Enumerating Graphlets with Amortized Time Complexity Independent of Graph Size

TL;DR

This paper introduces a novel algorithm for enumerating graphlets of size k in a graph with an amortized time per solution that depends only on k, making it efficient even for large graphs.

Contribution

The authors present the first algorithms for listing all k-graphlets with amortized cost depending solely on k, independent of the graph size or maximum degree.

Findings

Amortized cost for listing k-graphlets is O(k^2) per solution.

Edge k-graphlets can be listed in O(k) time per solution.

For bounded degree graphs, cost reduces to O(k) per solution.

Abstract

Graphlets of order $k$ in a graph $G$ are connected subgraphs induced by $k$ nodes (called $k$-graphlets) or by $k$ edges (called edge $k$-graphlets). They are among the interesting subgraphs in network analysis to get insights on both the local and global structure of a network. While several algorithms exist for discovering and enumerating graphlets, the cost per solution of such algorithms typically depends on the size of the graph $G$, or its maximum degree. In real networks, even the latter can be in the order of millions, whereas $k$ is typically required to be a small value. In this paper we provide the first algorithm to list all graphlets of order $k$ in a graph $G=(V,E)$ with an amortized cost per solution depending \emph{solely} on the order $k$, contrarily to previous approaches where the cost depends \emph{also} on the size of $G$ or its maximum degree. Specifically, we show that it is possible to list $k$-graphlets in $O(k^2)$ time per solution, and to list edge $k$-graphlets in $O(k)$ time per solution. Furthermore we show that, if the input graph has bounded degree, then the cost per solution for listing $k$-graphlets is reduced to $O(k)$. Whenever $k = O(1)$, as it is often the case in practical settings, these algorithms are the first to achieve constant time per solution.