376 research outputs found
Random networking: between order and chaos
Full text linkWith the arrival of the Internet, a good understanding of networks has become important for
everyone. Network theory, which originated in the eighteenth century with Euler, and in the
nineteenth century withMarkov, has until recently concentrated its attentionmainly on regular types of graphs. In his inaugural lecture, Remco van der Hofstad shows us a shift towards highly irregular graphs having vertices with extremely high degrees. He argues that this irregularity is a main characteristic of real life networks such as the Internet, social networks and networks describing biophysical phenomena. On January 1, 2005, Remco van der Hofstad was appointed full professor in Probability at the University of Eindhoven
Annealed Ising model on configuration models
Full text linkIn this paper, we study the annealed ferromagnetic Ising model on the configuration model. In an annealed system, we
take the average on both sides of the ratio defining the Boltzmann–Gibbs measure of the Ising model. In the configuration model, the degrees are specified. Remarkably, when the degrees are deterministic, the critical value of the annealed Ising model is the same as that for the quenched Ising model. For independent and identically distributed (i.i.d.) degrees, instead, the annealed critical value is strictly smaller than that of the quenched Ising model. This identifies the degree structure of the underlying graph as the main driver for the critical value. Furthermore, in both contexts (deterministic or random degrees), we provide the variational expression for the annealed pressure. Interestingly, our rigorous results establish that only part of the heuristic conjectures in the physics literature were correct
Annealed inhomogeneities in random ferromagnets
Full text linkWe consider spin models on complex networks frequently used to model social and technological systems. We study the annealed ferromagnetic Ising model for random networks with either independent edges (Erdős-Rényi) or prescribed degree distributions (configuration model). Contrary to many physical models, the annealed setting is poorly understood and behaves quite differently than the quenched system. In annealed networks with a fluctuating number of edges, the Ising model changes the degree distribution, an aspect previously ignored. For random networks with Poissonian degrees, this gives rise to three distinct annealed critical temperatures depending on the precise model choice, only one of which reproduces the quenched one. In particular, two of these annealed critical temperatures are finite even when the quenched one is infinite because then the annealed graph creates a giant component for all sufficiently small temperatures. We see that the critical exponents in the configuration model with deterministic degrees are the same as the quenched ones, which are the mean-field exponents if the degree distribution has finite fourth moment and power-law-dependent critical exponents otherwise. Remarkably, the annealing for the configuration model with random independent and identically distributed degrees washes away the universality class with power-law critical exponents
Quenched Central Limit Theorems for the Ising Model on Random Graphs
Full text linkThemain goal of the paper is to prove central limit theorems for the magnetization
rescaled by the square root of N for the Ising model on random graphs with N vertices.Both random quenched
and averaged quenched measures are considered.We work in the uniqueness regime β > βc
or β > 0 and B not equal to 0, where β is the inverse temperature, βc is the critical inverse temperature
and B is the external magnetic field. In the random quenched setting our results apply to
general tree-like random graphs (as introduced by Dembo, Montanari and further studied by
Dommers and the first and third author) and our proof follows that of Ellis in Z^d. For the
averaged quenched setting, we specialize to two particular random graph models, namely
the 2-regular configuration model and the configuration model with degrees 1 and 2. In
these cases our proofs are based on explicit computations relying on the solution of the one
dimensional Ising model
Annealed central limit theorems for the Ising model on random graphs
Full text linkThe 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 0le eta^{mathrm{an}}_c < infty and then prove our results in the uniqueness regime, i.e., the values of inverse temperature and external magnetic field for which either eta<eta^{mathrm{an}}_c and , or eta>0 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
Ising models on power-law random graphs
Full text linkWe study a ferromagnetic Ising model on random graphs with a power-law degree distribution and compute the thermodynamic limit of the pressure when the mean degree is finite (degree exponent τ>2), for which the random graph has a tree-like structure. For this, we closely follow the analysis by Dembo and Montanari (Ann. Appl. Probab. 20(2):565–592, 2010) which assumes finite variance degrees (τ>3), adapting it when necessary and also simplifying it when possible. Our results also apply in cases where the degree distribution does not obey a power law.We further identify the thermodynamic limits of various physical quantities, such as the magnetization and the internal energy
Large Deviations for the Annealed Ising Model on Inhomogeneous Random Graphs: Spins and Degrees
Full text linkWe prove a large deviations principle for the total spin and the number of edges under the annealed Ising measure on generalized random graphs. We also give detailed results on how the annealing over the Ising model changes the degrees of the vertices in the graph and show how it gives rise to interesting correlated random graphs
Universality for critical heavy-tailed network models: Metric structure of maximal components
No full textWe 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 (2018) [15]. We develop general principles under which the identical scaling limits as in [15] can be obtained. Of independent interest, we derive refined asymptotics for various susceptibility functions and the maximal diameter in the barely subcritical regime
Ising Critical Exponents on Random Trees and Graphs
Full text linkWe study the critical behavior of the ferromagnetic Ising model on random trees as well
as so-called locally tree-like random graphs. We pay special attention to trees and graphs
with a power-law offspring or degree distribution whose tail behavior is characterized by its
power-law exponent > 2. We show that the critical temperature of the Ising model equals
the inverse hyperbolic tangent of the inverse of the mean offspring or mean forward degree
distribution. In particular, the inverse critical temperature equals zero when ∈ (2, 3] where
this mean equals infinity.
We further study the critical exponents , and
, describing how the (root) magnetiza-
tion behaves close to criticality. We rigorously identify these critical exponents and show that
they take the values as predicted by Dorogovstev, et al. [9] and Leone et al. [17]. These values
depend on the power-law exponent , taking the mean-field values for > 5, but different
values for ∈ (3, 5)
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