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.

This result was presented by Fernando Alcalde at the conference Foliations 2014, that took place in Madrid, Spain, in September 2014.Corrigendum in "Discrete and continuous dynamical systems vol. 37 n° 8 (2017), 4585-4586 p. DOI: 10.3934/dcds.2017196International audienceWe consider a minimal compact lamination by hyperbolic surfaces. We prove that if it admits a leaf whose holonomy covering is not topologically trivial, then the horocycle flow on its unitary tangent bundle is minimal

Evolutionary dynamics has been classically studied for homogeneous populations, but now there is a growing interest in the non-homogenous case. One of the most important models has been proposed in [11], adapting to a weighted directed graph the process described in [12]. The Markov chain associated with the graph can be modified by erasing all non-trivial loops in its state space, obtaining the so-called Embedded Markov chain (EMC). The fixation probability remains unchanged, but the expected time to absorption (fixation or extinction) is reduced. In this paper, we shall use this idea to compute asymptotically the average fixation probability for complete bipartite graphs Kn,m. To this end, we firstly review some recent results on evolutionary dynamics on graphs trying to clarify some points. We also revisit the 'Star Theorem' proved in [11] for the star graphs K1,m. Theoretically, EMC techniques allow fast computation of the fixation probability, but in practice this is not always true. Thus, in the last part of the paper, we compare this algorithm with the standard Monte Carlo method for some kind of complex networks.

The evolutionary dynamics of a finite population where resident individuals are replaced by mutant ones depends on its spatial structure. Usually, the population adopts the form of an undirected graph where the place occupied by each individual is represented by a vertex and it is bidirectionally linked to the places that can be occupied by its 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 vertex. But there also are undirected graph structures acting as suppressors of selection where this probability is less than that of the same individual placed in a homogeneous population. Here, firstly, we present the distribution of these evolutionary regimes for all undirected graphs with N ≤ 10 vertices. Some of them exhibit transitions between different regimes when the mutant fitness increases. In particular, as it has been already observed for small-order random graphs, we show that most graphs of order N ≤ 10 are amplifiers of selection. Secondly, we describe examples of amplifiers of order 7 that become suppressors from some critical value. In fact, for graphs of order N ≤ 7, we apply computer-aided techniques to symbolically compute their fixation probability and then their evolutionary regime, as well as the critical values for which they change their regime. Thirdly, the same technique is applied to some families of highly symmetrical graphs as a mean to explore methods of suppressing selection. The existence of suppression mechanisms that reverse an amplification regime when fitness increases could have a great interest in biology and network science. Finally, the analysis of all graphs from order 8 to order 10 reveals a complex and rich evolutionary dynamics, with multiple transitions between different regimes, which have not been examined in detail until now.

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.