Statistics of Lead Changes in Popularity-Driven Systems

We study statistical properties of the highest degree, or most popular, nodes in growing networks. We show that the number of lead changes increases logarithmically with network size N, independent of the details of the growth mechanism. The probability that the first node retains the lead approaches a finite constant for popularity-driven growth, and decays as N^{-phi}(ln N)^{-1/2}, with phi=0.08607..., for growth with no popularity bias.Comment: 4 pages, 4 figures, 2 column revtex format. Minor changes in response to referee comments. For publication in PR

Power-law distributions from additive preferential redistributions

We introduce a non-growth model that generates the power-law distribution with the Zipf exponent. There are N elements, each of which is characterized by a quantity, and at each time step these quantities are redistributed through binary random interactions with a simple additive preferential rule, while the sum of quantities is conserved. The situation described by this model is similar to those of closed $N$-particle systems when conservative two-body collisions are only allowed. We obtain stationary distributions of these quantities both analytically and numerically while varying parameters of the model, and find that the model exhibits the scaling behavior for some parameter ranges. Unlike well-known growth models, this alternative mechanism generates the power-law distribution when the growth is not expected and the dynamics of the system is based on interactions between elements. This model can be applied to some examples such as personal wealths, city sizes, and the generation of scale-free networks when only rewiring is allowed.Comment: 12 pages, 4 figures; Changed some expressions and notations; Added more explanations and changed the order of presentation in Sec.III while results are the sam

A Group-Based Yule Model for Bipartite Author-Paper Networks

This paper presents a novel model for author-paper networks, which is based on the assumption that authors are organized into groups and that, for each research topic, the number of papers published by a group is based on a success-breeds-success model. Collaboration between groups is modeled as random invitations from a group to an outside member. To analyze the model, a number of different metrics that can be obtained in author-paper networks were extracted. A simulation example shows that this model can effectively mimic the behavior of a real-world author-paper network, extracted from a collection of 900 journal papers in the field of complex networks.Comment: 13 pages (preprint format), 7 figure

A Yule-Simon process with memory

The Yule-Simon model has been used as a tool to describe the growth of diverse systems, acquiring a paradigmatic character in many fields of research. Here we study a modified Yule-Simon model that takes into account the full history of the system by means of an hyperbolic memory kernel. We show how the memory kernel changes the properties of preferential attachment and provide an approximate analytical solution for the frequency distribution density as well as for the frequency-rank distribution.Comment: 7 pages, 5 figures; accepted for publication in Europhysics Letter

Power Law Distribution of Wealth in a Money-Based Model

A money-based model for the power law distribution (PLD) of wealth in an economically interacting population is introduced. The basic feature of our model is concentrating on the capital movements and avoiding the complexity of micro behaviors of individuals. It is proposed as an extension of the Equiluz and Zimmermann's (EZ) model for crowding and information transmission in financial markets. Still, we must emphasize that in EZ model the PLD without exponential correction is obtained only for a particular parameter, while our pattern will give it within a wide range. The Zipf exponent depends on the parameters in a nontrivial way and is exactly calculated in this paper.Comment: 5 pages and 4 figure

Critical and Near-Critical Branching Processes

Scale-free dynamics in physical and biological systems can arise from a variety of causes. Here, we explore a branching process which leads to such dynamics. We find conditions for the appearance of power laws and study quantitatively what happens to these power laws when such conditions are violated. From a branching process model, we predict the behavior of two systems which seem to exhibit near scale-free behavior--rank-frequency distributions of number of subtaxa in biology, and abundance distributions of genotypes in an artificial life system. In the light of these, we discuss distributions of avalanche sizes in the Bak-Tang-Wiesenfeld sandpile model.Comment: 9 pages LaTex with 10 PS figures. v.1 of this paper contains results from non-critical sandpile simulations that were excised from the published versio

Rank Statistics in Biological Evolution

We present a statistical analysis of biological evolution processes. Specifically, we study the stochastic replication-mutation-death model where the population of a species may grow or shrink by birth or death, respectively, and additionally, mutations lead to the creation of new species. We rank the various species by the chronological order by which they originate. The average population N_k of the kth species decays algebraically with rank, N_k ~ M^{mu} k^{-mu}, where M is the average total population. The characteristic exponent mu=(alpha-gamma)/(alpha+beta-gamma)$ depends on alpha, beta, and gamma, the replication, mutation, and death rates. Furthermore, the average population P_k of all descendants of the kth species has a universal algebraic behavior, P_k ~ M/k.Comment: 4 pages, 3 figure

Exact Scale Invariance in Mixing of Binary Candidates in Voting Model

We introduce a voting model and discuss the scale invariance in the mixing of candidates. The Candidates are classified into two categories $\mu\in \{0,1\}$ and are called as `binary' candidates. There are in total $N=N_{0}+N_{1}$ candidates, and voters vote for them one by one. The probability that a candidate gets a vote is proportional to the number of votes. The initial number of votes (`seed') of a candidate $\mu$ is set to be $s_{\mu}$. After infinite counts of voting, the probability function of the share of votes of the candidate $\mu$ obeys gamma distributions with the shape exponent $s_{\mu}$ in the thermodynamic limit $Z_{0}=N_{1}s_{1}+N_{0}s_{0}\to \infty$. Between the cumulative functions $\{x_{\mu}\}$ of binary candidates, the power-law relation $1-x_{1} \sim (1-x_{0})^{\alpha}$ with the critical exponent $\alpha=s_{1}/s_{0}$ holds in the region $1-x_{0},1-x_{1}<<1$. In the double scaling limit $(s_{1},s_{0})\to (0,0)$ and $Z_{0} \to \infty$ with $s_{1}/s_{0}=\alpha$ fixed, the relation $1-x_{1}=(1-x_{0})^{\alpha}$ holds exactly over the entire range $0\le x_{0},x_{1} \le 1$. We study the data on horse races obtained from the Japan Racing Association for the period 1986 to 2006 and confirm scale invariance.Comment: 19 pages, 8 figures, 2 table

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