Contact processes on random graphs with power law degree
distributions have critical value 0.
Shirshendu Chatterjee and Rick Durrett
Abstract.
If we consider the contact process with infection rate
l on
a random graph on n vertices with power law degree distributions,
mean field calculations suggest that the critical value
lc
of the infection rate is positive if the power a > 3.
Physicists seem to regard this as an established fact, since
the result has recently been generalized to bipartite graphs by
Gomez-Gardenes et al (2008) [PNAS 105, 1399-1404]. Here, we show that the critical value
lc
is zero for any value of a > 3, and
the contact process, starting from all vertices infected, maintains a
positive density of infected sites for time at least
exp(n 1- d )
for any d > 0. Using the last result, together with the contact process
duality, we can establish the existence of a quasi-stationary distribution in which
a randomly chosen vertex is occupied with probability
r(l).
It is expected that r(l) ≈
C lb
as l → 0.
Here we show that
a - 1 ≤ b ≤
2 a - 3, and so b > 2 for
a > 3. Thus even though the graph is locally tree-like,
b does not take the mean field critical value
b=1.
Preprint
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