Source: Adv. in Appl. Probab. Volume 42, Number 3
(2010), 706-738.
We study first passage percolation (FPP) on the configuration model (CM) having
power-law degrees with exponent τ ∈ [1, 2) and exponential edge
weights. We derive the distributional limit of the minimal weight of a path
between typical vertices in the network and the number of edges on the
minimal-weight path, both of which can be computed in terms of the
Poisson-Dirichlet distribution. We explicitly describe these limits via
construction of infinite limiting objects describing the FPP problem in the
densely connected core of the network. We consider two separate cases, the
original CM, in which each edge, regardless of its multiplicity,
receives an independent exponential weight, and the erased CM, for which
there is an independent exponential weight between any pair of direct
neighbors. While the results are qualitatively similar, surprisingly, the
limiting random variables are quite different. Our results imply that the flow
carrying properties of the network are markedly different from either the
mean-field setting or the locally tree-like setting, which occurs as
τ > 2, and for which the hopcount between typical vertices scales
as log n. In our setting the hopcount is tight and has an explicit
limiting distribution, showing that information can be transferred remarkably
quickly between different vertices in the network. This efficiency has a down
side in that such networks are remarkably fragile to directed attacks. These
results continue a general program by the authors to obtain a complete picture
of how random disorder changes the inherent geometry of various random network
models; see Aldous and Bhamidi (2010), Bhamidi (2008), and Bhamidi, van der
Hofstad and Hooghiemstra (2009).
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