We revisit Kemeny's constant in the context of Web navigation, also known as "surfing." We generalize the constant, derive upper and lower bounds on it, and give it a novel interpretation in terms of the number of links a random surfer will follow to reach his final destination

Levene, Mark

Loizou, George

English

Birkbeck Institutional Research Online

Kemeny’s Constant and the Random SurferMark Levene and George LoizouSchool of Computer Science and Information SystemsBirkbeck College, University of LondonLondon WC1E 7HX, U.K.{mark,george}@dcs.bbk.ac.ukAbstractWe revisit Kemeny’s constant in the context of Web navigation, also known as “surf-ing”. We generalize the constant, derive upper and lower bounds on it, and give it a novelinterpretation in terms of the number of links a random surfer will follow to reach hisfinal destination.1 IntroductionA Markov chain [11, Chapter 4] represents a stochastic process whereby the transition fromone state to another depends only on the current state the process is in. The probabilities of aMarkov chain are described using a transition matrix, which is nonnegative and row stochastic[5, Section 8.7]. Markov chains are used to model a wide number of physical phenomena [7,Chapter 7] and have been widely studied in the statistical literature since their inception byA.A. Markov in 1907. A Markov chain is finite ergodic if its transition matrix is finite, rowstochastic, and primitive [5, Definition 8.5.0].Kemeny’s constant [7, Corollary 4.3.6] gives an interesting quantity for finite ergodicMarkov chains. Kemeny [4, Section 11.5, Exercise 19] offered a prize for the first person tofind an intuitive interpretation for his constant; as far as we know, to date no intuitive andstraightforward explanation has yet been found.The aim of this paper is twofold. First, we derive a general formula for Kemeny’s constantinvolving the eigenvalues of the transition matrix that allows us to derive upper and lowerbounds on the constant. Second, we give an intuitive and straightforward interpretation ofKemeny’s constant in terms of user navigation through the World-Wide-Web.We now briefly set the scene for our Markov chain model of the Web. One of the mainactivities of users interacting with the Web is that of navigation (colloquially known as “surf-ing”) whereby users follow links and browse the destination Web pages. During the process ofnavigation users often experience disorientation and lose track of the context, causing them tobe unsure how to proceed to satisfy their original goal; this problem is known as the navigationproblem [9, 10]. Understanding user navigation patterns and their underlying distribution isimportant since it can lead to better Web site design [1] with the intention of saving users’effort by directly guiding them to the most relevant Web pages.We view the Web as a finite irreducible Markov chain, where the probabilities attachedto transitions denote the expected utility the user attains from following the corresponding1links. This view is naturally realized by considering the user’s home page as an artificialstarting point for all navigation sessions and assuming that there is a positive probability(however small) of jumping to any other relevant Web page. During a navigation session theuser follows links according to the transition probabilities and may eventually return to hishome page. Thus a navigation session amounts to a random walk through the Markov chainthat is terminated once the user reaches his final destination, i.e., he has no further incentiveto continue surfing.Imagine a random surfer who is following links according to the transition probabilities.At some stage our random surfer is “lost” and does not know the state he is at and wherehe is heading for. We show that in this context Kemeny’s constant can be interpreted as themean number of links the random surfer needs to follow before reaching his destination.2 General Bounds on Kemeny’s ConstantLet A be an n×n real or complex matrix. Assume that λ is a nonzero eigenvalue of A withalgebraic multiplicity 1, and let λ2, λ3, . . . , λn denote the remaining eigenvalues of A includingalgebraic multiplicities. Let x and yH , where yH denotes the conjugate transpose of y, be rightand left eigenvectors, respectively, corresponding to λ, so that Ax = λx and yHA = λyH .These two eigenvectors, which are unique up to a scalar multiple, are nonorthogonal, and cantherefore be normalized so that yHx = 1 [5, Lemma 6.3.10].We next consider the matrix A− λxyH , which has the same Jordan canonical form as Aexcept that its 1×1 block [λ] is replaced by the 1×1 block [0]. Moreover, the eigenvalues ofλI − (A− λxyH) are λ, λ− λ2, λ− λ3, . . . , λ− λn, so they are all nonzero. The fundamentalmatrix associated with A and λ isZ ≡(λI − (A− λxyH))−1. (1)The eigenvalues of Z are λ−1, (λ− λ2)−1, (λ− λ3)−1, . . . , (λ− λn)−1, sotrZ =1λ+n∑i=21λ− λi . (2)Since x is an eigenvector of Z−1, it is an eigenvector of Z, soZx =1λx and yHZx =1λyHx =1λ. (3)The number K(A, λ) ≡ trZ − yHZx is known as Kemeny’s constant [4, Section 11.5,Exercise 19]. Thus from (2) and (3) it follows thatK(A, λ) =n∑i=21λ− λi . (4)2From (4) it follows that|K(A, λ) | ≤ n− 1|λ− λˆ | , (5)where λˆ denotes an eigenvalue of A that is closest to λ.If A and λ are both real, then the right-hand side of (4) is also real; it is positive ifλ > Reλj for all j = 2, 3, . . . , n. (This is the situation when λ is the Perron eigenvalue of anirreducible nonnegative matrix A.)If all the eigenvalues of A are real, then from (4)(n− 1)min{1λ− λ2 ,1λ− λ3 , . . . ,1λ− λn}≤ K(A, λ). (6)In order for A to possess a real spectrum, it is sufficient that there be a positive definite ma-trix Q such that AH = QAQ−1. In this event, we have A ≡ Q1/2AQ−1/2 = Q−1/2AHQ1/2 =AH , so A is Hermitian and similar to A. (We make use of this sufficient condition in the nextsection but only for a positive diagonal Q.)Suppose that A and λ are both real, and λ ≥| λj | for all j = 2, 3, . . . , n. Then, sincenon-real eigenvalues occur in conjugate pairs, it follows from (4) that0 <n− 12λ≤ (n− 1) min2≤j≤nλ−Reλj|λ− λj |2 ≤ K(A, λ). (7)This bound is valid, for example, if A is irreducible and nonnegative, and λ is its Perroneigenvalue. If all the eigenvalues of A are equimodulus, then the lower bound is attained.If A is real, nonnegative, and irreducible, then its Perron eigenvalue [5, Theorem 8.4.4] λis positive and algebraically simple, and thus A and λ satisfy (4) and (7). Moreover, there arepositive left and right eigenvectors of A associated with λ. If, in addition, A is row stochastic,then (4) and (7) hold with λ = 1, and an associated eigenvector is h, the vector all of whoseentries are 1.3 Specific Bounds on Kemeny’s ConstantLet P be the n × n transition matrix of a finite irreducible Markov chain. Then λ = 1is an algebraically simple eigenvalue of P , Ph = h, and P has a positive left eigenvectorpi = (pi1, pi2, . . . , pin)T associated with λ, so that piTP = piT . If we normalize pi by requiringthat piTh = 1, we obtain the (unique) stationary probability vector of P .In [6, Theorem 1] the following result regarding a generalisation of the fundamental matrixof a finite irreducible Markov chain was presented.Theorem 3.1 Let β and g be any two vectors such that βTh and piT g are nonzero. ThenI − P + gβT is nonsingular.3Proof: If x 6= 0 and (I−P+gβT )x = 0, i.e., I−P+gβT is singular, then (I−P )x = −(βTx)g,which yields piT (I −P )x = 0 = −(βTx)piT g. Hence βTx = 0 and so (I −P )x = 0. Therefore,x = γh (γ being a nonzero constant) whence βTh = 0, since βTx = 0. We could also showthat piT g = 0. 2The matrix Z ≡ (I − P + gβT )−1 is an analogue of (1). For a finite irreducible Markovchain, we may take βT = piT and g = h, in which case we haveZ ≡ (I − P + hpiT )−1 (8)with piTh = 1, Zh = h and piTZ = piT .As a result of Theorem 3.1, (8) is only one of an infinite number of choices for Z, and aform of it that does not involve pi can be obtained. For example, let g = h and let βT be anyrow vector such that βTh = 1. ThenZ ≡ (I − P + hβT )−1 (9)with βTh = 1, Zh = h, and βTZ = piT .Since βT and g can be shown to be left and right eigenvectors of P associated with thealgebraically simple eigenvalue λ = 1, it follows from (2) that all these Z’s are associated withthe same Kemeny constant, namelyK(P ) = K(P, 1) =n∑i=211− λi , (10)where λi is the ith eigenvalue of P .A finite irreducible Markov chain is said to be reversible, if piiPij = pijPji for all i, j, thatis, P and the positive definite diagonal matrix Π = diag(pi1, pi2, . . . , pin) satisfy ΠP = P TΠ,so that P T = ΠPΠ−1. Since this condition ensures that all the eigenvalues of P are real, wecan order them so that− 1 ≤ λn ≤ λn−1 ≤ · · · ≤ λ2 < λ = 1. (11)If, additionally, the Markov chain is ergodic then −1 < λn as well. (See [3], for example,for upper bounds on λ2.) Thus, when a finite irreducible Markov chain is reversible, (10) and(11) yieldn− 12≤ K(P ) ≤ n− 11− λ2 ,and the lower bound is strict if the chain is ergodic.4 An Interpretation of Kemeny’s ConstantLet P be the transition matrix of a finite irreducible Markov chain. The expected numberof steps required to reach state sj , when starting a Markov chain from state si, is called themean first passage time from si to sj , and is denoted by mij ; by convention mii = 0. (Since4P is irreducible pii 6= 0 for all i.) Let M = (mij) be the mean first passage matrix [4, p. 455]for such a chain. Then, by [4, Theorem 11.16]mij =zjj − zijpij.Since Zh = h and piTh = 1,K(P ) = trZ − piTZh =n∑j=1zjj − 1=n∑j=1(zjj − zijpij)pij=n∑j=1pijmij . (See [4,p. 468].)In order to give our interpretation of Kemeny’s constant involving a random surfer, wecan rewrite K(P ) asK(P ) =n∑i=1piin∑j=1pij mij ,due to the fact that piTh = 1.The term∑j pij mij gives the mean first passage time from state i when the destinationstate is unknown; denote this term by Mi. It follows that Kemeny’s constant, given by∑i pii Mi, can be interpreted as the mean first passage time from an unknown starting stateto an unknown destination state.Imagine therefore a random surfer who is following links according to the transition prob-abilities. At some stage our random surfer is “lost” and does not know the state he is at andwhere he is heading for. In this context Kemeny’s constant can be interpreted as the meannumber of links the random surfer follows before reaching his destination. Thus the randomsurfer is not “lost” anymore, he just has to follow K(P ) random links and he can expect toarrive at his final destination. It is therefore in this context that the upper and lower boundsfor K(P ) in Sections 2 and 3 should be considered. For the Web, n is very large (greater than109, see [2] or [8]) and therefore a random walk whose length is of the order of the boundsthat we present is not feasible. However, if we can restrict the Web graph in some mannerto an irreducible (or ergodic) subset (i.e, a Web subgraph that represents an irreducible (orergodic) Markov chain), for example, by selecting only the pages that satisfy a user query,then the bounds we give may be useful.References[1] D.J. Aldous. Reorganizing large web sites. American Mathematical Monthly, 108:16–27,2001.[2] A. Broder, R. Kumar, F. Maghoul, P. Raghavan, A. Rajagopalan, R. Stata, A. Tomkins,and J. Wiener. Graph structure in the web. Computer Networks and ISDN Systems,30:309–320, 2000.5[3] P. Diaconis and D. Stroock. Geometric bounds for eigenvalues of Markov chains. Annalsof Applied Probability, 1:36–61, 1991.[4] C.M. Grinstead and J.L. Snell. Introduction to Probability. American MathematicalSociety, Providence, RI, 1997.[5] R.A. Horn and C.R. Johnson. Matrix Analysis. Cambridge University Press, Cambridge,U.K., 1985.[6] J.G. Kemeny. Generalization of a fundamental matrix. Linear Algebra and its Applica-tions, 38:193–206, 1981.[7] J.G. Kemeny and J.L. Snell. Finite Markov Chains. D. Van Nostrand, Princeton, NJ,1960.[8] S. Lawrence and C.L. Giles. Accessibility of information on the web. Nature, 400:107–109,1999.[9] M. Levene and G. Loizou. Navigation in hypertext is easy only sometimes. SIAM Journalon Computing, 29:728–760, 1999.[10] M. Levene and G. Loizou. A probabilistic approach to navigation in hypertext. Infor-mation Sciences, 114:165–186, 1999.[11] S.M. Ross. Stochastic Processes. John Wiley & Sons, New York, NY, 2nd edition, 1996.6

A probabilistic approach to navigation in hypertext.

Accessibility of information on the web.

Finite Markov Chains.

Generalization of a fundamental matrix. Linear Algebra and its Applications,

Geometric bounds for eigenvalues of Markov chains.

Graph structure in the web.

Introduction to Probability.

Matrix Analysis.

Navigation in hypertext is easy only sometimes.

Reorganizing large web sites.

Stochastic Processes.

Kemeny's constant and the random surfer

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Mark Levene

George Loizou

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American Mathematical Monthly

Kemeny's Constant and the Random Surfer

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Kemeny's constant and the random surfer

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