13,564 research outputs found
Critical behavior in inhomogeneous random graphs
We study the critical behavior of inhomogeneous random graphs where edges are
present independently but with unequal edge occupation probabilities. The edge
probabilities are moderated by vertex weights, and are such that the degree of
vertex i is close in distribution to a Poisson random variable with parameter
w_i, where w_i denotes the weight of vertex i. We choose the weights such that
the weight of a uniformly chosen vertex converges in distribution to a limiting
random variable W, in which case the proportion of vertices with degree k is
close to the probability that a Poisson random variable with random parameter W
takes the value k. We pay special attention to the power-law case, in which
P(W\geq k) is proportional to k^{-(\tau-1)} for some power-law exponent \tau>3,
a property which is then inherited by the asymptotic degree distribution.
We show that the critical behavior depends sensitively on the properties of
the asymptotic degree distribution moderated by the asymptotic weight
distribution W. Indeed, when P(W\geq k) \leq ck^{-(\tau-1)} for all k\geq 1 and
some \tau>4 and c>0, the largest critical connected component in a graph of
size n is of order n^{2/3}, as on the Erd\H{o}s-R\'enyi random graph. When,
instead, P(W\geq k)=ck^{-(\tau-1)}(1+o(1)) for k large and some \tau\in (3,4)
and c>0, the largest critical connected component is of the much smaller order
n^{(\tau-2)/(\tau-1)}.Comment: 26 page
Scaling limits for critical inhomogeneous random graphs with finite third moments
We identify the scaling limits for the sizes of the largest components at
criticality for inhomogeneous random graphs when the degree exponent
satisfies . We see that the sizes of the (rescaled) components converge
to the excursion lengths of an inhomogeneous Brownian motion, extending results
of \cite{Aldo97}. We rely heavily on martingale convergence techniques, and
concentration properties of (super)martingales. This paper is part of a
programme to study the critical behavior in inhomogeneous random graphs of
so-called rank-1 initiated in \cite{Hofs09a}.Comment: Final versio
Homogeneous and Scalable Gene Expression Regulatory Networks with Random Layouts of Switching Parameters
We consider a model of large regulatory gene expression networks where the
thresholds activating the sigmoidal interactions between genes and the signs of
these interactions are shuffled randomly. Such an approach allows for a
qualitative understanding of network dynamics in a lack of empirical data
concerning the large genomes of living organisms. Local dynamics of network
nodes exhibits the multistationarity and oscillations and depends crucially
upon the global topology of a "maximal" graph (comprising of all possible
interactions between genes in the network). The long time behavior observed in
the network defined on the homogeneous "maximal" graphs is featured by the
fraction of positive interactions () allowed between genes.
There exists a critical value such that if , the
oscillations persist in the system, otherwise, when it tends to
a fixed point (which position in the phase space is determined by the initial
conditions and the certain layout of switching parameters). In networks defined
on the inhomogeneous directed graphs depleted in cycles, no oscillations arise
in the system even if the negative interactions in between genes present
therein in abundance (). For such networks, the bidirectional edges
(if occur) influence on the dynamics essentially. In particular, if a number of
edges in the "maximal" graph is bidirectional, oscillations can arise and
persist in the system at any low rate of negative interactions between genes
(). Local dynamics observed in the inhomogeneous scalable regulatory
networks is less sensitive to the choice of initial conditions. The scale free
networks demonstrate their high error tolerance.Comment: LaTeX, 30 pages, 20 picture
Annealed central limit theorems for the Ising model on random graphs
The aim of this paper is to prove central limit theorems with respect to the
annealed measure for the magnetization rescaled by of Ising models
on random graphs. More precisely, we consider the general rank-1 inhomogeneous
random graph (or generalized random graph), the 2-regular configuration model
and the configuration model with degrees 1 and 2. For the generalized random
graph, we first show the existence of a finite annealed inverse critical
temperature and then
prove our results in the uniqueness regime, i.e., the values of inverse
temperature and external magnetic field for which either and , or and . In the case of the configuration model, the central limit theorem holds in
the whole region of the parameters and , because phase transitions
do not exist for these systems as they are closely related to one-dimensional
Ising models. Our proofs are based on explicit computations that are possible
since the Ising model on the generalized random graph in the annealed setting
is reduced to an inhomogeneous Curie-Weiss model, while the analysis of the
configuration model with degrees only taking values 1 and 2 relies on that of
the classical one-dimensional Ising model.Comment: 40 page
Degree correlations in scale-free null models
We study the average nearest neighbor degree of vertices with degree
. In many real-world networks with power-law degree distribution
falls off in , a property ascribed to the constraint that any two vertices
are connected by at most one edge. We show that indeed decays in in
three simple random graph null models with power-law degrees: the erased
configuration model, the rank-1 inhomogeneous random graph and the hyperbolic
random graph. We consider the large-network limit when the number of nodes
tends to infinity. We find for all three null models that starts to
decay beyond and then settles on a power law , with the degree exponent.Comment: 21 pages, 4 figure
A Two-populations Ising model on diluted Random Graphs
We consider the Ising model for two interacting groups of spins embedded in
an Erd\"{o}s-R\'{e}nyi random graph. The critical properties of the system are
investigated by means of extensive Monte Carlo simulations. Our results
evidence the existence of a phase transition at a value of the inter-groups
interaction coupling which depends algebraically on the dilution of
the graph and on the relative width of the two populations, as explained by
means of scaling arguments. We also measure the critical exponents, which are
consistent with those of the Curie-Weiss model, hence suggesting a wide
robustness of the universality class.Comment: 11 pages, 4 figure
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