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.

Figure 1. Subgraph examples.

In “XXXXXX”, every node is symmetric to every other node as, in a clique, all nodes can be swapped. The “XX00XX” graph is symmetric as it has rotation symmetry (all nodes can be shifted in the ring) and reflection symmetry (top two nodes can be switched with bottom two nodes for example). The “XZ00ZY” graph is also symmetric as the same configuration is obtained when node is switched with node and node with node . While the G4 graph has no symmetric properties, the tetrahedron and the Petersen graph have more complex symmetric structures.

Figure 2. Some permuted instances of the Petersen graph.

Figure 3. Example of subgraph refinement.

The top figure shows the initial partitioning in the “XZ00ZY” graph in which all nodes are in the same cell. However, nodes and have an outgoing edge while nodes 3 and 4 do not. This indicates the partition needs to be refined. Nodes and are put in a separate cell as shown in the bottom figure.

Figure 4. Subgraph symmetry analysis of “XX00XX”.

The branches are denoted with the applied coupling. The boxed numbers indicate the order of tree traversal, with a depth-first exploration, according to the smallest node first coupling. The initial partition has all nodes in the same cells and . The first coupling splits up the cells in both partitions in 3 cells (separation of cells denoted by |). When a permutation is found, the orbit partition is updated as shown. Orbit pruning is used to reduce the required computations as explained in text.