Foundations of Data Science

present and ask how correlated this percentage is with the presences or absence of edges
at some distance d. One is interested in whether the correlation drops off with distance.
To explore this concept we consider the Ising model studied in physics.
The Ising or ferromagnetic model is a pairwise random Markov field. The underlying
graph, usually a lattice, assigns a value of ±1, called spin, to the variable at each vertex.
The probability (Gibbs measure) of a given configuration of spins is proportional
to exp(β
∑ x i x j ) =
∏ e βx ix j
where x i = ±1 is the value associated with vertex i.
Thus
(i,j)∈E
(i,j)∈E
p (x 1 , x 2 , . . . , x n ) = 1 Z
where Z is a normalization constant.
∏
(i,j)∈E
exp(βx i x j ) = 1 Z e β ∑
x i x j
(i,j)∈E
The value of the summation is simply the difference in the number of edges whose
vertices have the same spin minus the number of edges whose vertices have opposite spin.
The constant β is viewed as inverse temperature. High temperature corresponds to a low
value of β and low temperature corresponds to a high value of β. At high temperature,
low β, the spins of adjacent vertices are uncorrelated whereas at low temperature adjacent
vertices have identical spins. The reason for this is that the probability of a configuration
is proportional to e β ∑ x i x j.
i∼j As β is increased, for configurations with a large number of
edges whose vertices have identical spins, e β ∑ x i x j
i∼j increases more than for configurations
whose edges have vertices with non identical spins. When the normalization constant 1 Z
is adjusted for the new value of β, the highest probability configurations are those where
adjacent vertices have identical spins.
Given the above probability distribution, what is the correlation between two variables
x i and x j . To answer this question, consider the probability that x i equals plus one as a
function of the probability that x j equals plus one. If the probability that x i equals plus
one is 1 2 independent of the value of the probability that x j equals plus one, we say the
values are uncorrelated.
Consider the special case where the graph G is a tree. In this case a phase transition
occurs at β 0 = 1 d+1
ln where d is the degree of the tree. For a sufficiently tall tree and for
2 d−1
β > β 0 , the probability that the root has value +1 is bounded away from 1 / 2 and depends
on whether the majority of leaves have value +1 or -1. For β < β 0 the probability that
the root has value +1 is 1 / 2 independent of the values at the leaves of the tree.
Consider a height one tree of degree d. If i of the leaves have spin +1 and d − i have
spin -1, then the probability of the root having spin +1 is proportional to
e iβ−(d−i)β = e (2i−d)β .
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