  Many social and economic systems can be represented as attributed networks
encoding the relations between entities who are themselves described by
different node attributes. Finding anomalies in these systems is crucial for
detecting abuses such as credit card frauds, web spams or network intrusions.
Intuitively, anomalous nodes are defined as nodes whose attributes differ
starkly from the attributes of a certain set of nodes of reference, called the
context of the anomaly. While some methods have proposed to spot anomalies
locally, globally or within a community context, the problem remain challenging
due to the multi-scale composition of real networks and the heterogeneity of
node metadata. Here, we propose a principled way to uncover outlier nodes
simultaneously with the context with respect to which they are anomalous, at
all relevant scales of the network. We characterize anomalous nodes in terms of
the concentration retained for each node after smoothing specific signals
localized on the vertices of the graph. Besides, we introduce a graph signal
processing formulation of the Markov stability framework used in community
detection, in order to find the context of anomalies. The performance of our
method is assessed on synthetic and real-world attributed networks and shows
superior results concerning state of the art algorithms. Finally, we show the
scalability of our approach in large networks employing Chebychev polynomial
approximations.
