  Monitoring of networks for anomaly detection has attracted a lot of attention
in recent years especially with the rise of connected devices and social
networks. This is of importance as anomaly detection could span a wide range of
application, from detecting terrorist cells in counter-terrorism efforts to
phishing attacks in social network circles. For this reason, numerous
techniques for anomaly detection have been introduced. However, application of
these techniques to more complex network models is hindered by various
challenges such as the size of the network being investigated, how much apriori
information is needed, the size of the anomalous graph, among others. A recent
technique introduced by Miller et al, which relies on a spectral framework for
anomaly detection, has the potential to address many of these challenges. In
their discussion of the spectral framework, three algorithms were proposed that
relied on the eigenvalues and eigenvectors of the residual matrix of a binary
network. The authors demonstrated the ability to detect anomalous subgraphs
that were less than 1% of the network size. However, to date, there is little
work that has been done to evaluate the statistical performance of these
algorithms. This study investigates the statistical properties of the spectral
methods, specifically the Chi-square and L1 norm algorithm proposed by Miller.
We will analyze the performance of the algorithm using simulated networks and
also extend the method's application to count networks. Finally we will make
some methodological improvements and recommendations to both algorithms.
