Distance-generalized Core Decomposition

The $k$-core of a graph is defined as the maximal subgraph in which every vertex is connected to at least $k$ other vertices within that subgraph. In this work we introduce a distance-based generalization of the notion of $k$-core, which we refer to as the $(k,h)$-core, i.e., the maximal subgraph in which every vertex has at least $k$ other vertices at distance $\leq h$ within that subgraph. We study the properties of the $(k,h)$-core showing that it preserves many of the nice features of the classic core decomposition (e.g., its connection with the notion of distance-generalized chromatic number) and it preserves its usefulness to speed-up or approximate distance-generalized notions of dense structures, such as $h$-club. Computing the distance-generalized core decomposition over large networks is intrinsically complex. However, by exploiting clever upper and lower bounds we can partition the computation in a set of totally independent subcomputations, opening the door to top-down exploration and to multithreading, and thus achieving an efficient algorithm

Cohesive Subgraph Detection and Search in Large Graphs

Graphs are widely used to model complex networks in real-world applications. The identification and search for cohesive subgraphs are fundamental tasks in graph analysis. Despite the many models proposed to measure subgraph cohesiveness and solve tasks, there are notable gaps in the research. Specifically, current studies often overlook the higher-order information while modeling and the search for stable communities in signed social networks. This thesis aims to bridge these gaps by exploring these two issues. Firstly, k-peak is a well-regarded cohesive subgraph model in graph analysis. However, the k-peak model only considers the direct neighbors of a vertex, consequently limiting its capacity to uncover higher-order structural information of the graph. To address this limitation, we propose a new model in this thesis, named (k,h)-peak, which incorporates higher-order (h-hops) neighborhood information of vertices. Employing the (k,h)-peak model, we explore the higher-order peak decomposition problem that calculates the vertex peakness for all conceivable k values given a particular h. To tackle this problem efficiently, we propose an advanced local computation based algorithm, which is parallelizable, and additionally, devise novel pruning strategies to mitigate unnecessary computation. Experiments as well as case studies are conducted on real-world datasets to evaluate the efficiency and effectiveness of our proposed solutions. Secondly, most existing studies of community search focus on unsigned graphs, \ie treating all relationships as positive. However, friend-and-foe relationships naturally exist in many real-world applications. Ignoring the signed information may lead to unstable communities. To make up for these deficiencies, in this thesis, we study a novel stable community search called Signed k-Truss Community Search (STCS), which leverages the properties of k-truss and the balanced triangle theorem. Given a signed graph and a query vertex, the STCS returns the community that is densely connected (ensured by the k-truss model), query-centered (smallest diameter), and eliminates all the unbalanced structures. We prove that the problem of identifying the maximum signed k-truss community is NP-hard. To answer the STCS, we develop both exact and approximate algorithms. Specifically, we proposed the bottom-up exact approach in a BFS manner by integrating the local framework, shrinking strategy, and breaking strategy. In addition, to further improve the search efficiency, we propose a 2-approximation algorithm. To deal with large graphs, novel pruning strategies and algorithms are developed. Finally, we conduct experiments on real-world signed networks to evaluate the performance of proposed techniques