35 research outputs found
Critical Percolation on Random Networks with Prescribed Degrees
Random graphs have played an instrumental role in modelling real-world
networks arising from the internet topology, social networks, or even
protein-interaction networks within cells. Percolation, on the other hand, has
been the fundamental model for understanding robustness and spread of epidemics
on these networks. From a mathematical perspective, percolation is one of the
simplest models that exhibits phase transition, and fascinating features are
observed around the critical point. In this thesis, we prove limit theorems
about structural properties of the connected components obtained from
percolation on random graphs at criticality. The results are obtained for
random graphs with general degree sequence, and we identify different
universality classes for the critical behavior based on moment assumptions on
the degree distribution.Comment: Ph.D. thesi
Phase transitions of extremal cuts for the configuration model
The -section width and the Max-Cut for the configuration model are shown
to exhibit phase transitions according to the values of certain parameters of
the asymptotic degree distribution. These transitions mirror those observed on
Erd\H{o}s-R\'enyi random graphs, established by Luczak and McDiarmid (2001),
and Coppersmith et al. (2004), respectively
Universality for critical heavy-tailed network models: Metric structure of maximal components
We study limits of the largest connected components (viewed as metric spaces)
obtained by critical percolation on uniformly chosen graphs and configuration
models with heavy-tailed degrees. For rank-one inhomogeneous random graphs,
such results were derived by Bhamidi, van der Hofstad, Sen [Probab. Theory
Relat. Fields 2018]. We develop general principles under which the identical
scaling limits as the rank-one case can be obtained. Of independent interest,
we derive refined asymptotics for various susceptibility functions and the
maximal diameter in the barely subcritical regime.Comment: Final published version. 47 pages, 6 figure
Global lower mass-bound for critical configuration models in the heavy-tailed regime
We establish the global lower mass-bound property for the largest connected
components in the critical window for the configuration model when the degree
distribution has an infinite third moment. The scaling limit of the critical
percolation clusters, viewed as measured metric spaces, was established in [7]
with respect to the Gromov-weak topology. Our result extends those scaling
limit results to the stronger Gromov-Hausdorff-Prokhorov topology under
slightly stronger assumptions on the degree distribution. This implies the
distributional convergence of global functionals such as the diameter of the
largest critical components. Further, our result gives a sufficient condition
for compactness of the random metric spaces that arise as scaling limits of
critical clusters in the heavy-tailed regime.Comment: 25 page
Critical percolation on scale-free random graphs: New universality class for the configuration model
In this paper, we study the critical behavior of percolation on a
configuration model with degree distribution satisfying an infinite
second-moment condition, which includes power-law degrees with exponent . It is well known that, in this regime, many canonical random graph
models, such as the configuration model, are robust in the sense that the giant
component is not destroyed when the percolation probability stays bounded away
from zero. Thus, the critical behavior is observed when the percolation
probability tends to zero with the network size, despite of the fact that the
average degree remains bounded.
In this paper, we initiate the study of critical random graphs in the
infinite second-moment regime by identifying the critical window for the
configuration model. We prove scaling limits for component sizes and surplus
edges, and show that the maximum diameter the critical components is of order
, which contrasts with the previous universality classes arising in the
literature. This introduces a third and novel universality class for the
critical behavior of percolation on random networks, that is not covered by the
multiplicative coalescent framework due to Aldous and Limic (1998). We also
prove concentration of the component sizes outside the critical window, and
that a unique, complex giant component emerges after the critical window. This
completes the picture for the percolation phase transition on the configuration
model.Comment: 43 pages, Proof of continuity of the largest excursion is update