I was reading though https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7811397/ and had a question I was hoping to get clarified.
In Section 3.1 it looks like one begins with nodes located at integer pairs $(a,b)$ with $1\leq a,b \leq L.$ Equations (29) and (30) govern the movement of walkers during each time step. Essentially, each coordinate just moves up or down by a random integer between $-h$ and $h.$
There must be some sort of conditions placed on $h$, correct? For example, I cannot have $L=3$ and $h=5$. This choice of $h$ would allow a walker to leave the $L \times L$ integer lattice.
Similarly, I believe one must impose some sort of boundary conditions. Is this right? For example, if $L=3$ and walker $i$ has position $(3,3)$ after $n$ steps, then the random integer $-h\leq j\leq h$ cannot take the value $1$, for example.
Also. You do not need to read the entire article. Section 3 is self-contained! Thanks!
Yes, something must be specified about the boundary, but since the paper does not specify anything, we cannot read the authors' minds to tell you what they meant.
I think the most plausible answer (given the emphasis on regular networks in section 2) is that the grid wraps around modulo $L$: if a walker is at position $(3,3)$ and moves by $(-5, -4)$, the walker ends up at $(L-2, L-1)$. This ensures that every node, even the nodes near the boundary, have degree $(2h+1)^2$, as claimed.