• John Baez, Networks and population biology (Part 4), 6 May 2011.

It’s about an attempt by Persi Diaconis to define random graphs where certain features show up. Here’s a teaser, which leaves out the shocking conclusion:

People have studied the Erdős–Rényi random graphs very intensively, so now people are eager to study random graphs with more interesting correlations. For example, consider the graph where we draw an edge between any two people who are friends. If you’re my friend and I’m friends with someone else, that improves the chances that you’re friends with them! In other words, friends tend to form ‘triangles’. But in an Erdős–Rényi random graph there’s no effect like that.

‘Exponential families’ of random graphs seem like a way around this problem. The idea here is to pick a specific collection of graphs and say how commonly we want these to appear in our random graph. If we only use one graph and we take this to be two vertices connected by an edge, we’ll get an Erdős–Rényi random graph. But, if we also want our graph to contain a lot of triangles, we can pick to be a triangle.

His approach seems very well-motivated by statistical mechanics. However, it doesn’t work! But the way it fails is itself interesting.

]]>• Alan M. Frieze, On the value of a random minimum spanning tree problem, *Discrete Applied Mathematics* **10** (1985), 47–56.

Perhaps this was before people knew about, or talked about, the ‘giant component’ of a random graph.

]]>First use all edges smaller than 2c/n. There will be about cn of them, so for c>1, there will be a giant component C of size bn where b + exp(-bc) – 1 = 0. Delete edges from C until it is a tree T. The sum of weights in T is about bn(c/n) = bc.

Next, use edges with weights in [2c/n,1] to join G-T to T, using the edge with smallest weight for each vertex in G-T. There are about bn choices for each one, so these weights add up to about (1-b)n (2c/n + 1/(bn)) = (1-b)(2c_1/b). The total is bc + 2c – 2cb + 1/b – 1 = 2c – 1 – bc – 1/b. For example, if c=2, b is about .8 and this is 3-1.6+1.25=2.65.

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