Universality of Zipf's Law

We introduce a simple and generic model that reproduces Zipf's law. By regarding the time evolution of the model as a random walk in the logarithmic scale, we explain theoretically why this model reproduces Zipf's law. The explanation shows that the behavior of the model is very robust and universal.Comment: 5 eps files included. To be published in J. Phys. Soc. Jp

Bug propagation and debugging in asymmetric software structures

Software dependence networks are shown to be scale-free and asymmetric. We then study how software components are affected by the failure of one of them, and the inverse problem of locating the faulty component. Software at all levels is fragile with respect to the failure of a random single component. Locating a faulty component is easy if the failures only affect their nearest neighbors, while it is hard if the failures propagate further.Comment: 4 pages, 4 figure

Emergence of Zipf's Law in the Evolution of Communication

Zipf's law seems to be ubiquitous in human languages and appears to be a universal property of complex communicating systems. Following the early proposal made by Zipf concerning the presence of a tension between the efforts of speaker and hearer in a communication system, we introduce evolution by means of a variational approach to the problem based on Kullback's Minimum Discrimination of Information Principle. Therefore, using a formalism fully embedded in the framework of information theory, we demonstrate that Zipf's law is the only expected outcome of an evolving, communicative system under a rigorous definition of the communicative tension described by Zipf.Comment: 7 pages, 2 figure

Universal scaling in sports ranking

Ranking is a ubiquitous phenomenon in the human society. By clicking the web pages of Forbes, you may find all kinds of rankings, such as world's most powerful people, world's richest people, top-paid tennis stars, and so on and so forth. Herewith, we study a specific kind, sports ranking systems in which players' scores and prize money are calculated based on their performances in attending various tournaments. A typical example is tennis. It is found that the distributions of both scores and prize money follow universal power laws, with exponents nearly identical for most sports fields. In order to understand the origin of this universal scaling we focus on the tennis ranking systems. By checking the data we find that, for any pair of players, the probability that the higher-ranked player will top the lower-ranked opponent is proportional to the rank difference between the pair. Such a dependence can be well fitted to a sigmoidal function. By using this feature, we propose a simple toy model which can simulate the competition of players in different tournaments. The simulations yield results consistent with the empirical findings. Extensive studies indicate the model is robust with respect to the modifications of the minor parts.Comment: 8 pages, 7 figure

Network properties of written human language

We investigate the nature of written human language within the framework of complex network theory. In particular, we analyse the topology of Orwell's \textit{1984} focusing on the local properties of the network, such as the properties of the nearest neighbors and the clustering coefficient. We find a composite power law behavior for both the average nearest neighbor's degree and average clustering coefficient as a function of the vertex degree. This implies the existence of different functional classes of vertices. Furthermore we find that the second order vertex correlations are an essential component of the network architecture. To model our empirical results we extend a previously introduced model for language due to Dorogovtsev and Mendes. We propose an accelerated growing network model that contains three growth mechanisms: linear preferential attachment, local preferential attachment and the random growth of a pre-determined small finite subset of initial vertices. We find that with these elementary stochastic rules we are able to produce a network showing syntactic-like structures

Interacting Individuals Leading to Zipf's Law

We present a general approach to explain the Zipf's law of city distribution. If the simplest interaction (pairwise) is assumed, individuals tend to form cities in agreement with the well-known statisticsComment: 4 pages 2 figure

Zipf's Law in Gene Expression

Using data from gene expression databases on various organisms and tissues, including yeast, nematodes, human normal and cancer tissues, and embryonic stem cells, we found that the abundances of expressed genes exhibit a power-law distribution with an exponent close to -1, i.e., they obey Zipf's law. Furthermore, by simulations of a simple model with an intra-cellular reaction network, we found that Zipf's law of chemical abundance is a universal feature of cells where such a network optimizes the efficiency and faithfulness of self-reproduction. These findings provide novel insights into the nature of the organization of reaction dynamics in living cells.Comment: revtex, 11 pages, 3 figures, submitted to Phys. Rev. Let

Time-Varying Priority Queuing Models for Human Dynamics

Queuing models provide insight into the temporal inhomogeneity of human dynamics, characterized by the broad distribution of waiting times of individuals performing tasks. We study the queuing model of an agent trying to execute a task of interest, the priority of which may vary with time due to the agent's "state of mind." However, its execution is disrupted by other tasks of random priorities. By considering the priority of the task of interest either decreasing or increasing algebraically in time, we analytically obtain and numerically confirm the bimodal and unimodal waiting time distributions with power-law decaying tails, respectively. These results are also compared to the updating time distribution of papers in the arXiv.org and the processing time distribution of papers in Physical Review journals. Our analysis helps to understand human task execution in a more realistic scenario.Comment: 8 pages, 6 figure

Zipf's law in Nuclear Multifragmentation and Percolation Theory

We investigate the average sizes of the $n$ largest fragments in nuclear multifragmentation events near the critical point of the nuclear matter phase diagram. We perform analytic calculations employing Poisson statistics as well as Monte Carlo simulations of the percolation type. We find that previous claims of manifestations of Zipf's Law in the rank-ordered fragment size distributions are not born out in our result, neither in finite nor infinite systems. Instead, we find that Zipf-Mandelbrot distributions are needed to describe the results, and we show how one can derive them in the infinite size limit. However, we agree with previous authors that the investigation of rank-ordered fragment size distributions is an alternative way to look for the critical point in the nuclear matter diagram.Comment: 8 pages, 11 figures, submitted to PR

Bidding process in online auctions and winning strategy:rate equation approach

Online auctions have expanded rapidly over the last decade and have become a fascinating new type of business or commercial transaction in this digital era. Here we introduce a master equation for the bidding process that takes place in online auctions. We find that the number of distinct bidders who bid $k$ times, called the $k$-frequent bidder, up to the $t$-th bidding progresses as $n_k(t)\sim tk^{-2.4}$. The successfully transmitted bidding rate by the $k$-frequent bidder is obtained as $q_k(t) \sim k^{-1.4}$, independent of $t$ for large $t$. This theoretical prediction is in agreement with empirical data. These results imply that bidding at the last moment is a rational and effective strategy to win in an eBay auction.Comment: 4 pages, 6 figure