Kevin E. Bassler

Optimal transport on complex networks.

We present a heuristic algorithm for the optimization of transport on complex networks. Previously proposed network transport 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 node congestion. By using the resulting routing, a network can sustain significantly higher traffic without jamming than in the case of shortest path routing.

Efficient and Exact Sampling of Simple Graphs with Given Arbitrary Degree Sequence

Uniform sampling from graphical realizations of a given degree sequence is a fundamental component in simulation-based measurements of network observables, with applications ranging from epidemics, through social networks to Internet modeling. Existing graph sampling methods are either link-swap based (Markov-Chain Monte Carlo algorithms) or stub-matching based (the Configuration Model). Both types are ill-controlled, with typically unknown mixing times for link-swap methods and uncontrolled rejections for the Configuration Model. Here we propose an efficient, polynomial time algorithm that generates statistically independent graph samples with a given, arbitrary, degree sequence. The algorithm provides a weight associated with each sample, allowing the observable to be measured either uniformly over the graph ensemble, or, alternatively, with a desired distribution. Unlike other algorithms, this method always produces a sample, without back-tracking or rejections. Using a central limit theorem-based reasoning, we argue, that for large , and for degree sequences admitting many realizations, the sample weights are expected to have a lognormal distribution. As examples, we apply our algorithm to generate networks with degree sequences drawn from power-law distributions and from binomial distributions.

Self-organized networks of competing boolean agents

A model of Boolean agents competing in a market is presented where each agent bases his action on information obtained from a small group of other agents. The agents play a competitive game that rewards those in the minority. After a long time interval, the poorest players strategy is changed randomly, and the process is repeated. Eventually the network evolves to a stationary but intermittent state where random mutation of the worst strategy can change the behavior of the entire network, often causing a switch in the dynamics between attractors of vastly different lengths.

Markov Processes, Hurst Exponents, and Nonlinear Diffusion Equations with application to finance

We show by explicit closed form calculations that a Hurst exponent H≠12 does not necessarily imply long time correlations like those found in fractional Brownian motion (fBm). We construct a large set of scaling solutions of Fokker–Planck partial differential equations (pdes) where H≠12. Thus Markov processes, which by construction have no long time correlations, can have H≠12. If a Markov process scales with Hurst exponent H≠12 then it simply means that the process has nonstationary increments. For the scaling solutions, we show how to reduce the calculation of the probability density to a single integration once the diffusion coefficient D(x,t) is specified. As an example, we generate a class of student-t-like densities from the class of quadratic diffusion coefficients. Notably, the Tsallis density is one member of that large class. The Tsallis density is usually thought to result from a nonlinear diffusion equation, but instead we explicitly show that it follows from a Markov process generated by a linear Fokker–Planck equation, and therefore from a corresponding Langevin equation. Having a Tsallis density with H≠12 therefore does not imply dynamics with correlated signals, e.g., like those of fBm. A short review of the requirements for fBm is given for clarity, and we explain why the usual simple argument that H≠12 implies correlations fails for Markov processes with scaling solutions. Finally, we discuss the question of scaling of the full Green function g(x,t;x′,t′) of the Fokker–Planck pdes.

Improved community structure detection using a modified fine-tuning strategy

The community structure of a complex network can be determined by finding the partitioning of its nodes that maximizes modularity. Many of the proposed algorithms for doing this work by recursively bisecting the network. We show that this unduely constrains their results, leading to a bias in the size of the communities they find and limiting their effectiveness. To solve this problem, we propose adding a step, which is a modification of the Kernighan-Lin algorithm, to the existing algorithms. This additional step does not increase the order of their computational complexity. We show that, if this step is combined with a commonly used method, the identified constraint and resulting bias are removed, and its ability to find the optimal partitioning is improved. The effectiveness of this combined algorithm is also demonstrated by using it on real-world example networks. For a number of these examples, it achieves the best results of any known algorithm.

All scale-free networks are sparse.

We study the realizability of scale-free networks with a given degree sequence, showing that the fraction of realizable sequences undergoes two first-order transitions at the values 0 and 2 of the power-law exponent. We substantiate this finding by analytical reasoning and by a numerical method, proposed here, based on extreme value arguments, which can be applied to any given degree distribution. Our results reveal a fundamental reason why large scale-free networks without constraints on minimum and maximum degree must be sparse.

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 and tested with numerical simulation 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.

Congestion-gradient driven transport on complex networks

We present a study of transport on complex networks with routing based on local information. Particles hop from one node of the network to another according to a set of routing rules with different degrees of congestion awareness, ranging from random diffusion to rigid congestion-gradient driven flow. Each node can be either source or destination for particles and all nodes have the same routing capacity, which are features of ad hoc wireless networks. It is shown that the transport capacity increases when a small amount of congestion awareness is present in the routing rules, and that it then decreases as the routing rules become too rigid when the flow becomes strictly congestion-gradient driven. Therefore, an optimum value of the congestion awareness exists in the routing rules. It is also shown that, in the limit of a large number of nodes, networks using routing based on local information jam at any nonzero load. Finally, we study the correlation between congestion at node level and a betweenness centrality measure.

Constructing and sampling directed graphs with given degree sequences

The interactions between the components of complex networks are often directed. Proper modeling of such systems frequently requires the construction of ensembles of digraphs with a given sequence of in- and out-degrees. As the number of simple labeled graphs with a given degree sequence is typically very large even for short sequences, sampling methods are needed for statistical studies. Currently, there are two main classes of methods that generate samples. One of the existing methods first generates a restricted class of graphs and then uses a Markov chain Monte-Carlo algorithm based on edge swaps to generate other realizations. As the mixing time of this process is still unknown, the independence of the samples is not well controlled. The other class of methods is based on the configuration model that may lead to unacceptably many sample rejections due to self-loops and multiple edges. Here we present an algorithm that can directly construct all possible realizations of a given bi-degree sequence by simple digraphs. Our method is rejection-free, guarantees the independence of the constructed samples and provides their weight. The weights can then be used to compute statistical averages of network observables as if they were obtained from uniformly distributed sampling or from any other chosen distribution.

Quantifying randomness in real networks.

Represented as graphs, real networks are intricate combinations of order and disorder. Fixing some of the structural properties of network models to their values observed in real networks, many other properties appear as statistical consequences of these fixed observables, plus randomness in other respects. Here we employ the dk-series, a complete set of basic characteristics of the network structure, to study the statistical dependencies between different network properties. We consider six real networks—the Internet, US airport network, human protein interactions, technosocial web of trust, English word network, and an fMRI map of the human brain—and find that many important local and global structural properties of these networks are closely reproduced by dk-random graphs whose degree distributions, degree correlations and clustering are as in the corresponding real network. We discuss important conceptual, methodological, and practical implications of this evaluation of network randomness, and release software to generate dk-random graphs.Represented as graphs, real networks are intricate combinations of order and disorder. Fixing some of the structural properties of network models to their values observed in real networks, many other properties appear as statistical consequences of these fixed observables, plus randomness in other respects. Here we employ the