Continuous Monitoring of Maximum Clique Over Dynamic Graphs

Abstract

The maximum clique problem ( <i>MCP</i> ) has various applications to reveal the structure and function of graphs. Graphs are constantly updated in the real life. However, no algorithm is specifically designed for dynamic graph. Although <i>MCP</i> in dynamic graphs can be solved by simply invoking a state-of-the-art static approach, such as <i>PMC</i> , when the graph is updated, such an approach of simply re-calculating from scratch is inefficient. The key issue with <i>MCP</i> algorithm is to find a large clique, namely a <i>seed</i> , as fast as possible. Thus, search space can be pruned based on the seed. Size of the seed greedily found by <i>PMC</i> cannot be guaranteed, as it fluctuates considerably. Moreover, the time required to find a seed under <i>PMC</i> is up to <inline-formula><tex-math notation="LaTeX">$O(| E| \cdot \Delta (G))$</tex-math></inline-formula> , where <inline-formula><tex-math notation="LaTeX">$\Delta (G)$</tex-math></inline-formula> is the highest degree in <i>G</i> . In this article, we intend to find a sizable seed by updating the previous maximum clique with the incident vertices of the inserted/deleted edge. Size of the seed now is guaranteed to be no less than <inline-formula><tex-math notation="LaTeX">$\omega (G^{\prime})\; - \;1$</tex-math></inline-formula> , where <inline-formula><tex-math notation="LaTeX">$\omega (G^{\prime})$</tex-math></inline-formula> is the size of the maximum clique on the updated graph. Moreover, the seed can be found in a time complexity of <inline-formula><tex-math notation="LaTeX">$O(\Delta (G)^{2})$</tex-math></inline-formula> . Two other crucial issues related to the <i>MCP</i> in dynamic graphs are refreshing rate and refreshing overhead. After a tight upper bound is imposed on <inline-formula><tex-math notation="LaTeX">$\omega (G^{\prime})$</tex-math></inline-formula> , the necessity of refreshing is evaluated by comparing the seed with its largest challenger, then unnecessary refreshing is wiped out effectively. The size of the largest challenger is judiciously estimated using a lazy growth strategy. Subsequently, the search space in refreshing is confined on a much smaller subgraph using a local refreshing strategy. Extensive experiments indicate that the proposed approach outperforms the baseline algorithm by approximately one order of magnitude.