Suppose we have a graph on $n$ nodes. We would like to assign to each node either a $+1$ or a $-1$. Call this a configuration $\sigma \in \{+1,-1\}^n$. The number of $+1$s that we have to assign is exactly $s$ (hence the number of $-1$s is $n-s$.) Given a configuration $\sigma$, we look at each node $i$ and sum the values assigned to its neighbors, call this $\xi_i(\sigma)$. We then count the number of nodes for which $\xi_i(\sigma)$ is nonnegative:

$$ N(\sigma) := \sum_{i=1}^n 1( \xi_i(\sigma) \ge 0). $$

The question is: what is the configuration $\sigma$ that maximizes $N(\sigma)$? Can we give a bound on $(\max N)/n$ in terms of $s/n$. If it helps, the graph can be assumed to be Erdős-Renyi.

exactnumber of edges to put down uniformly at random. It is not so surprising that the behaviour of these two models (putting down $pn(n-1)/2$ edges at random and putting in each edge with probability $p$) has very similar behaviour. – Anthony Quas May 13 '12 at 15:05