Truss-Based Structural Diversity Search in Large Graphs

Social decisions made by individuals are easily influenced by information from their social neighborhoods. A key predictor of social contagion is the multiplicity of social contexts inside the individual’s contact neighborhood, which is termed structural diversity. However, the existing models have limited decomposability for analyzing large-scale networks, and suffer from the inaccurate reflection of social context diversity. In this paper, we propose a truss-based structural diversity model to overcome the weak decomposability. Based on this model, we study a novel problem of truss-based structural diversity search in a graph <inline-formula><tex-math notation="LaTeX">$G$</tex-math><alternatives><mml:math><mml:mi>G</mml:mi></mml:math><inline-graphic xlink:href="huang-ieq1-3027950.gif"/></alternatives></inline-formula>, that is, to find the <inline-formula><tex-math notation="LaTeX">$r$</tex-math><alternatives><mml:math><mml:mi>r</mml:mi></mml:math><inline-graphic xlink:href="huang-ieq2-3027950.gif"/></alternatives></inline-formula> vertices with the highest truss-based structural diversity and return their social contexts. To tackle this problem, we propose an online structural diversity search algorithm in <inline-formula><tex-math notation="LaTeX">$O(\rho (m+\mathcal {T}))$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mi>ρ</mml:mi><mml:mo>(</mml:mo><mml:mi>m</mml:mi><mml:mo>+</mml:mo><mml:mi mathvariant="script">T</mml:mi><mml:mo>)</mml:mo><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="huang-ieq3-3027950.gif"/></alternatives></inline-formula> time, where <inline-formula><tex-math notation="LaTeX">$\rho$</tex-math><alternatives><mml:math><mml:mi>ρ</mml:mi></mml:math><inline-graphic xlink:href="huang-ieq4-3027950.gif"/></alternatives></inline-formula>, <inline-formula><tex-math notation="LaTeX">$m$</tex-math><alternatives><mml:math><mml:mi>m</mml:mi></mml:math><inline-graphic xlink:href="huang-ieq5-3027950.gif"/></alternatives></inline-formula>, and <inline-formula><tex-math notation="LaTeX">$\mathcal {T}$</tex-math><alternatives><mml:math><mml:mi mathvariant="script">T</mml:mi></mml:math><inline-graphic xlink:href="huang-ieq6-3027950.gif"/></alternatives></inline-formula> are respectively the arboricity, the number of edges, and the number of triangles in <inline-formula><tex-math notation="LaTeX">$G$</tex-math><alternatives><mml:math><mml:mi>G</mml:mi></mml:math><inline-graphic xlink:href="huang-ieq7-3027950.gif"/></alternatives></inline-formula>. To improve the efficiency, we design an elegant and compact index, called TSD-index, which keeps the structural diversity information for all individual vertices. We further optimize the structure of TSD-index into a highly compressed GCT-index. Our GCT-index-based structural diversity search utilizes the global triangle information for fast index construction and finds answers in <inline-formula><tex-math notation="LaTeX">$O(m)$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mi>m</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="huang-ieq8-3027950.gif"/></alternatives></inline-formula> time. Extensive experiments demonstrate the effectiveness and efficiency of our proposed model and algorithms, against state-of-the-art methods.