  We propose a notion of contraction function for a family of graphs and
establish its connection to the strong spatial mixing for spin systems. More
specifically, we show that for anti-ferromagnetic Potts model on families of
graphs characterized by a specific contraction function, the model exhibits
strong spatial mixing, and if further the graphs exhibit certain local sparsity
which are very natural and easy to satisfy by typical sparse graphs, then we
also have FPTAS for computing the partition function.
  This new characterization of strong spatial mixing of multi-spin system does
not require maximum degree of the graphs to be bounded, but instead it relates
the decay of correlation of the model to a notion of effective average degree
measured by the contraction of a function on the family of graphs. It also
generalizes other notion of effective average degree which may determine the
strong spatial mixing, such as the connective constant, whose connection to
strong spatial mixing is only known for very simple models and is not
extendable to general spin systems.
  As direct consequences: (1) we obtain FPTAS for the partition function of
$q$-state anti-ferromagnetic Potts model with activity $0\le\beta<1$ on graphs
of maximum degree bounded by $d$ when $q> 3(1-\beta)d+1$, improving the
previous best bound $\beta> 3(1-\beta)d$ and asymptotically approaching the
inapproximability threshold $q=(1-\beta)d$, and (2) we obtain an efficient
sampler (in the same sense of fully polynomial-time almost uniform sampler,
FPAUS) for the Potts model on Erd\H{o}s-R\'enyi random graph
$\mathcal{G}(n,d/n)$ with sufficiently large constant $d$, provided that $q>
3(1-\beta)d+4$. In particular when $\beta=0$, the sampler becomes an FPAUS for
for proper $q$-coloring in $\mathcal{G}(n,d/n)$ with $q> 3d+4$, improving the
current best bound $q> 5.5d$ for FPAUS for $q$-coloring in
$\mathcal{G}(n,d/n)$.
