Many complex networks in nature exhibit two properties that are seemingly at odds. They are clustered - neighbors of neighbors are very likely to be neighbors - and they are small worlds - any two nodes can typically be connected by a relatively short path. Watts and Strogatz [17] referred to this as the small world phenomenon and proposed a network model that was shown through simulation to capture the two properties. The model incorporates a parameter that interpolates between fully local and fully global regimes. As the parameter is varied the small world property is roused before the clustering property is lost

Higham, D.J.

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Finite differences in a small world

University of Strathclyde Institutional Repository

Strathprints Institutional RepositoryHigham, D.J. (2003) Finite differences in a small world. In: Proceedings of the 20th BiennialConference on Numerical Analysis, Dundee. University of Dundee, Dundee, pp. 81-84.Strathprints is designed to allow users to access the research output of the University of Strathclyde.Copyright c© and Moral Rights for the papers on this site are retained by the individual authorsand/or other copyright owners. You may not engage in further distribution of the material for anyprofitmaking activities or any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or study, educational, ornot-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to Strathprints administrator:mailto:strathprints@strath.ac.ukhttp://strathprints.strath.ac.uk/Higham NA03 Dundee 2003Finite Differences in a Small WorldDesmond J. Higham(1)Department of Mathematics,University of Strathclyde,Glasgow G1 1XH, Scotlanddjh@maths.strath.ac.uk1 IntroductionMany complex networks in nature exhibit two properties that are seemingly at odds. Theyare clustered—neighbors of neighbors are very likely to be neighbors—and they are smallworlds—any two nodes can typically be connected by a relatively short path. Watts andStrogatz [17] referred to this as the small world phenomenon and proposed a networkmodel that was shown through simulation to capture the two properties. The modelincorporates a parameter that interpolates between fully local and fully global regimes. Asthe parameter is varied the small world property is roused before the clustering property islost. These computations were later backed up by a semi-heuristic mean-field analysis in[12]. Motivated by [17] and the book [16], many authors have shown experimentally thatthe small world phenomenon can be found across a range of areas in science, medicine andtechnology [3, 8, 11, 10, 14, 15]. In this work we show that the small world phenomenonalso emerges through a local-to-global cutoff in the context of Markov chains. Our modelis based on a simple random walk and hence is relevant to many physical, sociological,epidemiological and computational applications [1, 2, 9, 13]. In particular, it may beregarded as a teleporting random walk, of the type used by the search engine Google tolocate pages on the web [6, 7], restricted to a simple ring network.An extremely attractive feature of our model is that it can be analyzed rigorously byusing tools from numerical analysis.Full details of the material summarized here can be found in [5]. Recent work that dealswith the question of navigation in a small world setting appears in [4].2 ModelOur model is a perturbed version of a period random walk on a ring. Imagine the numbers1, 2, 3, . . . , N arranged like the hours on a clock. Take a periodic random walk aroundthis clock—at each time level flip a fair coin; if it lands heads, move clockwise, and if itlands tails, move anticlockwise. Now alter the process so that before taking each step youtoss another, biased, coin that lands heads with small probability δ. In the event thatthe biased coins lands heads, then pick a number between 1 and N uniformly at randomand move there. Otherwise, follow the fair coin procedure above. In this model, mostof the time we follow a traditional periodic random walk, but occasionally (when thebiased coin lands heads) we take a teleporting jump across the network. This teleportingidea is used by search engines such as Google to locate pages on the web. (Traverse theweb by following links out of pages, but, to avoid reaching a cycle or a dead end, take(1)Supported by a Research Fellowship from the Leverhulme Trust81NA03 Dundee 2003 Highamthe occasional jump). Teleporting is very similar in spirit to the rewiring/shortcuttingprocess that has been observed to produce small world networks of the type described insection 1, and we will show that an analogous cutoff effect arises.3 ResultThe small world phenomenon for networks involves the pathlength between nodes. Anal-ogously, we may study the small world phenomenon for our random walk model bymeasuring the mean hitting time, that is, the average number of steps that it takes toreach B starting from A, where A,B ∈ {1, 2, . . . , N} are chosen uniformly at random.The problem of analysing the mean hitting time can be set up as a linear algebraic systemof dimension N−1. We are interested in behaviour for large systems, that is, the N →∞limit.The key to the analysis is to identify the mean hitting time system as a finite differenceapproximation to an underlying continuum limit. The continuum limit itself dependsupon N (i.e. the grid spacing), but careful analysis shows that convergence takes placein an appropriate sense, so that limiting behaviour may be captured.The analysis shows that it is natural to let the teleporting jump probability, δ, scale withN according toδ =KN2, for fixed K.In this regime, we find that, up to O(N−1) terms, the mean hitting time, normalized bythe mean hitting time when δ = 0, has the formy =6Kx, (3.7)wherex =√2K2 tanh√2K2− 1 (3.8)is an expression for the expected number of teleporting jumps taken per random walk.We may now examine the small world property of the model by plotting y against x asK is varied. This shows how the introduction of shortcuts (teleporting jumps) makes iteasier to get around the network. Figure 3.5 gives the picture. We see that there is anabrupt drop in the hitting time as x increases beyond x ≈ 10−1. This “order 1 shortcutcutoff effect” agrees with the fundamental observation of Watts and Strogatz [17] fortheir small world network model.References[1] H. C. Berg, Random Walks in Biology, Princeton University Press, 1983.[2] H. Caswell, Matrix Population Models, Sinauer Associates, Inc., Sunderland, MA, sec-ond ed., 2001.82Higham NA03 Dundee 200310−6 10−4 10−2 100 102 10400.10.20.30.40.50.60.70.80.91Expected Number of Teleporting JumpsNormalized Mean Hitting TimeFigure 3.5: x-axis is the average number of teleporting jumps per excursion, y-axis is themean hitting time (normalized by dividing by the mean hitting time with δ = 0). Thepicture was produced from (3.7)–(3.8) by varying K.[3] P. M. Gleiss, P. F. Stadler, A. Wagner, and D. A. Fell, Relevant cycles inchemical reaction networks, Advances in Complex Systems, 4 (2001), pp. 207–226.[4] D. J. Higham, Greedy pathlengths and small world graphs, Tech. Rep. 8, University ofStrathclyde, Glasgow, UK, 2002.[5] , A matrix perturbation view of the small world phenomenon, SIAM J. Matrix Anal.Appl., (to appear (2003)).[6] S. D. Kamvar, T. H. Haveliwala, C. D. Manning, and G. H. Golub, Extrapolationmethods for accelerating pagerank computations, Proceedings of the Twelfth InternationalWorld Wide Web Conference, (2003).[7] C. Moler, The world’s largest matrix computation, MATLAB News and Notes, October(2002).[8] J. M. Montoya and R. V. Sole´, Small world patterns in food webs, Journal of Theo-retical Biology, 214 (2002), pp. 405–412.[9] R. Motwani and P. Raghavan, Randomized Algorithms, Cambridge University Press,1995.[10] M. E. J. Newman, The structure of scientific collaboration networks, Proc. Natl. Acad.Sci., 98 (2001), pp. 404–409.[11] , The structure and function of complex networks, SIAM Review, 45 (2003), pp. 167–256.[12] M. E. J. Newman, C. Moore, and D. J. Watts, Mean-field solution of the small-worldnetwork model, Physical Review Letters, 84 (2000), pp. 3201–3204.83NA03 Dundee 2003[13] J. R. Norris, Markov Chains, Cambridge University Press, 1997.[14] S. H. Strogatz, Exploring complex networks, Nature, 410 (2001), pp. 268–276.[15] A. Wagner and D. A. Fell, The small world inside large metabolic networks, Proc.Roy. Soc. London, B., 268 (2001), pp. 1803–1810.[16] D. J. Watts, Small Worlds: The Dynamics of Networks between Order and Randomness,Princeton University Press, 1999.[17] D. J. Watts and S. H. Strogatz, Collective dynamics of ‘small-world’ networks, Na-ture, 393 (1998), pp. 440–442.84

0.9 Expected Number of Teleporting Jumps N o r m a l i z e d M e a n H i t t i n g T i m e

Random Walks in Biology,

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     Edinburgh Research Explorer                                      Finite differences in a small worldCitation for published version:Higham, DJ 2003, Finite differences in a small world. in DF Griffiths & GA Watson (eds), Proceedings of the20th Biennial Conference on Numerical Analysis, Dundee. The University of Dundee, Dundee, pp. 81-84.Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Peer reviewed versionPublished In:Proceedings of the 20th Biennial Conference on Numerical Analysis, DundeeGeneral rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 09. May. 2019Higham NA03 Dundee 2003Finite Differences in a Small WorldDesmond J. Higham(1)Department of Mathematics,University of Strathclyde,Glasgow G1 1XH, Scotlanddjh@maths.strath.ac.uk1 IntroductionMany complex networks in nature exhibit two properties that are seemingly at odds. Theyare clustered—neighbors of neighbors are very likely to be neighbors—and they are smallworlds—any two nodes can typically be connected by a relatively short path. Watts andStrogatz [17] referred to this as the small world phenomenon and proposed a networkmodel that was shown through simulation to capture the two properties. The modelincorporates a parameter that interpolates between fully local and fully global regimes. Asthe parameter is varied the small world property is roused before the clustering property islost. These computations were later backed up by a semi-heuristic mean-field analysis in[12]. Motivated by [17] and the book [16], many authors have shown experimentally thatthe small world phenomenon can be found across a range of areas in science, medicine andtechnology [3, 8, 11, 10, 14, 15]. In this work we show that the small world phenomenonalso emerges through a local-to-global cutoff in the context of Markov chains. Our modelis based on a simple random walk and hence is relevant to many physical, sociological,epidemiological and computational applications [1, 2, 9, 13]. In particular, it may beregarded as a teleporting random walk, of the type used by the search engine Google tolocate pages on the web [6, 7], restricted to a simple ring network.An extremely attractive feature of our model is that it can be analyzed rigorously byusing tools from numerical analysis.Full details of the material summarized here can be found in [5]. Recent work that dealswith the question of navigation in a small world setting appears in [4].2 ModelOur model is a perturbed version of a period random walk on a ring. Imagine the numbers1, 2, 3, . . . , N arranged like the hours on a clock. Take a periodic random walk aroundthis clock—at each time level flip a fair coin; if it lands heads, move clockwise, and if itlands tails, move anticlockwise. Now alter the process so that before taking each step youtoss another, biased, coin that lands heads with small probability δ. In the event thatthe biased coins lands heads, then pick a number between 1 and N uniformly at randomand move there. Otherwise, follow the fair coin procedure above. In this model, mostof the time we follow a traditional periodic random walk, but occasionally (when thebiased coin lands heads) we take a teleporting jump across the network. This teleportingidea is used by search engines such as Google to locate pages on the web. (Traverse theweb by following links out of pages, but, to avoid reaching a cycle or a dead end, take(1)Supported by a Research Fellowship from the Leverhulme Trust81NA03 Dundee 2003 Highamthe occasional jump). Teleporting is very similar in spirit to the rewiring/shortcuttingprocess that has been observed to produce small world networks of the type described insection 1, and we will show that an analogous cutoff effect arises.3 ResultThe small world phenomenon for networks involves the pathlength between nodes. Anal-ogously, we may study the small world phenomenon for our random walk model bymeasuring the mean hitting time, that is, the average number of steps that it takes toreach B starting from A, where A,B ∈ {1, 2, . . . , N} are chosen uniformly at random.The problem of analysing the mean hitting time can be set up as a linear algebraic systemof dimension N−1. We are interested in behaviour for large systems, that is, the N →∞limit.The key to the analysis is to identify the mean hitting time system as a finite differenceapproximation to an underlying continuum limit. The continuum limit itself dependsupon N (i.e. the grid spacing), but careful analysis shows that convergence takes placein an appropriate sense, so that limiting behaviour may be captured.The analysis shows that it is natural to let the teleporting jump probability, δ, scale withN according toδ =KN2, for fixed K.In this regime, we find that, up to O(N−1) terms, the mean hitting time, normalized bythe mean hitting time when δ = 0, has the formy =6Kx, (3.7)wherex =√2K2 tanh√2K2− 1 (3.8)is an expression for the expected number of teleporting jumps taken per random walk.We may now examine the small world property of the model by plotting y against x asK is varied. This shows how the introduction of shortcuts (teleporting jumps) makes iteasier to get around the network. Figure 3.5 gives the picture. We see that there is anabrupt drop in the hitting time as x increases beyond x ≈ 10−1. This “order 1 shortcutcutoff effect” agrees with the fundamental observation of Watts and Strogatz [17] fortheir small world network model.References[1] H. C. Berg, Random Walks in Biology, Princeton University Press, 1983.[2] H. Caswell, Matrix Population Models, Sinauer Associates, Inc., Sunderland, MA, sec-ond ed., 2001.82Higham NA03 Dundee 200310−6 10−4 10−2 100 102 10400.10.20.30.40.50.60.70.80.91Expected Number of Teleporting JumpsNormalized Mean Hitting TimeFigure 3.5: x-axis is the average number of teleporting jumps per excursion, y-axis is themean hitting time (normalized by dividing by the mean hitting time with δ = 0). Thepicture was produced from (3.7)–(3.8) by varying K.[3] P. M. Gleiss, P. F. Stadler, A. Wagner, and D. A. Fell, Relevant cycles inchemical reaction networks, Advances in Complex Systems, 4 (2001), pp. 207–226.[4] D. J. Higham, Greedy pathlengths and small world graphs, Tech. Rep. 8, University ofStrathclyde, Glasgow, UK, 2002.[5] , A matrix perturbation view of the small world phenomenon, SIAM J. Matrix Anal.Appl., (to appear (2003)).[6] S. D. Kamvar, T. H. Haveliwala, C. D. Manning, and G. H. Golub, Extrapolationmethods for accelerating pagerank computations, Proceedings of the Twelfth InternationalWorld Wide Web Conference, (2003).[7] C. Moler, The world’s largest matrix computation, MATLAB News and Notes, October(2002).[8] J. M. Montoya and R. V. Sole´, Small world patterns in food webs, Journal of Theo-retical Biology, 214 (2002), pp. 405–412.[9] R. Motwani and P. Raghavan, Randomized Algorithms, Cambridge University Press,1995.[10] M. E. J. Newman, The structure of scientific collaboration networks, Proc. Natl. Acad.Sci., 98 (2001), pp. 404–409.[11] , The structure and function of complex networks, SIAM Review, 45 (2003), pp. 167–256.[12] M. E. J. Newman, C. Moore, and D. J. Watts, Mean-field solution of the small-worldnetwork model, Physical Review Letters, 84 (2000), pp. 3201–3204.83NA03 Dundee 2003[13] J. R. Norris, Markov Chains, Cambridge University Press, 1997.[14] S. H. Strogatz, Exploring complex networks, Nature, 410 (2001), pp. 268–276.[15] A. Wagner and D. A. Fell, The small world inside large metabolic networks, Proc.Roy. Soc. London, B., 268 (2001), pp. 1803–1810.[16] D. J. Watts, Small Worlds: The Dynamics of Networks between Order and Randomness,Princeton University Press, 1999.[17] D. J. Watts and S. H. Strogatz, Collective dynamics of ‘small-world’ networks, Na-ture, 393 (1998), pp. 440–442.84

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Finite differences in a small world

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