The Index-Based Subgraph Matching Algorithm with General Symmetries (ISMAGS): Exploiting Symmetry for Faster Subgraph Enumeration

Abstract

Subgraph matching algorithms are used to find and enumerate specific interconnection structures in networks. By enumerating these specific structures/subgraphs, the fundamental properties of the network can be derived. More specifically in biological networks, subgraph matching algorithms are used to discover network motifs, specific patterns occurring more often than expected by chance. Finding these network motifs yields information on the underlying biological relations modelled by the network. In this work, we present the Index-based Subgraph Matching Algorithm with General Symmetries (ISMAGS), an improved version of the Index-based Subgraph Matching Algorithm (ISMA). ISMA quickly finds all instances of a predefined motif in a network by intelligently exploring the search space and taking into account easily identifiable symmetric structures. However, more complex symmetries (possibly involving switching multiple nodes) are not taken into account, resulting in superfluous output. ISMAGS overcomes this problem by using a customised symmetry analysis phase to detect all symmetric structures in the network motif subgraphs. These structures are then converted to symmetry-breaking constraints used to prune the search space and speed up calculations. The performance of the algorithm was tested on several types of networks (biological, social and computer networks) for various subgraphs with a varying degree of symmetry. For subgraphs with complex (multi-node) symmetric structures, high speed-up factors are obtained as the search space is pruned by the symmetry-breaking constraints. For subgraphs with no or simple symmetric structures, ISMAGS still reduces computation times by optimising set operations. Moreover, the calculated list of subgraph instances is minimal as it contains no instances that differ by only a subgraph symmetry. An implementation of the algorithm is freely available at https://github.com/mhoubraken/ISMAGS.

Higher-order organization of complex networks

“…However, since most real-world networks are sparse, we instead focus on the complexity of algorithms in terms of the number of edges and the maximum degree in the graph. For this case, there are several efficient practical algorithms for real networks with available software(42)(43)(44)(45)(46).Theoretically, motif counting is efficient. Here we consider four classes of motifs:(1) triangles, (2) wedges (connected, non-triangle three-node motifs), (3) four-node motifs, and (4)k-cliques.…”

A Cytoscape app for motif enumeration with ISMAGS

“…This eliminates the possibility of finding different motif instances with the same graph nodes, which decreases the running time of the algorithm. More detailed information can be found in Houbraken et al (2014).…”

“…To assess the application's running time, a number of motifs were searched in random graphs. As extensive runtime benchmarks of the ISMAGS algorithm can be found in Houbraken et al (2014) we focus here on benchmarking the Cytoscape app itself. All of these tests were performed on a computer with 4 Intel ® Core TM i5-3230M CPU @ 2.60 GHz CPUs and 7.7 GiB internal memory.…”

“…In this work, we focus on an application which can be run on a standard, off-the-shelf desktop or laptop without specialized hardware. ISMAGS (Houbraken et al, 2014) is an efficient Java implementation for the motif enumeration problem running on stock hardware.…”

An Algorithm to Automatically Generate the Combinatorial Orbit Counting Equations

“…A more formal definition: graph G is an instance of motif M if and only if Since only certain significant subgraphs are required, not all motifs are counted. ISMAGS [ 4 , 5 ] is one motif finding algorithm which lists each occurrence of a specific motif within an explored graph quickly. As there is no distinction between which graphlets are ‘significant’ and which not, counting all graphlets at the same time can be more useful than counting each of them apart.…”