Elegant six-page proof reveals the emergence of random structure
Maybe someone familiar with the graph theory math can help me.I don't understand why these properties aren't numerically accessible. Why is the estimation necessary?E.g. Assume N nodes, and edges E can be created at random with redundancy (picking same edge e twice just keeps the prior edge) by picking pairs of points.Assume N random edge picks are made, then the probability of e.g. a Hamiltonian cycle appearing in those N picks can be calculated.(My quick back of envelope sketch says P(H-cycle) = N! / N^N , but please consult the expert literature.)But one can also calculate the probability of a H-cycle given N+1 random picks, N+2 random picks, ... and that appears as 
 = P(H-cycle) * (1 + extra factors that account for increases in other edges not redundantly in the H-cycle)(Again, my back of envelope sketch for N + 2 picks gives:
    = P(H-cycle) * ( 1 + 2/(N-1)[N - 3 - 1/N]) , but please consult the expert literature.)These probabilities would seem to tell a user that given a graph with N nodes and E edges,  that if e.g. in the H-cycle case, E > N the user can get an explicit probability for the likelihood of a H-cycle being present.Are there graph properties that prevent this approach being viable?