Stochastic Block Models (SBMs) are a popular approach to modeling single real-world graphs. The key idea of SBMs is to partition the vertices of the graph into blocks with similar edge densities within, as well as between different blocks. However, what if we are given not one but multiple graphs that are unaligned and of different sizes? How can we find out if these graphs share blocks with similar connectivity structures? In this paper, we propose the shared stochastic block modeling (SSBM) problem, in which we model n graphs using SBMs that share parameters of s blocks. We show that fitting an SSBM is NP-hard, and consider two approaches to fit good models in practice. In the first, we directly maximize the likelihood of the shared model using a Markov chain Monte Carlo algorithm. In the second, we first fit an SBM for each graph and then select which blocks to share. We propose an integer linear program to find the optimal shared blocks and to scale to large numbers of blocks, we propose a fast greedy algorithm. Through extensive empirical evaluation on synthetic and real-world data, we show that our methods work well in practice.

Finding dense subgraphs is a core problem with numerous graph mining applications such as community detection in social networks and anomaly detection. However, in many real-world networks connections are not equal. One way to label edges as either strong or weak is to use strong triadic closure~(STC). Here, if one node connects strongly with two other nodes, then those two nodes should be connected at least with a weak edge. STC-labelings are not unique and finding the maximum number of strong edges is NP-hard. In this paper, we apply STC to dense subgraph discovery. More formally, our score for a given subgraph is the ratio between the sum of the number of strong edges and weak edges, weighted by a user parameter λ, and the number of nodes of the subgraph. Our goal is to find a subgraph and an STC-labeling maximizing the score. We show that for λ = 1, our problem is equivalent to finding the densest subgraph, while for λ = 0, our problem is equivalent to finding the largest clique, making our problem NP-hard. We propose an exact algorithm based on integer linear programming and four practical polynomial-time heuristics. We present an extensive experimental study that shows that our algorithms can find the ground truth in synthetic datasets and run efficiently in real-world datasets.

One of the most well-known and simplest models for diversity maximization is the Max-Min Diversification (MMD) model, which has been extensively studied in the data mining and database literature. In this paper, we initiate the study of the Asymmetric Max-Min Diversification (AMMD) problem. The input is a positive integer k and a complete digraph over n vertices, together with a nonnegative distance function over the edges obeying the directed triangle inequality. The objective is to select a set of k vertices, which maximizes the smallest pairwise distance between them. AMMD reduces to the well-studied MMD problem in case the distances are symmetric, and has natural applications to query result diversification, web search, and facility location problems. Although the MMD problem admits a simple 1/2-approximation by greedily selecting the next-furthest point, this strategy fails for AMMD and it remained unclear how to design good approximation algorithms for AMMD.

Abstract Finding densely connected groups of nodes in networks is a widely-used tool for analysis in graph mining. A popular choice for finding such groups is to find subgraphs with a high average degree. While useful, interpreting such subgraphs may be difficult. On the other hand, many real-world networks have additional information, and we are specifically interested in networks with labels on edges. In this paper, we study finding sets of labels that induce dense subgraphs. We consider two notions of density: average degree and the number of edges minus the number of nodes weighted by a parameter $$\alpha$$ α . There are many ways to induce a subgraph from a set of labels, and we study two cases: First, we study conjunctive-induced dense subgraphs, where the subgraph edges need to have all labels. Secondly, we study disjunctive-induced dense subgraphs, where the subgraph edges need to have at least one label. We show that both problems are NP-hard. Because of the hardness, we resort to greedy heuristics. We show that we can implement the greedy search efficiently: the respective running times for finding conjunctive-induced and disjunctive-induced dense subgraphs are in $$\mathcal {O} \mathopen {}\left( p \log k\right)$$ O p log k and $$\mathcal {O} \mathopen {}\left( p \log ^2 k\right)$$ O p log 2 k , where p is the number of edge-label pairs and k is the number of labels. Our experimental evaluation demonstrates that we can find the ground truth in synthetic graphs and that we can find interpretable subgraphs from real-world networks.