I Let $G$ be the set of simple graphs on $n$ nodes. Given a $g \in G$, we denote the number of triangles in $g$ with $n(g)$. Given some positive real-valued parameter $w$, we define the the function $Z(w)$ as follows: $$Z(w)= \sum_{g}\exp(w\cdot n(g))$$

I think its an intractable problem to compute $Z(w)$, but I don't know about approximations. Is there a known FPRAS for computing $Z(w)$?

  • $\begingroup$ @InuyashaYagami Partition function is a bad name here inspired from physics/statistics/Machine Learning. But it does not matter, I have edited the question to just have it as $Z(w)$. But if you are interested: en.wikipedia.org/wiki/… $\endgroup$
    – SagarM
    Commented Jun 22, 2023 at 13:42
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    $\begingroup$ I assume that $G$ is the set of all simple undirected graphs on $n$ nodes. I didn't find the solution, but maybe the following will help. It suffices to know the distribution of the number of triangles in an Erdos-Renyi graph with $p= \frac 12$. Basically, something like stats.stackexchange.com/questions/338267/…, but you need to know the whole distribution. $\endgroup$
    – Dmitry
    Commented Jun 22, 2023 at 14:09
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    $\begingroup$ So, basically, you are interested in Moment generating function for the number of triangles in Erdos-Renyi$(n, \frac 12)$. $\endgroup$
    – Dmitry
    Commented Jun 22, 2023 at 14:24


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