Ultrametricity and Memory in a Solvable Model of Self-Organized Criticality

Slowly driven dissipative systems may evolve to a critical state where long periods of apparent equilibrium are punctuated by intermittent avalanches of activity. We present a self-organized critical model of punctuated equilibrium behavior in the context of biological evolution, and solve it in the limit that the number of independent traits for each species diverges. We derive an exact equation of motion for the avalanche dynamics from the microscopic rules. In the continuum limit, avalanches propagate via a diffusion equation with a nonlocal, history-dependent potential representing memory. This nonlocal potential gives rise to a non-Gaussian (fat) tail for the subdiffusive spreading of activity. The probability for the activity to spread beyond a distance $r$ in time $s$ decays as $\sqrt{24\over\pi}s^{-3/2}x^{1/3} \exp{[-{3\over 4}x^{1/3}]}$ for $x={r^4\over s} \gg 1$. The potential represents a hierarchy of time scales that is dynamically generated by the ultrametric structure of avalanches, which can be quantified in terms of ``backward'' avalanches. In addition, a number of other correlation functions characterizing the punctuated equilibrium dynamics are determined exactly.Comment: 44 pages, Revtex, (12 ps-figures included

Asymptotic analysis of noisy fitness maximization, applied to metabolism and growth

We consider a population dynamics model coupling cell growth to a diffusion in the space of metabolic phenotypes as it can be obtained from realistic constraints-based modelling. In the asymptotic regime of slow diffusion, that coincides with the relevant experimental range, the resulting non-linear Fokker-Planck equation is solved for the steady state in the WKB approximation that maps it into the ground state of a quantum particle in an Airy potential plus a centrifugal term. We retrieve scaling laws for growth rate fluctuations and time response with respect to the distance from the maximum growth rate suggesting that suboptimal populations can have a faster response to perturbations.Comment: 24 pages, 6 figure

SQG-Differential Evolution for difficult optimization problems under a tight function evaluation budget

In the context of industrial engineering, it is important to integrate efficient computational optimization methods in the product development process. Some of the most challenging simulation-based engineering design optimization problems are characterized by: a large number of design variables, the absence of analytical gradients, highly non-linear objectives and a limited function evaluation budget. Although a huge variety of different optimization algorithms is available, the development and selection of efficient algorithms for problems with these industrial relevant characteristics, remains a challenge. In this communication, a hybrid variant of Differential Evolution (DE) is introduced which combines aspects of Stochastic Quasi-Gradient (SQG) methods within the framework of DE, in order to improve optimization efficiency on problems with the previously mentioned characteristics. The performance of the resulting derivative-free algorithm is compared with other state-of-the-art DE variants on 25 commonly used benchmark functions, under tight function evaluation budget constraints of 1000 evaluations. The experimental results indicate that the new algorithm performs excellent on the 'difficult' (high dimensional, multi-modal, inseparable) test functions. The operations used in the proposed mutation scheme, are computationally inexpensive, and can be easily implemented in existing differential evolution variants or other population-based optimization algorithms by a few lines of program code as an non-invasive optional setting. Besides the applicability of the presented algorithm by itself, the described concepts can serve as a useful and interesting addition to the algorithmic operators in the frameworks of heuristics and evolutionary optimization and computing

Differential Evolution and Combinatorial Search for Constrained Index Traking

Index tracking is a valuable low-cost alternative to active portfolio management. The implementation of a quantitative approach, however, is a major challenge from an optimization perspective. The optimal selection of a group of assets that can replicate the index of a much larger portfolio requires both to find the optimal subset of assets and to fine-tune their weights. The former is a combinatorial, the latter a continuous numerical problem. Both problems need to be addressed simultaneously, because whether or not a selection of assets is promising depends on the allocation weights and vice versa. Moreover, the problem is usually of high dimension. Typically, an optimal subset of 30-150 positions out of 100-600 need to be selected and their weights determined. Search heuristics can be a viable and valuable alternative to traditional methods, which often cannot deal with the problem. In this paper, we propose a new optimization method, which is partly based on Differential Evolution (DE) and on combinatorial search. The main advantage of our method is that it can tackle index tracking problem as complex as it is, generating accurate and robust results

Life at the front of an expanding population

Recent microbial experiments suggest that enhanced genetic drift at the frontier of a two-dimensional range expansion can cause genetic sectoring patterns with fractal domain boundaries. Here, we propose and analyze a simple model of asexual biological evolution at expanding frontiers to explain these neutral patterns and predict the effect of natural selection. Our model attributes the observed gradual decrease in the number of sectors at the leading edge to an unbiased random walk of sector boundaries. Natural selection introduces a deterministic bias in the wandering of domain boundaries that renders beneficial mutations more likely to escape genetic drift and become established in a sector. We find that the opening angle of those sectors and the rate at which they become established depend sensitively on the selective advantage of the mutants. Deleterious mutations, on the other hand, are not able to establish a sector permanently. They can, however, temporarily "surf" on the population front, and thereby reach unusual high frequencies. As a consequence, expanding frontiers are susceptible to deleterious mutations as revealed by the high fraction of mutants at mutation-selection balance. Numerically, we also determine the condition at which the wild type is lost in favor of deleterious mutants (genetic meltdown) at a growing front. Our prediction for this error threshold differs qualitatively from existing well-mixed theories, and sets tight constraints on sustainable mutation rates for populations that undergo frequent range expansions.Comment: Updat

Geometric origin of scaling in large traffic networks

Large scale traffic networks are an indispensable part of contemporary human mobility and international trade. Networks of airport travel or cargo ships movements are invaluable for the understanding of human mobility patterns\cite{Guimera2005}, epidemic spreading\cite{Colizza2006}, global trade\cite{Imo2006} and spread of invasive species\cite{Ruiz2000}. Universal features of such networks are necessary ingredients of their description and can point to important mechanisms of their formation. Different studies\cite{Barthelemy2010} point to the universal character of some of the exponents measured in such networks. Here we show that exponents which relate i) the strength of nodes to their degree and ii) weights of links to degrees of nodes that they connect have a geometric origin. We present a simple robust model which exhibits the observed power laws and relates exponents to the dimensionality of 2D space in which traffic networks are embedded. The model is studied both analytically and in simulations and the conditions which result with previously reported exponents are clearly explained. We show that the relation between weight strength and degree is $s(k)\sim k^{3/2}$, the relation between distance strength and degree is $s^d(k)\sim k^{3/2}$ and the relation between weight of link and degrees of linked nodes is $w_{ij}\sim(k_ik_j)^{1/2}$ on the plane 2D surface. We further analyse the influence of spherical geometry, relevant for the whole planet, on exact values of these exponents. Our model predicts that these exponents should be found in future studies of port networks and impose constraints on more refined models of port networks.Comment: 17 pages, 5 figures, 1 tabl