Jamming Model for the Extremal Optimization Heuristic

Extremal Optimization, a recently introduced meta-heuristic for hard optimization problems, is analyzed on a simple model of jamming. The model is motivated first by the problem of finding lowest energy configurations for a disordered spin system on a fixed-valence graph. The numerical results for the spin system exhibit the same phenomena found in all earlier studies of extremal optimization, and our analytical results for the model reproduce many of these features.Comment: 9 pages, RevTex4, 7 ps-figures included, as to appear in J. Phys. A, related papers available at http://www.physics.emory.edu/faculty/boettcher

Optimal routing on complex networks

We present a novel heuristic algorithm for routing optimization on complex networks. Previously proposed routing optimization algorithms aim at avoiding or reducing link overload. Our algorithm balances traffic on a network by minimizing the maximum node betweenness with as little path lengthening as possible, thus being useful in cases when networks are jamming due to queuing overload. By using the resulting routing table, a network can sustain significantly higher traffic without jamming than in the case of traditional shortest path routing.Comment: 4 pages, 5 figure

Optimizing at the Ergodic Edge

Using a simple, annealed model, some of the key features of the recently introduced extremal optimization heuristic are demonstrated. In particular, it is shown that the dynamics of local search possesses a generic critical point under the variation of its sole parameter, separating phases of too greedy (non-ergodic, jammed) and too random (ergodic) exploration. Comparison of various local search methods within this model suggests that the existence of the critical point is essential for the optimal performance of the heuristic.Comment: RevTex4, 17 pages, 3 ps-figures incl., for related information, see http://www.physics.emory.edu/faculty/boettcher/publications.htm

Transport optimization on complex networks

We present a comparative study of the application of a recently introduced heuristic algorithm to the optimization of transport on three major types of complex networks. The algorithm balances network traffic iteratively by minimizing the maximum node betweenness with as little path lengthening as possible. We show that by using this optimal routing, a network can sustain significantly higher traffic without jamming than in the case of shortest path routing. A formula is proved that allows quick computation of the average number of hops along the path and of the average travel times once the betweennesses of the nodes are computed. Using this formula, we show that routing optimization preserves the small-world character exhibited by networks under shortest path routing, and that it significantly reduces the average travel time on congested networks with only a negligible increase in the average travel time at low loads. Finally, we study the correlation between the weights of the links in the case of optimal routing and the betweennesses of the nodes connected by them.Comment: 19 pages, 7 figure

Algorithmic design of self-assembling structures

We study inverse statistical mechanics: how can one design a potential function so as to produce a specified ground state? In this paper, we show that unexpectedly simple potential functions suffice for certain symmetrical configurations, and we apply techniques from coding and information theory to provide mathematical proof that the ground state has been achieved. These potential functions are required to be decreasing and convex, which rules out the use of potential wells. Furthermore, we give an algorithm for constructing a potential function with a desired ground state.Comment: 8 pages, 5 figure

Obtaining Stiffness Exponents from Bond-diluted Lattice Spin Glasses

Recently, a method has been proposed to obtain accurate predictions for low-temperature properties of lattice spin glasses that is practical even above the upper critical dimension, $d_c=6$. This method is based on the observation that bond-dilution enables the numerical treatment of larger lattices, and that the subsequent combination of such data at various bond densities into a finite-size scaling Ansatz produces more robust scaling behavior. In the present study we test the potential of such a procedure, in particular, to obtain the stiffness exponent for the hierarchical Migdal-Kadanoff lattice. Critical exponents for this model are known with great accuracy and any simulations can be executed to very large lattice sizes at almost any bond density, effecting a insightful comparison that highlights the advantages -- as well as the weaknesses -- of this method. These insights are applied to the Edwards-Anderson model in $d=3$ with Gaussian bonds.Comment: corrected version, 10 pages, RevTex4, 12 ps-figures included; related papers available a http://www.physics.emory.edu/faculty/boettcher

Traffic Jams: Cluster Formation in Low-Dimensional Cellular Automata Models for Highway and City Traffic

Cellular automata (CA) models are quite popular in the field of traffic flow. They allow an effective implementation of real-time traffic computer-simulations. Therefore, various approaches based on CA models have been suggested in recent years. The first part of this thesis focuses on the so-called VDR (velocity-dependent randomization) model which is a modified version of the well known Nagel-Schreckenberg (NaSch) CA model. This choice is motivated by the fact that wide phase separated jams occur in the model. On the basis of random walk theory an analytical approach to the dynamics of these separated jam clusters is given. The predictions are in good agreement with the results of computer simulations and provide a deeper insight into the dynamics of wide jams which seem to be generic for CA approaches and are therefore of special interest. Furthermore, the impact of a localized defect in a periodic system is analyzed in the VDR model. It turns out that depending on the magnitude of the defect stop-and-go traffic can occur which can not be found in the VDR model without lattice defects. Finally, the VDR model is studied with open boundaries. The phase diagrams, obtained by Monte-Carlo simulations, reveal two jam phases with a stripped microscopic structure and for finite systems the existence of a new high-flow phase is shown. The second part of this thesis concentrates on CA models for city traffic with the focus on the Chowdhury-Schadschneider (ChSch) model. In the context of jam clusters the model reveals interesting features since two factors exert influence on the jamming behavior. On the one hand, jams are induced at crossings due to the traffic lights, i.e., cars are forced to stop at a ``red light', and, on the other hand, the dynamics of such induced jams is governed by the NaSch model rules. One part of the investigations covers global (fixed) traffic light strategies. These are found to lead to strong oscillations in the global flow except for the case of randomly switching lights. Furthermore, the impact of adaptive (local) traffic light control is analyzed. It is found that the autonomous strategies can nearly match the global optimum of the ChSch model. In order to provide a more realistic vehicle distribution, the ChSch model is enhanced by a stochastic turning of vehicles and by inhomogeneous densities. Here, the autonomous strategies can outperform the global ones in some cases

Entropy landscape and non-Gibbs solutions in constraint satisfaction problems

We study the entropy landscape of solutions for the bicoloring problem in random graphs, a representative difficult constraint satisfaction problem. Our goal is to classify which type of clusters of solutions are addressed by different algorithms. In the first part of the study we use the cavity method to obtain the number of clusters with a given internal entropy and determine the phase diagram of the problem, e.g. dynamical, rigidity and SAT-UNSAT transitions. In the second part of the paper we analyze different algorithms and locate their behavior in the entropy landscape of the problem. For instance we show that a smoothed version of a decimation strategy based on Belief Propagation is able to find solutions belonging to sub-dominant clusters even beyond the so called rigidity transition where the thermodynamically relevant clusters become frozen. These non-equilibrium solutions belong to the most probable unfrozen clusters.Comment: 38 pages, 10 figure