Local algorithms for distance-generalized core decomposition over large dynamic graphs

Abstract

The distance-generalized core, also called ( k , h )-core, is defined as the maximal subgraph in which every vertex has at least k vertices at distance no longer than h. Compared with k -core, ( k , h )-core can identify more fine-grained subgraphs and, hence, is more useful for the applications such as network analysis and graph coloring. The state-of-the-art algorithms for ( k , h )-core decomposition are peeling algorithms, which iteratively delete the vertex with the minimum h -degree (i.e., the least number of neighbors within h hops). However, they suffer from some limitations, such as low parallelism and incapability of supporting dynamic graphs. To address these limitations, in this paper, we revisit the problem of ( k , h )-core decomposition. First, we introduce two novel concepts of pairwise h-attainability index and n-order H-index based on an insightful observation. Then, we thoroughly analyze the properties of n -order H-index and propose a parallelizable local algorithm for ( k , h )-core decomposition. Moreover, several optimizations are presented to accelerate the local algorithm. Furthermore, we extend the proposed local algorithm to address the ( k , h )-core maintenance problem for dynamic graphs. Experimental studies on real-world graphs show that, compared to the best existing solution, our proposed algorithms can reduce the ( k , h )-core decomposition time by 1--3 orders of magnitude and save the maintenance cost by 1--2 orders of magnitude.

Neighborhood-based Hypergraph Core Decomposition

“…Due to those 'missing' hyperedges, the number of neighbors of some nodes in that subhypergraph may drop below 𝑘, violating the coreness condition. Thus a local approach used for computing the 𝑘-core [40,48,52] or (𝑘, ℎ)-core [47] in graphs results in incorrect neighborhood-based hypergraph cores ( §6.1).…”

“…A hypergraph can also be transformed into a bipartite graph by representing the hyperedges as nodes in the second partition and creating an edge between two cross-partition nodes if the hyperedge in the second partition contains a node in the first partition. Finally, distance-2 core decomposition [15,47] is applied (Figure 4). In distance-2 core-decomposition, nodes in 𝑘-core have at least 𝑘 2-hop neighbors in the subgraph.…”

“…Optimizations-I and II have not been used in earlier core-decomposition works for both graphs and hypergraphs. Our Optimizations-III and IV are motivated by [47], Let us denote 𝐸 + (𝑣) and 𝑁 + (𝑣) at 𝑘 as 𝐸 + (𝑣, 𝑘) and 𝑁 + (𝑣, 𝑘), respectively. Exploiting the index structure 𝐵, we have the following dynamic programming paradigm for efficiently computing 𝑁 + (𝑣, 𝑘).…”

“…Note that Liu et al [47] determine the redundancy of H computation for node 𝑣 based on the convergence of both ℎ…”

“…A 𝑘-uniform hypergraph with 𝑚 hyperedges causes its clique graph to have O (𝑚𝑘 2 ) edges. The bipartite graph representation also requires distance-2 core decomposition [15,47], which is more expensive due to inflated problem-size( §6).…”