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

Lack of consensus in social systems

We propose an exactly solvable model for the dynamics of voters in a two-party system. The opinion formation process is modeled on a random network of agents. The dynamical nature of interpersonal relations is also reflected in the model, as the connections in the network evolve with the dynamics of the voters. In the infinite time limit, an exact solution predicts the emergence of consensus, for arbitrary initial conditions. However, before consensus is reached, two different metastable states can persist for exponentially long times. One state reflects a perfect balancing of opinions, the other reflects a completely static situation. An estimate of the associated lifetimes suggests that lack of consensus is typical for large systems.Comment: 4 pages, 6 figures, submitted to Phys. Rev. Let

Coexistence of opposite opinions in a network with communities

The Majority Rule is applied to a topology that consists of two coupled random networks, thereby mimicking the modular structure observed in social networks. We calculate analytically the asymptotic behaviour of the model and derive a phase diagram that depends on the frequency of random opinion flips and on the inter-connectivity between the two communities. It is shown that three regimes may take place: a disordered regime, where no collective phenomena takes place; a symmetric regime, where the nodes in both communities reach the same average opinion; an asymmetric regime, where the nodes in each community reach an opposite average opinion. The transition from the asymmetric regime to the symmetric regime is shown to be discontinuous.Comment: 14 pages, 4 figure

Complete diagrammatics of the single ring theorem

Using diagrammatic techniques, we provide explicit functional relations between the cumulant generating functions for the biunitarily invariant ensembles in the limit of large size of matrices. The formalism allows to map two distinct areas of free random variables: Hermitian positive definite operators and non-normal R-diagonal operators. We also rederive the Haagerup-Larsen theorem and show how its recent extension to the eigenvector correlation function appears naturally within this approach.Comment: 18 pages, 6 figures, version accepted for publicatio

Turnover, account value and diversification of real traders: evidence of collective portfolio optimizing behavior

Despite the availability of very detailed data on financial market, agent-based modeling is hindered by the lack of information about real trader behavior. This makes it impossible to validate agent-based models, which are thus reverse-engineering attempts. This work is a contribution to the building of a set of stylized facts about the traders themselves. Using the client database of Swissquote Bank SA, the largest on-line Swiss broker, we find empirical relationships between turnover, account values and the number of assets in which a trader is invested. A theory based on simple mean-variance portfolio optimization that crucially includes variable transaction costs is able to reproduce faithfully the observed behaviors. We finally argue that our results bring into light the collective ability of a population to construct a mean-variance portfolio that takes into account the structure of transaction costsComment: 26 pages, 9 figures, Fig. 8 fixe

The helium atom in a strong magnetic field

We investigate the electronic structure of the helium atom in a magnetic field b etween B=0 and B=100a.u. The atom is treated as a nonrelativistic system with two interactin g electrons and a fixed nucleus. Scaling laws are provided connecting the fixed-nucleus Hamiltonia n to the one for the case of finite nuclear mass. Respecting the symmetries of the electronic Ham iltonian in the presence of a magnetic field, we represent this Hamiltonian as a matrix with res pect to a two-particle basis composed of one-particle states of a Gaussian basis set. The corresponding generalized eigenvalue problem is solved numerically, providing in the present paper results for vanish ing magnetic quantum number M=0 and even or odd z-parity, each for both singlet and triplet spin symmetry. Total electronic energies of the ground state and the first few excitations in each su bspace as well as their one-electron ionization energies are presented as a function of the magnetic fie ld, and their behaviour is discussed. Energy values for electromagnetic transitions within the M=0 sub space are shown, and a complete table of wavelengths at all the detected stationary points with respect to their field dependence is given, thereby providing a basis for a comparison with observed ab sorption spectra of magnetic white dwarfs.Comment: 21 pages, 4 Figures, acc.f.publ.in J.Phys.