $A$ are dormant, and the walker remains in $A$. (b)-(d) present the edge cover ratio against time in RRG, WSN, and BAN respectively, with $N=200$, $\lambda\in\{0.50, 2.00\}$, $\alpha=2.60$, and $k\in\{2,4,8\}$. Each data point is obtained by the average of 100 independent realizations.
• Figure 5: Vertex cover rates of random walks in real networks. We set $\lambda\in\{0.50, 2.00, 3.50\}$ and $\alpha=2.60$ for cross simulations. We observe the random walks during $t\in[0,2000]$. Each data point is obtained by the average of 100 independent runs with randomly selected initial vertex. The settings of real networks are the same as Fig. \ref{['fig: component in real networks']}.