The connectivity-shell hypothesis for the metric/topological picture of space-time

Let $ST$ be the supergraph whose grains are maximal complete subgraphs, or mss, of the underlying network, with neighborhood structure defined by intersection of cliques. For node sets or subgraphs $A$ and $B$, define their connectivity by

$$c_{AB}:=\frac{|A\sim B|}{|A\sim B|_m},\qquad 0\leq c_{AB}\leq 1,$$

where $|A\sim B|$ is the number of bonds joining $A$ to $B$ and $|A\sim B|_m$ is the maximal possible number. Let $S_0$ be a typical grain, and let $\{S^0_\nu\}$, $\{S^1_{\nu'}\}$, and so on denote successive shells around it in $ST$.

Connectivity-shell hypothesis. When the unfolding process is fully developed, the infinitesimal neighborhood of $S_0$ is densely connected to it, and connectivity decreases through successive shells:

$$c_{S_0S_0}=1,\qquad c_{S_0S^0_\nu}\lesssim 1,$$ $$1\geq c_{S_0S^0_\nu}\geq c_{S_0S^1_{\nu'}}\geq c_{S_0S^2_{\nu”}}\geq\cdots.$$

Moreover, this shell structure should be consistent with the neighborhood structure of $ST$: node distance on $ST$ should correspond, more or less, to decreasing connectivity.

The hypothesis is intended to provide a metric/topological interpretation of the emergent space-time supergraph. The paper motivates it by the observed regularity of random graphs and the peaked typical size of maximal complete subgraphs, but presents the consistency with the $ST$ neighborhood structure as an expectation rather than a proved result.