Stability Analysis of Time-varying Networks Based on Persistent Homology

Currently, there are many metrics used to describe the stability of social networks, with most of them being based on graph theory methods. Persistent homology is beneficial for assessing the stability of dynamic networks because it considers the high-dimensional topological information of the network. In time-varying social networks, studying stability becomes even more difficult due to the constant changes in edges and nodes. This paper defines the persistent homology rate (PH-r) based on the Wasserstein distance in persistent homology theory. PH-r is used to quantify the topological change between networks and to analyze the structural stability of time-varying social networks. An algorithm is proposed for analyzing the stability of time-varying networks based on PH-r, which effectively describes the stability of time-varying social networks. Secondly, the stability of communities in time-varying social networks is being studied because communities are useful for describing the stability of social relationships. A new algorithm for community division based on persistent homology is proposed. It effectively divides the community and provides a new perspective on community division. Finally, we consider each community as a node to obtain a new time-varying social network. Then two methods are proposed to describe the stability of communities in time-varying social networks. The first method uses PH-r to analyze the stability of the new time-varying social network in order to analyze the stability of the communities, and then proposes an algorithm for describing the stability of communities in time-varying social networks. The second approach uses persistence entropy to describe the stability of communities in time-varying social networks. The experimental results show that persistent homology can describe the stability of time-varying social networks and communities.