  Frequent pattern mining is a key area of study that gives insights into the
structure and dynamics of evolving networks, such as social or road networks.
However, not only does a network evolve, but often the way that it evolves,
itself evolves. Thus, knowing, in addition to patterns' frequencies, for how
long and how regularly they have occurred---i.e., their persistence---can add
to our understanding of evolving networks. In this work, we propose the problem
of mining activity that persists through time in continually evolving
networks---i.e., activity that repeatedly and consistently occurs. We extend
the notion of temporal motifs to capture activity among specific nodes, in what
we call activity snippets, which are small sequences of edge-updates that
reoccur. We propose axioms and properties that a measure of persistence should
satisfy, and develop such a persistence measure. We also propose PENminer, an
efficient framework for mining activity snippets' Persistence in Evolving
Networks, and design both offline and streaming algorithms. We apply PENminer
to numerous real, large-scale evolving networks and edge streams, and find
activity that is surprisingly regular over a long period of time, but too
infrequent to be discovered by aggregate count alone, and bursts of activity
exposed by their lack of persistence. Our findings with PENminer include
neighborhoods in NYC where taxi traffic persisted through Hurricane Sandy, the
opening of new bike-stations, characteristics of social network users, and
more. Moreover, we use PENminer towards identifying anomalies in multiple
networks, outperforming baselines at identifying subtle anomalies by 9.8-48% in
AUC.
