Graph Comparison via the Non-backtracking Spectrum

The comparison of graphs is a vitally important, yet difficult task which arises across a number of diverse research areas including biological and social networks. There have been a number of approaches to define graph distance however often these are not metrics (rendering standard data-mining techniques infeasible), or are computationally infeasible for large graphs. In this work we define a new metric based on the spectrum of the non-backtracking graph operator and show that it can not only be used to compare graphs generated through different mechanisms, but can reliably compare graphs of varying size. We observe that the family of Watts-Strogatz graphs lie on a manifold in the non-backtracking spectral embedding and show how this metric can be used in a standard classification problem of empirical graphs.Comment: 11 pages, 5 figures, 2 table

Non-backtracking Operators for Community Detection in Signed Networks

Community detection or clustering is crucial for understanding the structure of complex systems. In some networks, nodes are allowed to be linked by either 'positive' or 'negative' edges. Such networks are called signed networks. Discovering communities in signed networks is more challenging. In this article, we innovatively propose a non-backtracking matrix for signed networks, and theoretically derive a detectability threshold and prove the feasibility in community detection. Furthermore, we improve the operator by considering the balanced paths in the network (denoted as balanced non-backtracking operator). Simulation results demonstrate that the balanced non-backtracking matrix-based approach significantly outperforms the adjacency matrix-based and the signed non-backtracking matrix-based algorithm. It shows great potential to detect communities with or without overlap.Comment: 11 pages,9 figure

Node Immunization with Non-backtracking Eigenvalues

The non-backtracking matrix and its eigenvalues have many applications in network science and graph mining, such as node and edge centrality, community detection, length spectrum theory, graph distance, and epidemic and percolation thresholds. Moreover, in network epidemiology, the reciprocal of the largest eigenvalue of the non-backtracking matrix is a good approximation for the epidemic threshold of certain network dynamics. In this work, we develop techniques that identify which nodes have the largest impact on the leading non-backtracking eigenvalue. We do so by studying the behavior of the spectrum of the non-backtracking matrix after a node is removed from the graph. From this analysis we derive two new centrality measures: X-degree and X-non-backtracking centrality. We perform extensive experimentation with targeted immunization strategies derived from these two centrality measures. Our spectral analysis and centrality measures can be broadly applied, and will be of interest to both theorists and practitioners alike