In this paper the concept of page rank for the world wide web is discussed.
The possibility of describing the distribution of page rank by an exponential law is considered.
It is shown that the concept is essentially equal to that of status score, a centrality measure discussed already in 1953 by Katz. A structural classification of users in the web is given in terms of graph theoretical concepts

Hoede, C.

English

In this paper the concept of page rank for the world wide web is discussed. The possibility of describing the distribution of page rank by an exponential law is considered. It is shown that the concept is essentially equal to that of status score, a centrality measure discussed already in 1953 by Katz. A structural classification of users in the web is given in terms of graph theoretical concepts

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On Page Rank

University of Twente Research Information

On Page RankC. HoedeFaculty of Electrical Engineering, Mathematics and Computer ScienceUniversity of TwenteP.O.Box 2177500 AE Enschede, The NetherlandsAbstractIn this paper the concept of page rank for the world wide web is discussed.The possibility of describing the distribution of page rank by an exponential law isconsidered.It is shown that the concept is essentially equal to that of status score, a centralitymeasure discussed already in 1953 by Katz. A structural classification of users in theweb is given in terms of graph theoretical concepts.Key words: page rank, status score, graphAMS Subject Classifications 2000: 05C99, 68M10, 94C151 IntroductionThe concept of page rank was introduced by Brin and Page in 1998 [1]. The world wideweb, to be called just web from now on, can be described by a directed graph. For graphtheoretical terminology we refer to one of the many text books. As there is no uniformityof terminology we have to state that a directed graph ~G(V, A) consists of two sets, the setV of vertices and the set A of arcs. A is a subset of V × V , the Cartesian product of Vwith V, where V is a set of labeled elements. The arcs may be labeled as well. Labelsof vertices describe the sites in the web. If site v can connect to site w then arc (v,w) ispresent in ~G. The label l of (v,w) can be chosen to express the extent to which this isdone. In principle ~G is a complete directed graph. The arcs going out from a vertex v,including the loop (v,v), to the N vertices of ~G may carry labels li, chosen such thatN∑i=1li(v, vi) = Cv, (1)1where i refers to vertex vi in some numbering of V and Cv is a constant specific forvertex v.Based on the pattern of choices made by the sites, the problem is to determine a measurefor the centrality of a site. The measure should incorporate the fact that some sites aremore often visited than others. This is what the concept of page rank is about. We willcome back to the concept more explicitly in Section 3.2 Description of a measure distributionPresumably a minority will have a high rank and the number a(r) of sites with rank r, stillto be defined, will diminish from the minimum value Rm for r to the maximum value RMfor r, see Figure 1.rxa(r)R Rm MFigure 1: A measure distributionWe focus on the arced area, giving the total number T(x) of sites greater than or equalto x, where, as N is very large, we have gone over from a discrete description to a continuousone.T (x) =∫ RMxa(r)dr.In the literature for T(x) assumptions are made, in particular that T(x) behaves asT (x) = C1x−α + C2, (2)with α > 1.2We want to see which consequences such an assumption has, given α, Rm and RM , forthe average rank R̄.From T (RM) = 0 followsC2 = −C1.R−αMand from T (Rm) = N followsC1.R−αm + C2 = N,leading toC1 = N.1[R−αm − R−αM ]andC2 = −N.R−αM[R−αm − R−αM ].SoT (x) = N.[x−α − R−αM ][R−αm − R−αM ]. (3)Differentiating Equation (1) to x on both sides givesdT (x)dx= −a(x),from which we derivea(r) = N.α[R−αm − R−αM ].r−α−1.Now R̄ follows asR̄ =1N∫ RMRma(r).rdr =α[R−αm − R−αM ]∫ RMRmr−αdr = ... =αα − 1.Rm.1 − ( RmRM)α−11 − ( RmRM)α. (4)If Rm = f.RM we have3R̄ =αα − 1.Rm.1 − fα−11 − fα.If, for example, α = 2 and f = 12we find R̄ = 43.Rm andlimf→1R̄ =αα − 1.Rm.α − 1α= Rm,Equation (4) is an important consequence of the assumptions we made, in particularthat on the shape of the function T(x) in Equation (2).3 Page rank and centralityIn a directed graph ~G each vertex j has an outdegree od(j) and an indegree id(j), being thenumber of outgoing arcs and incoming arcs respectively. For the complete graph both areN. We will discuss this case first as, in principle, it covers all other cases.The labels on the arcs are assumed to satisfy Equation (1). We now interpret Cv as theintensity with which from site v sites are visited. ”Intensity” may be seen as a frequency,say the number of jumps to other, possibly also the same, sites per unit of time. The labelsli, i = 1,2,...,N, can then be seen as differentiating frequencies of jumps to these sites.As formulated by Volkovich, Litvak and Donato [7], Page Rank PR is a method forevaluating ” popularity of nodes in information networks”. It is therefore a kind of central-ity measure. The basic modeling given by Brin and Page [1] is by an equation for the pagerank values PR(i), i = 1,2,...,N. This equation is to capture the idea that the rank for asite depends on the ranks of the sites that choose the site to jump to and on the frequencywith which this is done. This basic modeling is expressed by Equation (5)PR(i) =N∑j=1lj(vj , vi).PR(j). (5)The important next step is, of course, the choice of the labels lj.First we can choose Cv = 1 for all vertices v of ~G. The labels can then be interpreted asprobabilities, the setting chosen in [7]. Second, we can, for each vertex v, determine a set Ovof outgoing arcs of which one is chosen to jump along. This with probability c, 0 ≤ c ≤ 1,whereas a jump following one of the other outgoing arcs happens with probability 1-c. Forthe labels this means that they are c|Ov|in the first case and 1−cN−|Ov|in the second case. Ovmight be called a set of preferred outgoing arcs. If Ov = ∅, then the vertex is called a”dangling” vertex in [7]. In that paper the jump is either along a preferred arc with totalprobability c or to any other vertex with total probability 1-c, meaning that also preferred4arcs are to be taken into consideration. If D is the set of dangling vertices this leads toEquation (6)PR(i) = c∑j→i1od(j).PR(j) +cN∑j∈DPR(j) +1 − cN, i = 1, ..., N, (6)given in [7]. The notation j → i here indicates that (j,i) is a preferred outgoing arc ofj. This modeling slightly deviates from the one given above. Other choices of the labelsare possible. A variety of page rank equations is therefore possible.For our purpose we will stick to the general Equation (5).The main point of this section is that the concept of page rank is not new. Centralityof actors in social networks has been considered in sociology as far back as 1953 by Katz[5]. The measure searched for was that for influence or status of an actor. Each actor issupposed to influence all other actors, and Katz’s approach was to calculate all weightedpaths going out from a vertex. That is, the social network was modeled as a completedirected graph ~G, with labels on the arcs, possibly zero, and the weight of a path fromvertex v was defined as the product of the weights on the arcs of the path.Now these weights are just labels and the sum of the weights of arcs going out from v maybe Cv. When we now remark that the paths from vertex v can all be seen as consistingof one first arc, to a neighbour, followed by a path from that neighbour, then defining S,status, as the sum of weights of all outgoing paths we obtainS(i) =N∑j=1li(vi, vj).S(j). (7)But now we see that Equation (5) is precisely Equation (7), when all arcs in the graph~G for the web are reversed, (v,w) becomes (w,v), while the labels are kept the same, i.e.puttingli(vi, vj) = lj(vj , vi).The conclusion is that the page rank equation is nothing but the status equation for the”inverted web”. This means that all the literature on that centrality measure is applicable.As a final remark we point out that both equations determine page rank respectivelystatus up to a constant factor.54 Page rank and in-degreeTo discuss the relationship between page rank and indegree in the webgraph, we firstdiscuss a status score introduced by Hoede [3] in an internal report, but contained in theGRADAP manual for software used in the social sciences [2].Let W be the adjacency matrix A of ~G but with entries 1 for Ai,j replaced by entriesWi,j. As is well-known (Al)i,j gives the number of walks of length l from vertex i to vertexj. (W l)i.j gives the sum of weights of walks of length l from vertex i to vertex j. Statusseen as sum of the weights of all paths starting in vertex i givesS(i) =∞∑l=1N∑j=1(W l)i,j (8)As(W l)i,j =N∑k=1(W l−1)i,k(W )k,jfor l > 1, we haveS(i) =N∑j=1(W )i,j +N∑k=1(W )i,k.S(k).We now put d(i) =∑Nj=1(W )i,j, the i-th component of a vector d called the directinfluence vector. S(i) is the i-th component of the status score vector. This vector satisfiesthe equationS = d + W.S. (9)However, for this the series of elements of powers of W should converge. This wasone of the main problems in the theory. A solution can be obtained by using the idea ofrenormalization as common in theoretical physics. Instead of attaching attenuation factorsto weights of arcs to control the convergence, as was e.g. done by Hubbell [4], we can takeEquation (9) as definition of S to obtainS = [I − W ]−1.dfor the status score. Only in case the matrix I - W is not invertible, which can becorrected by a minute change in W, this formula cannot be applied. Thus convergenceproblems are avoided.6We now recall that page rank is essentially status score for the inverted web. Develop-ment of [I − W ]−1 in Equation (10) shows thatS = I.d + W.d + W 2.d + .... (10)The first term in this development is I.d so, in that approximation, S(i) = d(i), thedirect influence of vertex i. But that is essentially the weighted outdegree in the invertedweb. So the weighted indegree in the web in first approximation determines the page rankin the web.A second way to arrive at this result is to consider Equation (5). Without any furtherinformation, a first approximate solution for the page rank is obtained by putting PR(j)equal to the average page rank PR, leading toPR(i) =N∑j=1lj(vj , vi).PR.But, apart from the factor PR, this is precisely the weighted indegree of vertex i.We refer to [7] for a more refined treatment of the relationship between page rank andindegree in the web, that turn out to have the same distribution, as might be expected.5 Structural classification of sitesThe centrality of a web site, its ”popularity” if one wants, is an obvious concept to consider.However, there are other interesting questions. Is there a way to distinguish between siteson the basis of structural aspects?In this case we consider a, not necessarily complete, directed graph ~G without any labelson the arcs. Generalization to a complete graph with labels can come later.A first distinction between vertices of ~G(V, A) can be made on the basis of outdegreeod(v) and indegree id(v), v ∈ V .A vertex with id(v)=0 is a source, whereas a vertex v with od(v)=0 is a sink. Clearly inthe web the sites that are sources are extremely impopular, whereas the sites that are sinksare the ones to which jumps are made, but as an end-destination for a path of jumps fromsite to site by some user of the web. A first natural feature to consider is the quotientQ(v) =id(v)od(v)for vertex v. The range of Q is from zero to infinity. Q partitions the vertex set V intoclasses of vertices with the same value. Within a class the actual values of id(v) or od(v)may distinguish between the class members. Consider Figure 2.7123456Figure 2: A graph ~GFor Q we find Q(1) = ∞, Q(2) = 0, Q(3) = 1, Q(4) = 2. Q(5) = 0, Q(6) = 1.In a large web different sites v may have identical values for id(v) and od(v). Yet theremay be a feature that distinguishes them. Consider the out-neighbours of the verticesv1 and v2 with od(v1) = od(v2). There may be significant difference between the waysthese neighbouring sets are connected. In Figure 3 we consider two vertices with the sameout-degree 4.The ”out-neighbourhood” of v1 contains only one arc, whereas that of v2 contains sevenarcs. Various measures to distinguish between these two situations are possible.One, simple, way to compare the ”complexity” of two out-neighbourhoods is to considerthe ”density” of the graph induced by the neighbouring vertices. In our example, we maycompare the numbers 1 and 7 with the maximum number of arcs possible. Including loopsthere are 42 = 16 possible arcs, which would lead to complexities 116and 716for the out-neighbourhoods of v1 and v2.Another way, more akin to the concept of complexity in graph theory, would be to considerall outgoing spanning trees on the vertex set consisting of a vertex v and all its out-neighbours, with root in v. For vertex v1 there would be two such trees, for vertex v2only one arc is obligatory and the loop can be left out of consideration. The problem isthen to count the outgoing spanning trees in the graph of Figure 4, where the obligatoryarc is drawn bold. The number may increase rapidly for larger neighbourhoods. For theundirected complete labeled graph on n vertices there are nn−2 different spanning trees.8v v1 2Figure 3: Two out-neighbourhoodsv 2Figure 4: Obligatory arc9For the ”in-neighbourhood” an analogous calculation can be made. Let Cin(v) andCout(v) denote these complexities. Then, next to id(v), od(v) and Q(v), we can distinguishvertices on the basis of Cin(v) and Cout(v). Clearly a vertex for which Cin(v) = Cout(v) = 0has a position in the web, different from one for a vertex w for which Cin(w) and Cout(w)are both high. Note that these numbers may be normed by dividing by the maximumvalue that they can assume, like we did for the simple measure.The generalization to labeled directed graphs is immediate by replacing indegree andoutdegree by weighted indegree and outdegree, possibly also allowing that ~G is completenow. For the complexities one might, for example, not count the number of arcs in anin-neighbourhood or an out-neighbourhood, but the sum of the weights of such arcs. Inall cases the interpretation should make sense.The quotient Q may be used, also in the weighted version, in the following way. Verticesof ~G0, the whole web graph, may be partitioned into two sets. A set for which Q ≥ 1 anda set of vertices for which Q < 1. The choice of the constant 1 is arbitrary here, one mightalso consider the set with minimum value of Q and the rest of the vertices. The verticeswith the higher values of Q may be said to get more attention from other vertices thanthey give attention to other vertices.Now we delete the vertices of the second set and their adjacent arcs and focus on the graph~G1 induced by these vertices. Note that the degrees respectively the weighted values ofindegrees and outdegrees now may have changed. A vertex that received much attentionfrom vertices of the second set, in ~G0, may receive little attention from the other verticesin ~G1. Again Q’s can be considered, now in ~G1, and a second partitioning takes place.The ”surviving” vertices, for which Q ≥ 1, then induce a graph ~G2. The procedure can berepeated to any desired extent.For the graph in Figure 2, ~G0, the first partition yields {1,3,4,6} for the first set and{2,5} for the second set. Figure 5 gives the graph ~G1.For ~G1 we find Q(1) = ∞, Q(3) = 0, Q(4) = 1 and Q(6) = 0. Consequently thepartitioning gives {1,4} and {3,6}. ~G2 is induced by {1,4} and consists of the single arc(4,1). In ~G2 Q(1) = ∞ and Q(4) = 0, partitioning V ( ~G2) into {1} and {4}. The outcomeof a simple procedure like this is a ranking:{2, 5} ≻ {3, 6} ≻ {4} ≻ {1}.As getting to the top of such a ranking requires incoming arcs from vertices that rankhigher and higher themselves, the basic idea of the page rank Equation (5), as well as thestatus score Equation(7) is captured. The procedure does not give a value for the ranking,although the values for Q calculated in the procedure may be used to distinguish betweendifferent vertices in a class for which Q < 1 and is therefore not considered anymore.101346Figure 5: ~G1References[1] Brin, S. and L. Page, The Anatomy of a Large-Scale Hypertextual Web Search Engine,Computer Networks and ISDN Systems 33, 107-117, (1998).[2] GRADAP (GRAph Definition and Analysis Package), Iec ProGAMMA, Groningen,ISBN 90-5144-004-9, (1989).[3] Hoede, C., A new status score for actors in a network, Memorandum nr.243, Depart-ment of Applied Mathematics, University of Twente, ISSN 0169-2690, (1979).[4] Hubbell, C.H., An Input-Output approach to clique identification, Sociometry, Vol.28,no.4, 377-399, (1965).[5] Katz, L., A new status score derived from sociometric analysis, Psychometrika, Vol.18,no.1, 39-43, (1953).[6] Litvak, N., W.R.W. Scheinhardt and Y. Volkovich, In-degree and Page Rank of Webpages: Why do they follow similar power laws, Memorandum nr.1807, Department ofApplied Mathematics, University of Twente, ISSN 1874-4850, (2006).[7] Volkovich, Y., N. Litvak and D. Donato, Determining factors behind the Page Ranklog-log plot, Memorandum nr.1823, Department of Applied Mathematics, Universityof Twente, ISSN 1874-4850, (2007).11

A new status score derived from sociometric analysis,

A new status score for actors in a network, Memorandum nr.243,

An Input-Output approach to clique identiﬁcation,

Deﬁnition and Analysis Package), Iec ProGAMMA,

Determining factors behind the Page Rank log-log plot,

In-degree and Page Rank of Web pages: Why do they follow similar power laws, Memorandum nr.1807,

The Anatomy of a Large-Scale Hypertextual Web Search Engine,

On Page Rank

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