Community search is the problem of finding a good community for a given set of query vertices. One of the most studied formulations of community search asks for a connected subgraph that contains all query vertices and maximizes the minimum degree. All existing approaches to min-degree-based community search suffer from limitations concerning efficiency, as they need to visit (large part of) the whole input graph, as well as accuracy, as they output communities quite large and not really cohesive. Moreover, some existing methods lack generality: they handle only single-vertex queries, find communities that are not optimal in terms of minimum degree, and/or require input parameters. In this work we advance the state of the art on community search by proposing a novel method that overcomes all these limitations: it is in general more efficient and effective-one/two orders of magnitude on average, it can handle multiple query vertices, it yields optimal communities, and it is parameterfree. These properties are confirmed by an extensive experimental analysis performed on various real-world graphs.

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

Signed networks contain edge annotations to indicate whether each interaction is friendly (positive edge) or antagonistic (negative edge). The model is simple but powerful and it can capture novel and interesting structural properties of real-world phenomena. The analysis of signed networks has many applications from modeling discussions in social media, to mining user reviews, and to recommending products in e-commerce sites.In this paper we consider the problem of discovering polarized communities in signed networks. In particular, we search for two communities (subsets of the network vertices) where within communities there are mostly positive edges while across communities there are mostly negative edges. We formulate this novel problem as a "discrete eigenvector" problem, which we show to be NP-hard. We then develop two intuitive spectral algorithms: one deterministic, and one randomized with quality guarantee √ n (where n is the number of vertices in the graph), tight up to constant factors.We validate our algorithms against non-trivial baselines on realworld signed networks. Our experiments confirm that our algorithms produce higher quality solutions, are much faster and can scale to much larger networks than the baselines, and are able to detect ground-truth polarized communities.

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