10.5. FINDING OVERLAPPING COMMUNITIES
383
Prior Probabilities
When we do an MLE analysis, we generally assume that the parameters
can take any value in their range, and there is no bias in favor of particular
values. However, if that is not the case, then we can multiply the formula
we get for the probability of the observed artifact being generated, as
a function of the parameter values, by the function that represents the
relative likelihood of those values of the parameter being the true values.
The exercises oﬀer examples of MLE with assumptions about the prior
distribution of the parameters.
The only way the right side can be 0 is if p is 0 or 1, or the last factor,
23(1 −p) −82p

is 0.
When p is 0 or 1, the value of the likelihood function p23(1 −p)82 is
minimized, not maximized, so it must be the last factor that is 0. That is, the
likelihood of generating the graph of Fig. 10.19 is maximized when
23 −23p −82p = 0
or p = 23/105.
That outcome is hardly a surprise. It says the most likely value for p is the
observed fraction of possible edges that are present in the graph. However, when
we use a more complicated mechanism to generate graphs or other artifacts,
the value of the parameters that produce the observed artifact with maximum
likelihood is far from obvious.
✷
10.5.3
The Aﬃliation-Graph Model
We shall now introduce a reasonable mechanism, called the aﬃliation-graph
model, to generate social graphs from communities. Once we see how the pa-
rameters of the model inﬂuence the likelihood of seeing a given graph, we can
address how one would solve for the values of the parameters that give the
maximum likelihood. The elements of the graph-generation mechanism are:
1. There is a given number of communities, and there is a given number of
individuals (nodes of the graph).
2. Each community can have any set of individuals as members. That is,
the memberships in the communities are parameters of the model.
3. Each community C has a probability pC associated with it, the probability
that two members of community C are connected by an edge because they
are both members of C. These probabilities are also parameters of the
model.
