Balanced Clique Computation in Signed Networks: Concepts and Algorithms

Clique is one of the most fundamental models for cohesive subgraph mining in network analysis. Existing clique model mainly focuses on unsigned networks. However, in real world, many applications are modeled as signed networks with positive and negative edges. As the signed networks hold their own properties different from the unsigned networks, the existing clique model is inapplicable for the signed networks. Motivated by this, we propose the balanced clique model that considers the most fundamental and dominant theory, structural balance theory, for signed networks. Following the balanced clique model, we study the <underline>m</underline>aximal <underline>b</underline>alanced <underline>c</underline>lique <underline>e</underline>numeration problem (<inline-formula><tex-math notation="LaTeX">${\mathsf {MBCE}}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">MBCE</mml:mi></mml:math><inline-graphic xlink:href="yuan-ieq1-3225562.gif"/></alternatives></inline-formula>) which computes all the maximal balanced cliques in a given signed network. Moreover, in some applications, users prefer a unique and representative balanced clique with maximum size rather than all balanced cliques. Thus, we also study the <underline>m</underline>aximum <underline>b</underline>alanced <underline>c</underline>lique <underline>s</underline>earch problem (<inline-formula><tex-math notation="LaTeX">${\mathsf {MBCS}}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">MBCS</mml:mi></mml:math><inline-graphic xlink:href="yuan-ieq2-3225562.gif"/></alternatives></inline-formula>) which computes the balanced clique with maximum size. We show that <inline-formula><tex-math notation="LaTeX">${\mathsf {MBCE}}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">MBCE</mml:mi></mml:math><inline-graphic xlink:href="yuan-ieq3-3225562.gif"/></alternatives></inline-formula> problem and <inline-formula><tex-math notation="LaTeX">${\mathsf {MBCS}}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">MBCS</mml:mi></mml:math><inline-graphic xlink:href="yuan-ieq4-3225562.gif"/></alternatives></inline-formula> problem are both NP-Hard. For the <inline-formula><tex-math notation="LaTeX">${\mathsf {MBCE}}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">MBCE</mml:mi></mml:math><inline-graphic xlink:href="yuan-ieq5-3225562.gif"/></alternatives></inline-formula> problem, a straightforward solution is to treat the signed network as two unsigned networks and leverage the off-the-shelf techniques for unsigned networks. However, such a solution is inefficient for large signed networks. To address this problem, in this paper, we first propose a new maximal balanced clique enumeration algorithm by exploiting the unique properties of signed networks. Based on the new proposed algorithm, we devise two optimization strategies to further improve the efficiency of the enumeration. For the <inline-formula><tex-math notation="LaTeX">${\mathsf {MBCS}}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">MBCS</mml:mi></mml:math><inline-graphic xlink:href="yuan-ieq6-3225562.gif"/></alternatives></inline-formula> problem, we first propose a baseline solution. To overcome the huge search space problem of the baseline solution, we propose a new search framework based on search space partition. To further improve the efficiency of the new framework, we propose multiple optimization strategies regarding to redundant search branches and invalid candidates. We conduct extensive experiments on large real datasets. The experimental results demonstrate the efficiency, effectiveness and scalability of our proposed algorithms for <inline-formula><tex-math notation="LaTeX">${\mathsf {MBCE}}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">MBCE</mml:mi></mml:math><inline-graphic xlink:href="yuan-ieq7-3225562.gif"/></alternatives></inline-formula> problem and <inline-formula><tex-math notation="LaTeX">${\mathsf {MBCS}}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">MBCS</mml:mi></mml:math><inline-graphic xlink:href="yuan-ieq8-3225562.gif"/></alternatives></inline-formula> problem.