I have a graph G of N nodes and M edges. Graph G is a real-world graph, which could be connected or disconnected. Still, its largest component has small-world properties: High clustering coefficient and low average shortest length.

Many random subgraphs can be generated from the graph G as follows. I choose randomly (uniformly) n nodes (n < N), where two nodes are connected if they are connected in the original graph. The total number of edges in one subgraph is denoted by m.

I have two questions:

• Should I expect the subgraphs to have some properties of the Erdos-Ranyi graphs?
• And what is the expected number of edges of the subgraphs (m)? Thank you.