Scaling in small-world resistor networks

Korniss, G.; Hastings, Matthew B.; Bassler, Kevin E.; Berryman, Matthew J.; Kozma, Balázs; Abbott, Derek

doi:10.1016/j.physleta.2005.09.081

Cited by 37 publications

(69 citation statements)

References 67 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Various conditions have been proposed to quantify this trade-off for sparse graphs, both in theoretical studies as well as in power network applications. The coupling is typically quantified by the algebraic connectivity λ 2 (L) (Wu and Kumagai, 1980;Pecora and Carroll, 1998;Nishikawa et al, 2003;Jadbabaie et al, 2004;Restrepo et al, 2005;Boccaletti et al, 2006;Arenas et al, 2008;Dörfler and Bullo, 2012b;Motter et al, 2013), the weighted nodal degree deg i = n j=1 a ij (Wu and Kumagai, 1982;Korniss et al, 2006;Gómez-Gardeñes et al, 2007;Buzna et al, 2009;Bullo, 2012b, 2013a;Skardal et al, 2013), or various metrics related to the notion of effective resistance (Wu and Kumagai, 1982;Korniss et al, 2006;Dörfler and Bullo, 2013a). The frequency dissimilarity is quantified either by absolute norms ω p or by incremental norms 11 B T ω p , for p ∈ N. Here, we specifically consider the three incremental norms:…”

Section: Survey Of Synchronization Metrics and Conditionsmentioning

confidence: 99%

Synchronization in complex networks of phase oscillators: A survey

2014

View full text Add to dashboard Cite

The emergence of synchronization in a network of coupled oscillators is a fascinating subject of multidisciplinary research. This survey reviews the vast literature on the theory and the applications of complex oscillator networks. We focus on phase oscillator models that are widespread in real-world synchronization phenomena, that generalize the celebrated Kuramoto model, and that feature a rich phenomenology. We review the history and the countless applications of this model throughout science and engineering. We justify the importance of the widespread coupled oscillator model as a locally canonical model and describe some selected applications relevant to control scientists, including vehicle coordination, electric power networks, and clock synchronization. We introduce the reader to several synchronization notions and performance estimates. We propose analysis approaches to phase and frequency synchronization, phase balancing, pattern formation, and partial synchronization. We present the sharpest known results about synchronization in networks of homogeneous and heterogeneous oscillators, with complete or sparse interconnection topologies, and in finite-dimensional and infinite-dimensional settings. We conclude by summarizing the limitations of existing analysis methods and by highlighting some directions for future research.

show abstract

Section: Survey Of Synchronization Metrics and Conditionsmentioning

confidence: 99%

Synchronization in complex networks of phase oscillators: A survey

2014

View full text Add to dashboard Cite

show abstract

“…Although it may appear somewhat simplistic (and, indeed prototypical), such problems are motivated by the dynamics and fluctuations in task completion landscapes in causally-constrained queuing networks [32], with applications in manufacturing supply chains, e-commerce-based services facilitated by interconnected servers [33], and certain distributed-computing schemes on computer networks [9,10,11,12,13]. This simplified problem is the Edwards-Wilkinson (EW) process [34] on the respective network [35,36,37,38,39].…”

Section: Introductionmentioning

confidence: 99%

“…Associating the weight/strength of each link with its cost, we ask what is the optimal allocation of the weights (in terms of β) in strongly heterogeneous networks, with a fixed total cost, in order to maximize synchronization in a noisy environment. For the EW process on any network, the natural observable is the width or spread of the synchronization landscape [35,36,37,38,39]. Then the task becomes minimizing the width as a function of β subject to a (cost) constraint.…”

Section: Introductionmentioning

confidence: 99%

Synchronization in weighted uncorrelated complex networks in a noisy environment: Optimization and connections with transport efficiency

Korniss

2007

Phys. Rev. E

Self Cite

111

View full text Add to dashboard Cite

Abstract. Motivated by synchronization problems in noisy environments, we study the Edwards-Wilkinson process on weighted uncorrelated scale-free networks. We consider a specific form of the weights, where the strength (and the associated cost) of a link is proportional to (k i k j ) β with k i and k j being the degrees of the nodes connected by the link. Subject to the constraint that the total network cost is fixed, we find that in the mean-field approximation on uncorrelated scale-free graphs, synchronization is optimal at β * =−1. Numerical results, based on exact numerical diagonalization of the corresponding network Laplacian, confirm the mean-field results, with small corrections to the optimal value of β * . Employing our recent connections between the Edwards-Wilkinson process and resistor networks, and some well-known connections between random walks and resistor networks, we also pursue a naturally related problem of optimizing performance in queue-limited communication networks utilizing local weighted routing schemes.

show abstract

“…The quantities to be transported can either be of a material nature such as power or goods, or of a non-material nature such as information packets, which are transported on the Internet, or influence, which is transported on social networks [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]. Optimization of network transport is thus an important problem for a variety of fields in science and technology.…”

Section: Introductionmentioning

confidence: 99%

Transport optimization on complex networks

Danila

Marsh

et al. 2007

Chaos: An Interdisciplinary Journal of Nonlinear Science

101

View full text Add to dashboard Cite

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.Keywords: complex networks, scaling laws, transport * dbogdan@mail.uh.edu † marshj@ainfosec.com ‡ bassler@uh.edu 1 One of the most important problems in the study of complex networks is how to best route transport on the networks. This problem is important because transport is the main function of many natural and human-made networks. Often, the transport routes used on networks are the so-called shortest-path routes, which are the routes with the minimum number of hops between any two nodes. However, this approach, which is currently used to route transport of information packets on the Internet, typically leads to congestion and eventually jamming of highly connected nodes of the networks called hubs. For this reason and in light of recent research, interest has developed in finding the routing rules that allow a given network to bear the maximum possible traffic. Specifically, the problem can be stated as follows. Given a complex network and a set of processing power and traffic demand constraints for its nodes, find the set of routing rules which allow the network to bear the highest possible amount of traffic without jamming. This problem is known to be NP -hard, meaning that the time required for the computation of an exact solution increases with the number of nodes faster than any polynomial. In this paper we argue that a heuristic transport routing optimization algorithm recently published by us achieves near-optimal transport routing in polynomial time and show this to be true for three important types of complex networks. Of course, any optimized routing when compared to shortest-path routing occurs at the expense of increasing the average number of hops between the nodes. We show that with our algorithm the average number of hops after optimization increases with the number of nodes no faster than logarithmically and that optimization significantly decreases the average travel time on congested networks.

show abstract

Scaling in small-world resistor networks

Cited by 37 publications

References 67 publications

Synchronization in complex networks of phase oscillators: A survey

Synchronization in complex networks of phase oscillators: A survey

Synchronization in weighted uncorrelated complex networks in a noisy environment: Optimization and connections with transport efficiency

Transport optimization on complex networks

Contact Info

Product

Resources

About