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 $q$-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 $q=2$ (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

Dynamic Scaling, Data-collapse and Self-Similarity in Mediation-Driven Attachment Networks

Recently, we have shown that if the $i$th node of the Barab\'{a}si-Albert (BA) network is characterized by the generalized degree $q_i(t)=k_i(t)t_i^\beta/m$, where $k_i(t)\sim t^\beta$ and $m$ are its degree at current time $t$ and at birth time $t_i$, then the corresponding distribution function $F(q,t)$ 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 $\beta$ of the MDA model assumes a spectrum of value $1/2\leq \beta \leq 1$. Moreover, we find that the scaling curves for small $m$ are significantly different from those of the larger $m$ 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 $m$.Comment: 8 pages, 8 captioned figure

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

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 $\eta$ 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 $\eta < 2$, and admit tractable inference algorithms; we draw on previous results to show that $\eta > 2$ 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 $\eta$ 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 $\xi(t) \sim t^{1/z}$ with $z$ 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