Fig. 3: Structure of a basic symmetry motif (BSM) for an arbitrary network measure F.
a Every orbit in a BSM is an (α, β)-uniform graph\({K}_{n}^{\alpha ,\beta }\), the graph with n vertices and adjacency matrix A = (aij) with aij = α = F(i, j) if i ≠ j and aii = β = F(i, i) for some constants α and β. Here, we show an example of one orbit with four nodes and edges labelled by their weights. b The connectivity between two orbits Δ1 and Δ2 in the same BSM (after a suitable relabelling Δ1 = {v1, …, vn}, Δ2 = {w1, …, wn}) is given by γ = F(vi, wj) for i ≠ j, and δ = F(vi, wi), the (δ, γ)-uniform join of the two orbits. Here we show an example of two orbits (shown by colour) with three vertices each and edges labelled by their weights. c In the quotient, the BSM orbit becomes a single vertex with a self-loop weighted by (n − 1)α + β, and d the two orbits are joined by an edge weighted by (n − 1)γ + δ. Here, we show the quotients (c) and d of the previous BSMs (a) and b, respectively. Note that, by annotating each orbit in the quotient by n and α (or β), and each intra-motif edge by γ (or δ), we can recover each BSM from such annotated quotient.
