Despite the fact that many important problems (including clustering) can be described using hypergraphs, theoretical foundations as well as practical algorithms using hypergraphs are not well developed yet. In this paper, we propose a hypergraph modularity function that generalizes its well established and widely used graph counterpart measure of how clustered a network is. In order to define it properly, we generalize the Chung-Lu model for graphs to hypergraphs. We then provide the theoretical foundations to search for an optimal solution with respect to our hypergraph modularity function. A simple heuristic algorithm is described and applied to a few illustrative examples. We show that using a strict version of our proposed modularity function often leads to a solution where a smaller number of hyperedges get cut as compared to optimizing modularity of 2-section graph of a hypergraph.

We propose an ensemble clustering algorithm for graphs (ECG), which is based on the Louvain algorithm and the concept of consensus clustering. We validate our approach by replicating a recently published study comparing graph clustering algorithms over artificial networks, showing that ECG outperforms the leading algorithms from that study. We also illustrate how the ensemble obtained with ECG can be used to quantify the presence of community structure in the graph.

We recently proposed a new ensemble clustering algorithm for graphs (ECG) based on the concept of consensus clustering. We validated our approach by replicating a study comparing graph clustering algorithms over benchmark graphs, showing that ECG outperforms the leading algorithms. In this paper, we extend our comparison by considering a wider range of parameters for the benchmark, generating graphs with different properties. We provide new experimental results showing that the ECG algorithm alleviates the well-known resolution limit issue, and that it leads to better stability of the partitions. We also illustrate how the ensemble obtained with ECG can be used to quantify the presence of community structure in the graph, and to zoom in on the sub-graph most closely associated with seed vertices. Finally, we illustrate further applications of ECG by comparing it to previous results for community detection on weighted graphs, and community-aware anomaly detection.Most networks that arise in nature exhibit complex structure [1, 2] with subsets of vertices densely interconnected relative to the rest of the network, which we call communities or clusters. Binary relational data-sets are typically represented as graphs G = (V, E), where vertices v ∈ V represent the entities, and edges e ∈ E represent the relations between pairs of entities. Graph clustering aims at finding a partition of the vertices V = C 1 ∪ . . . ∪ C l into good clusters. This is an ill-posed problem [3], as there is no universal definition of good clusters, leading to a wide variety of graph clustering algorithms [1, 4-10], with different objective functions. In a recent study [11], several state-of-the art algorithms implemented in the igraph [12] package were compared over a wide range of artificial networks generated via the LFR benchmark [13]. We recently introduced a new ensemble clustering algorithm for graphs (ECG), which compared favorably with leading algorithms from that study [14].The ECG algorithm is based on the concept of co-association consensus clustering. It is similar to other consensus clustering algorithms, in particular [15], but differs in two major points: (1) the choice of an algorithm that alleviates the resolution limit issue for the generation step, and (2) the restriction to endpoints of edges for co-occurrences of vertex pairs, which keeps low computational complexity.The rest of the paper is organized as follows. We briefly describe the ECG algorithm in Section 2, where we also recall some results from the previous comparison study. New results are included in the following three sections.In Section 3, we extend our study to a wider variety of graphs by varying the power law exponents of the LFR benchmark. Some of the advantages of ECG are its stability, and its ability to alleviate the well known resolution limit issue. We illustrate those properties in Section 4. We also take a closer look at the edge weights generated by the ECG algorithm, showing that they can be good indicators of the presence (or absence) of comm...

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