arXiv subjects

Gerald Paul

Publications and source records attributed to Gerald Paul.

At least 19 recordsLinked to original sources

arXiv2012

A GPU implementation of the Simulated Annealing Heuristic for the Quadratic Assignment Problem

The quadratic assignment problem (QAP) is one of the most difficult combinatorial optimization problems. An effective heuristic for obtaining approximate solutions to the QAP is simulated annealing (SA). Here we describe an SA implementation for the QAP which runs on a graphics processing unit (GPU). GPUs are composed of low cost commodity graphics chips which in combination provide a powerful platform for general purpose parallel computing. For SA runs with large numbers of iterations, we find performance 50-100 times better than that of a recent non-parallel but very efficient implementation of SA for the QAP

cs.DC↗

arXiv2011

An efficient implementation of the simulated annealing heuristic for the quadratic assignment problem

The quadratic assignment problem (QAP) is one of the most difficult combinatorial optimization problems. One of the most powerful and commonly used heuristics to obtain approximations to the optimal solution of the QAP is simulated annealing (SA). We present an efficient implementation of the SA heuristic which performs more than 100 times faster then existing implementations for large problem sizes and a large number of SA iterations.

cs.NE↗

arXiv2011

The Random Quadratic Assignment Problem

Optimal assignment of classes to classrooms \cite{dickey}, design of DNA microarrays \cite{carvalho}, cross species gene analysis \cite{kolar}, creation of hospital layouts cite{elshafei}, and assignment of components to locations on circuit boards \cite{steinberg} are a few of the many problems which have been formulated as a quadratic assignment problem (QAP). Originally formulated in 1957, the QAP is one of the most difficult of all combinatorial optimization problems. Here, we use statistical mechanical methods to study the asymptotic behavior of problems in which the entries of at least one of the two matrices that specify the problem are chosen from a random distribution $P$. Surprisingly, this case has not been studied before using statistical methods despite the fact that the QAP was first proposed over 50 years ago \cite{Koopmans}. We find simple forms for $C_{\rm min}$ and $C_{\rm max}$, the costs of the minimal and maximum solutions respectively. Notable features of our results are the symmetry of the results for $C_{\rm min}$ and $C_{\rm max}$ and the dependence on $P$ only through its mean and standard deviation, independent of the details of $P$. After the asymptotic cost is determined for a given QAP problem, one can straightforwardly calculate the asymptotic cost of a QAP problem specified with a different random distribution $P$.

cond-mat.stat-mech↗

arXiv2010

Comparative Performance of Tabu Search and Simulated Annealing Heuristics for the Quadratic Assignment Problem

For almost two decades the question of whether tabu search (TS) or simulated annealing (SA) performs better for the quadratic assignment problem has been unresolved. To answer this question satisfactorily, we compare performance at various values of targeted solution quality, running each heuristic at its optimal number of iterations for each target. We find that for a number of varied problem instances, SA performs better for higher quality targets while TS performs better for lower quality targets.

cs.DS↗

arXiv2010

An Efficient Implementation of the Robust Tabu Search Heuristic for Sparse Quadratic Assignment Problems

We propose and develop an efficient implementation of the robust tabu search heuristic for sparse quadratic assignment problems. The traditional implementation of the heuristic applicable to all quadratic assignment problems is of O(N^2) complexity per iteration for problems of size N. Using multiple priority queues to determine the next best move instead of scanning all possible moves, and using adjacency lists to minimize the operations needed to determine the cost of moves, we reduce the asymptotic complexity per iteration to O(N log N ). For practical sized problems, the complexity is O(N).

cs.DS↗

arXiv2009

A Variable Depth Sequential Search Heuristic for the Quadratic Assignment Problem

We develop a variable depth search heuristic for the quadratic assignment problem. The heuristic is based on sequential changes in assignments analogous to the Lin-Kernighan sequential edge moves for the traveling salesman problem. We treat unstructured problem instances of sizes 60 to 400. When the heuristic is used in conjunction with robust tabu search, we measure performance improvements of up to a factor of 15 compared to the use of robust tabu alone. The performance improvement increases as the problem size increases.

cs.DS↗

arXiv2009

Catastrophic cascade of failures in interdependent networks

Many systems, ranging from engineering to medical to societal, can only be properly characterized by multiple interdependent networks whose normal functioning depends on one another. Failure of a fraction of nodes in one network may lead to a failure in another network. This in turn may cause further malfunction of additional nodes in the first network and so on. Such a cascade of failures, triggered by a failure of a small faction of nodes in only one network, may lead to the complete fragmentation of all networks. We introduce a model and an analytical framework for studying interdependent networks. We obtain interesting and surprising results that should significantly effect the design of robust real-world networks. For two interdependent Erdos-Renyi (ER) networks, we find that the critical average degree below which both networks collapse is =2.445, compared to =1 for a single ER network. Furthermore, while for a single network a broader degree distribution of the network nodes results in higher robustness to random failure, for interdependent networks, the broader the distribution is, the more vulnerable the networks become to random failure.

cond-mat.dis-nn↗

arXiv2007

Betweenness Centrality of Fractal and Non-Fractal Scale-Free Model Networks and Tests on Real Networks

We study the betweenness centrality of fractal and non-fractal scale-free network models as well as real networks. We show that the correlation between degree and betweenness centrality $C$ of nodes is much weaker in fractal network models compared to non-fractal models. We also show that nodes of both fractal and non-fractal scale-free networks have power law betweenness centrality distribution $P(C)\sim C^{-δ}$. We find that for non-fractal scale-free networks $δ= 2$, and for fractal scale-free networks $δ= 2-1/d_{B}$, where $d_{B}$ is the dimension of the fractal network. We support these results by explicit calculations on four real networks: pharmaceutical firms (N=6776), yeast (N=1458), WWW (N=2526), and a sample of Internet network at AS level (N=20566), where $N$ is the number of nodes in the largest connected component of a network. We also study the crossover phenomenon from fractal to non-fractal networks upon adding random edges to a fractal network. We show that the crossover length $\ell^{*}$, separating fractal and non-fractal regimes, scales with dimension $d_{B}$ of the network as $p^{-1/d_{B}}$, where $p$ is the density of random edges added to the network. We find that the correlation between degree and betweenness centrality increases with $p$.

physics.soc-ph↗

arXiv2007

Graph Partitioning Induced Phase Transitions

We study the percolation properties of graph partitioning on random regular graphs with N vertices of degree $k$. Optimal graph partitioning is directly related to optimal attack and immunization of complex networks. We find that for any partitioning process (even if non-optimal) that partitions the graph into equal sized connected components (clusters), the system undergoes a percolation phase transition at $f=f_c=1-2/k$ where $f$ is the fraction of edges removed to partition the graph. For optimal partitioning, at the percolation threshold, we find $S \sim N^{0.4}$ where $S$ is the size of the clusters and $\ell\sim N^{0.25}$ where $\ell$ is their diameter. Additionally, we find that $S$ undergoes multiple non-percolation transitions for $f<f_c$.

cond-mat.stat-mech↗

arXiv2006

Percolation theory applied to measures of fragmentation in social networks

We apply percolation theory to a recently proposed measure of fragmentation $F$ for social networks. The measure $F$ is defined as the ratio between the number of pairs of nodes that are not connected in the fragmented network after removing a fraction $q$ of nodes and the total number of pairs in the original fully connected network. We compare $F$ with the traditional measure used in percolation theory, $P_{\infty}$, the fraction of nodes in the largest cluster relative to the total number of nodes. Using both analytical and numerical methods from percolation, we study Erdős-Rényi (ER) and scale-free (SF) networks under various types of node removal strategies. The removal strategies are: random removal, high degree removal and high betweenness centrality removal. We find that for a network obtained after removal (all strategies) of a fraction $q$ of nodes above percolation threshold, $P_{\infty}\approx (1-F)^{1/2}$. For fixed $P_{\infty}$ and close to percolation threshold ($q=q_c$), we show that $1-F$ better reflects the actual fragmentation. Close to $q_c$, for a given $P_{\infty}$, $1-F$ has a broad distribution and it is thus possible to improve the fragmentation of the network. We also study and compare the fragmentation measure $F$ and the percolation measure $P_{\infty}$ for a real social network of workplaces linked by the households of the employees and find similar results.

cond-mat.dis-nn↗

arXiv2006

A Complexity O(1) Priority Queue for Event Driven Molecular Dynamics Simulations

We propose and implement a priority queue suitable for use in event driven molecular dynamics simulations. All operations on the queue take on average O(1) time per collision. In comparison, previously studied queues for event driven molecular dynamics simulations require O(log $N$) time per collision for systems of $N$ particles.

physics.comp-ph↗

arXiv2005

Optimization of Network Robustness to Random Breakdowns

We study network configurations that provide optimal robustness to random breakdowns for networks with a given number of nodes $N$ and a given cost--which we take as the average number of connections per node $\kav$. We find that the network design that maximizes $f_c$, the fraction of nodes that are randomly removed before global connectivity is lost, consists of $q=[(\kav-1)/\sqrt\kav]\sqrt N$ high degree nodes (``hubs'') of degree $\sqrt{\kav N}$ and $N-q$ nodes of degree 1. Also, we show that $1-f_c$ approaches 0 as $1/\sqrt N$--faster than any other network configuration including scale-free networks. We offer a simple heuristic argument to explain our results.

cond-mat.stat-mech↗

arXiv2005

Resilience of Complex Networks to Random Breakdown

Using Monte Carlo simulations we calculate $f_c$, the fraction of nodes which are randomly removed before global connectivity is lost, for networks with scale-free and bimodal degree distributions. Our results differ with the results predicted by an equation for $f_c$ proposed by Cohen, et al. We discuss the reasons for this disagreement and clarify the domain for which the proposed equation is valid.

cond-mat.stat-mech↗

arXiv2002

Fractal Dimension of 3-Blocks in 4d, 5d, and 6d Percolation Systems

Using Monte Carlo simulations we study the distributions of the 3-block mass $N_3$ in 4d, 5d, and 6d percolation systems. Because the probability of creating large 3-blocks in these dimensions is very small, we use a ``go with the winners'' method of statistical enhancement to simulate configurations having probability as small as $10^{-30}$. In earlier work, the fractal dimensions of 3-blocks, $d_3$, in 2d and 3d were found to be $1.20\pm 0.1$ and $1.15\pm 0.1$, respectively, consistent with the possibility that the fractal dimension might be the same in all dimensions. We find that the fractal dimension of 3-blocks decreases rapidly in higher dimensions, and estimate $d_3=0.7\pm 0.2$ (4d) and $0.5\pm 0.2$ (5d). At the upper critical dimension of percolation, $d_c=6$, our simulations are consistent with $d_3=0$ with logarithmic corrections to power-law scaling.

cond-mat.dis-nn↗

arXiv2002

Discrepancy between Monte-Carlo Results and Analytic Values for the Average Excluded Volume of Rectangular Prisms

We perform Monte Carlo simulations to determine the average excluded volume of randomly oriented rectangular prisms, randomly oriented ellipsoids and randomly oriented capped cylinders in 3-D. There is agreement between the analytically obtained and the results of simulations for randomly oriented ellipsoids and randomly oriented capped cylinders. However, we find that the for randomly oriented prisms obtained from the simulations differs from the analytically obtained results. In particular, for cubes, the percentage difference is 3.92, far exceeding the bounds of statistical error in our simulation.{\bf Added in Revision 2: We recently found the cause of the discrepancy between the simulation result and the analytic value of the excluded volume to be the effect of an error in our simulation code. Upon rectification of the simulation code, the simulation yields $ 11.00 \pm 0.002 $ as the excluded volume of a pair of randomly oriented cubes of unit volume. The simulation also yields results as predicted by the analytic formula for all other cases of rectangular prisms that we study.}

cond-mat.dis-nn↗

arXiv2002

The Approximate Invariance of the Average Number of Connections for the Continuum Percolation of Squares at Criticality

We perform Monte Carlo simulations to determine the average excluded area $ $ of randomly oriented squares, randomly oriented widthless sticks and aligned squares in two dimensions. We find significant differences between our results for randomly oriented squares and previous analytical results for the same. The sources of these differences are explained. Using our results for $ $ and Monte Carlo simulation results for the percolation threshold, we estimate the mean number of connections per object $B_c$ at the percolation threshold for squares in 2-D. We study systems of squares that are allowed random orientations within a specified angular interval. Our simulations show that the variation in $B_c$ is within 1.6% when the angular interval is varied from 0 to $π/2$.

cond-mat.dis-nn↗

arXiv2002

The continuum percolation threshold for interpenetrating squares and cubes

Monte Carlo simulations are performed to determine the critical percolation threshold for interpenetrating square objects in two dimensions and cubic objects in three dimensions. Simulations are performed for two cases: (i) objects whose edges are aligned parallel to one another and (ii) randomly oriented objects. For squares whose edges are aligned, the critical area fraction at the percolation threshold phi_c=0.6666 +/- 0.0004, while for randomly oriented squares phi_c=0.6254 +/- 0.0002, 6% smaller. For cubes whose edges are aligned, the critical volume fraction at the percolation threshold phi_c=0.2773 +/- 0.0002, while for randomly oriented cubes phi_c=0.2236 +/- 0.0002, 24% smaller.

cond-mat.stat-mech↗

arXiv2002

Fractal Behavior of the Shortest Path Between Two Lines in Percolation Systems

Using Monte-Carlo simulations, we determine the scaling form for the probability distribution of the shortest path, $\ell$, between two lines in a 3-dimensional percolation system at criticality; the two lines can have arbitrary positions, orientations and lengths. We find that the probability distributions can exhibit up to four distinct power law regimes (separated by cross-over regimes) with exponents depending on the relative orientations of the lines. We explain this rich fractal behavior with scaling arguments.

cond-mat.stat-mech↗