On Directed Densest Subgraph Discovery

Given a directed graph G , the directed densest subgraph (DDS) problem refers to the finding of a subgraph from G , whose density is the highest among all the subgraphs of G . The DDS problem is fundamental to a wide range of applications, such as fraud detection, community mining, and graph compression. However, existing DDS solutions suffer from efficiency and scalability problems: on a 3,000-edge graph, it takes three days for one of the best exact algorithms to complete. In this article, we develop an efficient and scalable DDS solution. We introduce the notion of [ x , y ]-core, which is a dense subgraph for G , and show that the densest subgraph can be accurately located through the [ x , y ]-core with theoretical guarantees. Based on the [ x , y ]-core, we develop exact and approximation algorithms. We further study the problems of maintaining the DDS over dynamic directed graphs and finding the weighted DDS on weighted directed graphs, and we develop efficient non-trivial algorithms to solve these two problems by extending our DDS algorithms. We have performed an extensive evaluation of our approaches on 15 real large datasets. The results show that our proposed solutions are up to six orders of magnitude faster than the state-of-the-art.

Efficient Directed Densest Subgraph Discovery

Given a directed graph G, the directed densest subgraph (DDS) problem refers to the finding of a subgraph from G, whose density is the highest among all the subgraphs of G. The DDS problem is fundamental to a wide range of applications, such as fraud detection, community mining, and graph compression. However, existing DDS solutions suffer from efficiency and scalability problems: on a threethousand- edge graph, it takes three days for one of the best exact algorithms to complete. In this paper, we develop an efficient and scalable DDS solution. We introduce the notion of [x, y]-core, which is a dense subgraph for G, and show that the densest subgraph can be accurately located through the [x, y]-core with theoretical guarantees. Based on the [x, y]-core, we develop both exact and approximation algorithms. We have performed an extensive evaluation of our approaches on eight real large datasets. The results show that our proposed solutions are up to six orders of magnitude faster than the state-of-the-art.

Technical Perspective of Efficient Directed Densest Subgraph Discovery

The problem is useful in graph mining because dense subgraphs often represent patterns deserving special attention. They could indicate, for example, an authoritative community in a social network, a building brick of more complex biology structures, or even a type of malicious behavior such as spamming. See [1, 3] and the references therein for an extensive discussion on the applications of DDS.

KClist++

The problem of finding densest subgraphs has received increasing attention in recent years finding applications in biology, finance, as well as social network analysis. The k -clique densest subgraph problem is a generalization of the densest subgraph problem, where the objective is to find a subgraph maximizing the ratio between the number of k -cliques in the subgraph and its number of nodes. It includes as a special case the problem of finding subgraphs with largest average number of triangles ( k = 3), which plays an important role in social network analysis. Moreover, algorithms that deal with larger values of k can effectively find quasi-cliques. The densest subgraph problem can be solved in polynomial time with algorithms based on maximum flow, linear programming or a recent approach based on convex optimization. In particular, the latter approach can scale to graphs containing tens of billions of edges. While finding a densest subgraph in large graphs is no longer a bottleneck, the k -clique densest subgraph remains challenging even when k = 3. Our work aims at developing near-optimal and exact algorithms for the k -clique densest subgraph problem on large real-world graphs. We give a surprisingly simple procedure that can be employed to find the maximal k -clique densest subgraph in large-real world graphs. By leveraging appealing properties of existing results, we combine it with a recent approach for listing all k -cliques in a graph and a sampling scheme, obtaining the state-of-the-art approaches for the aforementioned problem. Our theoretical results are complemented with an extensive experimental evaluation showing the effectiveness of our approach in large real-world graphs.