11,716 research outputs found
Topics in coarsening phenomena
These lecture notes give a very short introduction to coarsening phenomena
and summarize some recent results in the field. They focus on three aspects:
the super-universality hypothesis, the geometry of growing structures, and
coarsening in the spiral kinetically constrained model.Comment: Lecture notes. Fundamental Problems in Statistical Physics XII,
Leuven, Aug 30 - Sept 12, 200
Curvature-driven coarsening in the two dimensional Potts model
We study the geometric properties of polymixtures after a sudden quench in
temperature. We mimic these systems with the -states Potts model on a square
lattice with and without weak quenched disorder, and their evolution with Monte
Carlo simulations with non-conserved order parameter. We analyze the
distribution of hull enclosed areas for different initial conditions and
compare our results with recent exact and numerical findings for (Ising)
case. Our results demonstrate the memory of the presence or absence of
long-range correlations in the initial state during the coarsening regime and
exhibit super-universality properties.Comment: 12 pages, 16 figure
Scale-free networks with a large- to hypersmall-world transition
Recently there have been a tremendous interest in models of networks with a
power-law distribution of degree -- so called "scale-free networks." It has
been observed that such networks, normally, have extremely short path-lengths,
scaling logarithmically or slower with system size. As en exotic and
unintuitive example we propose a simple stochastic model capable of generating
scale-free networks with linearly scaling distances. Furthermore, by tuning a
parameter the model undergoes a phase transition to a regime with extremely
short average distances, apparently slower than log log N (which we call a
hypersmall-world regime). We characterize the degree-degree correlation and
clustering properties of this class of networks.Comment: errors fixed, one new figure, to appear in Physica
Maximal Cliques in Scale-Free Random Graphs
We investigate the number of maximal cliques, i.e., cliques that are not
contained in any larger clique, in three network models: Erd\H{o}s-R\'enyi
random graphs, inhomogeneous random graphs (also called Chung-Lu graphs), and
geometric inhomogeneous random graphs. For sparse and not-too-dense
Erd\H{o}s-R\'enyi graphs, we give linear and polynomial upper bounds on the
number of maximal cliques. For the dense regime, we give super-polynomial and
even exponential lower bounds. Although (geometric) inhomogeneous random graphs
are sparse, we give super-polynomial lower bounds for these models. This comes
form the fact that these graphs have a power-law degree distribution, which
leads to a dense subgraph in which we find many maximal cliques. These lower
bounds seem to contradict previous empirical evidence that (geometric)
inhomogeneous random graphs have only few maximal cliques. We resolve this
contradiction by providing experiments indicating that, even for large
networks, the linear lower-order terms dominate, before the super-polynomial
asymptotic behavior kicks in only for networks of extreme size
Scaling behaviours in the growth of networked systems and their geometric origins
abstract: Two classes of scaling behaviours, namely the super-linear scaling of links or activities, and the sub-linear scaling of area, diversity, or time elapsed with respect to size have been found to prevail in the growth of complex networked systems. Despite some pioneering modelling approaches proposed for specific systems, whether there exists some general mechanisms that account for the origins of such scaling behaviours in different contexts, especially in socioeconomic systems, remains an open question. We address this problem by introducing a geometric network model without free parameter, finding that both super-linear and sub-linear scaling behaviours can be simultaneously reproduced and that the scaling exponents are exclusively determined by the dimension of the Euclidean space in which the network is embedded. We implement some realistic extensions to the basic model to offer more accurate predictions for cities of various scaling behaviours and the Zipf distribution reported in the literature and observed in our empirical studies. All of the empirical results can be precisely recovered by our model with analytical predictions of all major properties. By virtue of these general findings concerning scaling behaviour, our models with simple mechanisms gain new insights into the evolution and development of complex networked systems.The final version of this article, as published in Scientific Reports, can be viewed online at: http://dx.doi.org/10.1038/srep0976
Dynamic Scaling, Data-collapse and Self-Similarity in Mediation-Driven Attachment Networks
Recently, we have shown that if the th node of the Barab\'{a}si-Albert
(BA) network is characterized by the generalized degree
, where and are its degree
at current time and at birth time , then the corresponding
distribution function exhibits dynamic scaling. Applying the same idea
to our recently proposed mediation-driven attachment (MDA) network, we find
that it too exhibits dynamic scaling but, unlike the BA model, the exponent
of the MDA model assumes a spectrum of value .
Moreover, we find that the scaling curves for small are significantly
different from those of the larger and the same is true for the BA networks
albeit in a lesser extent. We use the idea of the distribution of inverse
harmonic mean (IHM) of the neighbours of each node and show that the number of
data points that follow the power-law degree distribution increases as the
skewness of the IHM distribution decreases. Finally, we show that both MDA and
BA models become almost identical for large .Comment: 8 pages, 8 captioned figure
Sampling and Inference for Beta Neutral-to-the-Left Models of Sparse Networks
Empirical evidence suggests that heavy-tailed degree distributions occurring
in many real networks are well-approximated by power laws with exponents
that may take values either less than and greater than two. Models based on
various forms of exchangeability are able to capture power laws with , and admit tractable inference algorithms; we draw on previous results to
show that cannot be generated by the forms of exchangeability used
in existing random graph models. Preferential attachment models generate power
law exponents greater than two, but have been of limited use as statistical
models due to the inherent difficulty of performing inference in
non-exchangeable models. Motivated by this gap, we design and implement
inference algorithms for a recently proposed class of models that generates
of all possible values. We show that although they are not exchangeable,
these models have probabilistic structure amenable to inference. Our methods
make a large class of previously intractable models useful for statistical
inference.Comment: Accepted for publication in the proceedings of Conference on
Uncertainty in Artificial Intelligence (UAI) 201
A morphological study of cluster dynamics between critical points
We study the geometric properties of a system initially in equilibrium at a
critical point that is suddenly quenched to another critical point and
subsequently evolves towards the new equilibrium state. We focus on the
bidimensional Ising model and we use numerical methods to characterize the
morphological and statistical properties of spin and Fortuin-Kasteleyn clusters
during the critical evolution. The analysis of the dynamics of an out of
equilibrium interface is also performed. We show that the small scale
properties, smaller than the target critical growing length with the dynamic exponent, are characterized by equilibrium at the
working critical point, while the large scale properties, larger than the
critical growing length, are those of the initial critical point. These
features are similar to what was found for sub-critical quenches. We argue that
quenches between critical points could be amenable to a more detailed
analytical description.Comment: 26 pages, 13 figure
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