Heiko Bauke

Parameter estimation for power-law distributions by maximum likelihood methods

Abstract.Distributions following a power-law are an ubiquitous phenomenon. Methodsn for determining the exponent of a power-law tail by graphical means are often used in practice but are intrinsically unreliable. Maximum likelihood estimators for the exponent are a mathematically sound alternative to graphical methods. nn

Random numbers for large-scale distributed Monte Carlo simulations.

Monte Carlo simulations are one of the major tools in statistical physics, complex system science, and other fields, and an increasing number of these simulations is run on distributed systems like clusters or grids. This raises the issue of generating random numbers in a parallel, distributed environment. In this contribution we demonstrate that multiple linear recurrences in finite fields are an ideal method to produce high quality pseudorandom numbers in sequential and parallel algorithms. Their known weakness (failure of sampling points in high dimensions) can be overcome by an appropriate delinearization that preserves all desirable properties of the underlying linear sequence.

Universality in the level statistics of disordered systems

Energy spectra of disordered systems share a common feature: If the entropy of the quenched disorder is larger than the entropy of the dynamical variables, the spectrum is locally that of a random energy model and the correlation between energy and configuration is lost. We demonstrate this effect for the Edwards-Anderson model, but we also discuss its universality.

Number partitioning as a random energy model

Number partitioning is a classical problem from combinatorial optimization. In physical terms it corresponds to a long range anti-ferromagnetic Ising spin glass. It has been rigorously proven that the low lying energies of number partitioning behave like uncorrelated random variables. We claim that neighbouring energy levels are uncorrelated almost everywhere on the energy axis, and that energetically adjacent configurations are uncorrelated, too. Apparently there is no relation between geometry (configuration) and energy that could be exploited by an optimization algorithm. This local random energy picture of number partitioning is corroborated by numerical simulations and heuristic arguments.

Entropy of pseudo-random-number generators

Since the work of Phys. Rev. Lett. 69, 3382 (1992)]] some pseudo-random-number generators are known to yield wrong results in cluster Monte Carlo simulations. In this contribution the fundamental mechanism behind this failure is discussed. Almost all random-number generators calculate a new pseudo-random-number x(i) from preceding values, x(i) =f( x(i-1), x(i-2), ..., x(i-q) ). Failure of these generators in cluster Monte Carlo simulations and related experiments can be attributed to the low entropy of the production rule f() conditioned on the statistics of the input values x(i-1), ..., x(i-q). Being a measure only of the arithmetic operations in the generator rule, the conditional entropy is independent of the lag in the recurrence or the period of the sequence. In that sense it measures a more profound quality of a random-number generator than empirical tests with their limited horizon.

Pseudo Random Coins Show More Heads Than Tails

Tossing a coin is the most elementary Monte-Carlo experiment. In a computer the coin is replaced by a pseudo random number generator. It can be shown analytically and by exact enumerations that popular random number generators are not capable of imitating a fair coin: pseudo random coins show more “heads” than “tails.” This bias explains the empirically observed failure of some random number generators in random walk experiments. It can be traced down to the special role of the value zero in the algebra of finite fields.

Phase transition in multiprocessor scheduling

An easy-hard phase transition is shown to characterize the multiprocessor scheduling problem in which one has to distribute the workload on a parallel computer such as to minimize the overall run time. The transition can be analyzed in detail by mapping it on a mean-field antiferromagnetic Potts model. The static phase transition, characterized by a vanishing ground state entropy, corresponds to a transition in the performance of practical scheduling algorithms.

Topological phase transition in a network model with preferential attachment and node removal

AbstractnPreferential attachment is a popular model of growing networks. We consider a generalizednmodel with random node removal, and a combination of preferential and random attachment.nUsing a high-degree expansion of the master equation, we identify a topological phasentransition depending on the rate of node removal and the relative strength of preferentialnvs. random attachment, where the degree distribution goes from a power law to one with annexponential tail.n