Partitioning Breaks Communities

Considering a clique as a conservative definition of community structure, we examine how graph partitioning algorithms interact with cliques. Many popular community-finding algorithms partition the entire graph into non-overlapping communities. We show that on a wide range of empirical networks, from different domains, significant numbers of cliques are split across separate partitions, as produced by such algorithms. We examine the largest connected component of the sub graph formed by retaining only edges in cliques, and apply partitioning strategies that explicitly minimise the number of cliques split. We conclude that, due to the connectedness of many networks, any community finding algorithm that produces partitions must fail to find at least some significant structures. Moreover, contrary to traditional intuition, in some empirical networks, strong ties and cliques frequently do cross community boundaries.