Maximizing the Cohesion is NP-hard

We show that the problem of finding a set with maximum cohesion in an undirected network is NP-hard. Key-words: social networks, complex networks, cohesion, np-complete, complexity in ria -0 06 21 06 5, v er si on 2 9 O ct 2 01 1 Maximiser la Cohésion est NP-dur Résumé : Nous montrons que le problème de trouver un ensemble de cohésion maximum dans un graphe non orienté est NP-dur. Mots-clés : réseaux sociaux, réseaux complexes, coésion, np-complet, compléxité in ria -0 06 21 06 5, v er si on 2 9 O ct 2 01 1 Maximizing the Cohesion is NP-hard 3 Introduction In [1], we have introduced a new metric called the cohesion which rates the community-ness of a group of people in a social network from a sociological point of view. Through a large scale experiment on Facebook, we have established that the cohesion is highly correlated to the subjective user perception of the communities. In this article, we show that finding a set of vertices with maximum cohesion is NP-hard. Notations Let G = (V,E) be a graph with vertex set V and edge set E of size n = |V | ≥ 4. For all vertices u ∈ V , we write dG(u) the degree of u, or more simply d(u) . A triangle in G is a triplet of pairwise connected vertices. For all sets of vertices S ⊆ V , let G[S] = (S,ES) be the subgraph induced by S on G. We write m(S) = |ES | the number of edges in G[S], and i(S) = |{(u, v, w) ∈ S : (uv, vw, uw) ∈ E}| the number of triangles in G[S]. We define o(S) = |{(u, v, w), (u, v) ∈ S, w ∈ V \ S : (uv, vw, uw) ∈ E}|, the number of outbound triangles of S, that is: triangles in G which have exactly two vertices in S. Moreover, for all (u, v) in E, let △(uv) = |{w ∈ V : (uw, vw) ∈ E}| be the number of triangles the edge uv belongs to in G. Finally, we recall the definition of the cohesion of a set S in G: C(S) = i(S)