0
$\begingroup$

Consider a square lattice (random) graph $G$ embedded on a toroidal surface. Each edge $(i, j)$ of the graph has an associated likelihood probability $p_{ij}$. The probabilities $p_{ij}$ come from a probability distribution with some general parameters like mean $\mu$, variance $\sigma$, etc.

I'd like to study the likelihood ratio of homologically non-trivial cycles to homologically trivial cycles in this setting and ideally set some reasonable upper bounds on this likelihood ratio. Is there any relevant research related to this?

I've found that people have studied inhomogenous percolation models in random graphs where they use a method based on generating functions to derive the critical percolation threshold. This is not the same problem but it at least smells similar.

Improve this question
$\endgroup$
3
  • 1
    $\begingroup$ Can you give a more specific example of a question you would like to consider? Suppose we have an $n\times n$ lattice torus - if I understand the terminology right, a homologically non-trivial cycle must contain at least $n$ edges, while a homologically trivial cycle could contain just 4 edges? Or are you interested more in comparing cycles of similar lengths? $\endgroup$
    – James Martin
    Commented Aug 10, 2022 at 14:44
  • $\begingroup$ @JamesMartin Thanks very much for pointing this out. In the scenario I'm considering, I'm interested specifically in cycles that have at least $n$ edges, whether homologically trivial or not. But the first scenario that you mention would also be interesting to know about. $\endgroup$
    – Sanchayan Dutta
    Commented Aug 11, 2022 at 4:36
  • $\begingroup$ If there are any results about the second question you mention, where the cycles have similar lengths or the same length, I'd be glad to know about that. Even when $p_{ij} = p$ is a constant. :-) $\endgroup$
    – Sanchayan Dutta
    Commented Aug 11, 2022 at 4:43

0

Reset to default

You must log in to answer this question.