DOCSLIB.ORG

Ultrafast Consensus in Small-World Networks

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Ultrafast Consensus in Small-World Networks 2005 American Control Conference ThB03.5 June 8-10, 2005. Portland, OR, USA Ultrafast Consensus in Small-World Networks Reza Olfati-Saber Control and Dynamical Systems California Institute of Technology [email protected] Abstract— In this paper, we demonstrate a phase transition The small-world model of Watts & Strogatz initiated a phenomenon in algebraic connectivity of small-world net- tremendous amount of interest among researchers from mul- works. Algebraic connectivity of a graph is the second smallest tiple fields to study topological properties of complex net- eigenvalue of its Laplacian matrix and a measure of speed of solving consensus problems in networks. We demonstrate works. These properties include degree distribution, char- that it is possible to dramatically increase the algebraic acteristic length, clustering coefficient (see [42], [28]), ro- connectivity of a regular complex network by 1000 times or bustness to node failure, and search issues. The researchers more without adding new links or nodes to the network. This who have most contributed to this effort came from fields implies that a consensus problem can be solved incredibly such as statistical physics, computer science, economics, fast on certain small-world networks giving rise to a network design algorithm for ultrafast information networks. Our mathematical biology, communication networks, and power study relies on a procedure called “random rewiring” due to networks. Watts & Strogatz (Nature, 1998). Extensive numerical results In most engineering and biological complex systems, are provided to support our claims and conjectures. We prove the nodes have a dynamics—they are not labels or names that the mean of the bulk Laplacian spectrum of a complex of actors. In other words, “real-life” engineering networks network remains invariant under random rewiring. The same property only asymptotically holds for scale-free networks. A are interconnection of dynamic systems. The same applies relationship between increasing the algebraic connectivity of to broad examples of biological networks including gene complex networks and robustness to link and node failures networks and coupled neural oscillators. From the perspec- is also shown. This is an alternative approach to the use of tive of systems & control theory, the stability properties percolation theory for analysis of network robustness. We also of collective dynamics of networks of dynamic agents is of show some connections between our conjectures and certain open problems in the theory of random matrices. interest. This motivates exploration of spectral properties of Keywords: small-world networks, networked systems, complex networks. In the past, the study of spectral proper- consensus algorithms, phase transition, graph Laplacians, ties of random networks has been given little attention. This algebraic connectivity, network robustness, random matrices paper is a first step towards understanding the behavior of Laplacian spectra of complex networks and its application I. INTRODUCTION in design of ultrafast information networks. Complex networks are abundant in large-scale engineer- We use consensus problems [35], [33] as a framework to ing, biological, and social systems. Some examples include convey our ideas regarding the connections between spectral power networks, metabolic and gene networks [17], co- properties of complex networks and ultrafast solution to dis- authorship network of scientists [27], biological network of tributed decision-making problems for interacting groups of oscillators [44], [19], [18], [24], [39], economic networks agents. Distributed computation based on solving consensus [15], sensor networks [10], [30], [7], [38], swarms of problems has direct implications on sensor networks & data networked unmanned autonomous vehicles (UAVs) [32], fusion [38], load balancing [21], and swarms/flocks [32], [36], [6], and self-organizing biological swarms [25], [41]. [36]. Moreover, synchronization of coupled oscillators [44], For recent surveys on complex networks, the reader can [19], [18], [24] that has received tremendous attention over refer to [40], [28]. the past 35 years is a special case of a nonlinear consensus In 1998, Watts & Strogatz [42] introduced a network problem in networks [35] regarding the frequency of oscil- model called small-world network that was capable of inter- lation of all nodes2. For this case, algebraic connectivity is polating between a regular network and a random network (locally) a measure of speed of synchronization. using a single parameter. A small-world is a network with One of the first class of complex networks are random 1 a relatively small characteristic length . In a small-world, graphs introduced by Erdos&R¨ enyi´ (ER) [9] nearly 50 any two nodes can be linked using a few steps despite the years ago. A random graph can be constructed by connect- large size of the network. For example, the world-wide web p 8 ing any pairs of vertices of the graph with probability . (www) with n =8× 10 nodes has a characteristic length Random graphs are perhaps the most well-studied model of 18.5 [3]. of random networks. The unique feature of the ER model 1The average distance between two nodes in the network over all pairs of distinct nodes. The distance is the length of the shortest path connecting 2Further details regarding the connection between the two problems in two nodes. the subject of an upcoming article. 0-7803-9098-9/05/$25.00 ©2005 AACC 2371 is that beyond a critical value p>pc ≈ 1/n, a giant Note that the Laplacian matrix always has a zero eigen- connected component forms in the network [9], [40]. This value λ1 =0corresponding to the aligned state x = T phenomenon is the first known example of a combinatorial (1, 1,...,1) . In addition, if G is connected, then λ2 > 0 phase transition. Other forms of combinatorial and algo- [13]. Apparently, the analysis of consensus problems in rithmic phase transition phenomena were later discovered networks reduces to spectral analysis of Laplacian of the in discrete mathematics and computer science. network topology. Particularly, λ2 is the measure of speed The next generation of random networks consist of three of convergence (or performance) of the consensus algorithm models: i) small-world networks by Watts & Strogatz (WS) in (1) [33]. λ2 is named the algebraic connectivity of the [42], ii) semiregular small-world networks by Newman, graph by Fiedler [12] due to the following inequality: Moore, & Watts (NMW) [29], and iii) scale-free networks λ (G) ≤ ν(G) ≤ η(G) (e.g. www) by Barabasi´ & Albert (BA) [1]. We will describe 2 (3) the WS and NMW network models in details and leave the where ν(G) and η(G) are node-connectivity and edge discussion of the BA model to a future occasion. The ER connectivity of a graph, respectively (see [4] for definitions). model is inadequate for our study since it is not guaranteed According to this inequality, a network with a relatively to be connected unless the graph is relatively dense. That high algebraic connectivity is necessarily robust to both is why we focus on spectral properties of WS and NMW node-failures and edge-failures. A lower bound on this small-world models. degree of robustness is λ2. Here is an outline of the paper: Some background on For a network with communication time-delays, the con- consensus problems from systems & control point of view sensus algorithm takes the following form [35], [33]: is presented in Section II. Small-world networks and their semiregular version are described in Section III. Random x˙ i(t)= {xj(t − τ) − xi(t − τ)} (4) rewiring procedure for general networks is given in Sec- j∈Ni tion IV. Our main numerical results and conjectures on with a collective dynamics Laplacian spectral properties of small-world networks and network resilience are presented in Section V. The connec- x˙ = −Lx(t − τ). (5) tions between Laplacian of random networks and random We assume that the time-delay in all links is equal to τ matrix theory is discussed in Section VI. Finally, concluding (see [35] for the general case). A necessary and sufficient remarks are made in Section VII. condition for stability of system (5) is given in [33] as II. CONSENSUS PROBLEMS IN NETWORKS π τ<τmax = . (6) 2λn Consider a network of integrator agents x˙ i = ui with topology G =(V,E) in which each agent only com- Thus, λn is a measure of robustness to delay for reaching municates with its neighboring agents Ni = {j ∈ V : a consensus in a network. {i, j}∈E} on G =(V,E). Here, V = {1, 2,...,n} and The main result of this paper is that λ2 for a regular E ⊂ [V ]2 (the set of 2-element subsets of V ) denote the network can be increased multiple orders of magnitude set of nodes and edges/links of the network, respectively. In via changing the inter-agent information flow G without [35], [33], Olfati-Saber & Murray show that the following increasing the total number of the links of the network. linear dynamic system Moreover, this change has a negligible effect on λn (the system remains robustness to delay). x˙ (t)= (x (t) − x (t)) i j i (1) Some variations of consensus problems on graphs include j∈Ni the following areas: networks with switching topology [16], solves a consensus problem. More precisely, let [33], [26], [34], consensus on digraphs [33], [26], and a1,...,an ∈ R be n constants, then with the set of initial asynchronous consensus [14]. states xi(0) = ai, the state of all agents asymptotically III. SMALL-WORLD NETWORKS converges to the average value a¯ =1/n i ai provided that the network is connected. The collective dynamics of Small-world phenomenon is a feature of certain complex the agents in (1) can be expressed as networks in which any two arbitrary nodes can be connected using a few links [23]. This means that the average distance x˙(t)=−Lx(t) (2) between two nodes (i.e.
Load more

Recommended publications
  • Emerging Topics in Internet Technology: a Complex Networks Approach
    1 Emerging Topics in Internet Technology: A Complex Networks Approach Bisma S. Khan 1, Muaz A. Niazi *,2 COMSATS Institute of Information Technology, Islamabad, Pakistan Abstract —Communication networks, in general, and internet technology, in particular, is a fast-evolving area of research. While it is important to keep track of emerging trends in this domain, it is such a fast-growing area that it can be very difficult to keep track of literature. The problem is compounded by the fast-growing number of citation databases. While other databases are gradually indexing a large set of reliable content, currently the Web of Science represents one of the most highly valued databases. Research indexed in this database is known to highlight key advancements in any domain. In this paper, we present a Complex Network-based analytical approach to analyze recent data from the Web of Science in communication networks. Taking bibliographic records from the recent period of 2014 to 2017 , we model and analyze complex scientometric networks. Using bibliometric coupling applied over complex citation data we present answers to co-citation patterns of documents, co-occurrence patterns of terms, as well as the most influential articles, among others, We also present key pivot points and intellectual turning points. Complex network analysis of the data demonstrates a considerably high level of interest in two key clusters labeled descriptively as “social networks” and “computer networks”. In addition, key themes in highly cited literature were clearly identified as “communication networks,” “social networks,” and “complex networks”. Index Terms —CiteSpace, Communication Networks, Network Science, Complex Networks, Visualization, Data science.
    [Show full text]
  • Simplified Computational Model for Generating Bio- Logical Networks
    Simplified Computational Model for Generating Bio- logical Networks† Matthew H J Bailey,∗ David Ormrod Morley,∗ and Mark Wilson∗ A method to generate and simulate biological networks is discussed. An expanded Wooten- Winer-Weaire bond switching methods is proposed which allows for a distribution of node degrees in the network while conserving the mean average node degree. The networks are characterised in terms of their polygon structure and assortativities (a measure of local ordering). A wide range of experimental images are analysed and the underlying networks quantified in an analogous manner. Limitations in obtaining the network structure are discussed. A “network landscape” of the experimentally observed and simulated networks is constructed from the underlying metrics. The enhanced bond switching algorithm is able to generate networks spanning the full range of experimental observations. Two dimensional random networks are observed in a range of 17 contexts across considerably different length scales in nature: to be limited to avoid damaging the delicate networks . How- from nanometres, in the form of amorphous graphene; to me- ever, even when a high-quality image is obtained, there is further tres, in the form of the Giant’s causeway; to tens of kilometres, in difficulty in analysis. For example, each edge in a network is a the form of geopolitical borders 1–3. A framework for describing complex molecule made up of tens of thousands of atoms which these continuous random networks for chemical systems was first may not lie strictly in a single plane. In addition, the most inter- introduced by Zachariasen to describe silica-like glasses, and has esting dynamic behaviour often occurs over very long timescales proved to be extremely versatile in the years since 4, being used to – potentially decades.
    [Show full text]
  • Biological Network Approaches and Applications in Rare Disease Studies
    G C A T T A C G G C A T genes Review Biological Network Approaches and Applications in Rare Disease Studies Peng Zhang 1,* and Yuval Itan 2,3 1 St. Giles Laboratory of Human Genetics of Infectious Diseases, Rockefeller Branch, The Rockefeller University, New York, NY 10065, USA 2 The Charles Bronfman Institute for Personalized Medicine, Icahn School of Medicine at Mount Sinai, New York, NY 10029, USA; [email protected] 3 Department of Genetics and Genomic Sciences, Icahn School of Medicine at Mount Sinai, New York, NY 10029, USA * Correspondence: [email protected]; Tel.: +1-646-830-6622 Received: 3 September 2019; Accepted: 10 October 2019; Published: 12 October 2019 Abstract: Network biology has the capability to integrate, represent, interpret, and model complex biological systems by collectively accommodating biological omics data, biological interactions and associations, graph theory, statistical measures, and visualizations. Biological networks have recently been shown to be very useful for studies that decipher biological mechanisms and disease etiologies and for studies that predict therapeutic responses, at both the molecular and system levels. In this review, we briefly summarize the general framework of biological network studies, including data resources, network construction methods, statistical measures, network topological properties, and visualization tools. We also introduce several recent biological network applications and methods for the studies of rare diseases. Keywords: biological network; bioinformatics; database; software; application; rare diseases 1. Introduction Network biology provides insights into complex biological systems and can reveal informative patterns within these systems through the integration of biological omics data (e.g., genome, transcriptome, proteome, and metabolome) and biological interactome data (e.g., protein-protein interactions and gene-gene associations).
    [Show full text]
  • Complex Networks in Computer Vision
    Complex networks in computer vision Pavel Voronin Shape to Network [Backes 2009], [Backes 2010a], [Backes 2010b] Weights + thresholds Feature vectors Features = characteris>cs of thresholded undirected binary networks: degree or joint degree (avg / min / max), avg path length, clustering coefficient, etc. Mul>scale Fractal Dimension Properes • Only uses distances => rotaon invariant • Weights normalized by max distance => scale invariant • Uses pixels not curve elements => robust to noise and outliers => applicable to skeletons, mul>ple contours Face recogni>on [Goncalves 2010], [Tang 2012a] Mul>ple binarizaon thresholds Texture to Network [Chalumeau 2008], [Backes 2010c], [Backes 2013] Thresholds + features Features: degree, hierarchical degree (avg / min / max) Invariance, robustness Graph structure analysis [Tang 2012b] Graph to Network Thresholds + descriptors Degree, joint degree, clustering-distance Saliency Saccade eye movements + different fixaon >me = saliency map => model them as walks in networks [Harel 2006], [Costa 2007], [Gopalakrishnan 2010], [Pal 2010], [Kim 2013] Network construc>on Nodes • pixels • segments • blocks Edges • local (neighbourhood) • global (most similar) • both Edge weights = (distance in feature space) / (distance in image space) Features: • relave intensity • entropy of local orientaons • compactness of local colour Random walks Eigenvector centrality == staonary distribu>on for Markov chain == expected >me a walker spends in the node Random walk with restart (RWR) If we have prior info on node importance,
    [Show full text]
  • Analysis of Statistical and Structural Properties of Complex Networks with Random Networks
    Appl. Math. Inf. Sci. 11, No. 1, 137-146 (2017) 137 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.18576/amis/110116 Analysis of Statistical and Structural Properties of Complex networks with Random Networks S. Vairachilai1,∗, M. K. Kavitha Devi2 and M. Raja 1 1 Department of CSE, Faculty of Science and Technology, IFHE Hyderabad, Telangana -501203, India. 2 Department of CSE, ThiagarajarCollege of Engineering, Madurai,Tamil Nadu-625015, India. Received: 2 Sep. 2016, Revised: 25 Oct. 2016, Accepted: 29 Oct. 2016 Published online: 1 Jan. 2017 Abstract: Random graphs are extensive, in addition, it is used in several functional areas of research, particularly in the field of complex networks. The study of complex networks is a useful and active research areas in science, such as electrical power grids and telecommunication networks, collaboration and citation networks of scientists,protein interaction networks, World-Wide Web and Internet Social networks, etc. A social network is a graph in which n vertices and m edges are selected at random, the vertices represent people and the edges represent relationships between them. In network analysis, the number of properties is defined and studied in the literature to identify the important vertex in a network. Recent studies have focused on statistical and structural properties such as diameter, small world effect, clustering coefficient, centrality measure, modularity, community structure in social networks like Facebook, YouTube, Twitter, etc. In this paper, we first provide a brief introduction to the complex network properties. We then discuss the complex network properties with values expected for random graphs.
    [Show full text]
  • Arxiv:2006.02870V1 [Cs.SI] 4 Jun 2020
    The why, how, and when of representations for complex systems Leo Torres Ann S. Blevins [email protected] [email protected] Network Science Institute, Department of Bioengineering, Northeastern University University of Pennsylvania Danielle S. Bassett Tina Eliassi-Rad [email protected] [email protected] Department of Bioengineering, Network Science Institute and University of Pennsylvania Khoury College of Computer Sciences, Northeastern University June 5, 2020 arXiv:2006.02870v1 [cs.SI] 4 Jun 2020 1 Contents 1 Introduction 4 1.1 Definitions . .5 2 Dependencies by the system, for the system 6 2.1 Subset dependencies . .7 2.2 Temporal dependencies . .8 2.3 Spatial dependencies . 10 2.4 External sources of dependencies . 11 3 Formal representations of complex systems 12 3.1 Graphs . 13 3.2 Simplicial Complexes . 13 3.3 Hypergraphs . 15 3.4 Variations . 15 3.5 Encoding system dependencies . 18 4 Mathematical relationships between formalisms 21 5 Methods suitable for each representation 24 5.1 Methods for graphs . 24 5.2 Methods for simplicial complexes . 25 5.3 Methods for hypergraphs . 27 5.4 Methods and dependencies . 28 6 Examples 29 6.1 Coauthorship . 29 6.2 Email communications . 32 7 Applications 35 8 Discussion and Conclusion 36 9 Acknowledgments 38 10 Citation diversity statement 38 2 Abstract Complex systems thinking is applied to a wide variety of domains, from neuroscience to computer science and economics. The wide variety of implementations has resulted in two key challenges: the progenation of many domain-specific strategies that are seldom revisited or questioned, and the siloing of ideas within a domain due to inconsistency of complex systems language.
    [Show full text]
  • Complex Design Networks: Structure and Dynamics
    Complex Design Networks: Structure and Dynamics Dan Braha New England Complex Systems Institute, Cambridge, MA 02138, USA University of Massachusetts, Dartmouth, MA 02747, USA Email: [email protected] Why was the $6 billion FAA air traffic control project scrapped? How could the 1977 New York City blackout occur? Why do large scale engineering systems or technology projects fail? How do engineering changes and errors propagate, and how is that related to epidemics and earthquakes? In this paper we demonstrate how the rapidly expanding science of complex design networks could provide answers to these intriguing questions. We review key concepts, focusing on non-trivial topological features that often occur in real-world large-scale product design and development networks; and the remarkable interplay between these structural features and the dynamics of design rework and errors, network robustness and resilience, and design leverage via effective resource allocation. We anticipate that the empirical and theoretical insights gained by modeling real-world large-scale product design and development systems as self-organizing complex networks will turn out to be the standard framework of a genuine science of design. 1. Introduction: the road to complex networks in engineering design Large-scale product design and development is often a distributed process, which involves an intricate set of interconnected tasks carried out in an iterative manner by hundreds of designers (see Fig. 1), and is fundamental to the creation of complex man- made systems [1—16]. This complex network of interactions and coupling is at the heart of large-scale project failures as well as of large-scale engineering and software system failures (see Table 1).
    [Show full text]
  • Optimizing the Topology of Complex Neural Networks
    Optimising the topology of complex neural networks Fei Jiang1;2, Hugues Berry1, and Marc Schoenauer2 1Project-team Alchemy and 2Project-team TAO, INRIA Futurs, Parc Club Orsay Universite, 91893 Orsay Cedex France, [email protected] Summary. In this paper, we study instances of complex neural networks, i.e. neural networks with complex topologies. We use Self-Organizing Map neural networks whose neighbourhood relationships are defined by a complex network, to classify handwritten digits. We show that topology has a small impact on performance and robustness to neuron failures, at least at long learning times. Performance may however be increased (by almost 10%) by artificial evolution of the network topology. In our experimental conditions, the evolved networks are more random than their parents, but display a more heterogeneous degree distribution. 1 Introduction The connectivity structure of complex networks (i.e. their topology) is a cru- cial determinant of information transfer in large networks (internet, social networks, metabolic networks...) [1]. Hence, the computation made by com- plex neural networks, i.e. neural networks with complex connectivity struc- ture, could be dependent on their topology. For instance, recent studies have shown that introducing a small-world topology in a multilayer perceptron increases its performance [13, 2]. However, other studies have inspected the performance of Hopfield [5, 10, 9, 14] or Echo state networks [3] with small- world or scale-free topologies and reported more contrasted results. Using artificial evolutionary algorithms to modify the topology of neu- ral networks so as to optimise their performance has become widespread in the artificial neural networks community for several years [11, 16].
    [Show full text]
  • Applications of Network Science to Education Research: Quantifying Knowledge and the Development of Expertise Through Network Analysis
    education sciences Commentary Applications of Network Science to Education Research: Quantifying Knowledge and the Development of Expertise through Network Analysis Cynthia S. Q. Siew Department of Psychology, National University of Singapore, Singapore 117570, Singapore; [email protected] Received: 31 January 2020; Accepted: 3 April 2020; Published: 8 April 2020 Abstract: A fundamental goal of education is to inspire and instill deep, meaningful, and long-lasting conceptual change within the knowledge landscapes of students. This commentary posits that the tools of network science could be useful in helping educators achieve this goal in two ways. First, methods from cognitive psychology and network science could be helpful in quantifying and analyzing the structure of students’ knowledge of a given discipline as a knowledge network of interconnected concepts. Second, network science methods could be relevant for investigating the developmental trajectories of knowledge structures by quantifying structural change in knowledge networks, and potentially inform instructional design in order to optimize the acquisition of meaningful knowledge as the student progresses from being a novice to an expert in the subject. This commentary provides a brief introduction to common network science measures and suggests how they might be relevant for shedding light on the cognitive processes that underlie learning and retrieval, and discusses ways in which generative network growth models could inform pedagogical strategies to enable meaningful long-term conceptual change and knowledge development among students. Keywords: education; network science; knowledge; learning; expertise; development; conceptual representations 1. Introduction Cognitive scientists have had a long-standing interest in quantifying cognitive structures and representations [1,2]. The empirical evidence from cognitive psychology and psycholinguistics demonstrates that the structure of cognitive systems influences the psychological processes that operate within them.
    [Show full text]
  • Denoising Large-Scale Biological Data Using Network Filters
    Kavran and Clauset BMC Bioinformatics (2021) 22:157 https://doi.org/10.1186/s12859‑021‑04075‑x RESEARCH ARTICLE Open Access Denoising large‑scale biological data using network flters Andrew J. Kavran1,2 and Aaron Clauset2,3,4* *Correspondence: [email protected] Abstract 2 BioFrontiers Institute, Background: Large-scale biological data sets are often contaminated by noise, which University of Colorado, Boulder, CO, USA can impede accurate inferences about underlying processes. Such measurement Full list of author information noise can arise from endogenous biological factors like cell cycle and life history vari- is available at the end of the ation, and from exogenous technical factors like sample preparation and instrument article variation. Results: We describe a general method for automatically reducing noise in large- scale biological data sets. This method uses an interaction network to identify groups of correlated or anti-correlated measurements that can be combined or “fltered” to better recover an underlying biological signal. Similar to the process of denoising an image, a single network flter may be applied to an entire system, or the system may be frst decomposed into distinct modules and a diferent flter applied to each. Applied to synthetic data with known network structure and signal, network flters accurately reduce noise across a wide range of noise levels and structures. Applied to a machine learning task of predicting changes in human protein expression in healthy and cancer- ous tissues, network fltering prior to training increases accuracy up to 43% compared to using unfltered data. Conclusions: Network flters are a general way to denoise biological data and can account for both correlation and anti-correlation between diferent measurements.
    [Show full text]
  • Immunization for Complex Network Based on the Effective Degree Of
    Immunization for complex network based on the effective degree of vertex Ke Hu and Yi Tang* Department of Physics and Institute of Modern Physics, Xiangtan University, Xiangtan 411105, Hunan, China The basic idea of many effective immunization strategies is first to rank the importance of vertices according to the degrees of vertices and then remove the vertices from highest importance to lowest until the network becomes disconnected. Here we define the effective degrees of vertex, i.e., the number of its connections linking to un-immunized nodes in current network during the immunization procedure, to rank the importance of vertex, and modify these strategies by using the effective degrees of vertices. Simulations on both the scale-free network models with various degree correlations and two real networks have revealed that the immunization strategies based on the effective degrees are often more effective than those based on the degrees in the initial network. Keywords: Immunization, epidemic spreading, scale-free network, effective degree PACS number(s): 89.75.Hc, 87.23.Ge 1. Introduction The epidemic dynamic on complex network is largely benefited from the need of understanding the spreading of both computer viruses and human sexual diseases[1-3], which has recently become a subject of *Corresponding author. Email address: [email protected] 1 intensive investigations.[4-15] Since most of real networks involving the Internet and the web of sexual contacts appear to be of large heterogeneity in the connectivity properties,[16-19] epidemic modeling deals with the complex systems with heterogeneous connectivity to approach a more realistic situation. This highly heterogeneous topology of the networks is mainly reflected in the small average path lengths among any two nodes (small-world property)[20], and in a power law connectivity distribution (scale-free property)[19].
    [Show full text]
  • Complex Networks in Climate Science: Progress, Opportunities and Challenges
    COMPLEX NETWORKS IN CLIMATE SCIENCE: PROGRESS, OPPORTUNITIES AND CHALLENGES KARSTEN STEINHAEUSER1,2, NITESH V. CHAWLA1, AND AUROOP R. GANGULY2 Abstract. Networks have been used to describe and model a wide range of complex systems, both natural as well as man-made. One particularly interesting application in the earth sciences is the use of complex networks to represent and study the global climate system. In this paper, we motivate this general approach, explain the basic methodology, report on the state of the art (including our contributions), and outline open questions and opportunities for future research. 1. Introduction Datasets and systems that can be represented as interaction networks (or graphs), broadly defined as any collection of interrelated objects or entities, have received considerable attention both from a theoretical viewpoint [1, 2, 6, 8, 13, 31] as well as various application domains; examples include the analysis of social networks [30], chemical interactions between proteins [26], the behavior of financial markets [12], and many others. Recently, the study of complex networks – that is, networks which exhibit non-trivial topological properties – has permeated numerous fields and disciplines spanning the physical, social, and computational sciences. So why do networks enjoy such broad appeal? Briefly, it is their ability to serve at once as a data representation, as an analysis framework, and as a visualization tool. The analytic capabilities in particular are quite powerful, as networks can uncover structure and patterns at multiple scales, ranging from local properties to global phenomena, and thus help better understand the characteristics of complex systems. We focus on one particular application of networks in the earth sciences, namely, the construction and analysis of climate networks [25].
    [Show full text]