Hubs, lanes, and junction cliques

For every $k\in {\mathbb{Z}}^2$, introduce a hub site $H_k$. For every $k\in {\mathbb{Z}}^2$, $p\in P_n$, and $i=0,\dots ,N_n$, introduce a lane site

All these labels denote distinct sites.

The lane indexed by $(k,p)$ has graph edges

Its nominal endpoints are the hubs $H_k$ and $H_{k+p}$, but its endpoint sites are not identified with the hubs.

At hub $H_k$, define its junction set

Add all graph edges between distinct sites of $C_k$, making it a clique. Different junction sets are disjoint.

Let ${\Lambda }_n$ be this graph's vertex set and $d_n$ its unweighted shortest-path metric.