Characterization of L1-norm statistic for anomaly detection in Erdős Rényi graphs

We describe a test statistic based on the Lnorm of the eigenvectors of a modularity matrix to detect the presence of an embedded Erdős Rényi (ER) subgraph inside a larger ER random graph. An embedded subgraph may model a hidden community in a large network such as a social network or a computer network. We make use of the properties of the asymptotic distribution of eigenvectors of random graphs to derive the distribution of the test statistic under certain conditions on the subgraph size and edge probabilities. We show that the distributions differ sufficiently for well defined ranges of subgraph sizes and edge probabilities of the background graph and the subgraph. This method can have applications where it is sufficient to know whether there is an anomaly in a given graph without the need to infer its location. The results we derive on the distribution of the components of the eigenvector may also be useful to detect the subgraph nodes.