A New Measure of Network Robustness: Network Cluster Entropy

Abstract

Online social networks have gained tremendous popularity and have dramatically changed the way we communicate in recent years. It is a challenging problem, however, due to the difficulties of handling complex social network topologies and conducting accurate assessment in these topologies. Therefore, the robustness analysis of network topologies has been a hot research topic in recent years. To characterize the structure feature of complex social network quantificationally, we propose a new measure for network robustness, namely, network cluster entropy, which takes the impact of cluster density on the network structure into consideration. Besides, the relationship between the network cluster entropy and network connectivity reliability is established. To show the effectiveness of the proposed method, we compute the network cluster entropy of the Zachary’s Karate Club network with two existing indices under disparate divisions, and we also compare the results by measuring the ability of each index to characterize network heterogeneity. Both of experimental results and empirical analysis show that our proposed method has the more excellent performance compared with two existing methods. Therefore, it is indicated that the cluster entropy and network connectivity reliability will be an important tool to study the online social network.

Supported by the National Natural Science Foundation of China (Nos. 61977016 and 61572010), Natural Science Foundation of Fujian Province (Nos. 2020J01164, 2017J01738).

Appendix: The proof of Theorem 2.

Proof

Let G be a network with k connected subnetworks, denoted as $G_{1}$, $G_{2}$,...,$G_{k}$. Without loss of generality, we suppose that $G^{'}$ is obtained by adding some edges to G. Next we distinguish between two cases as follows.

Case 1. The number of components is unchanged.

Assume that $G_{j}^{'}$ is generated by adding x edges from the j-th subnetwork $G_{j}$. Then the size of $G_{j}^{'}$ is equal to $m_{j}+x$. Hence, the clustering density of the j-th subnetwork is $LCD_{j}^{'}=\frac{m_{j}+x-(n_{j}-1)}{C_{n_{j}}^{2}}$. Therefore,

$$E_{C}^{'}=-(\sum _{i=1, \ i\ne j}^{k}LCD_{i}lnLCD_{i}+LCD_{j}^{'}lnLCD_{j}^{'} ).$$

Since $f(x)=xlnx \ (x\ge 1)$ is a monotonic increasing function,

$$\begin{aligned} \begin{aligned}&~~~~~E_{C}^{'}-E_{C}\\&=-(\sum _{i=1, \ i\ne j}^{k}LCD_{i}lnLCD_{i}+LCD_{j}^{'}lnLCD_{j}^{'})-(-\sum _{i=1}^{k}LCD_{i}lnLCD_{i})\\&=-\frac{m_{j}+x-(n_{j}-1)}{c_{n_{j}}^{2}}ln\frac{m_{j}+x-(n_{j}-1)}{c_{n_{j}}^{2}}- (-\frac{m_{j}-(n_{j}-1)}{c_{n_{j}}^{2}}ln\frac{m_{j}-(n_{j}-1)}{c_{n_{j}}^{2}})\\&=\frac{m_{j}-(n_{j}-1)}{c_{n_{j}}^{2}}ln\frac{m_{j}-(n_{j}-1)}{c_{n_{j}}^{2}}- \frac{m_{j}+x-(n_{j}-1)}{c_{n_{j}}^{2}}ln\frac{m_{j}+x-(n_{j}-1)}{c_{n_{j}}^{2}}\\&< 0. \end{aligned} \end{aligned}$$

So $E_{C}^{'}<E_{C}$, which implies the assertion.

Case 2. The number of components diminishes.

Assume that we add one or more edges to G such that the number of subnetwork reduces. Without loss of generality, we suppose that the $(k-1)$-th subnetwork and k-th subnetwork is connected by some edge, which generates a new subnetwork $G^{'}$. One can easily check that $G^{'}$ is of $(n_{k-1}+n_{k})$ vertices as well as $(m_{k-1}$ + $m_{k}+1)$ edges. Then,

$$E_{C}^{'}=-(\sum _{i=1}^{k-2}LCD_{i}lnLCD_{i}+LCD_{k-1}^{'}lnLCD_{k-1}^{'} ).$$

Hence,

$$\begin{aligned} \begin{aligned}&~~~~~E_{C}^{'}-E_{C}\\&=-(\sum _{i=1}^{k-2}LCD_{i}lnLCD_{i}+LCD_{k-1}^{'}lnLCD_{k-1}^{'})-(-\sum _{i=1}^{k}LCD_{i}lnLCD_{i})\\&=LCD_{k-1}lnLCD_{k-1}+LCD_{k}lnLCD_{k}-LCD_{k}^{'}lnLCD_{k}^{'}\\&=\frac{m_{k-1}-(n_{k-1}-1)}{c_{n_{k-1}}^{2}-(n_{k-1}-1)} ln\frac{m_{k-1}-(n_{k-1}-1)}{c_{n_{k-1}}^{2}-(n_{k-1}-1)}+ \frac{m_{k}-(n_{k}-1)}{c_{n_{k}}^{2}-(n_{k}-1)} ln\frac{m_{k}-(n_{k}-1)}{c_{n_{k}}^{2}-(n_{k}-1)}\\&~~~~-\frac{m_{k}^{'}-(n_{k}^{'}-1)}{c_{n_{k}^{'}}^{2}-(n_{k}^{'}-1)} ln\frac{m_{k}^{'}-(n_{k}^{'}-1)}{c_{n_{k}^{'}}^{2}-(n_{k}^{'}-1)}\\&< 0, \end{aligned} \end{aligned}$$

which yields the desired results.