arXiv subjects

Winfried Just

Publications and source records attributed to Winfried Just.

18 recordsLinked to original sources

arXiv2018

Open-minded imitation can achieve near-optimal vaccination coverage

Studies of voluntary vaccination decisions by rational individuals predict that the population will reach a Nash equilibrium with vaccination coverage below the societal optimum. Human decision-making involves mechanisms in addition to rational calculations of self-interest, such as imitation of successful others. Previous research had shown that imitation alone cannot achieve better results. Under realistic choices of the parameters it may lead to equilibrium vaccination coverage even below the Nash equilibrium. However, these findings rely on the widely accepted use of Fermi functions for modeling the probabilities of switching to another strategy. We consider here a more general functional form of the switching probabilities. It is consistent with functions that give best fits for empirical data in a widely cited psychological experiment and involves one additional parameter $α$. This parameter can be loosely interpreted as a degree of open-mindedness. We found both by means of simulations and analytically that sufficiently high values of $α$ will drive the equilibrium vaccination coverage arbitrarily close to the societal optimum.

physics.soc-ph↗

arXiv2017

On the role of limsup in the definition of topological entropy via spanning or separation numbers. Part I: Basic examples

The notion of topological entropy can be conceptualized in terms of the number of forward trajectories that are distinguishable at resolution $\varepsilon$ within $T$ time units. It can then be formally defined as a limit of a limit superior that involves either covering numbers, or separation numbers, or spanning numbers. If covering numbers are used, the limit superior reduces to a limit. While it has been generally believed that the latter may not necessarily be the case when the definition is based on separation or spanning numbers, no actual counterexamples appear to have been previously known. Here we fill this gap in the literature by constructing such counterexamples.

math.DS↗

arXiv2017

Oscillations in epidemic models with spread of awareness

We study ODE models of epidemic spreading with a preventive behavioral response that is triggered by awareness of the infection. Previous studies of such models have mostly focused on the impact of the response on the initial growth of an outbreak and the existence and location of endemic equilibria. Here we study the question whether this type of response is sufficient to prevent future flare-ups from low endemic levels if awareness is assumed to decay over time. In the ODE context, such flare-ups would translate into sustained oscillations with significant amplitudes. Our results show that such oscillations are ruled out in Susceptible-Aware-Infectious-Susceptible models with a single compartment of aware hosts, but can occur if we consider two distinct compartments of aware hosts who differ in their willingness to alert other susceptible hosts.

q-bio.PE↗

arXiv2014

Lengths of attractors and transients in neuronal networks with random connectivities

We study how the dynamics of a class of discrete dynamical system models for neuronal networks depends on the connectivity of the network. Specifically, we assume that the network is an Erdős-R\'{enyi} random graph and analytically derive scaling laws for the average lengths of the attractors and transients under certain restrictions on the intrinsic parameters of the neurons, that is, their refractory periods and firing thresholds. In contrast to earlier results that were reported in \cite{TAWJ}, here we focus on the connection probabilities near the phase transition where the most complex dynamics is expected to occur.

math.DS↗

arXiv2013

Two classes of ODE models with switch-like behavior

In cases where the same real-world system can be modeled both by an ODE system $\bD$ and a Boolean system $\bB$ it is of interest to identify conditions under which the two systems will be consistent, that is, will make qualitatively equivalent predictions. In this note we introduce two broad classes of relatively simple models that provide a convenient framework for studying such questions. In contrast to the widely known class of Glass networks, the right-hand sides of our ODEs are Lipschitz-continuous. We prove that under suitable assumptions about $\bB$ consistency between $\bD$ and $\bB$ will be implied by sufficient separation of time scales in one class of our models while it may fail in the other class. These results appear to point to more general structure properties that favor consistency between ODE and Boolean models.

math.DS↗

arXiv2013

Cooperative Boolean systems with generically long attractors II

We prove that cooperativity in Boolean networks precludes a strong notion of sensitive dependence on initial conditions. Weaker notions of sensitive dependence are shown to be consistent with cooperativity, but if each regulatory functions is binary AND or binary OR, in N-dimensional networks they impose an upper bound of approximately sqrt(3)^N on the lengths of attractors that can be reached from a fraction p approaching 1 of initial conditions. The upper bound is shown to be sharp.

math.DS↗

arXiv2013

Cooperative Boolean systems with generically long attractors I

We study the class of cooperative Boolean networks whose only regulatory functions are COPY, binary AND, and binary OR. We prove that for all sufficiently large N and c < 2 there exist Boolean networks in this class that have an attractor of length > c^N whose basin of attraction comprises an arbitrarily large fraction of the state space. The existence of such networks contrasts with results on various other types of dynamical systems that show nongenericity or absence of non-steady state attractors under the assumption of cooperativity.

math.DS↗

arXiv2008

Extremely chaotic Boolean networks

It is an increasingly important problem to study conditions on the structure of a network that guarantee a given behavior for its underlying dynamical system. In this paper we report that a Boolean network may fall within the chaotic regime, even under the simultaneous assumption of several conditions which in randomized studies have been separately shown to correlate with ordered behavior. These properties include using at most two inputs for every variable, using biased and canalyzing regulatory functions, and restricting the number of negative feedback loops. We also prove for n-dimensional Boolean networks that if in addition the number of outputs for each variable is bounded and there exist periodic orbits of length c^n for c sufficiently close to 2, any network with these properties must have a large proportion of variables that simply copy previous values of other variables. Such systems share a structural similarity to a relatively small Turing machine acting on one or several tapes.

q-bio.MN↗

arXiv2008

Large attractors in cooperative bi-quadratic Boolean networks. Part II

Boolean networks have been the object of much attention, especially since S. Kauffman proposed them in the 1960's as models for gene regulatory networks. These systems are characterized by being defined on a Boolean state space and by simultaneous updating at discrete time steps. Of particular importance for biological applications are networks in which the indegree for each variable is bounded by a fixed constant, as was stressed by Kauffman in his original papers. An important question is which conditions on the network topology can rule out exponentially long periodic orbits in the system. In this paper we consider cooperative systems, i.e. systems with positive feedback interconnections among all variables, which in a continuous setting guarantees a very stable dynamics. In Part I of this paper we presented a construction that shows that for an arbitrary constant 0<c<2 and sufficiently large n there exist n-dimensional Boolean cooperative networks in which both the indegree and outdegree of each for each variable is bounded by two (bi-quadratic networks) and which nevertheless contain periodic orbits of length at least c^n. In this part, we prove an inverse result showing that for sufficiently large n and for 0<c<2 sufficiently close to 2, any n-dimensional cooperative, bi-quadratic Boolean network with a cycle of length at least c^n must have a large proportion of variables with indegree 1. Such systems therefore share a structural similarity to the systems constructed in Part I.

q-bio.MN↗

arXiv2007

Large attractors in cooperative bi-quadratic Boolean networks. Part I

Boolean networks have been the object of much attention, especially since S. Kauffman proposed them in the 1960's as models for gene regulatory networks. These systems are characterized by being defined on a Boolean state space and by simultaneous updating at discrete time steps. Of particular importance for biological applications are networks in which the indegree for each variable is bounded by a fixed constant, as was stressed by Kauffman in his original papers. An important question is which conditions on the network topology can rule out exponentially long periodic orbits in the system. In this paper, we consider systems with positive feedback interconnections among all variables (known as cooperative systems), which in a continuous setting guarantees a very stable dynamics. We show that for an arbitrary constant 0<c<2 and sufficiently large n there exist n-dimensional cooperative Boolean networks in which both the indegree and outdegree of each variable is bounded by two, and which nevertheless contain periodic orbits of length at least c^n. In Part II of this paper we will prove an inverse result showing that any system with such a dynamic behavior must in a sense be similar to the example described.

q-bio.MN↗

arXiv2007

Efficiently computing Groebner bases of ideals of points

We present an algorithm for computing Groebner bases of vanishing ideals of points that is optimized for the case when the number of points in the associated variety is less than the number of indeterminates. The algorithm first identifies a set of essential variables, which reduces the time complexity with respect to the number of indeterminates, and then uses PLU decompositions to reduce the time complexity with respect to the number of points. This gives a theoretical upper bound for its time complexity that is an order of magnitude lower than the known one for the standard Buchberger-Moeller algorithm if the number of indeterminates is much larger than the number of points. Comparison of implementations of our algorithm and the standard Buchberger-Moeller algorithm in Macaulay 2 confirm the theoretically predicted speedup. This work is motivated by recent applications of Groebner bases to the problem of network reconstruction in molecular biology.

math.AC↗

arXiv2007

Analogues of the Smale and Hirsch Theorems for Cooperative Boolean and Other Discrete Systems

Discrete dynamical systems defined on the state space {0,1,...,p-1}^n have been used in multiple applications, most recently for the modeling of gene and protein networks. In this paper we study to what extent well-known theorems by Smale and Hirsch, which form part of the theory of (continuous) monotone dynamical systems, generalize or fail to do so in the discrete case. We show that that arbitrary m-dimensional systems cannot necessarily be embedded into n-dimensional cooperative systems for n=m+1, as in the Smale theorem for the continuous case, but we show that this is possible for n=m+2 as long as p is sufficiently large. We also prove that a natural discrete analogue of strong cooperativity implies nontrivial bounds on the lengths of periodic orbits and imposes a condition akin to Lyapunov stability on all attractors. Finally, we explore several natural candidates for definitions of irreducibility of a discrete system. While some of these notions imply the strong cooperativity of a given cooperative system and impose even tighter bounds on the lengths of periodic orbits than strong cooperativity alone, other plausible definitions allow the existence of exponentially long periodic orbits.

math.DS↗

arXiv2006

Computing Groebner bases of ideals of few points in high dimensions

A contemporary and exciting application of Groebner bases is their use in computational biology, particularly in the reverse engineering of gene regulatory networks from experimental data. In this setting, the data are typically limited to tens of points, while the number of genes or variables is potentially in the thousands. As such data sets vastly underdetermine the biological network, many models may fit the same data and reverse engineering programs often require the use of methods for choosing parsimonious models. Grobner bases have recently been employed as a selection tool for polynomial dynamical systems that are characterized by maps in a vector space over a finite field. While there are numerous existing algorithms to compute Groebner bases, to date none has been specifically designed to cope with large numbers of variables and few distinct data points. In this paper, we present an algorithm for computing Grobner bases of zero-dimensional ideals that is optimized for the case when the number m of points is much smaller than the number n of indeterminates. The algorithm identifies those variables that are essential, that is, in the support of the standard monomials associated to a polynomial ideal, and computes the relations in the Groebner bases in terms of these variables. When n is much larger than m, the complexity is dominated by nm^3. The algorithm has been implemented and tested in the computer algebra system Macaulay 2. We provide a comparison of its performance to the Buchberger-Moeller algorithm, as built into the system.

math.AC↗

arXiv2003

The number and probability of canalizing functions

Canalizing functions have important applications in physics and biology. For example, they represent a mechanism capable of stabilizing chaotic behavior in Boolean network models of discrete dynamical systems. When comparing the class of canalizing functions to other classes of functions with respect to their evolutionary plausibility as emergent control rules in genetic regulatory systems, it is informative to know the number of canalizing functions with a given number of input variables. This is also important in the context of using the class of canalizing functions as a constraint during the inference of genetic networks from gene expression data. To this end, we derive an exact formula for the number of canalizing Boolean functions of n variables. We also derive a formula for the probability that a random Boolean function is canalizing for any given bias p of taking the value 1. In addition, we consider the number and probability of Boolean functions that are canalizing for exactly k variables. Finally, we provide an algorithm for randomly generating canalizing functions with a given bias p and any number of variables, which is needed for Monte Carlo simulations of Boolean networks.

math-ph↗

arXiv2000

The automorphism tower problem revisited

It is well-known that the automorphism towers of infinite centreless groups of cardinality kappa terminate in less than (2^{kappa})^+ steps. But an easy counting argument shows that (2^{kappa})^+ is not the best possible bound. However, in this paper, we will show that it is impossible to find an explicit better bound using ZFC.

math.LO↗

arXiv1998

Strongly almost disjoint sets and weakly uniform bases

A combinatorial principle CECA is formulated and its equivalence with GCH+ certain weakenings of Box_lambda for singular lambda is proved. CECA is used to show that certain ``almost point- < tau'' families can be refined to point- < tau families by removing a small set from each member of the family. This theorem in turn is used to show the consistency of ``every first countable T_1-space with a weakly uniform base has a point-countable base.''

math.LO↗

arXiv1995

The combinatorics of open covers (II)

We continue to investigate various diagonalization properties for sequences of open covers of separable metrizable spaces introduced in Part I. These properties generalize classical ones of Rothberger, Menger, Hurewicz, and Gerlits-Nagy. In particular, we show that most of the properties introduced in Part I are indeed distinct. We characterize two of the new properties by showing that they are equivalent to saying all finite powers have one of the classical properties above (Hurewicz property in one case and in the Menger property in other). We consider for each property the smallest cardinality of metric space which fails to have that property. In each case this cardinal turns out to equal another well-known cardinal less than the continuum. We also disprove (in ZFC) a conjecture of Hurewicz which is analogous to the Borel conjecture. Finally, we answer several questions from Part I concerning partition properties of covers.

math.LO↗

arXiv1992

Covering games and the Banach-Mazur game: k-tactics

Given a free ideal J of subsets of a set X, we consider games where player ONE plays an increasing sequence of elements of the sigma completion of J, and TWO tries to cover the union of this sequence by playing one set at a time from J. We describe various conditions under which player TWO has has a winning strategy that uses only information about the most recent k moves of ONE, and apply some of these results to the Banach-Mazur game.

math.LO↗