In this paper, we investigate a graph-theoretical model of social networks. The dot product model assumes that two individuals are connected in the social network if their attributes or opinions are similar. In the model, a d-dimensional vector a(v) represents the extent to which individual v has each of a set of d attributes or opinions. Then two individuals u and v are assumed to be friends, that is, they are connected in the graph model, if and only if a(u) . a(v) >= t, for some fixed, positive threshold t. The resulting graph is called a d-dot product graph. We consider diversity and clustering in social networks by using a d-dot product graph model for the network. Diversity is considered through the size of the largest independent set of the graph, and clustering through the size of the largest clique. We present both positive and negative results on the potential of this model. We obtain a tight result for the diversity problem, namely that it is polynomial-time solvable for d = 2, but NP-hard for d >= 3. We show that the clustering problem is polynomial-time solvable for d = 2. To our knowledge, these results are also the first on the computational complexity of combinatorial optimization problems on dot product graphs. We also give new insights into the structure of dot product graphs. We also consider the situation when two individuals u and v are connected if and only if their preferences are not antithetical, that is, if and only if a(u) . a(v) >= 0, and the situation when two individuals u and v are connected if and only if their preferences are neither antithetical nor "orthogonal", that is, if and only if a(u) . a(v) >= 0. For these two cases we prove that the diversity problem is polynomial-time solvable for any fixed d and that the clustering problem is polynomial-time solvable for d <= 3. (C) 2015 Elsevier B.V. All rights reserved.
