Exploratory Behavior, Trap Models and Glass Transitions

A random walk is performed on a disordered landscape composed of $N$ sites randomly and uniformly distributed inside a $d$-dimensional hypercube. The walker hops from one site to another with probability proportional to $\exp [- \beta E(D)]$, where $\beta = 1/T$ is the inverse of a formal temperature and $E(D)$ is an arbitrary cost function which depends on the hop distance $D$. Analytic results indicate that, if $E(D) = D^{d}$ and $N \to \infty$, there exists a glass transition at $\beta_d = \pi^{d/2}/\Gamma(d/2 + 1)$. Below $T_d$, the average trapping time diverges and the system falls into an out-of-equilibrium regime with aging phenomena. A L\'evy flight scenario and applications to exploratory behavior are considered.Comment: 4 pages, 1 figure, new versio

Escaping from cycles through a glass transition

A random walk is performed over a disordered media composed of $N$ sites random and uniformly distributed inside a $d$-dimensional hypercube. The walker cannot remain in the same site and hops to one of its $n$ neighboring sites with a transition probability that depends on the distance $D$ between sites according to a cost function $E(D)$. The stochasticity level is parametrized by a formal temperature $T$. In the case $T = 0$, the walk is deterministic and ergodicity is broken: the phase space is divided in a ${\cal O}(N)$ number of attractor basins of two-cycles that trap the walker. For $d = 1$, analytic results indicate the existence of a glass transition at $T_1 = 1/2$ as $N \to \infty$. Below $T_1$, the average trapping time in two-cycles diverges and out-of-equilibrium behavior appears. Similar glass transitions occur in higher dimensions choosing a proper cost function. We also present some results for the statistics of distances for Poisson spatial point processes.Comment: 11 pages, 4 figure

Extremal Optimization: Methods Derived from Co-Evolution

We describe a general-purpose method for finding high-quality solutions to hard optimization problems, inspired by self-organized critical models of co-evolution such as the Bak-Sneppen model. The method, called Extremal Optimization, successively eliminates extremely undesirable components of sub-optimal solutions, rather than ''breeding'' better components. In contrast to Genetic Algorithms which operate on an entire ''gene-pool'' of possible solutions, Extremal Optimization improves on a single candidate solution by treating each of its components as species co-evolving according to Darwinian principles. Unlike Simulated Annealing, its non-equilibrium approach effects an algorithm requiring few parameters to tune. With only one adjustable parameter, its performance proves competitive with, and often superior to, more elaborate stochastic optimization procedures. We demonstrate it here on two classic hard optimization problems: graph partitioning and the traveling salesman problem

Coloring geographical threshold graphs

We propose a coloring algorithm for sparse random graphs generated by the geographical threshold graph (GTG) model, a generalization of random geometric graphs (RGG). In a GTG, nodes are distributed in a Euclidean space, and edges are assigned according to a threshold function involving the distance between nodes as well as randomly chosen node weights. The motivation for analyzing this model is that many real networks (e.g., wireless networks, the Internet, etc.) need to be studied by using a “richer ” stochastic model (which in this case includes both a distance between nodes and weights on the nodes). Here, we analyze the GTG coloring algorithm together with the graph’s clique number, showing formally that in spite of the differences in structure between GTG and RGG, the asymptotic behavior of the ln n chromatic number is identical: χ = (1 + o(1)). Finally, ln ln n we consider the leading corrections to this expression, again using the coloring algorithm and clique number to provide bounds on the chromatic number. We show that the gap between the lower and upper bound is within C ln n/(ln ln n) 2, and specify the constant C.