Dense subgraph discovery is an important primitive in graph mining, which has a wide variety of applications in diverse domains. In the densest subgraph problem, given an undirected graph G = (V, E) with an edge-weight vector w, we aim to find a subset of vertices S that maximizes the density, i.e., w(S) |S|, where w(S) is the sum of the weights of the edges in the subgraph induced by S. Although the densest subgraph problem is one of the most well-studied optimization problems for dense subgraph discovery, there is an implicit strong assumption; it is assumed that the weights of all the edges are known exactly as input. In real-world applications, there are often cases where we have only uncertain information of the edge weights. In this study, we provide a framework for dense subgraph discovery under the uncertainty of edge weights. Specifically, we address such an uncertainty issue using the theory of robust optimization. First, we formulate our fundamental problem, the robust densest subgraph problem, and present a simple algorithm. We then formulate the robust densest subgraph problem with sampling oracle that models dense subgraph discovery using an edge-weight sampling oracle, and present an algorithm with a strong theoretical performance guarantee. Computational experiments using both synthetic graphs and popular real-world graphs demonstrate the effectiveness of our proposed algorithms.
In addition to the variant on uncertain graphs, the densest subgraph problem has numerous noteworthy problem variations. Examples include the size-constraint variants @cite_10 @cite_5 @cite_16 @cite_1 @cite_11 and the variants generalizing the term @math in the density @cite_24 @cite_8 @cite_27 and the term @math in the density @cite_29 . Furthermore, a large body of work has been devoted to the streaming or dynamic settings of the densest subgraph problem @cite_13 @cite_31 @cite_30 @cite_32 @cite_14 @cite_17 . Some literatures have considered the densest subgraph problem on hypergraphs @cite_32 @cite_20 or on multilayer networks @cite_33 .
@cite_13 @cite_30 @cite_14 @cite_33 @cite_8 @cite_29 @cite_1 @cite_32 @cite_17 @cite_24 @cite_27 @cite_5 @cite_31 @cite_16 @cite_10 @cite_20 @cite_11
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In this paper we study the densest subgraph problem, which plays a key role in many graph mining applications. The goal of the problem is to find a subset of nodes that induces a graph with maximum average degree. The problem has been extensively studied in the past few decades under a variety of different settings. Several exact and approximation algorithms were proposed. However, as normal graph can only model objects with pairwise relationships, the densest subgraph problem fails in identifying communities under relationships that involve more than 2 objects, e.g., in a network connecting authors by publications. We consider in this work the densest subgraph problem in hypergraphs, which generalizes the problem to a wider class of networks in which edges might have different cardinalities and contain more than 2 nodes. We present two exact algorithms and a near-linear time r-approximation algorithm for the problem, where r is the maximum cardinality of an edge in the hypergraph. We also consider the dynamic version of the problem, in which an adversary can insert or delete an edge from the hypergraph in each round and the goal is to maintain efficiently an approximation of the densest subgraph. We present two dynamic approximation algorithms in this paper with amortized polog update time, for any e > 0. For the case when there are only insertions, the approximation ratio we maintain is r(1+e), while for the fully dynamic case, the ratio is r2(1+e). Extensive experiments are performed on large real datasets to validate the effectiveness and efficiency of our algorithms. In the densest subgraph problem, given an edge-weighted undirected graph (G=(V,E,w) ), we are asked to find (S V ) that maximizes the density, i.e., w(S) |S|, where w(S) is the sum of weights of the edges in the subgraph induced by S. This problem has often been employed in a wide variety of graph mining applications. However, the problem has a drawback; it may happen that the obtained subset is too large or too small in comparison with the size desired in the application at hand. In this study, we address the size issue of the densest subgraph problem by generalizing the density of (S V ). Specifically, we introduce the f-density of (S V ), which is defined as w(S) f(|S|), where (f: Z _ 0 R _ 0 ) is a monotonically non-decreasing function. In the f-densest subgraph problem (f-DS), we aim to find (S V ) that maximizes the f-density w(S) f(|S|). Although f-DS does not explicitly specify the size of the output subset of vertices, we can handle the above size issue using a convex concave size function f appropriately. For f-DS with convex function f, we propose a nearly-linear-time algorithm with a provable approximation guarantee. On the other hand, for f-DS with concave function f, we propose an LP-based exact algorithm, a flow-based (O(|V|^3) )-time exact algorithm for unweighted graphs, and a nearly-linear-time approximation algorithm. Numerous graph mining applications rely on detecting subgraphs which are large near-cliques. Since formulations that are geared towards finding large near-cliques are 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 limitsS ⊆ 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. Given an undirected graph G = (V ,E ), the density of a subgraph on vertex set S is defined as @math , where E (S ) is the set of edges in the subgraph induced by nodes in S . Finding subgraphs of maximum density is a very well studied problem. One can also generalize this notion to directed graphs. For a directed graph one notion of density given by Kannan and Vinay [12] is as follows: given subsets S and T of vertices, the density of the subgraph is @math , where E (S ,T ) is the set of edges going from S to T . Without any size constraints, a subgraph of maximum density can be found in polynomial time. When we require the subgraph to have a specified size, the problem of finding a maximum density subgraph becomes NP -hard. In this paper we focus on developing fast polynomial time algorithms for several variations of dense subgraph problems for both directed and undirected graphs. When there is no size bound, we extend the flow based technique for obtaining a densest subgraph in directed graphs and also give a linear time 2-approximation algorithm for it. When a size lower bound is specified for both directed and undirected cases, we show that the problem is NP-complete and give fast algorithms to find subgraphs within a factor 2 of the optimum density. We also show that solving the densest subgraph problem with an upper bound on size is as hard as solving the problem with an exact size constraint, within a constant factor.