Constant set size
Constructing graphs from genetic encodings
Sci Rep, Jun 24, 2021; DOI: 10.1038/s41598-021-92577-2

We previously considered how we can map edges between sets to fully connected groups of participating nodes. Here, we consider how network metrics are impacted by this process. We begin with an ER model at the set level, consisting L=r links between 2b-x sets. In this section, each set contains the same number of nodes, 2x. In this way, each edge between classes corresponds to a wiring rule connecting 2x nodes to another 2x nodes on the resulting network of 2b nodes, which we term GM(ER). Given L=r links between 2b-x sets, at the ER level we have a density of

given the sparse assumption of r<<22(b-x). This is simply the density of an ER graph with r links, allowing for self-loops. Performing the GM(ER) mapping from the set-level ER network to the node-level network with N=2b and L=r22x, we find that the GM(ER) has a density of

Yet, if we compare the density of the ER and GM(ER) networks, we find that they are equivalent:

where the approximation holds as long as 2b>>1 and 2b-x>>1.

Thus, the mapping of an ER network using constant set sizes does not alter the density of the system. Further, since the clustering coefficient can be defined as the probability that a third edge is present in a triangle, we can assume the clustering will be constant, and ρ in the node-level GM(ER) system as well.

Finally, we derive the degree the distribution of the GM(ER) system. Since no overlaps between rules occur, the only allowed degree sizes are multiples of the number of links a node gains from a single rule, 2x. Thus, the degree distribution becomes: