Counting small subgraphs in multi-layer networks

Motivated by the prevalence of multi-layer network structures in biological and social systems, we investigate the problem of counting the number of occurrences of (small) subgraphs or motifs in multi-layer graphs in which each layer of the graph has useful structural properties. Making use of existing meta-theorems, we focus on the parameterised complexity of motif-counting problems, giving conditions on the layers of a graph that yield fixed-parameter tractable algorithms for motif-counting in the overall graph. We give a dichotomy showing that, under some restricting assumptions, either the problem of counting the number of motifs is fixed-parameter tractable, or the corresponding decision problem is already W[1]-hard.