Maximizing $(k,L)$-Core With Edge Augmentation in Multilayer Graphs

While most previous work pays attention on extracting dense subgraphs, such as $k$-cores, we argue that augmenting the graph to maximize the size of dense subgraphs is also very important and finds many applications. Therefore, in this article, we study the dense subgraph augmentation problem in multilayer graphs. Specifically, we propose the notion of $(k,L)$-core to model the dense subgraphs in multilayer graphs and propose a new research problem, budgeted maximal $(k,L)$-core augmentation (BMA) problem, which adds at most $b$ edges in the multilayer graphs to maximize the size of $(k,L)$-core. We prove the NP-hardness of the general BMA problem when $k\geq 2$ and devise a polynomial-time algorithm to find the optimal solution for a special case of BMA, i.e., $(2,1)$-BMA. We then devise an effective algorithm, named search for optimum and reorder adaptively (SORA), with various performance-improving strategies to tackle the general BMA problem. We evaluate the performance of the proposed approaches on multiple large-scale datasets and compare them with the state-of-the-art baselines. Experimental results indicate that our proposed approaches significantly outperform the baselines in terms of solution quality and efficiency.

I. Introduction

In graph mining, various measures have been studied to extract dense subgraphs for extracting knowledge from various data represented as graphs. Among them, -core has been actively studied due to its simple definition and tractability, which aims at finding the maximal induced subgraph where each vertex has at least neighbors. The notion of -core helps identify a group of vertices which have high cohesiveness. The extraction of -cores has a wide spectrum of applications, such as social network visualization [1], anomalies detection [2], identifying influential spreaders [3], and predict development of biological systems [4].