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Network properties under non-overlapping sets
This protocol is extracted from research article:
Constructing graphs from genetic encodings
Sci Rep, Jun 24, 2021;

Procedure

In Fig. Fig.1a1a the active and passive bits were separate: the first three bits were active and the last two were passive in defining both Set A and Set B. If this separation is the same for all rules (all rules are defined by the expression of the first three bits), then each node is contained by only a single set (although sets may be chosen multiple times to participate in a wiring rule). Given a set-level connectivity network (top level of Figs. Figs.1a1a and and3a),3a), we can produce a node-level network by inserting links between nodes if the sets they belong to had a link between them (bottom level of Figs. Figs.1a1a and and3a).3a). We can consider the resulting network to be a function of the original one, such as in “Constant set size”, we have ER networks at the set level, and GM(ER) networks at the node level (read GM(ER) as “node-level Genetic Model of set-level Erdős–Rényi network”). Although this process greatly amplifies the number of nodes and links between the class-level and the neuronal-level networks, we find that the network metrics remain largely unchanged in the process, even if the class sizes are non-uniform (if x, or the number of passive bits, varies between cell classes, Methods 1.2.2).

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 $L=r∗22x$, 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:

In this section we expand the simplifying assumption of mapping networks with constant set sizes, allowing now for sizes to be distributed according to q(s). The method in the previous section, where we mapped with fixed set sizes, is a special case of this with $q(s)=δ(s-s0)$.

Given an edge between two sets of sizes $si$ and $sj$ drawn from q(s), $si∗sj$ edges are introduced into the node-level network. The expected number of links at the node level is therefore the number of rules multiplied by a factor of $⟨s1s2⟩=⟨s⟩2$. The expected density can then be expressed as

when $Nset⟨s⟩>>1$. Thus, the overall density is not expected to differ from the ER network, even with variable set sizes.

To derive the degree distribution, we consider a set with degree $kj$ at the set-level network. Such a set contains $sj$ nodes, each with the same degree $Kj=∑i=1ksi$. Depending on q(s) and the degree distribution $pk$ of the set-level network, the resulting expected degree distribution of the node-level network $pK$ is given by

According to the law of total expectation and the law of total variance,

and

If the set-level model is an ER network, having Poisson degree distribution with $σ2(k)=⟨k⟩$, (12) simplifies to $σ2(K)=⟨k⟩⟨s2⟩=⟨K⟩⟨s2⟩⟨s⟩$. The GM(ER) model under variable set size therefore has a degree distribution with Fano-factor (as a measure of dispersion) $σ2(K)⟨K⟩=⟨s2⟩⟨s⟩$.

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