Xiaowei Ye, Rong-Hua Li, Qiangqiang Dai, Hongzhi Chen, and Guoren Wang. 2022. Lightning Fast and Space Efficient k-clique Counting. In Proceedings of the ACM Web Conference 2022 (WWW â€˜22)

### Introduction

• Analyzing cliques in graph data is critical in many applications, but exact counting or enumeration of these structures can be computationally costly.
• Problem: Given a graph $G$ and $k \in \N$, estimate the number of $k$-cliques in $G$.

#### Sampling Algorithm for Counting Cliques

• Sampling is often the way to go when the objective is to count some structure in a large graph.
• The aim is to efficiently gather samples from a sample space which encapsulates the set weâ€™re interested in.
• Assuming that the set of interest is $\mathcal{A}$ and the sample space is $\Omega$. If it is possible to obtain uniform random samples from $\Omega$, it is natural to take $t$ samples, and count the number of samples that are in $\mathcal{A}$.
• For this simple algorithm, the Chernoffâ€™s Bound ensures a probabilisitic guarantee. Chernoffâ€™s Bound for Sampling
Let $\rho = \abs{\mathcal{A}} / \abs{\Omega}$. A uniform sampling algorithm returns a $1 - \epsilon$ approximation of $\abs{\mathcal{A}}$ with probability $1 - 2\sigma$ if more than $\frac{3}{\rho\epsilon^2}\log(1/\sigma)$ samples are taken uniformly at random.
• Hence, the ultimate aim is to maximize $\rho$, which essentially means finding a sample space that closely mirrors $\mathcal{A}$.

### Key Ideas

This paper develops an efficient algorithm for $k$-clique estimation via uniform sampling. By employing a greedy coloring strategy, the algorithm initially reduces the sample space to the set of $k$-colored sets/paths, which are structures that have a high likelihood of being cliques. The counting of the number of $k$-colored sets/paths is achieved through dynamic programming.

For sparse graphs, PIVOTER (WSDM 20) already performs remarkably well. The authors thereby propose a framework where given graph is split into sparse and dense region, and run PIVOTER on sparse region, while the dense region is dealt with the sampling algorithm authors propose.

#### DP-Based Colored Set Sampling (DPColor)

• Consider the proper graph coloring (no edge should connect vertices with same color).
• A $k$-colored set (set of $k$ vertices with distinct color) is a good candidate for cliques! Correctness (Unbiasedness) of Sample Space
If a set $\set{v_1, \dots, v_k}$ is a $k$-clique in $G$, it must have distinct colors for any given proper coloring.
• To use small number of colors, use degeneracy-ordered greedy coloring
• To sample $k$-colored set, we shall count the number of $k$-colored set via dynamic programming.
• Let $a_i$ be the number of nodes with color $i$, and $F(i, j)$ be the number of $j$-colored sets, considering the vertices with color only up to $i$. The $F(i, j)$ follows the following recurrence. $$F(i, j) = a_i \times F(i-1, j-1) + F(i-1, j)$$
• Using this as weights, uniform random sampling can be easily done.

#### DP-Based Colored Path Sampling (DPColorPath)

• How to improve further? Instead of $k$-colored set, consider $k$-colored paths.
• Choose a center node $u$ arbitrarily. From $N(u)$, count and sample $k$-colored path.
• This is much more likely to be a clique than $k$-colored sets.
• Similar to $k$-colored sets, $k$-colored paths (locally on $N(u)$) can be counted via dynamic programming.

### Results

• The DPColorPath method demonstrates significant speed (an order of magnitude faster than the state-of-the-art methods) and negligible (0.1%) error on large-scale real-world graph datasets, including social networks and citation networks.
• The $\rho$ value for $k$-colored paths are much higher than $k$-colored sets

Overall, impressive results (accuracy and speed) with relatively simple algorithm. Implementation seems also reasonably doable. Giving more structure on the graph via proper coloring to reduce the sample space seems like a really nice idea.