Graphettes: Constant-time determination of graphlet and orbit identity including (possibly disconnected) graphlets up to size 8

Abstract

Graphlets are small connected induced subgraphs of a larger graph $G$. Graphlets are now commonly used to quantify local and global topology of networks in the field. Methods exist to exhaustively enumerate all graphlets (and their orbits) in large networks as efficiently as possible using orbit counting equations. However, the number of graphlets in $G$ is exponential in both the number of nodes and edges in $G$. Enumerating them all is already unacceptably expensive on existing large networks, and the problem will only get worse as networks continue to grow in size and density. Here we introduce an efficient method designed to aid statistical sampling of graphlets up to size $k=8$ from a large network. We define graphettes as the generalization of graphlets allowing for disconnected graphlets. Given a particular (undirected) graphette $g$, we introduce the idea of the canonical graphette $\mathcal K(g)$ as a representative member of the isomorphism group $Iso(g)$ of $g$. We compute the mapping $\mathcal K$, in the form of a lookup table, from all $2^{k(k-1)/2}$ undirected graphettes $g$ of size $k\le 8$ to their canonical representatives $\mathcal K(g)$, as well as the permutation that transforms $g$ to $\mathcal K(g)$. We also compute all automorphism orbits for each canonical graphette. Thus, given any $k\le 8$ nodes in a graph $G$, we can in constant time infer which graphette it is, as well as which orbit each of the $k$ nodes belongs to. Sampling a large number $N$ of such $k$-sets of nodes provides an approximation of both the distribution of graphlets and orbits across $G$, and the orbit degree vector at each node.