  A wide variety of application domains are concerned with data consisting of
entities and their relationships or connections, formally represented as
graphs. Within these diverse application areas, a common problem of interest is
the detection of a subset of entities whose connectivity is anomalous with
respect to the rest of the data. While the detection of such anomalous
subgraphs has received a substantial amount of attention, no
application-agnostic framework exists for analysis of signal detectability in
graph-based data. In this paper, we describe a framework that enables such
analysis using the principal eigenspace of a graph's residuals matrix, commonly
called the modularity matrix in community detection. Leveraging this analytical
tool, we show that the framework has a natural power metric in the spectral
norm of the anomalous subgraph's adjacency matrix (signal power) and of the
background graph's residuals matrix (noise power). We propose several
algorithms based on spectral properties of the residuals matrix, with more
computationally expensive techniques providing greater detection power.
Detection and identification performance are presented for a number of signal
and noise models, including clusters and bipartite foregrounds embedded into
simple random backgrounds as well as graphs with community structure and
realistic degree distributions. The trends observed verify intuition gleaned
from other signal processing areas, such as greater detection power when the
signal is embedded within a less active portion of the background. We
demonstrate the utility of the proposed techniques in detecting small, highly
anomalous subgraphs in real graphs derived from Internet traffic and product
co-purchases.
