Provably Fast Inference of Latent Features from Networks

Abstract

Numerous graph mining applications rely on detecting sub-graphs which are large near-cliques. Since formulations that are geared towards finding large near-cliques are NP-hard and frequently inapproximable due to connections with the Maximum Clique problem, the poly-time solvable densest subgraph problem which maximizes the average degree over all possible subgraphs "lies at the core of large scale data mining" [10]. However, frequently the densest subgraph problem fails in detecting large near-cliques in networks. In this work, we introduce the k-clique densest subgraph problem, k ≥ 2. This generalizes the well studied densest subgraph problem which is obtained as a special case for k = 2. For k = 3 we obtain a novel formulation which we refer to as the triangle densest subgraph problem: given a graph G(V, E), find a subset of vertices S * such that τ (S *) = max S⊆V t(S) |S| , where t(S) is the number of triangles induced by the set S. On the theory side, we prove that for any k constant, there exist an exact polynomial time algorithm for the k-clique densest subgraph problem. Furthermore, we propose an efficient 1 k-approximation algorithm which generalizes the greedy peeling algorithm of Asahiro and Charikar [8, 18] for k = 2. Finally, we show how to implement efficiently this peeling framework on MapReduce for any k ≥ 3, generalizing the work of Bahmani, Kumar and Vassilvitskii for the case k = 2 [10]. On the empirical side, our two main findings are that (i) the triangle densest subgraph is consistently closer to being a large near-clique compared to the densest subgraph and (ii) the peeling approximation algorithms for both k = 2 and k = 3 achieve on real-world networks approximation ratios closer to 1 rather than the pessimistic 1 k guarantee. An interesting consequence of our work is that triangle counting, a well-studied computational problem in the context of social network analysis can be used to detect large near-cliques. Finally, we evaluate our proposed method on a popular graph mining application.

Paired threshold graphs

“…In what follows, we provide a brief analysis of (a) the diameter of PT graphs, capturing relevant connectivity properties of networks; (b) the intersection number of PT graphs, which is of relevance for latent feature modeling and inference in social networks [6,27,28]; and (c) the clustering coefficient, providing a normalized count of the number of triangles in the graphs.…”

“…In these settings, one often assumes the existence of attachment and preference rules for network formation, or imposes constraints on subgraph structures as well as vertex and edge features that govern the creation of network communities [1,2,3,4,5]. Models of this type have been used to predict network dynamics and topology fluctuations, infer network community properties and preferences, determine the bottlenecks and rates of spread of information and commodities and elucidate functional and structural properties of individual network modules [6,7,8].…”