Fernando Alcalde Cuesta

Transversely Cantor laminations as inverse limits

We demonstrate that any minimal transversely Cantor compact lamination of dimension p and class C 1 without holonomy is an inverse limit of compact branched manifolds of dimension p. To prove this result, we extend the triangulation theorem for C 1 manifolds to transversely Cantor C 1 laminations. In fact, we give a simple proof of this classical theorem based on the existence of C 1 -compatible differentiable structures of class C ∞ .

Exploring the topological sources of robustness against invasion in biological and technological networks.

For a network, the accomplishment of its functions despite perturbations is called robustness. Although this property has been extensively studied, in most cases, the network is modified by removing nodes. In our approach, it is no longer perturbed by site percolation, but evolves after site invasion. The process transforming resident/healthy nodes into invader/mutant/diseased nodes is described by the Moran model. We explore the sources of robustness (or its counterpart, the propensity to spread favourable innovations) of the US high-voltage power grid network, the Internet2 academic network, and the C. elegans connectome. We compare them to three modular and non-modular benchmark networks, and samples of one thousand random networks with the same degree distribution. It is found that, contrary to what happens with networks of small order, fixation probability and robustness are poorly correlated with most of standard statistics, but they depend strongly on the degree distribution. While community detection techniques are able to detect the existence of a central core in Internet2, they are not effective in detecting hierarchical structures whose topological complexity arises from the repetition of a few rules. Box counting dimension and Rent’s rule are applied to show a subtle trade-off between topological and wiring complexity.

An Accurate Database of the Fixation Probabilities for All Undirected Graphs of Order 10 or Less

We present a extremely precise database of the fixation probabilities of mutant individuals in a non-homogeneous population which are spatially arranged on a small graph. We explore what features of a graph increase the chances of a beneficial allele of a gene to spread over a structured population.

Suppressors of selection

Inspired by recent works on evolutionary graph theory, an area of growing interest in mathematical and computational biology, we present examples of undirected structures acting as suppressors of selection for any fitness value r > 1. This means that the average fixation probability of an advantageous mutant or invader individual placed at some node is strictly less than that of this individual placed in a well-mixed population. This leads the way to study more robust structures less prone to invasion, contrary to what happens with the amplifiers of selection where the fixation probability is increased on average for advantageous invader individuals. A few families of amplifiers are known, although some effort was required to prove it. Here, we use computer aided techniques to find an exact analytical expression of the fixation probability for some graphs of small order (equal to 6, 8 and 10) proving that selection is effectively reduced for r > 1. Some numerical experiments using Monte Carlo methods are also performed for larger graphs and some variants.Motivated by growing interest in evolutionary network theory, we construct the first examples of undirected graph structures acting as reducers of selection. This means that the average fixation probability of an advantageous invader individual placed at some node is strictly less than that of this individual placed in a well-mixed population. Computer aided techniques are used to obtain an exact analytical expression of the fixation probability proving that selection is effectively reduced.

Sur l'intégration symplectique de la structure de poisson singuliére Λ = (x2 + y2) ∂/∂x Λ ∂/∂y de R2

The purpose of this note is to give an example of a singular Poisson structure on R2 which admits a symplectic realization by a Lie groupoid.

A method for validating Rent's rule for technological and biological networks

Rent’s rule is empirical power law introduced in an effort to describe and optimize the wiring complexity of computer logic graphs. It is known that brain and neuronal networks also obey Rent’s rule, which is consistent with the idea that wiring costs play a fundamental role in brain evolution and development. Here we propose a method to validate this power law for a certain range of network partitions. This method is based on the bifurcation phenomenon that appears when the network is subjected to random alterations preserving its degree distribution. It has been tested on a set of VLSI circuits and real networks, including biological and technological ones. We also analyzed the effect of different types of random alterations on the Rentian scaling in order to test the influence of the degree distribution. There are network architectures quite sensitive to these randomization procedures with significant increases in the values of the Rent exponents.

Nombre de branchement d’un pseudogroupe

Lyons has defined an average number of branches per vertex of an infinite locally finite rooted tree. This number has an important role in several probabilistic processes such as random walk and percolation. In this paper, we extend the notion of branching number to any measurable graphed pseudogroup of finite type acting on a probability space. We prove that such a pseudogroup is Liouvillian (i.e. almost every orbit does not admit non-constant bounded harmonic functions) if its branching number is equal to 1. In order to prove that this actually generalizes results of C. Series and V. Kaimanovich on equivalence relations with polynomial and subexponential growth, we describe an example of minimal lamination whose holonomy pseudogroup has exponential growth and branching number equal to 1.

Phase transitions in evolutionary dynamics

The evolutionary dynamics of a finite population where resident individuals are replaced by mutant ones depends on its spatial structure. The population adopts the form of an undirected graph where the place occupied by each individual is represented by a node and it is bidirectionally linked to the places that can be occupied by its clonal offspring. There are undirected graph structures that act as amplifiers of selection increasing the probability that the offspring of an advantageous mutant spreads through the graph reaching any node. But there also are undirected graph structures acting as suppressors of selection where, on the contrary, the fixation probability of an advantageous mutant is less than that of the same individual placed in a homogeneous population. Here, we show that some undirected graphs exhibit phase transitions between both evolutionary regimes when the mutant fitness varies. Firstly, as was already observed for small order random graphs, we show that most graphs of order 10 or less are amplifiers of selection or suppressors that become amplifiers from a unique transition phase. Secondly, we give examples of amplifiers of order 7 that become suppressors from some critical value. For graphs of order 6 and 7, we apply computer aided techniques to exactly determine their fixation probabilities and then their evolutionary regimes, as well as the exact critical values for which each graph changes its regime. A similar technique is used to explore a general method to suppress selection in bigger orders, namely from 8 to 15, up to some critical fitness value. The analysis of all graphs from order 8 to order 10 reveals a complex and rich evolutionary dynamics, which have not been examined in detail until now, and poses some new challenges in computing fixation probabilities and times of evolutionary graphs.

Insertion-tolerance and repetitiveness of random graphs

Bond percolation on Cayley graphs provides examples of random graphs. Other examples arise from the dynamical study of proper repetitive subgraphs of Cayley graphs. In this paper we demonstrate that these two families have mutually singular laws as a corollary of a general lemma about countable Borel equivalence relations on first countable Hausdorff spaces.