126,739 research outputs found
The Critical Phase for Random Graphs with a Given Degree Sequence
We consider random graphs with a fixed degree sequence. Molloy and Reed [11, 12] studied how the size of the giant component changes according to degree conditions. They showed that there is a phase transition and investigated the order of components before and after the critical phase. In this paper we study more closely the order of components at the critical phase, using singularity analysis of a generating function for a branching process which models the random graph with a given degree sequence
Generalized Threshold-Based Epidemics in Random Graphs: the Power of Extreme Values
Bootstrap percolation is a well-known activation process in a graph,
in which a node becomes active when it has at least active neighbors.
Such process, originally studied on regular structures, has been recently
investigated also in the context of random graphs, where it can serve as a simple
model for a wide variety of cascades, such as the
spreading of ideas, trends, viral contents, etc. over large social networks.
In particular, it has been shown that in the final active set
can exhibit a phase transition for a sub-linear number of seeds.
In this paper, we propose a unique framework to study similar
sub-linear phase transitions for a much broader class of graph models
and epidemic processes. Specifically, we consider i) a generalized version
of bootstrap percolation in with random activation thresholds
and random node-to-node influences; ii) different random graph models,
including graphs with given degree sequence and graphs with
community structure (block model). The common thread of our work is to
show the surprising sensitivity of the critical seed set size
to extreme values of distributions, which makes some systems dramatically
vulnerable to large-scale outbreaks. We validate our results running simulation on
both synthetic and real graphs
Percolation by cumulative merging and phase transition for the contact process on random graphs
Given a weighted graph, we introduce a partition of its vertex set such that
the distance between any two clusters is bounded from below by a power of the
minimum weight of both clusters. This partition is obtained by recursively
merging smaller clusters and cumulating their weights. For several classical
random weighted graphs, we show that there exists a phase transition regarding
the existence of an infinite cluster.
The motivation for introducing this partition arises from a connection with
the contact process as it roughly describes the geometry of the sets where the
process survives for a long time. We give a sufficient condition on a graph to
ensure that the contact process has a non trivial phase transition in terms of
the existence of an infinite cluster. As an application, we prove that the
contact process admits a sub-critical phase on d-dimensional random geometric
graphs and on random Delaunay triangulations. To the best of our knowledge,
these are the first examples of graphs with unbounded degrees where the
critical parameter is shown to be strictly positive.Comment: 50 pages, many figure
Vacant sets and vacant nets: Component structures induced by a random walk
Given a discrete random walk on a finite graph , the vacant set and vacant
net are, respectively, the sets of vertices and edges which remain unvisited by
the walk at a given step .%These sets induce subgraphs of the underlying
graph. Let be the subgraph of induced by the vacant set of the
walk at step . Similarly, let be the subgraph of
induced by the edges of the vacant net. For random -regular graphs , it
was previously established that for a simple random walk, the graph
of the vacant set undergoes a phase transition in the sense of the phase
transition on Erd\H{os}-Renyi graphs . Thus, for there is an
explicit value of the walk, such that for ,
has a unique giant component, plus components of size ,
whereas for all the components of are of
size . We establish the threshold value for a phase
transition in the graph of the vacant net of a simple
random walk on a random -regular graph. We obtain the corresponding
threshold results for the vacant set and vacant net of two modified random
walks. These are a non-backtracking random walk, and, for even, a random
walk which chooses unvisited edges whenever available. This allows a direct
comparison of thresholds between simple and modified walks on random
-regular graphs. The main findings are the following: As increases the
threshold for the vacant set converges to in all three walks. For
the vacant net, the threshold converges to for both the simple
random walk and non-backtracking random walk. When is even, the
threshold for the vacant net of the unvisited edge process converges to ,
which is also the vertex cover time of the process.Comment: Added results pertaining to modified walk
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
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