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.

Dense Sub-networks Discovery in Temporal Networks

Temporal networks have been successfully applied to analyse dynamics of networks. In this paper we focus on an approach recently introduced to identify dense subgraphs in a temporal network and we present a heuristic, based on the local search technique, for the problem. The experimental results we present on synthetic and real-world datasets show that our heuristic provides mostly better solutions (denser solutions) and that the heuristic is fast (comparable with the fastest method in literature, which is outperformed in terms of quality of the solutions). We present also experimental results of two variants of our method based on two different subroutines to compute a dense subgraph of a given graph.

Mining explainable local and global subgraph patterns with surprising densities

The connectivity structure of graphs is typically related to the attributes of the vertices. In social networks for example, the probability of a friendship between any pair of people depends on a range of attributes, such as their age, residence location, workplace, and hobbies. The high-level structure of a graph can thus possibly be described well by means of patterns of the form ‘the subgroup of all individuals with certain properties X are often (or rarely) friends with individuals in another subgroup defined by properties Y’, ideally relative to their expected connectivity. Such rules present potentially actionable and generalizable insight into the graph. Prior work has already considered the search for dense subgraphs (‘communities’) with homogeneous attributes. The first contribution in this paper is to generalize this type of pattern to densities between a pair of subgroups, as well as between all pairs from a set of subgroups that partition the vertices. Second, we develop a novel information-theoretic approach for quantifying the subjective interestingness of such patterns, by contrasting them with prior information an analyst may have about the graph’s connectivity. We demonstrate empirically that in the special case of dense subgraphs, this approach yields results that are superior to the state-of-the-art. Finally, we propose algorithms for efficiently finding interesting patterns of these different types.

Top-k overlapping densest subgraphs: approximation algorithms and computational complexity

A central problem in graph mining is finding dense subgraphs, with several applications in different fields, a notable example being identifying communities. While a lot of effort has been put in the problem of finding a single dense subgraph, only recently the focus has been shifted to the problem of finding a set of densest subgraphs. An approach introduced to find possible overlapping subgraphs is the problem. Given an integer $$k \ge 1$$ k ≥ 1 and a parameter $$\lambda > 0$$ λ > 0 , the goal of this problem is to find a set of k dense subgraphs that may share some vertices. The objective function to be maximized takes into account the density of the subgraphs, the parameter $$\lambda $$ λ and the distance between each pair of subgraphs in the solution. The problem has been shown to admit a $$\frac{1}{10}$$ 1 10 -factor approximation algorithm. Furthermore, the computational complexity of the problem has been left open. In this paper, we present contributions concerning the approximability and the computational complexity of the problem. For the approximability, we present approximation algorithms that improve the approximation factor to $$\frac{1}{2}$$ 1 2 , when k is smaller than the number of vertices in the graph, and to $$\frac{2}{3}$$ 2 3 , when k is a constant. For the computational complexity, we show that the problem is NP-hard even when $$k=3$$ k = 3 .