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When analyzing temporal networks, a fundamental task is the identification ofdense structures (i.e., groups of vertices that exhibit a large number oflinks), together with their temporal span (i.e., the period of time for whichthe high density holds). We tackle this task by introducing a notion oftemporal core decomposition where each core is associated with its span: wecall such cores span-cores.  As the total number of time intervals is quadratic in the size of thetemporal domain $T$ under analysis, the total number of span-cores is quadraticin $|T|$ as well. Our first contribution is an algorithm that, by exploitingcontainment properties among span-cores, computes all the span-coresefficiently. Then, we focus on the problem of finding only the maximalspan-cores, i.e., span-cores that are not dominated by any other span-core byboth the coreness property and the span. We devise a very efficient algorithmthat exploits theoretical findings on the maximality condition to directlycompute the maximal ones without computing all span-cores.  Experimentation on several real-world temporal networks confirms theefficiency and scalability of our methods. Applications on temporal networks,gathered by a proximity-sensing infrastructure recording face-to-faceinteractions in schools, highlight the relevance of the notion of (maximal)span-core in analyzing social dynamics and detecting/correcting anomalies inthe data.

Mining (maximal) span-cores from temporal networks