An extended formalism for preferential attachment in heterogeneous complex networks

In this paper we present a framework for the extension of the preferential attachment (PA) model to heterogeneous complex networks. We define a class of heterogeneous PA models, where node properties are described by fixed states in an arbitrary metric space, and introduce an affinity function that biases the attachment probabilities of links. We perform an analytical study of the stationary degree distributions in heterogeneous PA networks. We show that their degree densities exhibit a richer scaling behavior than their homogeneous counterparts, and that the power law scaling in the degree distribution is robust in presence of heterogeneity

Scaling-violation phenomena and fractality in the human posture control systems

By analyzing the movements of quiet standing persons by means of wavelet statistics, we observe multiple scaling regions in the underlying body dynamics. The use of the wavelet-variance function opens the possibility to relate scaling violations to different modes of posture control. We show that scaling behavior becomes close to perfect, when correctional movements are dominated by the vestibular system.Comment: 12 pages, 4 figures, to appear in Phys. Rev.

Fractal multi-level organisation of human groups in a virtual world

Humans are fundamentally social. They form societies which consist of hierarchically layered nested groups of various quality, size, and structure. The anthropologic literature has classified these groups as support cliques, sympathy groups, bands, cognitive groups, tribes, linguistic groups, and so on. Anthropologic data show that, on average, each group consists of approximately three subgroups. However, a general understanding of the structural dependence of groups at different layer is largely missing. We extend these early findings to a very large high-precision large-scale internet-based social network data. We analyze the organizational structure of a complete, multi-relational, large social multiplex network of a human society consisting of about 400,000 odd players of an open-ended massive multiplayer online game for which we know all about their various group memberships at different layers. Remarkably, the online players' society exhibits the same type of structured hierarchical layers as found in hunter-gatherer societies. Our findings suggest that the hierarchical organization of human society is deeply nested in human psychology

Understanding frequency distributions of path-dependent processes with non-multinomial maximum entropy approaches

Path-dependent stochastic processes are often non-ergodic and observables can no longer be computed within the ensemble picture. The resulting mathematical difficulties pose severe limits to the analytical understanding of path-dependent processes. Their statistics is typically non-multinomial in the sense that the multiplicities of the occurrence of states is not a multinomial factor. The maximum entropy principle is tightly related to multinomial processes, non-interacting systems, and to the ensemble picture; it loses its meaning for path-dependent processes. Here we show that an equivalent to the ensemble picture exists for path-dependent processes, such that the non-multinomial statistics of the underlying dynamical process, by construction, is captured correctly in a functional that plays the role of a relative entropy. We demonstrate this for self-reinforcing Pólya urn processes, which explicitly generalize multinomial statistics. We demonstrate the adequacy of this constructive approach towards non-multinomial entropies by computing frequency and rank distributions of Pólya urn processes. We show how microscopic update rules of a path-dependent process allow us to explicitly construct a non-multinomial entropy functional, that, when maximized, predicts the time-dependent distribution function

How driving rates determine the statistics of driven non-equilibrium systems with stationary distributions

Sample space reducing (SSR) processes offer a simple analytical way to understand the origin and ubiquity of power-laws in many path-dependent complex systems. SRR processes show a wide range of applications that range from fragmentation processes, language formation to search and cascading processes. Here we argue that they also offer a natural framework to understand stationary distributions of generic driven non-equilibrium systems that are composed of a driving- and a relaxing process. We show that the statistics of driven non-equilibrium systems can be derived from the understanding of the nature of the underlying driving process. For constant driving rates exact power-laws emerge with exponents that are related to the driving rate. If driving rates become state-dependent, or if they vary across the life-span of the process, the functional form of the state-dependence determines the statistics. Constant driving rates lead to exact power-laws, a linear state-dependence function yields exponential or Gamma distributions, a quadratic function produces the normal distribution. Logarithmic and power-law state dependence leads to log-normal and stretched exponential distribution functions, respectively. Also Weibull, Gompertz and Tsallis-Pareto distributions arise naturally from simple state-dependent driving rates. We discuss a simple physical example of consecutive elastic collisions that exactly represents a SSR process

Regional diversity in the murine cortical vascular network is revealed by synchrotron X-ray tomography and is amplified with age

Cortical bone is permeated by a system of pores, occupied by the blood supply and osteocytes. With ageing, bone mass reduction and disruption of the microstructure are associated with reduced vascular supply. Insight into the regulation of the blood supply to the bone could enhance the understanding of bone strength determinants and fracture healing. Using synchrotron radiation-based computed tomography, the distribution of vascular canals and osteocyte lacunae was assessed in murine cortical bone and the influence of age on these parameters was investigated. The tibiofibular junction from 15-week- and 10-month-old female C57BL/6J mice were imaged post-mortem. Vascular canals and three-dimensional spatial relationships between osteocyte lacunae and bone surfaces were computed for both age groups. At 15 weeks, the posterior region of the tibiofibular junction had a higher vascular canal volume density than the anterior, lateral and medial regions. Intracortical vascular networks in anterior and posterior regions were also different, with connectedness in the posterior higher than the anterior at 15 weeks. By 10 months, cortices were thinner, with cortical area fraction and vascular density reduced, but only in the posterior cortex. This provided the first evidence of age-related effects on murine bone porosity due to the location of the intracortical vasculature. Targeting the vasculature to modulate bone porosity could provide an effective way to treat degenerative bone diseases, such as osteoporosis

Traffic on complex networks: Towards understanding global statistical properties from microscopic density fluctuations

We study the microscopic time fluctuations of traffic load and the global statistical properties of a dense traffic of particles on scale-free cyclic graphs. For a wide range of driving rates R the traffic is stationary and the load time series exhibits antipersistence due to the regulatory role of the superstructure associated with two hub nodes in the network. We discuss how the superstructure affects the functioning of the network at high traffic density and at the jamming threshold. The degree of correlations systematically decreases with increasing traffic density and eventually disappears when approaching a jamming density Rc. Already before jamming we observe qualitative changes in the global network-load distributions and the particle queuing times. These changes are related to the occurrence of temporary crises in which the network-load increases dramatically, and then slowly falls back to a value characterizing free flow

When do generalized entropies apply? How phase space volume determines entropy

We show how the dependence of phase space volume $\Omega(N)$ of a classical system on its size $N$ uniquely determines its extensive entropy. We give a concise criterion when this entropy is not of Boltzmann-Gibbs type but has to assume a {\em generalized} (non-additive) form. We show that generalized entropies can only exist when the dynamically (statistically) relevant fraction of degrees of freedom in the system vanishes in the thermodynamic limit. These are systems where the bulk of the degrees of freedom is frozen and is practically statistically inactive. Systems governed by generalized entropies are therefore systems whose phase space volume effectively collapses to a lower-dimensional 'surface'. We explicitly illustrate the situation for binomial processes and argue that generalized entropies could be relevant for self organized critical systems such as sand piles, for spin systems which form meta-structures such as vortices, domains, instantons, etc., and for problems associated with anomalous diffusion.Comment: 5 pages, 2 figure