Journal ArticleDOI

# A Matrix Perturbation View of the Small World Phenomenon

01 Jan 2007-Siam Review (Society for Industrial and Applied Mathematics)-Vol. 49, Iss: 1, pp 91-108

TL;DR: This work uses techniques from applied matrix analysis to study small world cutoff in a Markov chain, consisting of a periodic random walk plus uniform jumps, and measures the small world property by expressing the mean hitting time, averaged over all states, in terms of the expected number of shortcuts per random walk.

AbstractWe use techniques from applied matrix analysis to study small world cutoff in a Markov chain. Our model consists of a periodic random walk plus uniform jumps. This has a direct interpretation as a teleporting random walk, of the type used by search engines to locate web pages, on a simple ring network. More loosely, the model may be regarded as an analogue of the original small world network of Watts and Strogatz [Nature, 393 (1998), pp. 440-442]. We measure the small world property by expressing the mean hitting time, averaged over all states, in terms of the expected number of shortcuts per random walk. This average mean hitting time is equivalent to the expected number of steps between a pair of states chosen uniformly at random. The analysis involves nonstandard matrix perturbation theory and the results come with rigorous and sharp asymptotic error estimates. Although developed in a different context, the resulting cutoff diagram agrees closely with that arising from the mean-field network theory of Newman, Moore, and Watts [Phys. Rev. Lett., 84 (2000), pp. 3201-3204].

Topics: Random walk (61%), Random graph (57%), Hitting time (57%), Expected value (54%), Small-world network (53%)

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##### Citations
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Journal ArticleDOI
TL;DR: A new class of evolving range-dependent random graphs is introduced that gives a tractable framework for modelling and simulation and can be used computationally and analytically to investigate the scenario where an evolutionary process takes place on an evolving network.
Abstract: Applications such as neuroscience, telecommunication, online social networking, transport and retail trading give rise to connectivity patterns that change over time. In this work, we address the resulting need for network models and computational algorithms that deal with dynamic links. We introduce a new class of evolving range-dependent random graphs that gives a tractable framework for modelling and simulation. We develop a spectral algorithm for calibrating a set of edge ranges from a sequence of network snapshots and give a proof of principle illustration on some neuroscience data. We also show how the model can be used computationally and analytically to investigate the scenario where an evolutionary process, such as an epidemic, takes place on an evolving network. This allows us to study the cumulative effect of two distinct types of dynamics.

89 citations

Journal ArticleDOI
TL;DR: A new formula for the Kemeny constant is presented and several perturbation results for the constant are developed, including conditions under which it is a convex function and for chains whose transition matrix has a certain directed graph structure.
Abstract: A quantity known as the Kemeny constant, which is used to measure the expected number of links that a surfer on the World Wide Web, located on a random web page, needs to follow before reaching his/her desired location, coincides with the more well known notion of the expected time to mixing, i.e., to reaching stationarity of an ergodic Markov chain. In this paper we present a new formula for the Kemeny constant and we develop several perturbation results for the constant, including conditions under which it is a convex function. Finally, for chains whose transition matrix has a certain directed graph structure we show that the Kemeny constant is dependent only on the common length of the cycles and the total number of vertices and not on the specific transition probabilities of the chain.

30 citations

### Cites methods from "A Matrix Perturbation View of the S..."

• ...We mention that condition numbers for Markov chains are used to measure the error in the computation of its stationary distribution vector....

[...]

Proceedings ArticleDOI
01 Dec 2009
TL;DR: It is shown that the general combinatorial problem can be relaxed to a convex optimization problem and the special case of adding a shortcut to a given structure is solved and insights for derivation of heuristics for the general case are provided.
Abstract: Distributed decision making in networked systems depends critically on the timely availability of critical fresh information. Performance of networked systems, from the perspective of achieving goals and objectives in a timely and efficient manner is constrained by their collaboration and communication structures and their interplay with the networked system's dynamics. In most cases achieving the system objectives requires many agent to agent communications. A reasonable measure for system robustness to communication topology change is the number of spanning trees in the graph abstraction of the communication system. We address the problem of network formation with robustness and connectivity constraints. Solutions to this problem have also applications in trust and the relationship of trust to control. We show that the general combinatorial problem can be relaxed to a convex optimization problem. We solve the special case of adding a shortcut to a given structure and provide insights for derivation of heuristics for the general case. We also analyze the small world effect in the context of abrupt increases in the number of spanning trees as a result of adding a few shortcuts to a base lattice in the Watts-Strogatz framework and thereby relate efficient topologies to small world and expander graphs.

27 citations

### Cites methods from "A Matrix Perturbation View of the S..."

• ...Here we adopt the same framework as in [13], [2] and study the increase of the number of spanning trees in the graph as a result of adding shortcuts with small weights....

[...]

• ...In our previous works, we developed a new method for investigating the effects of small world topologies by building on the probabilistic models of Higham [13], that established an equivalent representation of small world topologies as rare transitions among non-neighboring states in the Markov chain associated with a graph....

[...]

Proceedings ArticleDOI
01 Dec 2008
TL;DR: It is observed that adding a few long range edges to certain graph topologies can significantly increase the rate of convergence for consensus algorithms, and a probabilistic framework for analyzing this effect is provided.
Abstract: It has been observed that adding a few long range edges to certain graph topologies can significantly increase the rate of convergence for consensus algorithms. A notable example is the class of ring-structured Watts-Strogatz small world graphs. Building on probabilistic methods for analyzing `small-world phenomena?, developed in our earlier work, we provide here a probabilistic framework for analyzing this effect. We investigate what graph characteristics lead to such a significant improvement and develop bounds to analyze consensus problems on randomly varying graphs.

27 citations

### Cites methods from "A Matrix Perturbation View of the S..."

• ...In the earlier paper [1] Baras and Hovareshti developed a new method for investigating the effects of small world topologies by building on the probabilistic models of Higham [14], that established an equivalent representation of small world topologies as rare transitions among non-neighboring states in the Markov chain associated with a graph....

[...]

• ...In [14] small world phenomena were analyzed using these slightly randomly perturbed Markov chains and a ‘mean field’ approach for the Markov random field associated with the Markov chain....

[...]

Journal ArticleDOI
TL;DR: This work focuses on mean hitting time behavior in a thermodynamic limit and concludes by pointing out that studying hitting times allows the Markov jump and stochastic differential equation regimes to be compared in a manner that avoids avoidable errors.
Abstract: Markov jump processes can provide accurate models in many applications, notably chemical and biochemical kinetics, and population dynamics. Stochastic differential equations offer a computationally efficient way to approximate these processes. It is therefore of interest to establish results that shed light on the extent to which the jump and diffusion models agree. In this work we focus on mean hitting time behavior in a thermodynamic limit. We study three simple types of reactions where analytical results can be derived, and we find that the match between mean hitting time behavior of the two models is vastly different in each case. In particular, for a degradation reaction we find that the relative discrepancy decays extremely slowly, namely, as the inverse of the logarithm of the system size. After giving some further computational results, we conclude by pointing out that studying hitting times allows the Markov jump and stochastic differential equation regimes to be compared in a manner that avoids ...

20 citations

### Cites background from "A Matrix Perturbation View of the S..."

• ...Related issues arose in the small world analysis of [15, 16]....

[...]

##### References
More filters

Journal ArticleDOI
04 Jun 1998-Nature
TL;DR: Simple models of networks that can be tuned through this middle ground: regular networks ‘rewired’ to introduce increasing amounts of disorder are explored, finding that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs.
Abstract: Networks of coupled dynamical systems have been used to model biological oscillators, Josephson junction arrays, excitable media, neural networks, spatial games, genetic control networks and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks 'rewired' to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them 'small-world' networks, by analogy with the small-world phenomenon (popularly known as six degrees of separation. The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.

35,972 citations

### "A Matrix Perturbation View of the S..." refers background or methods or result in this paper

• ...For this scaling, we obtain a cutoff diagram that illustrates the small world phenomenon and may be compared with that of the Watts–Strogatz network model....

[...]

• ...Although the correspondence is not exact, we originally developed this model by analogy with the randomized network approach of Watts and Strogatz [25]....

[...]

• ...5 shortcuts per network are needed to give a reduction of 12 in the average path length (consistent with the simulations of [25])....

[...]

• ...It is possible to compare the behavior of this model with that of the k = 1 version of the Newman–Moore–Watts network model [19], which is closely related to the corresponding Watts–Strogatz model [25]....

[...]

• ...In the case of networks, the small world phenomenon has been characterized by expressing some measure of the average path length in terms of the expected number of shortcuts added [19, 25]....

[...]

Journal ArticleDOI
TL;DR: Developments in this field are reviewed, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.
Abstract: Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.

16,520 citations

### "A Matrix Perturbation View of the S..." refers methods in this paper

• ...SIAM has been quick to react, with Newman’s comprehensive and accessible review [25] being a key resource....

[...]

Journal ArticleDOI
08 Mar 2001-Nature
TL;DR: This work aims to understand how an enormous network of interacting dynamical systems — be they neurons, power stations or lasers — will behave collectively, given their individual dynamics and coupling architecture.
Abstract: The study of networks pervades all of science, from neurobiology to statistical physics. The most basic issues are structural: how does one characterize the wiring diagram of a food web or the Internet or the metabolic network of the bacterium Escherichia coli? Are there any unifying principles underlying their topology? From the perspective of nonlinear dynamics, we would also like to understand how an enormous network of interacting dynamical systems-be they neurons, power stations or lasers-will behave collectively, given their individual dynamics and coupling architecture. Researchers are only now beginning to unravel the structure and dynamics of complex networks.

7,139 citations

### "A Matrix Perturbation View of the S..." refers background in this paper

• ...Although it is simplistic, we believe that this model is relevant to many physical, sociological, epidemiological, and computational applications, as it combines the traditional notion of diffusion on a lattice [3, 4, 16, 20] with the type of partially random connectivity that has recently been used to describe complex, real-life networks [6, 12, 15, 17, 18, 22, 23, 24]....

[...]

Book
01 Jan 1995
TL;DR: This book introduces the basic concepts in the design and analysis of randomized algorithms and presents basic tools such as probability theory and probabilistic analysis that are frequently used in algorithmic applications.
Abstract: For many applications, a randomized algorithm is either the simplest or the fastest algorithm available, and sometimes both. This book introduces the basic concepts in the design and analysis of randomized algorithms. The first part of the text presents basic tools such as probability theory and probabilistic analysis that are frequently used in algorithmic applications. Algorithmic examples are also given to illustrate the use of each tool in a concrete setting. In the second part of the book, each chapter focuses on an important area to which randomized algorithms can be applied, providing a comprehensive and representative selection of the algorithms that might be used in each of these areas. Although written primarily as a text for advanced undergraduates and graduate students, this book should also prove invaluable as a reference for professionals and researchers.

4,409 citations

Journal ArticleDOI
TL;DR: It is shown that these collaboration networks form "small worlds," in which randomly chosen pairs of scientists are typically separated by only a short path of intermediate acquaintances.
Abstract: The structure of scientific collaboration networks is investigated. Two scientists are considered connected if they have authored a paper together and explicit networks of such connections are constructed by using data drawn from a number of databases, including MEDLINE (biomedical research), the Los Alamos e-Print Archive (physics), and NCSTRL (computer science). I show that these collaboration networks form “small worlds,” in which randomly chosen pairs of scientists are typically separated by only a short path of intermediate acquaintances. I further give results for mean and distribution of numbers of collaborators of authors, demonstrate the presence of clustering in the networks, and highlight a number of apparent differences in the patterns of collaboration between the fields studied.

4,204 citations

### "A Matrix Perturbation View of the S..." refers background in this paper

• ...Although it is simplistic, we believe that this model is relevant to many physical, sociological, epidemiological, and computational applications, as it combines the traditional notion of diffusion on a lattice [3, 4, 16, 20] with the type of partially random connectivity that has recently been used to describe complex, real-life networks [6, 12, 15, 17, 18, 22, 23, 24]....

[...]