Temporal network modeling with online and hidden vertices based on the birth and death process

TL;DR

The paper addresses intermittent social interactions by modeling temporal networks with vertices that switch between online and hidden states via a birth-death process, analyzed with a continuous-time Markov framework.The NOH model yields closed-form stationary distributions for the number of online neighbors and the online network size, highlighting a binomial form that depends on the rates $\lambda$ and $\mu$ and the initial topology, with explicit expressions for their means and variances.Through simulations on small-world and scale-free networks, the authors validate the theoretical results, analyze degree distributions, and demonstrate that the model can fit real networks, notably showing a strong fit to an Amazon co-purchase dataset via KL divergence.The work provides a tractable and interpretable framework for temporal networks with online/offline dynamics, with implications for understanding social processes and informing applications such as epidemic modeling and network fitting.

Abstract

Complex networks have played an important role in describing real complex systems since the end of the last century. Recently, research on real-world data sets reports intermittent interaction among social individuals. In this paper, we pay attention to this typical phenomenon of intermittent interaction by considering the state transition of network vertices between online and hidden based on the birth and death process. By continuous-time Markov theory, we show that both the number of each vertex's online neighbors and the online network size are stable and follow the homogeneous probability distribution in a similar form, inducing similar statistics as well. In addition, all propositions are verified via simulations. Moreover, we also present the degree distributions based on small-world and scale-free networks and find some regular patterns by simulations. The application in fitting real networks is discussed.

Figures (9)

• Figure 1: An example of the NOH model. This figure presents an evolution example of the network model with online and hidden vertices. The example network is composed of 6 vertices. Orange and grey durations denote the online and hidden duration of each vertex respectively. The time range is set as $[0,80]$, and we observe the network at $t=$20 and 60. When $t=20$, the vertices $B$, $E$, $F$ are online (colored in orange), and $A$, $C$, $D$ are hidden. When $t=80$, the vertices $B$, $C$, $E$ and $F$ are online (colored in orange), and $A$, $D$ are hidden. Connections between two online vertices are marked in orange solid lines. Connections that are temporarily cut are marked in grey dotted lines. (color online)
• Figure 2: System diagram of the proposed model. This figure presents an illustration of one vertex's state transition and online network size transition. The state transition of a large number of vertices constitutes our model. (color online)
• Figure 3: Snapshots of NOHs. The online network structure is stable as time evolves and different as $\lambda$ and $\mu$ change. We present snapshots at time $t=2\times10^3,4\times10^3,6\times10^3,8\times10^3,10^4$ with fixed $\lambda=0.005$ and $\mu=0.005$ for Figs. \ref{['fig:snapshots/1_1']}-\ref{['fig:snapshots/1_5']}, and $\mu=0.010$ for Figs. \ref{['fig:snapshots/2_1']}-\ref{['fig:snapshots/2_5']}. The initial network is set as SW with 100 vertices (small number for a better presentation), $K=10$ and $p=0.30$. Vertices in orange are online and in grey are hidden. Solid links in orange are for two online individuals, and dashed links in grey are for at least one hidden vertex. (color online)
• Figure 4: The number of each vertex's online neighbors as functions of time. The numbers of each vertex's online neighbors along with the expectation are stable as time evolves. Each subplot presents the numbers of all vertices' online neighbors for $t\leq10^3$ (blue plots) and the average number of each vertex's online neighbors (purple plots) with initial sizes $N(0)=2\times10^3$ and $\lambda$s$=0.010$. Network types and $\mu$s are set as \ref{['fig:1/SF_1.00_0.50']} SF, $\mu=0.005$, \ref{['fig:1/SF_1.00_1.00']} SF, $\mu=0.010$, \ref{['fig:1/SF_1.00_1.50']} SF, $\mu=0.015$, \ref{['fig:1/SW_1.00_0.50']} SW, $\mu=0.005$, \ref{['fig:1/SW_1.00_1.00']} SW, $\mu=0.010$, \ref{['fig:1/SW_1.00_1.50']} SW, $\mu=0.015$. The $x$-axis is set as the time $t$ in the range $[10^{-1}, 10^3]$. Besides, for SFs, we set $m=5$ and the $y$-axis as the number of online neighbors $k_t$ and its range as $[0, 200]$. For SWs, we set $K=20$, $p=0.3$ and the $y$-axis range as $[0, 30]$. (color online)
• Figure 5: Network sizes as functions of time. The online network sizes are stable as time evolves and estimable by $N(0)$, $\lambda$, and $\mu$. The volatility is relatively acceptable. We show the evolution process of network sizes under different $\lambda$s, $\mu$s and $N(0)$s. We set $N(0)=2\times10^3$ (red plots), $4\times10^3$ (green plots), $6\times10^3$ (blue plots) and $\lambda$s and $\mu$s as \ref{['fig:2/1']}$\lambda=0.010$, $\mu=0.005$, \ref{['fig:2/2']}$\lambda=0.010$, $\mu=0.010$, \ref{['fig:2/3']}$\lambda=0.010$, $\mu=0.015$, \ref{['fig:2/4']}$\lambda=0.020$, $\mu=0.005$, \ref{['fig:2/5']}$\lambda=0.020$, $\mu=0.010$, \ref{['fig:2/6']}$\lambda=0.020$, $\mu=0.015$. We set the $y$-axis as the network size $N$ and its range as $[0, 6.5\times10^3]$. We observe the evolution process of each network's size in the time interval $[0, 10^4]$. We emphasize again that the network size is not affected by the initial network type. Therefore, we do not set the initial network type as a variable. (color online)

Theorems & Definitions (12)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Lemma 1
• Lemma 2
• Theorem 1
• Theorem 2