Fast Eigen-Functions Tracking on Dynamic Graphs

Abstract

Many important graph parameters can be expressed as eigenfunctions of its adjacency matrix. Examples include epidemic threshold, graph robustness, etc. It is often of key importance to accurately monitor these parameters. For example, knowing that Ebola virus has already been brought to the US continent, to avoid the virus from spreading away, it is important to know which emerging connections among related people would cause great reduction on the epidemic threshold of the network. However, most, if not all, of the existing algorithms computing these measures assume that the input graph is static, despite the fact that almost all real graphs are evolving over time. In this paper, we propose two online algorithms to track the eigen-functions of a dynamic graph with linear complexity wrt the number of nodes and number of changed edges in the graph. The key idea is to leverage matrix perturbation theory to efficiently update the top eigen-pairs of the underlying graph without recomputing them from scratch at each time stamp. Experiment results demonstrate that our methods can reach up to 20× speedup with precision more than 80% for fairly long period of time. Keywords: dynamic graph; connectivity; graph spectrum; attribution analysis 1 Introduction Among others, node importance and graph connectivity are of key importance to understand fundamental characteristics of a network (such as social networks, power grid, transportation network, etc). To date, many different parameters of the graph have been invented to measure those properties from different perspective. One most commonly used parameter for node importance estimation is eigenvector centrality [15], which is effective on identifying influential nodes over the whole network. As for connectivity in the graph, important parameters include epidemic threshold ([28, 4, 20]), clustering coefficient [29] and graph robustness ([2, 8, 5]). One interesting observation is that many of those parameters can be calculated or estimated accurately by certain functions of the eigen-pairs of the graph. For example, it has been found that for an arbitrary graph, the tipping point for the dissemination process is controlled by the leading eigenvalue of certain system matrix associated with the graph [28,20]. As for the clustering coefficient computation, instead of doing Most of the algorithms that compute the above parameters/eigen-functions are based on static graphs. However in real world networks, the graph structure keeps evolving over time. It is of great importance to keep track of those measurements since subtle changes on graph structure may lead to sharp changes on the overall properties. For example, in epidemic process, some emerging connections between people may increase the leading eigenvalue a lot, and thus reduce the epidemic threshold, which in turn makes the virus easier to spread through the network. By monitoring related parameters in the network over time, we would be able to know when the change happens and identify its cause/attribution timely. ...

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