How to construct a good student
We will investigate the problem of designing a suitable student for learning from a given teacher. Both student and teacher, are represented by feed-forward neural networks, thus the problem of designing the student presents itself at two levels, the hardware (the architecture) and the software (learning algorithm). We will discuss how the problem can be solved analytically and what are the advantages (and disadvantages) of such an approach.
Learning curves for Gaussian process regression on random graphs
We investigate how well Gaussian process regression can learn functions defined on graphs, using large random graphs (regular and Poisson) as paradigmatic examples. We focus on the random-walk based kernel and look at the learning curves of the Gaussian process, which give the Bayes error vs. training set size. We find the learning curves generically depend on $n/V$, the number of training examples per vertex. We also find that learning curves exhibit two learning regimes with a transition period between the two. We show that previous approximate predictions for the learning curves using the covariance function spectrum (Sollich 1999), suitably extended from Euclidean spaces to graphs, give a good prediction for the two learning regimes but fail to quantitatively predict the transition between the two. Finally using belief propagation (or equivalently the cavity method) we derive improved predictions for the learning curves that should become exact in the limit of large graphs. We show how these perform against the spectrum based predictions.
Community Detection in Complex Networks by Message Passing
The talk will present the later work in collaboration with Prof. David Saad and Dr. Joerg Reichardt on using belief propagation for detecting community structures on bipartite graphs. The probabilistic dependence of the problem variable presents a mixture of dense and sparse connections which requires the introduction of new techniques for message passing. Some results will be presented and discussed.
Large deviations, metastable states and glass transitions
As liquids are cooled towards their glass transitions, their dynamics slow down dramatically and their relaxation becomes increasingly heterogeneous. By considering fluctuations of dynamical activity in glassy model systems, we have shown how this behaviour can be related to the existence of novel phase transitions. These phase transitions take place in ensembles of trajectories, and we study them through large deviation functions. I will discuss recent results for such transitions in kinetically constrained models, atomistic systems and spin glasses.
Complex dynamics of molecular liquids
We analyse molecular phase space trajectories. We investigate the dynamics in the framework of symbolic dynamics, that is the properties of a symbolic sequence obtained from the continuous trajectory. The statistics on the symbols carry information about the full dimensional phase space. In particular, a quantity ``statistical complexity'', defined as the Shannon entropy of a Markov type model for the observed symbolic subsequences, is used to measure temporal patterns present in the trajectory. We show that statistical complexity is effective in distinguishing chaotic signals from random ones with identical correlation properties. We also demonstrate that non-trivial temporal patterns in the chaotic trajectory, such as pseudo periodic motion, cause non-Markov property of the symbolic subsequences and imply time dependencies over very long intervals.
Juan P. Garrahan
Molecular random tilings
Recently we have shown that small organic molecules when adsorbed on graphite self-assemble into two-dimensional rhombus random tilings. These tilings are close to ideal, with long range correlations punctuated by sparse localised tiling defects. They are rare examples of molecular systems displaying "Coulomb" phase behaviour and fractional excitations. I will discuss the static, dynamic and growth properties of these kind of random tilings, and explore analogies with other dynamically arrested systems.
Dispersive Particle Swarm Optimization
We discuss premature convergence in the particle swarm optimization (PSO) algorithm and investigate a variant algorithm designed to correct this problem, attractive and repulsive PSO (ARPSO). We show that ARPSO is capable of improving when stuck in a low quality local minima, but that it has no systematic way of improving on good quality solutions. In order to combat this problem, and motivated by similar principles to ARPSO, we introduce a new PSO variant, dispersive PSO (DiPSO). We show that DiPSO significantly improves the robustness of PSO on problems which it was previously unable to solve systematically, and that it is compatible with one of the best current PSO variants.
Eytan Katzav and Isaac Perez Castillo
Large Deviation of the Minimum Eigenvalue of the Wishart-Laguerre Ensemble
We present analytical results on the probability of rare events where the minimal eigenvalue of a random matrix is much smaller than its typical value. The large deviation function that characterizes this probability is computed explicitly for the Wishart ensemble using the Coulomb-gas approach. We also find an interesting new scaling regime for almost square matrices that was not identified previously. Our analytical predictions are verified by extensive numerical simulations and some available exact results.
Derivatives and Credit Contagion in Financial Networks
We examine the impact of credit default swaps (CDS) on defaults and losses in a stylized economic system, composed of interconnected heterogeneous networks of corporates, banks and insurers. We analyse such a system using a stochastic setting, which allows us to exploit limit theorems to exactly solve the contagion dynamics for the entire system. The analysis shows that CDS, when used to expand banksâ loan books (arguing that CDS would offload the additional risks from banksâs balance sheets), can actually lead to greater instability of the entire network in times of economic stress, by creating additional con- tagion channels. This can lead to considerably enhanced probabilities for the occurrence of very large losses and very high default rates in the system.
Mathematics in immunology
In this work, we have calculated the strength of an eukaryotic immune response by evaluating the cell:cell correlation of the bond formed between the body defense cell (T-cell) and the external pathogen (Antigen Presenting Cell: APC). Our calculations show that the average contact area between TCR-APC is nanometric in size with contact times from a few minutes to hours. The results from our theory match available experimental observations and make predictions on future (experimental) measurements. The theoretical method developed in the process has a general scheme that can be applied to a range of diverse topics, including financial mathematics.
Tomaso Aste and Tiziana Di Matteo
The use of networks to capture and to model both local and cross-scale properties of complex systems
One of the typical features of real-world complex system is the simultaneous presence of specialized activities at local level coexisting with cross-scale activities at global level. This simultaneous presence of relevant information at both the local and global levels poses real challenges for the experimental study of these systems making them intrinsically non-separable. Indeed, on one hand one aims to simplify complexity by reducing the systems into clusters and communities associated with local specialized activities. On the other hand this introduces characteristics-scales that hide the cross-scale phenomena associated with emerging global organization. Here we present an approach that is effective in capturing both the local specialized clustering and the global emerging dynamics overcoming such a dichotomy. This approach combines graph theory and statistical physics methods yielding to a threshold-free hierarchical clustering. Applications to financial and biological systems are introduced and discussed.