Efficient <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="li-ieq1-3314643.gif"/></alternatives></inline-formula>-Clique Counting on Large Graphs: The Power of Color-Based Sampling Approaches

<inline-formula><tex-math notation="LaTeX">$K$</tex-math><alternatives><mml:math><mml:mi>K</mml:mi></mml:math><inline-graphic xlink:href="li-ieq3-3314643.gif"/></alternatives></inline-formula>-clique counting is a fundamental problem in network analysis which has attracted much attention in recent years. Computing the count of <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="li-ieq4-3314643.gif"/></alternatives></inline-formula>-cliques in a graph for a large <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="li-ieq5-3314643.gif"/></alternatives></inline-formula> (e.g., <inline-formula><tex-math notation="LaTeX">$k=8$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>8</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="li-ieq6-3314643.gif"/></alternatives></inline-formula>) is often intractable as the number of <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="li-ieq7-3314643.gif"/></alternatives></inline-formula>-cliques increases exponentially w.r. t. (with respect to) <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="li-ieq8-3314643.gif"/></alternatives></inline-formula>. Existing exact <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="li-ieq9-3314643.gif"/></alternatives></inline-formula>-clique counting algorithms are often hard to handle large dense graphs, while sampling-based solutions either require a huge number of samples or consume very high storage space to achieve a satisfactory accuracy. To overcome these limitations, we propose a new framework to estimate the number of <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="li-ieq10-3314643.gif"/></alternatives></inline-formula>-cliques which integrates both the exact <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="li-ieq11-3314643.gif"/></alternatives></inline-formula>-clique counting technique and three novel color-based sampling techniques. The key insight of our framework is that we only apply the exact algorithm to compute the <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="li-ieq12-3314643.gif"/></alternatives></inline-formula>-clique counts in the <italic>sparse regions</italic> of a graph, and use the proposed color-based sampling approaches to estimate the number of <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="li-ieq13-3314643.gif"/></alternatives></inline-formula>-cliques in the <italic>dense regions</italic> of the graph. Specifically, we develop three novel dynamic programming based <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="li-ieq14-3314643.gif"/></alternatives></inline-formula>-color set sampling techniques to efficiently estimate the <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="li-ieq15-3314643.gif"/></alternatives></inline-formula>-clique counts, where a <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="li-ieq16-3314643.gif"/></alternatives></inline-formula>-color set contains <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="li-ieq17-3314643.gif"/></alternatives></inline-formula> nodes with <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="li-ieq18-3314643.gif"/></alternatives></inline-formula> different colors. Since a <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="li-ieq19-3314643.gif"/></alternatives></inline-formula>-color set is often a good approximation of a <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="li-ieq20-3314643.gif"/></alternatives></inline-formula>-clique in the dense regions of a graph, our sampling-based solutions are extremely efficient and accurate. Moreover, the proposed sampling techniques are space efficient which use near-linear space w.r. t. graph size. We conduct extensive experiments to evaluate our algorithms using 8 real-life graphs. The results show that our best algorithm is at least one order of magnitude faster than the state-of-the-art sampling-based solutions (with the same relative error 0.1%) and can be up to three orders of magnitude faster than the state-of-the-art exact algorithm on large graphs.