Independent sets in random graphs from the weighted second moment method

We prove new lower bounds on the likely size of a maximum independent set in a random graph with a given average degree. Our method is a weighted version of the second moment method, where we give each independent set a weight based on the total degree of its vertices.

💡 Research Summary

The paper investigates the typical size of a maximum independent set in the Erdős–Rényi random graph G(n, d/n) where the average degree d is fixed. Classical applications of the second moment method count all independent sets of a given size equally, which works well for very sparse graphs but becomes ineffective as d grows because the degree distribution introduces significant variance among independent sets. To overcome this limitation, the authors introduce a weighted version of the second moment method.

For any independent set S they define a weight
 wλ(S) = exp(‑λ ∑_{v∈S} deg(v)),
where λ>0 is a tunable parameter. This weight penalizes sets that contain high‑degree vertices, thereby reducing the contribution of “atypical” independent sets to the second moment while preserving enough mass in the first moment to keep the expectation large.

Let Zk = Σ_{S∈I_k} wλ(S) be the weighted count of independent sets of size k (I_k denotes all such sets). The authors compute the first moment E