Local majority dynamics on preferential attachment graphs

Suppose in a graph $G$ vertices can be either red or blue. Let $k$ be odd. At each time step, each vertex $v$ in $G$ polls $k$ random neighbours and takes the majority colour. If it doesn't have $k$ neighbours, it simply polls all of them, or all less one if the degree of $v$ is even. We study this protocol on the preferential attachment model of Albert and Barab\'asi, which gives rise to a degree distribution that has roughly power-law $P(x) \sim \frac{1}{x^{3}}$, as well as generalisations which give exponents larger than $3$. The setting is as follows: Initially each vertex of $G$ is red independently with probability $\alpha < \frac{1}{2}$, and is otherwise blue. We show that if $\alpha$ is sufficiently biased away from $\frac{1}{2}$, then with high probability, consensus is reached on the initial global majority within $O(\log_d \log_d t)$ steps. Here $t$ is the number of vertices and $d \geq 5$ is the minimum of $k$ and $m$ (or $m-1$ if $m$ is even), $m$ being the number of edges each new vertex adds in the preferential attachment generative process. Additionally, our analysis reduces the required bias of $\alpha$ for graphs of a given degree sequence studied by the first author (which includes, e.g., random regular graphs)

Semi-Markov Graph Dynamics

In this paper, we outline a model of graph (or network) dynamics based on two ingredients. The first ingredient is a Markov chain on the space of possible graphs. The second ingredient is a semi-Markov counting process of renewal type. The model consists in subordinating the Markov chain to the semi-Markov counting process. In simple words, this means that the chain transitions occur at random time instants called epochs. The model is quite rich and its possible connections with algebraic geometry are briefly discussed. Moreover, for the sake of simplicity, we focus on the space of undirected graphs with a fixed number of nodes. However, in an example, we present an interbank market model where it is meaningful to use directed graphs or even weighted graphs.Comment: 25 pages, 4 figures, submitted to PLoS-ON

Modeling the scaling properties of human mobility

While the fat tailed jump size and the waiting time distributions characterizing individual human trajectories strongly suggest the relevance of the continuous time random walk (CTRW) models of human mobility, no one seriously believes that human traces are truly random. Given the importance of human mobility, from epidemic modeling to traffic prediction and urban planning, we need quantitative models that can account for the statistical characteristics of individual human trajectories. Here we use empirical data on human mobility, captured by mobile phone traces, to show that the predictions of the CTRW models are in systematic conflict with the empirical results. We introduce two principles that govern human trajectories, allowing us to build a statistically self-consistent microscopic model for individual human mobility. The model not only accounts for the empirically observed scaling laws but also allows us to analytically predict most of the pertinent scaling exponents

Dimension reduction for systems with slow relaxation

We develop reduced, stochastic models for high dimensional, dissipative dynamical systems that relax very slowly to equilibrium and can encode long term memory. We present a variety of empirical and first principles approaches for model reduction, and build a mathematical framework for analyzing the reduced models. We introduce the notions of universal and asymptotic filters to characterize `optimal' model reductions for sloppy linear models. We illustrate our methods by applying them to the practically important problem of modeling evaporation in oil spills.Comment: 48 Pages, 13 figures. Paper dedicated to the memory of Leo Kadanof

Transition probabilities for general birth-death processes with applications in ecology, genetics, and evolution

A birth-death process is a continuous-time Markov chain that counts the number of particles in a system over time. In the general process with $n$ current particles, a new particle is born with instantaneous rate $\lambda_n$ and a particle dies with instantaneous rate $\mu_n$. Currently no robust and efficient method exists to evaluate the finite-time transition probabilities in a general birth-death process with arbitrary birth and death rates. In this paper, we first revisit the theory of continued fractions to obtain expressions for the Laplace transforms of these transition probabilities and make explicit an important derivation connecting transition probabilities and continued fractions. We then develop an efficient algorithm for computing these probabilities that analyzes the error associated with approximations in the method. We demonstrate that this error-controlled method agrees with known solutions and outperforms previous approaches to computing these probabilities. Finally, we apply our novel method to several important problems in ecology, evolution, and genetics

Simpson's Paradox, Lord's Paradox, and Suppression Effects are the same phenomenon – the reversal paradox

This article discusses three statistical paradoxes that pervade epidemiological research: Simpson's paradox, Lord's paradox, and suppression. These paradoxes have important implications for the interpretation of evidence from observational studies. This article uses hypothetical scenarios to illustrate how the three paradoxes are different manifestations of one phenomenon – the reversal paradox – depending on whether the outcome and explanatory variables are categorical, continuous or a combination of both; this renders the issues and remedies for any one to be similar for all three. Although the three statistical paradoxes occur in different types of variables, they share the same characteristic: the association between two variables can be reversed, diminished, or enhanced when another variable is statistically controlled for. Understanding the concepts and theory behind these paradoxes provides insights into some controversial or contradictory research findings. These paradoxes show that prior knowledge and underlying causal theory play an important role in the statistical modelling of epidemiological data, where incorrect use of statistical models might produce consistent, replicable, yet erroneous results

Survival and selection biases in early animal evolution and a source of systematic overestimation in molecular clocks

Important evolutionary events such as the Cambrian Explosion have inspired many attempts at explanation: why do they happen when they do? What shapes them, and why do they eventually come to an end? However, much less attention has been paid to the idea of a ‘null hypothesis’—that certain features of such diversifications arise simply through their statistical structure. Such statistical features also appear to influence our perception of the timing of these events. Here, we show in particular that study of unusually large clades leads to systematic overestimates of clade ages from some types of molecular clocks, and that the size of this effect may be enough to account for the puzzling mismatches seen between these molecular clocks and the fossil record. Our analysis of the fossil record of the late Ediacaran to Cambrian suggests that it is likely to be recording a true evolutionary radiation of the bilaterians at this time, and that explanations involving various sorts of cryptic origins for the bilaterians do not seem to be necessary

Lyapunov exponents and phase diagrams reveal multi-factorial control over TRAIL-induced apoptosis

Kinetic modeling, phase diagrams analysis, and quantitative single-cell experiments are combined to investigate how multiple factors, including the XIAP:caspase-3 ratio and ligand concentration, regulate receptor-mediated apoptosis

Scaling properties of protein family phylogenies

One of the classical questions in evolutionary biology is how evolutionary processes are coupled at the gene and species level. With this motivation, we compare the topological properties (mainly the depth scaling, as a characterization of balance) of a large set of protein phylogenies with a set of species phylogenies. The comparative analysis shows that both sets of phylogenies share remarkably similar scaling behavior, suggesting the universality of branching rules and of the evolutionary processes that drive biological diversification from gene to species level. In order to explain such generality, we propose a simple model which allows us to estimate the proportion of evolvability/robustness needed to approximate the scaling behavior observed in the phylogenies, highlighting the relevance of the robustness of a biological system (species or protein) in the scaling properties of the phylogenetic trees. Thus, the rules that govern the incapability of a biological system to diversify are equally relevant both at the gene and at the species level.Comment: Replaced with final published versio

Drivers of Cape Verde archipelagic endemism in keyhole limpets

Oceanic archipelagos are the ideal setting for investigating processes that shape species assemblages. Focusing on keyhole limpets, genera Fissurella and Diodora from Cape Verde Islands, we used an integrative approach combining molecular phylogenetics with ocean transport simulations to infer species distribution patterns and analyse connectivity. Dispersal simulations, using pelagic larval duration and ocean currents as proxies, showed a reduced level of connectivity despite short distances between some of the islands. It is suggested that dispersal and persistence driven by patterns of oceanic circulation favouring self-recruitment played a primary role in explaining contemporary species distributions. Mitochondrial and nuclear data revealed the existence of eight Cape Verde endemic lineages, seven within Fissurella, distributed across the archipelago, and one within Diodora restricted to Boavista. The estimated origins for endemic Fissurella and Diodora were 10.2 and 6.7 MY, respectively. Between 9.5 and 4.5 MY, an intense period of volcanism in Boavista might have affected Diodora, preventing its diversification. Having originated earlier, Fissurella might have had more opportunities to disperse to other islands and speciate before those events. Bayesian analyses showed increased diversification rates in Fissurella possibly promoted by low sea levels during Plio-Pleistocene, which further explain differences in species richness between both genera.FCT - Portuguese Science Foundation [SFRH/BPD/109685/2015, SFRH/BPD/111003/2015]; Norte Portugal Regional Operational Program (NORTE), under the PORTUGAL Partnership Agreement, through the European Regional Development Fund (ERDF) [MARINFO - NORTE-01-0145-FEDER-000031]info:eu-repo/semantics/publishedVersio