Accelerating All 5-Vertex Subgraphs Counting Using GPUs

Abstract

The subgraph counting problem is the problem of counting the number of occurrences of graph patterns in the target graph and is widely used as a fundamental technique for network analyses in different domains. The computational cost of subgraph counting grows drastically as the size of the pattern increases; it takes much time even with the state-of-the-art algorithms when counting 5-vertex patterns. To this problem, this paper proposes a subgraph counting method using GPUs. More precisely, we employ one of the state-of-the-art algorithms for 5-vertex subgraph counting and extend it so that counting is executed in parallel using massive threads. We conducted experiments for evaluating the performance of our proposed method by using real-world datasets, and the results demonstrate that our proposed method is about 4x to 10x and about 3$\times$ to 5$\times$ times faster than the original method in computing 5-vertex and 4-vertex subgraphs, respectively.

Appendices

Appendix

A Counting Other Subgraphs

Table 3. Additional Notation

For other subgraphs, we can use simple formulas as described below. We can easily parallelize them.

$$\begin{aligned} F_{3}&= \sum _{(i,j \in E}(d(i)-)(d(j)-1) - 3F_{1}, F_{4} = \sum _{i \in V}\left( {\begin{array}{c}d(i)\\ 3\end{array}}\right) , F_{5} = \sum _{i \in V}t(i)(d(i)-2),\\[-0.5em] F_{6}&= \sum _{i \in V}\sum _{j \prec i}\left( {\begin{array}{c}W_{++}(i,j)+W_{+-}(i,j)\\ 2\end{array}}\right) , F_{7} = \sum _{(i,n) \in E}\left( {\begin{array}{c}|T(i,j)|\\ 2\end{array}}\right) , F_{9} = \sum _{i \in V}\left( {\begin{array}{c}d(i)\\ 4\end{array}}\right) \\[-0.5em] F_{10}&= \sum _{(i,j) \in E}[(d(i)-1)\left( {\begin{array}{c}(d(j)-1)\\ 2\end{array}}\right) + (d(j)-1)\left( {\begin{array}{c}(d(i)-1)\\ 2\end{array}}\right) ] - 2F_{5}\\[-2.0em]\\ F_{11}&= \sum _{i \in V}\sum _{(i,j) \in E}(d(j)-1) - 4F_{6} - F_{5} - 3F_{1}, F_{12} = \sum _{i \in V}t(i)(d(i)-2)\\[-0.5em] F_{13}&= \sum _{(i,j) \in E}((d(i)-1)t(j)+(d(j)-1)t(i)) - 4F_{7} - 6F_{1} - 2F_{5}\\[-0.5em] F_{14}&= \sum _{(i,j) \in E}|T(i,j)|(d(i)-2)(d(j)-2) - 2F_{7}, F_{15} = \sum _{i \in V}C_{4}(i)(d(i)-2) -2F_{7}\\[-0.7em] F_{17}&= \sum _{i \in V}\left( {\begin{array}{c}t_{i}\\ 2\end{array}}\right) - 2F_{7}, F_{18} = \sum _{(i,j) \in E}\sum _{k \in T(i,j)}(d(k) -2)(|T(i,j)| - 1) - 12F_{8}\\[-1.0em] F_{19}&= \sum _{(i,j) \in E}\left( {\begin{array}{c}|T(i,j)|\\ 2\end{array}}\right) (d(i) + d(j) - 6), F_{20} = \sum _{(i,j) \in E}C_{4}(i,j)|T(i,j)| - 4F_{7}\\[-1em] F_{21}&= \sum _{i \in V}\sum _{i \prec j}\left( {\begin{array}{c}W(i,j)\\ 3\end{array}}\right) , F_{22}= \sum _{(i,j) \in E}\left( {\begin{array}{c}|T(i,j)|\\ 3\end{array}}\right) , F_{23} = \sum _{i \in V}K_{4}(i)(d(i)-3)\\[-0.5em] F_{24}&= \sum _{(i,j) \in E}\sum _{k \in T(i,j)}((|T(i,j)| - 1)(|T(i,k)| - 1)\\[-0.5em]&+ (|T(i,j)|-1)(|T(j,k)|-1) + (|T(j,k)| - 1)(|T(i,k)|-1))\\[-0.5em] F_{26}&= \sum _{u < |diamondValue[u]|} \left( {\begin{array}{c}diamonValue[u]\\ 2\end{array}}\right) , F_{27} = \sum _{(i,j) \in E}K_{4}(i,j)(|T(i,j)|-2) \end{aligned}$$