Title: Concentration of Multilinear Functions of the Ising Model with Applications to Network Data,
Abstract: We prove near-tight concentration of measure for polynomial functions of the
Ising model under high temperature. For any degree $d$, we show that a
degree-$d$ polynomial of a $n$-spin Ising model exhibits exponential tails that
scale as $\exp(-r^{2/d})$ at radius $r=\tilde{\Omega}_d(n^{d/2})$. Our
concentration radius is optimal up to logarithmic factors for constant $d$,
improving known results by polynomial factors in the number of spins. We
demonstrate the efficacy of polynomial functions as statistics for testing the
strength of interactions in social networks in both synthetic and real world
data.