Power laws, Pareto distributions and Zipf's law

When the probability of measuring a particular value of some quantity varies inversely as a power of that value, the quantity is said to follow a power law, also known variously as Zipf's law or the Pareto distribution. Power laws appear widely in physics, biology, earth and planetary sciences, economics and finance, computer science, demography and the social sciences. For instance, the distributions of the sizes of cities, earthquakes, solar flares, moon craters, wars and people's personal fortunes all appear to follow power laws. The origin of power-law behaviour has been a topic of debate in the scientific community for more than a century. Here we review some of the empirical evidence for the existence of power-law forms and the theories proposed to explain them.Comment: 28 pages, 16 figures, minor corrections and additions in this versio

On Convergence and Threshold Properties of Discrete Lotka-Volterra Population Protocols

In this work we focus on a natural class of population protocols whose dynamics are modelled by the discrete version of Lotka-Volterra equations. In such protocols, when an agent $a$ of type (species) $i$ interacts with an agent $b$ of type (species) $j$ with $a$ as the initiator, then $b$'s type becomes $i$ with probability $P\_{ij}$. In such an interaction, we think of $a$ as the predator, $b$ as the prey, and the type of the prey is either converted to that of the predator or stays as is. Such protocols capture the dynamics of some opinion spreading models and generalize the well-known Rock-Paper-Scissors discrete dynamics. We consider the pairwise interactions among agents that are scheduled uniformly at random. We start by considering the convergence time and show that any Lotka-Volterra-type protocol on an $n$-agent population converges to some absorbing state in time polynomial in $n$, w.h.p., when any pair of agents is allowed to interact. By contrast, when the interaction graph is a star, even the Rock-Paper-Scissors protocol requires exponential time to converge. We then study threshold effects exhibited by Lotka-Volterra-type protocols with 3 and more species under interactions between any pair of agents. We start by presenting a simple 4-type protocol in which the probability difference of reaching the two possible absorbing states is strongly amplified by the ratio of the initial populations of the two other types, which are transient, but "control" convergence. We then prove that the Rock-Paper-Scissors protocol reaches each of its three possible absorbing states with almost equal probability, starting from any configuration satisfying some sub-linear lower bound on the initial size of each species. That is, Rock-Paper-Scissors is a realization of a "coin-flip consensus" in a distributed system. Some of our techniques may be of independent value

Fluctuation scaling in complex systems: Taylor's law and beyond

Complex systems consist of many interacting elements which participate in some dynamical process. The activity of various elements is often different and the fluctuation in the activity of an element grows monotonically with the average activity. This relationship is often of the form "$fluctuations \approx const.\times average^\alpha$", where the exponent $\alpha$ is predominantly in the range $[1/2, 1]$. This power law has been observed in a very wide range of disciplines, ranging from population dynamics through the Internet to the stock market and it is often treated under the names \emph{Taylor's law} or \emph{fluctuation scaling}. This review attempts to show how general the above scaling relationship is by surveying the literature, as well as by reporting some new empirical data and model calculations. We also show some basic principles that can underlie the generality of the phenomenon. This is followed by a mean-field framework based on sums of random variables. In this context the emergence of fluctuation scaling is equivalent to some corresponding limit theorems. In certain physical systems fluctuation scaling can be related to finite size scaling.Comment: 33 pages, 20 figures, 2 tables, submitted to Advances in Physic

Bibliometric data in clinical cardiology revisited. The case of 37 Dutch professors

In this paper, we assess the bibliometric parameters of 37 Dutch professors in clinical cardiology. Those are the Hirsch index (h-index) based on all papers, the h-index based on first authored papers, the number of papers, the number of citations and the citations per paper. A top 10 for each of the five parameters was compiled. In theory, the same 10 professors might appear in each of these top 10s. Alternatively, each of the 37 professors under assessment could appear one or more times. In practice, we found 22 out of these 37 professors in the 5 top 10s. Thus, there is no golden parameter. In addition, there is too much inhomogeneity in citation characteristics even within a relatively homogeneous group of clinical cardiologists. Therefore, citation analysis should be applied with great care in science policy. This is even more important when different fields of medicine are compared in university medical centres. It may be possible to develop better parameters in the future, but the present ones are simply not good enough. Also, we observed a quite remarkable explosion of publications per author which can, paradoxical as it may sound, probably not be interpreted as an increase in productivity of scientists, but as the effect of an increase in the number of co-authors and the strategic effect of networks

Nodal dynamics, not degree distributions, determine the structural controllability of complex networks

Structural controllability has been proposed as an analytical framework for making predictions regarding the control of complex networks across myriad disciplines in the physical and life sciences (Liu et al., Nature:473(7346):167-173, 2011). Although the integration of control theory and network analysis is important, we argue that the application of the structural controllability framework to most if not all real-world networks leads to the conclusion that a single control input, applied to the power dominating set (PDS), is all that is needed for structural controllability. This result is consistent with the well-known fact that controllability and its dual observability are generic properties of systems. We argue that more important than issues of structural controllability are the questions of whether a system is almost uncontrollable, whether it is almost unobservable, and whether it possesses almost pole-zero cancellations.Comment: 1 Figures, 6 page

Coexistence of competing stage-structured populations

This paper analyzes the stability of a coexistence equilibrium point of a model for competition between two stage-structured populations. In this model, for each population, competition for resources may affect any one of the following population parameters: reproduction, juvenile survival, maturation rate, or adult survival. The results show that the competitive strength of a population is affected by (1) the ratio of the population parameter influenced by competition under no resource limitation (maximum compensatory capacity) over the same parameter under a resource limitation due to competition (equilibrium rate) and (2) the ratio of interspecific competition over intraspecific competition; this ratio was previously shown to depend on resource-use overlap. The former ratio, which we define as fitness, can be equalized by adjusting organisms' life history strategies, thereby promoting coexistence. We conclude that in addition to niche differentiation among populations, the life history strategies of organisms play an important role in coexistence

Evolutionary Games

International audienceEvolutionary games constitute the most recent major mathematical tool for understanding, modelling and predicting evolution in biology and other fields. They complement other well establlished tools such as branching processes and the Lotka-Volterra [6] equations (e.g. for the predator - prey dynamics or for epidemics evolution). Evolutionary Games also brings novel features to game theory. First, it focuses on the dynam- ics of competition rather than restricting attention to the equilibrium. In particular, it tries to explain how an equilibrium emerges. Second, it brings new de nitions of stability, that are more adapted to the context of large populations. Finally, in contrast to standard game theory, players are not assumed to be \rational" or \knowledgeable" as to anticipate the other players' choices. The objective of this article, is to present founda- tions as well as recent advances in evolutionary games, highlight the novel concepts that they introduce with respect to game theory as formulated by John Nash, and describe through several examples their huge potential as tools for modeling interactions in complex systems

Ross, Macdonald, and a Theory for the Dynamics and Control of Mosquito-Transmitted Pathogens

Ronald Ross and George Macdonald are credited with developing a mathematical model of mosquito-borne pathogen transmission. A systematic historical review suggests that several mathematicians and scientists contributed to development of the Ross-Macdonald model over a period of 70 years. Ross developed two different mathematical models, Macdonald a third, and various “Ross-Macdonald” mathematical models exist. Ross-Macdonald models are best defined by a consensus set of assumptions. The mathematical model is just one part of a theory for the dynamics and control of mosquito-transmitted pathogens that also includes epidemiological and entomological concepts and metrics for measuring transmission. All the basic elements of the theory had fallen into place by the end of the Global Malaria Eradication Programme (GMEP, 1955–1969) with the concept of vectorial capacity, methods for measuring key components of transmission by mosquitoes, and a quantitative theory of vector control. The Ross-Macdonald theory has since played a central role in development of research on mosquito-borne pathogen transmission and the development of strategies for mosquito-borne disease prevention