E-CLoG: Counting edge-centric local graphlets

In recent years, graphlet counting has emerged as an important task in topological graph analysis. However, the existing works on graphlet counting obtain the graphlet counts for the entire network as a whole. These works capture the key graphical patterns that prevail in a given network but they fail to meet the demand of the majority of real-life graph related prediction tasks such as link prediction, edge/node classification, etc., which require to build features for an edge (or a vertex) of a network. To meet the demand for such applications, efficient algorithms are needed for counting local graphlets within the context of an edge (or a vertex). In this work, we propose an efficient method, titled E-CLOG, for counting all 3,4 and 5 size local graphlets with the context of a given edge for its all different edge orbits. We also provide a shared-memory, multi-core implementation of E-CLOG, which makes it even more scalable for very large real-world networks. In particular, We obtain strong scaling on a variety of graphs (14x-20x on 36 cores). We provide extensive experimental results to demonstrate the efficiency and effectiveness of the proposed method. For instance, we show that E-CLOG is faster than existing work by multiple order of magnitudes; for the Wordnet graph E-CLOG counts all 3,4 and 5-size local graphlets in 1.5 hours using a single thread and in only a few minutes using the parallel implementation, whereas the baseline method does not finish in more than 4 days. We also show that local graphlet counts around an edge are much better features for link prediction than well-known topological features; our experiments show that the former enjoys between 10% to 45% of improvement in the AUC value for predicting future links in three real-life social and collaboration networks

Motif counting beyond five nodes

Counting graphlets is a well-studied problem in graph mining and social network analysis. Recently, several papers explored very simple and natural algorithms based on Monte Carlo sampling of Markov Chains (MC), and reported encouraging results. We show, perhaps surprisingly, that such algorithms are outperformed by color coding (CC) [2], a sophisticated algorithmic technique that we extend to the case of graphlet sampling and for which we prove strong statistical guarantees. Our computational experiments on graphs with millions of nodes show CC to be more accurate than MC; furthermore, we formally show that the mixing time of the MC approach is too high in general, even when the input graph has high conductance. All this comes at a price however. While MC is very efficient in terms of space, CC’s memory requirements become demanding when the size of the input graph and that of the graphlets grow. And yet, our experiments show that CC can push the limits of the state-of-the-art, both in terms of the size of the input graph and of that of the graphlets

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

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 [Formula: see text] as a representative member of the isomorphism group Iso(g) of g. We compute the mapping [Formula: see text], in the form of a lookup table, from all 2k(k - 1)/2 undirected graphettes g of size k ≤ 8 to their canonical representatives [Formula: see text], as well as the permutation that transforms g to [Formula: see text]. 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

Combinatorial algorithm for counting small induced graphs and orbits

Graphlet analysis is an approach to network analysis that is particularly popular in bioinformatics. We show how to set up a system of linear equations that relate the orbit counts and can be used in an algorithm that is significantly faster than the existing approaches based on direct enumeration of graphlets. The algorithm requires existence of a vertex with certain properties; we show that such vertex exists for graphlets of arbitrary size, except for complete graphs and $C_4$, which are treated separately. Empirical analysis of running time agrees with the theoretical results