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 as a representative member of the isomorphism group Iso(g) of g. We compute the mapping , in the form of a lookup table, from all 2k(k − 1)/2 undirected graphettes g of size k ≤ 8 to their canonical representatives , as well as the permutation that transforms g to . We also compute all automorphism orbits for each canonical graphette. Thus, given any k ≤ 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.

Edge crossings in random linear arrangements

“…We have developed efficient algorithms to calculate V * [C] in general graphs and also in forests. Since the calculation of V * [C] reduces to a problem of counting the number of subgraphs of a certain type (Equation 2 and Table 1), one could also apply algorithms for counting graphlets or graphettes [26,27]. Graphlets are connected subgraphs while graphettes are a generalization of graphlets to potentially disconnected subgraphs.…”

“…The subgraphs for calculating the number of products of each types are indeed graphettes; only for types 03 and 04 the subgraphs are graphlets ( Figure 2). As the subgraphs we are interested in have between 4 and 6 vertices ( Figure 2; recall that types 00 and 01 do not matter), we could generate all subsets of 4, 5 and 6 vertices and use the look-up table provided in [27] to classify the corresponding subgraphs in constant time for each subset. However, that would produce and algorithm that runs in Θ(n 6 ) time while ours runs in O n 5 (recall it runs in o nm 2 , Table 3).…”

Optimising orbit counting of arbitrary order by equation selection

“…Originally, graphlets were restricted to 5 or fewer nodes because of limitations in computing power. Later research [5, 6] increased that number of nodes. Regardless of the exact computing power available, the exponentially exploding number of graphlets enforces the use of some cut-off on the number of nodes in a graphlet.…”

“…Other techniques to count graphlets have been developed: ranging from combinatorial counting that uses graphlets’ symmetries and substructures to count them without finding any graphlet, which is currently limited to graphlets on 4 nodes [11], to sampling techniques that do not count all graphlets but identify randomly sampled subgraphs [6], or incremental counting [12].…”

BLANT—fast graphlet sampling tool

“…Using a pre-computed lookup table allows graphlet isomorphism to be done in constant time for k-graphlets up to size k = 8 (Hasan et al, 2017). BLANT (Basic Local Alignment for Networks Tool) leverages this speed, and rather than taking 18 hours, it can produce output statistically indistinguishable from ORCA's in minutes.…”