Higher-Order Truss Decomposition in Graphs

<inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="yuan-ieq1-3137955.gif"/></alternatives></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><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="yuan-ieq2-3137955.gif"/></alternatives></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><alternatives><mml:math><mml:mrow><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="yuan-ieq3-3137955.gif"/></alternatives></inline-formula>-truss that considers the higher-order neighborhood (<inline-formula><tex-math notation="LaTeX">$\tau$</tex-math><alternatives><mml:math><mml:mi>τ</mml:mi></mml:math><inline-graphic xlink:href="yuan-ieq4-3137955.gif"/></alternatives></inline-formula> hop) information of an edge. Based on the <inline-formula><tex-math notation="LaTeX">$(k, \tau)$</tex-math><alternatives><mml:math><mml:mrow><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="yuan-ieq5-3137955.gif"/></alternatives></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><alternatives><mml:math><mml:mrow><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="yuan-ieq6-3137955.gif"/></alternatives></inline-formula>-trusses for all possible <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="yuan-ieq7-3137955.gif"/></alternatives></inline-formula> values regarding a given <inline-formula><tex-math notation="LaTeX">$\tau$</tex-math><alternatives><mml:math><mml:mi>τ</mml:mi></mml:math><inline-graphic xlink:href="yuan-ieq8-3137955.gif"/></alternatives></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><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="yuan-ieq9-3137955.gif"/></alternatives></inline-formula> values to compute the corresponding <inline-formula><tex-math notation="LaTeX">$(k, \tau)$</tex-math><alternatives><mml:math><mml:mrow><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="yuan-ieq10-3137955.gif"/></alternatives></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.