Higher-Order Truss Decomposition in Graphs

Zi Chen, Long Yuan, Li Han, Zhengping Qian

Abstract: inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -truss model is a typical cohesive subgraph model and has been received considerable attention recently. However, the <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -truss model only considers the direct common neighbors of an edge, which restricts its ability to reveal fine-grained structure information of the graph. Motivated by this, in this paper, we propose a new model named <inline-formula><tex-math notation="LaTeX">$(k, \tau)$</tex-math></inline-formula> -truss that considers the higher-order neighborhood ( <inline-formula><tex-math notation="LaTeX">$\tau$</tex-math></inline-formula> hop) information of an edge. Based on the <inline-formula><tex-math notation="LaTeX">$(k, \tau)$</tex-math></inline-formula> -truss model, we study the higher-order truss decomposition problem which computes the <inline-formula><tex-math notation="LaTeX">$(k, \tau)$</tex-math></inline-formula> -trusses for all possible <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> values regarding a given <inline-formula><tex-math notation="LaTeX">$\tau$</tex-math></inline-formula> . Higher-order truss decomposition can be used in the applications such as community detection and search, hierarchical structure analysis, and graph visualization. To address this problem, we first propose a bottom-up decomposition paradigm in the increasing order of <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> values to compute the corresponding <inline-formula><tex-math notation="LaTeX">$(k, \tau)$</tex-math></inline-formula> -truss. Based on the bottom-up decomposition paradigm, we further devise three optimization strategies to reduce the unnecessary computation. We evaluate our proposed algorithms on real datasets and synthetic datasets, the experimental results demonstrate the efficiency, effectiveness and scalability of our proposed algorithms.