generate-synthetic-networks.rst

Generate synthetic networks

It is necessary to benchmark our algorithm via synthetic networks, where we have perfect confidence on the model parameters. Here we can control the nodes' block membership b, the edge count matrix e_rs, and the degree distribution d. Once we specified these 3 parameters, we call Graph-tool's generate_sbm function to generate a graph-tool graph instance, and then use that instance as an input for our biSBM.optimalks.OptimalKs class.

Block membership b

Edge count matrix e_rs

Easy cases

The easy cases are bipartite version of the planted partition models. It is a (K_a, K_b)-bipartite structure with e_rs = pc/K + (1 − p)(1 − c)/K(K − 1), where K_a = K_b = K and p = c_out/c_in is a parameter that controls how crisp the structure is. When p = 0, we expect no connections at all between nodes of different groups, i.e., e_rs = Eδ_rs/K.

A easy case with ⟨k⟩ = 10 and p = 0.1, planted as (K_a, K_b) = (5, 5) equally-sized communities.

Hard cases

The hard cases are planted networks where K_a ≠ K_b, but the edge count matrix is still designed in a way that allow the control of network structural strength.

A hard case with ⟨k⟩ = 10 and p = 0.1, planted as (K_a, K_b) = (5, 31) equally-sized communities.

Even harder cases

A even harder case is simply a random draw out of the ensemble of edge count matrices where we only fix K_a and K_b.

A harder case with ⟨k⟩ = 10, planted as (K_a, K_b) = (5, 31) equally-sized communities. This is the strongest structure out of 10⁶ random e_rs draws.

Degree distribution d