Correlation clustering on networks

Abstract

Random networks with co-existing positive and negative links are studied from the viewpoint of the NP hard correlation clustering problem. The task is to produce a clustering of the vertices which maximizes the number of positive edges within clusters and the number of negative edges between clusters. Simulated annealing, Monte Carlo renormalization and molecular dynamics optimization are used to find the optimal cluster structure. Recently, this problem was studied for globally coupled systems and an interesting phase-transition-like phenomenon was predicted: in the thermodynamic limit the relative size of the largest cluster, r, exhibits a step-like behavior as a function of the density of positive links q (r = 0 if q < 1/2 and r = 1 if q > 1/2). Here we prove that when considering random networks with a constant bond density, the same phase transition is expected. A totally different result emerges however, when networks with a fixed average number of connections per node are considered. In such cases a nontrivial spin-glass-type behavior is found, where the location of the critical point shifts toward q > 1/2 values. The results also suggest that instead of the simple step-like behavior, the r(q) curve has a more complex shape, which depends on the specific topology of the considered network.