23 research outputs found
Percolation on sparse random graphs with given degree sequence
We study the two most common types of percolation process on a sparse random
graph with a given degree sequence. Namely, we examine first a bond percolation
process where the edges of the graph are retained with probability p and
afterwards we focus on site percolation where the vertices are retained with
probability p. We establish critical values for p above which a giant component
emerges in both cases. Moreover, we show that in fact these coincide. As a
special case, our results apply to power law random graphs. We obtain rigorous
proofs for formulas derived by several physicists for such graphs.Comment: 20 page
Critical percolation on random regular graphs
We show that for all the size of the largest
component of a random -regular graph on vertices around the percolation
threshold is , with high probability. This extends
known results for fixed and for , confirming a prediction of
Nachmias and Peres on a question of Benjamini. As a corollary, for the largest
component of the percolated random -regular graph, we also determine the
diameter and the mixing time of the lazy random walk. In contrast to previous
approaches, our proof is based on a simple application of the switching method.Comment: 10 page
An old approach to the giant component problem
In 1998, Molloy and Reed showed that, under suitable conditions, if a
sequence of degree sequences converges to a probability distribution , then
the size of the largest component in corresponding -vertex random graph is
asymptotically , where is a constant defined by the
solution to certain equations that can be interpreted as the survival
probability of a branching process associated to . There have been a number
of papers strengthening this result in various ways; here we prove a strong
form of the result (with exponential bounds on the probability of large
deviations) under minimal conditions.Comment: 24 pages; only minor change
Locality of critical percolation on expanding graph sequences
We study the locality of critical percolation on finite graphs: let be
a sequence of finite graphs, converging locally weakly to a (random, rooted)
infinite graph . Consider Bernoulli edge percolation: does the critical
probability for the emergence of an infinite component on coincide with the
critical probability for the emergence of a linear-sized component on ? In
this short article we give a positive answer provided the graphs satisfy
an expansion condition, and the limiting graph has finite expected root
degree. The main result of Benjamini, Nachmias, and Peres (2011), where this
question was first formulated, showed the result assuming the satisfy a
uniform degree bound and uniform expansion condition, and converge to a
deterministic limit . Later work of Sarkar (2021) extended the result to
allow for a random limit , but still required a uniform degree bound and
uniform expansion for . Our result replaces the degree bound on with
the (milder) requirement that must have finite expected root degree. Our
proof is a modification of the previous results, using a pruning procedure and
the second moment method to control unbounded degrees
A finite-state model of botnets’ desinfection and removal
Existing multi-agent systems (either for implementing distributed security threats or for countering them in the Internet) either do not maintain agent graph connectivity or maintain it with network redundancy, which significantly increases the overheads. The paper analyzes a finite-state model of adaptive behavior in a multi-agent system. This model uses a d-regular agent graph and some methods to maintain network connectivity, which provides sustainable system performance in aggressive environment
On giant components and treewidth in the layers model
Given an undirected -vertex graph and an integer , let
denote the random vertex induced subgraph of generated by ordering
according to a random permutation and including in those
vertices with at most of their neighbors preceding them in this order.
The distribution of subgraphs sampled in this manner is called the \emph{layers
model with parameter} . The layers model has found applications in studying
-degenerate subgraphs, the design of algorithms for the maximum
independent set problem, and in bootstrap percolation.
In the current work we expand the study of structural properties of the
layers model.
We prove that there are -regular graphs for which with high
probability has a connected component of size . Moreover,
this connected component has treewidth . This lower bound on the
treewidth extends to many other random graph models. In contrast, is
known to be a forest (hence of treewidth~1), and we establish that if is of
bounded degree then with high probability the largest connected component in
is of size . We also consider the infinite two-dimensional
grid, for which we prove that the first four layers contain a unique infinite
connected component with probability
Preferential attachment without vertex growth: emergence of the giant component
We study the following preferential attachment variant of the classical
Erdos-Renyi random graph process. Starting with an empty graph on n vertices,
new edges are added one-by-one, and each time an edge is chosen with
probability roughly proportional to the product of the current degrees of its
endpoints (note that the vertex set is fixed). We determine the asymptotic size
of the giant component in the supercritical phase, confirming a conjecture of
Pittel from 2010. Our proof uses a simple method: we condition on the vertex
degrees (of a multigraph variant), and use known results for the configuration
model.Comment: 20 page
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
The phase transition in the configuration model
Let be a random graph with a given degree sequence , such as a
random -regular graph where is fixed and . We study
the percolation phase transition on such graphs , i.e., the emergence as
increases of a unique giant component in the random subgraph obtained by
keeping edges independently with probability . More generally, we study the
emergence of a giant component in itself as varies. We show that a
single method can be used to prove very precise results below, inside and above
the `scaling window' of the phase transition, matching many of the known
results for the much simpler model . This method is a natural extension
of that used by Bollobas and the author to study , itself based on work
of Aldous and of Nachmias and Peres; the calculations are significantly more
involved in the present setting.Comment: 37 page