4

I am looking for a formula for determining the expected number of independent sets of size $k$ (for arbitrary $k$) in a random graph $G(n,p)$. Here $n$ is the number of vertices and each edge is included with independent probability $p$. I would like to be able to calculate this for arbitrary $p$ if possible.

I have come across the article [1] which provides a formula for the special case $p = 0.5$.

I have also come across the article [2] which in p. 12 provides a value for some cases other than $p = 0.5$, so I would assume it is known for some $p$ values other than $0.5$.

My questions are:

  1. Do you know how one shows, or could you provide a reference for the formula shown in [1] for the case $p = 0.5$. The paper gives some references but they are about clique problems and I am not sure how I could arrive from them to the result shown in that paper.
  2. Is there a known formula for arbitrary $p$? If not, for what $p$ values is a formula known and where could I find such formulas?

[1] Chromatic and Independence Numbers of $G_{{n}, {{1}\over{2}}}$

[1] Feo, Thomas A., Mauricio GC Resende, and Stuart H. Smith. “A Greedy Randomized Adaptive Search Procedure for Maximum Independent Set.” Operations Research 42, no. 5 (1994): 860–78.

fiftyeight
  • 401
  • 3
  • 8

2 Answers2

7

The probability that a specific set of size $k$ is independent is exactly $(1-p)^{\binom{k}{2}}$ (why?). Linearity of expectation shows that the expected number of independent sets of size $k$ is $\binom{n}{k} (1-p)^{\binom{k}{2}}$ (why?).

If you can't follow this calculation, please follow Denis Pankratov's advice and look up linearity of expectation and indicator random variables.

Yuval Filmus
  • 280,205
  • 27
  • 317
  • 514
2

This is an old thread, but I thought it was worth adding: there actually is at least one paper that derives this result (or to be precise, the equivalent result of the expected number of cliques of size $k$; cliques in $G(n,p)$ correspond to independent sets in $G(n,1-p)$). It is a conference paper, "On the Complete Subgraphs of a Random Graph" by David Matula, from 1970, available here, on the author's website. It's not a hard result to derive, but someone thought it was worth publishing at least once!

Gwen McKinley
  • 21
  • 1