Speaker
Nikos Kouvaris

Description Recently, much attention has been attracted to non-equilibrium pattern formation in nonlinear systems organized on complex networks. A well known self-organized pattern is the Turing pattern which has been analyzed in ecological systems where activators and inhibitors diffuse in the same network with different mobility rates. However, in many ecological systems, different species may diffuse within their own private paths in order to keep their purities. Following this idea, we study an ecological system organized on multiplex networks, where activators and inhibitors occupy nodes in different networks/layers. They diffuse within their own layer through the paths defined by the intra-layer architecture, while they react across the layers with their counterparts defined by the inter-layer architecture. This separation of links in different layers brings about a significant difference from the case of a single network. Particularly, if the ratio between the mean degrees of the inhibitors and the activators layers exceeds a certain threshold, the multiplex system undergoes a Turing instability even if the rates of the diffusional mobility are the same for both species. This collective behaviour has been studied numerically by simulations on the actual multiplex networks as well as analytically by employing a stability analysis on the multiplex system. A mean-field theory has also been used to describe the full developed Turing patterns. As has been revealed from the analytical studies the instability threshold can be found by analyzing the correlation between the degrees of the nodes in the two layers.

Speaker
Diego Pazó

Description Clapping audiences, pedestrians on the London’s millennium bridge, and flashing fireflies are spectacular examples of collective synchronization in living systems. This ubiquitous tendency towards synchrony also occurs at the microscopic level, where thousands of heart pacemaker cells self-organize their rhythmic activities to initiate the heartbeat. I will review theoretical attempts to describe these phenomena in a common framework, and present recent advances on the mathematical analysis of populations of pulse-coupled oscillators.

Speaker
Tomas Alarcon, Alvaro Corral, Alex Roxin

Description Description This course aims to give a practical guide to the concepts of Statistical Physics to students unfamiliar with this subject. The course starts by introducing the essentials of thermodynamics, information theory, and the microcanonical and canonical ensembles. Once the basic concepts have been covered, the Ising model is introduced, which serves as a guide to more advanced subjects: Critical phenomena and renormalisation group, dynamics of critical phenomena (Glauber and Kawasaki dynamics), and disordered systems and applications to neuroscience. Schedule. We expect to have three sessions of two hours each for three weeks. This may be subject to changes as the course progresses. First week "Introduction to Statistical Physics, equilibrium ensembles and critical phenomena", lectures by Alvaro Corral Monday May 5th, 14:00-16:00 Wednesday May 7th, 14:00-16:00 Thursday May 8th, 14:00-16:00 Second week "Dynamics of the Ising model with conserved (Kawasaki) and non-conserved (Glauber) order parameter", lectures by Tomas Alarcón Monday May 12th, 14:00-16:00 Wednesday May 14th, 14:00-16:00 Thursday May 15th, 14:00-16:00 Third week "Applications to neuroscience: Attractor models of memory", lectures by Alex Roxin Monday May 19th, 14:00-16:00 Wednesday May 21th, 14:00-16:00 Thursday May 22nd, 14:00-16:00 Contact People interested in attending this short course should send an email to ufdcrm@crm.cat Updates in the planning of the course will be available at http://www.crm.cat/en/Activities/Curs_2013-2014/Pages/StatisticalPhysics.aspx

Speaker
Sébastien Bubeck

Description We are interested in the following question: suppose we generate a large graph according to the linear preferential attachment model---can we say anything about the initial (seed) graph? A precise answer to this question could lead to new insights for the diverse applications of the preferential attachment model. In this work we focus on the case of trees grown according to the preferential attachment model. We first show that the seed has no effect from a weak local limit point of view. On the other hand, we conjecture that different seeds lead to different distributions of limiting trees from a total variation point of view. We prove this result for seeds with different degree sequences.

Speaker
Marc Noy

Description The theory of random graphs was born in 1960 with the fundamental work of Erdos and Renyi. Since then, the field has grown enormously and has become one of the central topics in combinatorics. The model most analyzed is the G(n, p) model, consisting of n labelled vertices and in which the edges are chosen independently with probability p. The value p = p(n) is allowed to be a function of n, and one can show that many important properties have a threshold function. In particular, p = 1/n is the threshold for the emergence of the giant component, where a fundamental phase transition occurs (already discovered by Erdos and Renyi). In recent years, alternative models have been introduced in order to capture properties of real life networks. In a different direction, we are also interested in analyzing random graphs from a constrained class of graphs defined by a global property, such as being regular, triangle-free, acyclic or planar, where the distribution is uniform among all graphs with the same number of vertices. In this context we cannot draw edges independently and we have to rely on counting. The first task is to estimate how many graphs are there with a given number of vertices, and this can be done using a variety of enumerative and probabilistic tools. After reviewing classical models, we will discuss several key examples and different approaches. Then we will focus on random graphs with a topological flavour, where there has been much progress recently. We will also touch briefly on recent developments on logical limit laws. The talk is addressed to a wide audience including undergraduate and graduate students. The exposition will be mostly self-contained and will not assume familiarity with random graphs.