Convex optimization for the densest subgraph and densest submatrix problems

We consider the densest $k$-subgraph problem, which seeks to identify the $k$-node subgraph of a given input graph with maximum number of edges. This problem is well-known to be NP-hard, by reduction to the maximum clique problem. We propose a new convex relaxation for the densest $k$-subgraph problem, based on a nuclear norm relaxation of a low-rank plus sparse decomposition of the adjacency matrices of $k$-node subgraphs to partially address this intractability. We establish that the densest $k$-subgraph can be recovered with high probability from the optimal solution of this convex relaxation if the input graph is randomly sampled from a distribution of random graphs constructed to contain an especially dense $k$-node subgraph with high probability. Specifically, the relaxation is exact when the edges of the input graph are added independently at random, with edges within a particular $k$-node subgraph added with higher probability than other edges in the graph. We provide a sufficient condition on the size of this subgraph $k$ and the expected density under which the optimal solution of the proposed relaxation recovers this $k$-node subgraph with high probability. Further, we propose a first-order method for solving this relaxation based on the alternating direction method of multipliers, and empirically confirm our predicted recovery thresholds using simulations involving randomly generated graphs, as well as graphs drawn from social and collaborative networks.

Exact Clustering of Weighted Graphs via Semidefinite Programming

As a model problem for clustering, we consider the densest k-disjoint-clique problem of partitioning a weighted complete graph into k disjoint subgraphs such that the sum of the densities of these subgraphs is maximized. We establish that such subgraphs can be recovered from the solution of a particular semidefinite relaxation with high probability if the input graph is sampled from a distribution of clusterable graphs. Specifically, the semidefinite relaxation is exact if the graph consists of k large disjoint subgraphs, corresponding to clusters, with weight concentrated within these subgraphs, plus a moderate number of outliers. Further, we establish that if noise is weakly obscuring these clusters, i.e, the between-cluster edges are assigned very small weights, then we can recover significantly smaller clusters. For example, we show that in approximately sparse graphs, where the between-cluster weights tend to zero as the size n of the graph tends to infinity, we can recover clusters of size polylogarithmic in n. Empirical evidence from numerical simulations is also provided to support these theoretical phase transitions to perfect recovery of the cluster structure.