Ranking Systems: The PageRank Axioms

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1 Rnking Systes: The PgeRnk Axios Alon Altn nd Moshe Tennenholtz Fculty of Industril Engineering nd Mngeent Technion Isrel Institute of Technology Hif Isrel 31st My 2005 Astrct This pper initites reserch on the foundtions of rnking systes, fundentl ingredient of sic e-coerce nd Internet Technologies In order to understnd the essence nd the exct rtionle of pge rnking lgoriths we suggest the xiotic pproch tken in the forl theory of socil choice In this pper we del with PgeRnk, the ost fous pge rnking lgorith We present set of siple (grph-theoretic, ordinl) xios tht re stisfied y PgeRnk, nd oreover ny pge rnking lgorith tht does stisfy the ust coincide with PgeRnk This is the first representtion theore of tht kind, ridging the gp etween pge rnking lgoriths nd the theticl theory of socil choice 1 Introduction The rnking of gents sed on other gents input is fundentl to e-coerce nd ulti-gent systes (see eg [4, 16]) Moreover, the rnking of gents sed on other gents input hve ecoe centrl ingredient of vriety of Internet sites, where perhps the ost fous exples re Google s PgeRnk lgorith[11] nd ey s reputtion syste[15] One iportnt set of such rnking systes re pge rnking systes It is well known tht pge rnking is fundentl for serch technology, s well s for other pplictions A jor prole therefore is the study of the rtionle of using prticulr pge rnking lgorith Wht re the properties of prticulr pge rnking lgorith tht chrcterize nd differentite it fro other pge rnking lgoriths? In order to ddress this chllenge we introduce nd dpt the xiotic pproch, dopted in the theticl theory of socil choice, into the context of pge rnking If we tret the Internet s grph, where the nodes/pges re gents, nd the links originting fro node/pge p define the preferences of the corresponding gent (ie pge tht p links to is preferle to pge tht p does not link to) then the pge rnking prole ecoes the prole of ggregting individul rnkings into glol (socil) rnking Hence, the prole of pge rnking ecoes (novel) prole of socil choice In the clssicl theory of socil choice, s nifested y Arrow[1], set of gents/voters is clled to rnk set of lterntives Given the gents input, ie the gents individul rnkings, socil rnking of the lterntives is generted The theory studies desired properties of the ggregtion of gents rnkings into socil rnking In prticulr, Arrow s celerted ipossiility theore[1] shows tht there is no ggregtion rule tht stisfies soe inil requireents, while y relxing ny of these requireents pproprite socil ggregtion rules cn e defined The novel feture of the pge rnking setting is tht the set of gents nd the set of lterntives coincide Therefore, in such setting one y need to consider the trnsitive effects of voting For exple, if gent (ie pge) reports on the iportnce of (ie links to) 1

2 pge then this y influence the crediility of report y on the iportnce of gent c; these indirect effects should e considered when we wish to ggregte the infortion provided y the gents into socil rnking The theory of socil choice is n xiotic theory, nd consists of two copleentry perspectives: The nortive perspective: devise set of requireents tht socil ggregtion rule should stisfy, nd try to find whether there is socil ggregtion rule tht stisfies these requireents The descriptive perspective: given prticulr lgorith r for the ggregtion of individul rnkings into socil rnking, then r stisfies ny properties; the ojective is to find sll set of siple properties(k xios) tht re stisfied y r nd hs the dditionl feture tht every lgorith tht stisfies these properties ust coincide with r A result showing such set of properties is tered representtion theore nd cptures the exct essence of(nd ssuptions ehind) the use of the prticulr lgorith An excellent exple for the nortive perspective is Arrow s ipossiility theore entioned ove In [19] we presented such n pproch for rnking systes Mny efforts hve een invested in the descriptive pproch in the frework of the clssicl theory of socil choice In tht setting, representtion theores hve een presented for clssicl voting rules such s the jority rule over two lterntives[8] (see [9] for n overview) Tckling the descriptive pproch in the new Internet context, where the set of voters nd the set of lterntives coincide (ie the pge rnking context) reined n open jor chllenge In our work we ddress the ove chllenge y introducing representtion theore for PgeRnk Needless to sy tht PgeRnk[11] is the ost fous pge rnking procedure In prticulr, PgeRnk is the sis for Google s serch technology 1 [2]If we tret the Internet s strongly connected grph, where the nodes re the pges nd the edges re links etween pges, then PgeRnk cn e defined s the liit proility distriution reched in rndo wlk on tht grph Roughly speking, pge p 1 will e rnked higher thn pge p 2 if the proility of reching p 1 is greter thn the proility of reching p 2 We will show severl siple properties (clled xios) one y require pge rnking lgorith to stisfy nd prove tht the PgeRnk lgorith does stisfy these xios Then, we prove our in result: ny pge rnking lgorith tht does stisfy these xios ust coincide with PgeRnk! The only work tht we re filir with which dels with relted xiotiztion is the recent work on the xiotiztion of cittion indexes [12] This work dels however with the cse of nueric inputs (eg the inputs re not only grphs, s in pge rnking, ut include lso nueric esures for the nuer of cittions y ech node, nd y ech node for ech other node), nd (ost iportntly) the xios considered re nueric s well (eg when defining the xios we re llowed for coputtions such s division or trix ultipliction) Our i is quite different: we re fter ordinl, grph-theoretic requireents tht will provide sound nd coplete xiotiztion for PgeRnk This cretes ost significnt chllenge: while the PgeRnk lgorith is nueric nd is sed on the coputtion of eigenvectors, we re fter siple grph-theoretic properties tht will fully chrcterize the relted rnking procedure The clssicl theory of socil choice ly the foundtions to lrge prt of the rigorous work on the design nd nlysis of socil interctions Indeed, the ost clssicl results in the theory of echnis design (eg the Gird-Stterthwite [5, 17] theores) re pplictions of the theory of socil choice While econoic echnis design hd ecoe n extensive line of study in coputer science (see eg [10]) nd electronic coerce (see eg [7, 13, 3]), our work introduces nother connection etween lgoriths nd Internet technologies to the theticl theory of socil choice In the next section we define our setting nd soe preliinries, including the PgeRnk rnking syste In Section 3 we introduce five xios one y require to hold for ny pge rnking procedure, nd cli tht 1 In fct, rnking sed on siilr ides cn e found in other contexts s well See [14] for the use of PgeRnk-like procedure in the coprison of journls ipct 2

3 PgeRnk does stisfy these xios In Section 4 we show soe useful properties iplied y the xios In Section 5 we use these properties for proving tht ny pge rnking procedure tht does stisfy the xios should coincide with PgeRnk Further discussion of the pproch tken in this pper is presented in Section 6 This pper is suppleented y n ppendix which includes proofs of the theores 2 Pge Rnking The current prctice of the rnking of Internet pges is sed on the ide of coputing the liit sttionry proility distriution of rndo wlk on the Internet grph, where the nodes re pges, nd the edges re links ong the pges In order for the result of tht process will e well defined, we restrict our ttention to strongly connected grphs: Definition 21 A directed grph G = (V, E) is clled strongly connected if for ll vertices v 1, v 2 V there exists pth fro v 1 to v 2 in E The output of pge rnking procedure cn e viewed s liner ordering of set of lterntives: Definition 22 Let A e soe set A reltion R A A is clled n ordering on A if it is reflexive, trnsitive, coplete nd nti-syetric Let L(A) denote the set of orderings on A Nottion: Let e n ordering, then is the equlity predicte of Forlly, if nd only if nd Given the ove we cn define wht rnking syste is: Definition 23 Let G V e the set of ll strongly connected grphs with vertex set V A rnking syste F is functionl tht for every finite vertex set V ps every strongly connected grph G G V to n ordering F G L(V ) In order to define the PgeRnk rnking syste, we first recll the following stndrd definitions: Definition 24 Let G = (V, E) e directed grph, nd let v V e vertex in G Then: The successor set of v is S G (v) = {u (v, u) E}, nd the predecessor set of v is P G (v) = {u (u, v) E} We now define the PgeRnk trix which is the trix which cptures the rndo wlk creted y the PgeRnk procedure Nely, in this process we strt in rndo pge, nd itertively ove to one of the pges tht re linked to y the current pge, ssigning equl proilities to ech such pge Definition 25 Let G = (V, E) e directed grph, nd ssue V = {v 1, v 2,, v n } the PgeRnk Mtrix A G (of diension n n) is defined s: { 1/ S G (v j ) (v j, v i ) E [A G ] i,j = 0 Otherwise The PgeRnk procedure will rnk pges ccording to the sttionry proility distriution otined in the liit of the ove rndo wlk; this is forlly defined s follows: Definition 26 Let G = (V, E) e soe strongly connected grph, nd ssue V = {v 1, v 2,, v n } Let r e the unique solution of the syste A G r = r where r 1 = 1 The PgeRnk P R G (v i ) of vertex v i V is defined s P R G (v i ) = r i The PgeRnk rnking syste is rnking syste tht for the vertex set V ps G to P R G, where P R G is defined s: for ll v i, v j V : v i P R G v j if nd only if P R G (v i ) P R G (v j ) The ove defines powerful heuristic for the rnking of Internet pges, s dopted y serch engines[11] This is however prticulr nueric procedure, nd our i is to tret it fro n xiotic socil choice perspective, providing grph-theoretic, ordinl representtion theore for PgeRnk 3

4 c c () Vote y Coittee () Collpsing = = x (c) Proxy Figure 1: Sketch of severl xios 3 The Axios Fro the perspective of the theory of socil choice, ech pge in the Internet grph is viewed s n gent, where this gent prefers the pges (ie gents) it links to upon pges it does not link to The prole of finding socil ggregtion rule will ecoe therefore the prole of pge rnking The ide is to serch for siple xios, ie requireents we wish the pge rnking syste to stisfy Most of these requireents will hve the following structure: pge is preferle to pge when the grph is G if nd only if is preferle to when the grph is G Our i is to serch for sll set of xios tht cn e shown to e stisfied y PgeRnk The xios need to e siple grph-theoretic, ordinl properties, which do not refer to nueric coputtions In explining soe of the xios we will refer to Figure 1 For siplicity, while the xios re stted s if nd only if stteents, we will soetie ephsize in the intuitive explntion of n xio only one of the directions (in ll cses siilr intuitions hold for the other direction) The first xio is strightforwrd: Axio 31 (Isoorphis) A rnking syste F stisfies isoorphis if for every isoorphis function ϕ : V 1 V 2, nd two isoorphic grphs G G V1, ϕ(g) G V2 : F ϕ(g) = ϕ( F G ) The isoorphis xio tells us tht the rnking procedure should e independent of the nes we choose for the vertices The second xio is lso quite intuitive It tells us tht if is rnked t lest s high s if the grph is G, where in G does not link to itself, then should e rnked higher thn if ll tht we dd to G is link fro to itself Moreover, the reltive rnking of other vertices in the new grph should rein s efore Forlly, we hve the following nottion nd xio: 2 2 One y cli tht this xio kes no sense if we do not llow self loops This is however only siple technicl issue 4

5 Nottion: Let G = (V, E) G V e grph st (v, v) / E Let G = (V, E {(v, v)}) Let us denote SelfEdge(G, v) = G nd SelfEdge 1 (G, v) = G Note tht SelfEdge 1 (G, v) is well defined Axio 32 (Self edge) Let F e rnking syste F stisfies the self edge xio if for every vertex set V nd for every vertex v V nd for every grph G = (V, E) G V st (v, v) / E, nd for every v 1, v 2 V \ {v}: Let G = SelfEdge(G, v) If v 1 F G v then v F G v 1; nd v 1 F G v 2 iff v 1 F G v 2 The following, third xio (titled Vote y coittee) cptures the following ide, which is illustrted in Figure 1() If pge links to pges nd c, then the reltive rnking of ll pges should e the se s in the cse where the direct links fro to nd c re replced y links fro to new set of pges, which link (only) to nd c The ide here is tht the ount of iportnce provides to nd c y linking to the, should not chnge due to the fct tht ssigns its power through coittee of (new) representtives, ll of which ehve s More generlly, nd ore forlly, we hve the following: Axio 33 (Vote y coittee) Let F e rnking syste F stisfies vote y coittee if for every vertex set V, for every vertex v V, for every grph G = (V, E) G V, for every v 1, v 2 V, nd for every N: Let G = (V {u 1, u 2,, u }, E \ {(v, x) x S G (v)} {(v, u i ) i = 1,, } {(u i, x) x S G (v), i = 1,, }), where {u 1, u 2,, u } V = Then, v 1 F G v 2 iff v 1 F G v 2 The 4th xio, tered collpsing is illustrted in Figure 1() The ide of this xio is tht if there is pir of pges, sy nd, where oth nd link to the se set of pges, ut the sets of pges tht link to nd re disjoint, then if we collpse nd into singleton, sy, where ll links to ecoe now links to, then the reltive rnking of ll pges, excluding nd of course, should rein s efore The intuition here is tht if there re two voters (ie pges), nd, who vote siilrly (ie hve the se outgoing links), nd the power of ech one of the stes fro the fct set of other voters hve voted for hi, where the sets of voters for nd for re disjoint, then if ll voters for nd would vote only for (dropping ) then should provide the se iportnce to other gents s nd did together This of course relies on hving nd voting for the se individuls As result, the following xio is quite intuitive: Axio 34 (collpsing) Let F e rnking syste F stisfies collpsing if for every vertex set V, for every v, v V, for every v 1, v 2 V \ {v, v }, nd for every grph G = (V, E) G V for which S G (v) = S G (v ), P G (v) P G (v ) =, nd [P G (v) P G (v )] {v, v } = : Let G = (V \{v }, E\{(v, x) x S G (v )}\{(x, v ) x P G (v )} {(x, v) x P G (v )}) Then, v 1 F G v 2 iff v 1 F G v 2 The lst xio we introduce, tered the proxy xio, is illustrted in Figure 1(c) Roughly speking, this xio tells us tht if there is set of k pges, ll hving the se iportnce, which link to, where itself links to k pges, then if we drop nd connect directly, nd in 1-1 fshion, the pges which linked to to the pges tht linked to, then the reltive rnking of ll pges (excluding ) should rein the se This xio cptures equl distriution of iportnce The iportnce of is received fro k pges, ll with the se power, nd is split ong k pges; lterntively, the pges tht link to could pss directly the iportnce to pges tht link to, without using s proxy for distriution More forlly, nd ore generlly, we hve the following: Axio 35 (proxy) Let F e rnking syste F stisfies proxy if for every vertex set V, for every vertex v V, for every v 1, v 2 V \ {v}, nd for every grph G = (V, E) G V for which P G (v) = S G (v), for ll p P G (v): S G (p) = {v}, nd for ll p, p P G (v): p F G p : Assue P G (v) = {p 1, p 2,, p } nd S G (v) = {s 1, s 2,, s } Let G = (V \ {v}, E \ {(x, v), (v, x) x V } {(p i, s i ) i {1,, }}) Then, v 1 F G v 2 iff v 1 F G v 2 If we do not llow self loops then the xio should e replced y new one, where the ddition of self-loop to is replced y the ddition of new pge,, where links to nd where links only to Our results will rein siilr 5

6 31 Soundness Although we hve provided soe intuitive explntion for the xios, one y rgue tht prticulr xio(s) re not tht resonle As it turns out however, ll the ove xios re stisfied y the PgeRnk procedure The proof of the following sic (soundness) proposition ppers in the ppendix In Section 5 we show tht the ove xios re not only stisfied y PgeRnk, ut lso copletely nd uniquely chrcterize the PgeRnk procedure Proposition 36 The PgeRnk rnking syste P R stisfies isoorphis, self edge, vote y coittee, collpsing, nd proxy 4 Severl Useful Properties In this section we prove three technicl properties which re iplied y our xios As result, these three properties re stisfied y the PgeRnk rnking syste The purpose of presenting the is rther technicl: they will e used in the next section, when we show tht the PgeRnk rnking syste is the only one tht stisfies our xios Nottion: Let V e vertex set nd let v V e vertex Let G = (V, E) G V e grph where S(v) = {s}, P (v) = {p}, nd (s, p) / E We will use Del(G, v) to denote the grph G = (V, E ) defined y: V = V \ {v} E = E \ {(p, v), (v, s)} {(p, s)} The Del(, ) opertor siply reoves vertex fro the grph tht hs n in-degree nd out-degree of 1, replcing it y n edge fro its predecessor to its successor The following le sys tht when our xios re stisfied then this opertor does not chnge the reltive rnking of ll (reining) pges The proof of this le ppers in the ppendix Definition 41 Let F e rnking syste F hs the wek deletion property if for every vertex set V, for every vertex v V nd for ll vertices v 1, v 2 V \{v}, nd for every grph G = (V, E) G V st S(v) = {s}, P (v) = {p}, nd (s, p) / E: Let G = Del(G, v) Then, v 1 F G v 2 iff v 1 F G v 2 Le 42 Let F e rnking syste tht stisfies isoorphis, vote y coittee nd proxy Then, F hs the wek deletion property We now ove to second deletion property stisfied y the xios Nottion: Let V e vertex set nd let v V e vertex Let G = (V, E) G V e grph where S(v) = {s 1, s 2,, s t } nd P (v) = {p i j j = 1,, t; i = 0,, }, nd S(pi j ) = {v} for ll j {1, t} nd i {0,, } We will use Delete(G, v, {(s 1, {p i 1 i = 0, }),, (s t, {p i t i = 0, })}) to denote the grph G = (V, E ) defined y: V = V \ {v} E = E \ {(p i j, v), (v, s j) i = 0,, ; j = 1,, t} {(p i j, s j) i = 0,, ; j = 1,, t} When the grouping of the predecessors is trivil or understood fro context, we will sloppily use Delete(G, v) A sketch of the Delete opertor cn e found in Figure 2 In this figure we see tht node x which links to three other nodes, nd hs two sets of three predecessors, where the nodes in ech such set re of the se 6

7 = = x = = Figure 2: Sketch of Delete(G, x) c c d d Figure 3: Sketch of Duplicte(G,, 3) iportnce The Delete opertor will drop x nd connect exctly one eleent fro ech of the predecessor sets to exctly one node in the successor set The following le sys tht when our xios re stisfied then this opertor does not chnge the reltive rnking of ll (reining) pges The proof of this le ppers in the ppendix Definition 43 Let F e rnking syste F hs the strong deletion property if for every vertex set V, for every vertex v V, for ll v 1, v 2 V \ {v}, nd for every grph G = (V, E) G V st S(v) = {s 1, s 2,, s t }, P (v) = {p i j j = 1,, t; i = 0,, }, S(pi j ) = {v} for ll j {1, t} nd i {0,, }, nd pi j F G pi k for ll i {0,, } nd j, k {1, t}: Let G = Delete(G, v, {(s 1, {p i 1 i = 0, }), (s t, {p i t i = 0, })}) Then, v 1 F G v 2 iff v 1 F G v 2 Le 44 Let F e rnking syste tht stisfies collpsing nd proxy Then, F hs the strong deletion property We conclude with third property which is lso stisfied y the xios Nottion: Let V e vertex set nd let G = (V, E) G V e grph Let S(v) = {s 0 1, s0 2,, s0 t } We will use Duplicte(G, v, ) to denote the grph G = (V, E ) defined y: V = V {s i j i = 1,, 1; j = 1, t} E = E {(v, s i j) i = 1,, 1; j = 1, t} {(s i j, u) i = 1,, 1; j = 1, t; u S G (s 0 j)} A sketch of the Duplicte opertor cn e found in Figure 3 In this figure we see tht links to two nodes, ech of which hs its own successor set Then, ech node in the successor set of is duplicted y fctor of three, ie for ech node in the successor set of we dd two new nodes to the successor set of, ech of which with the se successor set s The following le sys tht when our xios re stisfied then this opertor does not chnge the reltive rnking of the pges, excluding the ones which hve een duplicted The proof ppers in the Appendix 7

8 Definition 45 Let F e rnking syste F hs the edge dupliction property if for every vertex set V, for ll vertices v, v 1, v 2 V, for every N, nd for every grph G = (V, E) G V : Let S(v) = {s 0 1, s 0 2,, s 0 t }, nd let G = Duplicte(G, v, ) Then, v 1 F G v 2 iff v 1 F G v 2 Le 46 Let F e rnking syste tht stisfies isoorphis, vote y coittee, collpsing, nd proxy Then, F hs the edge dupliction property 5 Copleteness We re now redy to show tht tht our xios fully chrcterize the PgeRnk rnking syste We cn prove: Theore 51 A rnking syste F stisfies isoorphis, self edge, vote y coittee, collpsing, nd proxy if nd only if F is the PgeRnk rnking syste Given Proposition 36, it is enough to prove the following: Proposition 52 Let F 1 nd F 2 e rnking systes tht hve the wek deletion, strong deletion, nd edge dupliction properties, nd stisfy the self edge nd isoorphis xios Then, F 1 nd F 2 re the se rnking syste (nottion: F 1 F 2 ) The proof of Proposition 52 is in the ppendix We shll now descrie sketch of the proof The sic ide of the proof is to egin with grph G = (V, E) nd two ritrry vertices nd in V, nd nipulte G y pplying Del(, ), Delete(,, ), Duplicte(,, ), nd SelfEdge(, ) to chieve new grph G n for which F 1 nd F 2 rnk nd the se s in G (Forlly F G n F G for F {F 1, F 2 }) Afterwrds, G n is further nipulted to generte G n+δ for which F G n+δ, ut F G n F G n+δ for F {F 1, F 2 } or vice vers (with nd replced) So, we conclude tht F1 G n F2 G n, nd thus F1 G F2 G The steps required to generte G n fro G, nd then G n+δ fro G n y e descried lgorithiclly These steps re illustrted in Figure 4: 1 Add new vertex on every edge on the initil grph (Figure 4), thus splitting ech originl edge into two new edges These vertices do not chnge the reltive rnking of nd due to the wek deletion property 2 If no originl vertices exist in the grph except nd, go to step 8 Otherwise, select n originl vertex x / {, } (in Figure 4 we strt y selecting c) 3 Reove ll vertices tht re oth predecessors nd successors of x nd ll edges connected to these vertices All of these re new vertices, which hve n in-degree nd out-degree of 1 Bsiclly, this step reoves ll self-edges of x (with n dded vertex on the) These deletions do not chnge the reltive rnking of nd due to the wek deletion property nd the self edge xio 4 Duplicte ll predecessors of predecessors of x y x s out-degree This does not chnge the reltive rnking of nd due to the dupliction property (Figure 4c) Note tht ll the vertices we duplicte re originl ones (possily or, ut not x), so to dd dditionl in-etween vertices efore x, king the in-degree of x ultiple of its out degree, split into groups of isoorphic, nd thus eqully rnked, vertices 5 Delete x using Delete(G, x) (Figure 4d) 8

9 c c c d d d d () Initil grph () After dding vertices (c) After dupliction of c s predecessors (d) After deletion of c d (e) After deletion of c s interedite successors (f) After deletion of d (g) After dupliction of (h) Finl isoorphic grph Figure 4: Exple run of the copleteness 9 lgorith Here

10 6 Delete the successors of x (new vertices) to retin the stte of one new vertex etween ech pir of originl vertices (Figure 4e) These deletions do not chnge the reltive rnking of nd due to the strong deletion property 7 Go to step 2 (Figure 4f illustrtes the second itertion, where d is selected) 8 Now, nd re the only originl vertices reining in the grph, nd the grph could e defined y the nuer of vertices (with edges) etween nd, etween nd, etween nd, nd etween nd 9 Duplicte y the nuer of edges with vertices fro to nd vice vers, thus equlizing the nuer of edges with vertices fro to the nuer fro to (Figure 4g) This reltive rnking etween nd is retined due to the dupliction property 10 Now, dd self edges (with vertices) to the vertex v {, } with fewer self-edges (with vertices), until the nuer of self edges is equl etween nd (Figure 4h) Let v = {, } \ {v} By the self edge xio nd the wek deletion property, if v F v efore dding the self edges, then now v F v for F {F 1, F 2 } 11 By the isoorphis xio, in this grph,, therefore in the grph fter step 9, v F v for F {F 1, F 2 } But s the reltive rnking of nd did not chnge until step 10, v F G v for F {F 1, F 2 }, nd thus F1 G F2 G 6 Discussion Representtion theores re the forl theticl tool for the justifiction of decision nd choice rules We hve lredy entioned the forl theory of socil choice, ut representtion theores lso ly theticl foundtions for other rnches of decision nd choice theory For exple, the crowning chieveent of the theory of (single-gent) choice is Svge s representtion theore [18], which provides sound nd coplete xiotiztion for the expected utility xiiztion decision criterion Here lso one looks for ordinl requireents, which do not refer to nueric coputtions, under which n gent cn e viewed s n expected utility xiizer This is siilr to our work, where we considered only grph-theoretic ordinl xios to justify the nueric coputtions done y PgeRnk Although PgeRnk is proly the ost populr pge rnking procedure, it y e interesting to ttept nd provide xiotiztion for other pge rnking procedures, such s Hus nd Authorities [6] Once such xiotiztion is found the different xiotic systes cn e copred s sis for rigorous evlution We elieve tht the prole of rnking of Internet pges is indeed fundentl prole We see the fct tht this centrl prole is new type of socil choice prole s especilly intriguing In order to provide theticl foundtions to pge rnking systes we therefore need to serch for sic representtion theores tht will provide ordinl, grph theoretic xiotiztions for sic heuristics nd pproches for pge rnking Representtion theores isolte the essence of prticulr rnking systes, nd provide ens for the evlution (nd potentilly coprison) of such systes In this pper we initited work on this topic y introducing such representtion theore for PgeRnk We hope tht others will join us in exploring the connections etween pge rnking lgoriths nd the theticl theory of socil choice 10

11 References [1] KJ Arrow Socil Choice nd Individul Vlues (2nd Ed) Yle University Press, 1963 [2] Sergey Brin nd Lwrence Pge The ntoy of lrge-scle hypertextul We serch engine Coputer Networks nd ISDN Systes, 30(1 7): , 1998 [3] Vincent Conitzer, Jérôe Lng, nd Tuos Sndhol How ny cndidtes re needed to ke elections hrd to nipulte? In Proceedings of the 9th conference on Theoreticl spects of rtionlity nd knowledge, pges ACM Press, 2003 [4] C Dellrocs Efficiency through feedck-contingent fees nd rewrds in uction rketplces with dverse selection nd orl hzrd In 3rd ACM Conference on Electronic Coerce (EC-03), pges 11 18, 2003 [5] A Gird Mnipultion of voting schees Econoetric, 41: , 1973 [6] Jon M Kleinerg Authorittive sources in hyperlinked environent Journl of the ACM (JACM), 46(5): , 1999 [7] Dniel J Lehnn, Liden It O Cllghn, nd Yov Shoh Truth reveltion in pproxitely efficient cointoril uctions In ACM Conference on Electronic Coerce, pges , 1999 [8] Kenneth O My A set of independent, necessry nd sufficient conditions for siple jority decision Econoetric, 20(4):680 84, 1952 [9] H Moulin Axios of Coopertive Decision Mking Cridge University Press, 1991 [10] N Nisn nd A Ronen Algorithic echnis design Proceedings of STOC-99, 1999 [11] L Pge, S Brin, R Motwni, nd T Winogrd The pgernk cittion rnking: Bringing order to the we Technicl Report, Stnford University, 1998 [12] I Plcios-Huert nd O Volij The esureent of intellectul influence Econoetric, 73(3), 2004 [13] Dvid C Prkes Itertive Cointoril Auctions: Achieving Econoic nd Coputtionl Efficiency PhD thesis, Deprtent of Coputer nd Infortion Science, University of Pennsylvni, My 2001 [14] G Pinski nd F Nrin Cittion influence for journl ggregtes of scientific pulictions: Theory, with pplictions to the literture of physics Infortion Processing nd Mngeent, pges , 1976 [15] P Resnick nd R Zeckhuser Trust ong strngers in internet trnsctions: Epiricl nlysis of ey s reputtion syste Working Pper for the NBER workshop on epiricl studies of electronic coerce, Jnury 2001 [16] P Resnick, R Zeckhuser, R Friedn, nd E Kuwr Reputtion systes Counictions of the ACM, 43(12):45 48, 2000 [17] MA Stterthwite Strtey proofness nd rrow s conditions: Existence nd correspondence theores for voting procedures nd socil welfre functions Journl of Econoic Theory, 10: , 1975 [18] LJ Svge The Foundtions of Sttistics John Wiley nd Sons, New York, 1954 Revised nd enlrged edition, Dover, New York, 1972 [19] M Tennenholtz Reputtion systes: An xiotic pproch In Proceedings of the 20th conference on uncertinity in Artificil Intelligence (UAI-04),

12 A Proofs This section includes our proofs These proofs re not prt of the extended strct, ut y e used y the interested reviewer A1 Proof of Proposition 36 Proof The isoorphis xio is stisfied directly fro the definition y the ssuption tht V = {v 1, v 2,, v n } For the vote y coittee xio, let V = {v 1, v 2,, v n } e vertex set, let G = (V, E) G V e grph, nd let v s, v t V e vertices nd let N e nturl nuer Assue v s P G R v t Let G = (V {v n+1, v n+2,, v n+ }, E \{(v 1, x) x S G (v 1 )} {(v 1, v n+j ) j = 1,, } {(v n+j, x) x S G (v 1 ), j = 1,, }) Let r e the solution of A G r = r, where r 1 = 1 Let r e the following vector: r = r 1 r n r 1 / r 1 / We will now prove tht A G r = r Note tht y definition of G, the trix A G A G = 0 1,2 1,n 1,1 1,1 0 n,2 n,n n,1 n,1 1/ 0 1/ is If we ultiply, we get: for i {1, n}: [A G r ] i = i,j r j + i,1 r 1 / = i,j r j = r i, j=2 nd for i {n + 1, n + }, [A G r ] i = 1/ r 1, s required Also r 1 = r 1 = 1, so P R G (v j ) = r j for ll j {1,, n + } Now, P R G (v s ) = r s = r s = P R G (v s ) P R G (v t ) = r t = r t = P R G (v t), s required For the collpsing xio, let V = {v 1, v 2,, v n }, nd let G = (V, E) G V Assue S(v n ) = S(v n 1 ) nd P (v n ) P (v n 1 ) = Let v k, v l V e vertices (k, l < n 1) Assue v k P R G v l Let G = (V \ {v n }, E \ {(v n, x) x S G (v n )} \ {(x, v n ) x P G (v n )} {(x, v n 1 ) x P G (v n )}) Let r e the solution of A G r = r, where r 1 = 1 Let r e the following vector: r 1 r = r n 2 r n 1 + r n i

13 We will now prove tht A G r = r Note tht y definition of G, the trix A G is 1,1 1,2 1,n 1 A G = n 2,1 n 2,2 n 2,n 1 n 1,1 + n,1 n 1,2 + n,2 0 If we ultiply, we get for i {1, n 2}: n 2 n 2 [A G r ] i = i,n 1 (r n + r n 1 ) + i,j r j = i,n 1 r n + i,n 1 r n 1 + i,j r j Note tht i,n = i,n 1 = 1 S(v n), so [A G r ] i = [A G r ] n 1 = n 2 i,j r j + i,n 1 r n 1 + i,n r n = i,j r j = r i n 2 n 2 n 2 ( n 1,j + n,j )r j = n 1,j r j + n,j r j Note tht n 1,n 1 = n 1,n = n,n 1 = n,n = 0, so [A G r ] n 1 = n 1,j r j + n,j r j = r n 1 + r n So, we get A G r = r s required Also r 1 = r 1 = 1, so P R G (v j ) = r j for ll j {1,, n 1} Now, P R G (v k ) = r k = r k = P R G (v k ) P R G (v l ) = r l = r l = P R G (v l), s required For the proxy xio, let V = {v 1, v 2,, v n }, nd let G = (V, E) G V Assue P (v n ) = {v 1, v 2,, v }, v 1 v 2 v, nd S(v n ) = {v t+1, v t+2,, v t+ }, where t {0,, } Let v k, v l V e vertices (k, l < n) Assue v k P R G v l Let G = (V \ {v n }, E \ {(x, v n ), (v n, x) x V } {(v i, v t+i ) i {1,, }}) Let r e the solution of A G r = r, where r 1 = 1 Since v 1 v 2 v, we hve r 1 = r 2 = = r, nd note tht ecuse P G (v n ) = {v 1, v 2,, v } nd S(v i ) = {v n } for ll i {1,, }: r n = n,i r i = r 1 + r r = r 1 = i=1 Let r = r n By definition of G, the trix A G is ,+1 1,+2 1,n t,+1 t,+2 t,n t+1,+1 t+1,+2 t+1,n t+2,+1 t+2,+2 t+2,n 1 A G = t+,+1 t+,+2 t+,n t++1,+1 t++1,+2 t++1,n n 1,+1 n 1,+2 n 1,n 1 ii

14 We ultiply cn now ultiply, nd since i,n = 0 for ll i {1, t, t + + 1,, n 1} (ecuse S(v n ) = {t + 1,, t + }) nd i,j = 0 for ll i {1,, n 1} nd j {1,, } (ecuse S(v j ) = {v n }), we get for i {1, t, t + + 1,, n 1}: [A G r ] i = n 1 j=+1 i,j r j = i,j r j = r i nd for i {t + 1,, t + }: [A G r ] i = = n 1 j=+1 n 1 n 1 i,j r j + r i t = i,j r j + 1 = i,j r j + 1 r n = n 1 i,j r j + i,n r n = i,j r j = r i So, we get A G r = r s required Also r 1 = r 1 = 1, so P R G (v j ) = r j for ll j {1,, n 1} Now, P R G (v k ) = r k = r k = P R G (v k ) P R G (v l ) = r l = r l = P R G (v l), s required For the self edge xio, let V = {v 1, v 2,, v n }, nd let G = (V, E) G V Assue (v 1, v 1 ) / E Let r e the solution of A G r = r, where r 1 = 1 Let G = (V, E {(v 1, v 1 )}) nd let = S G (v 1 ) Let r e the following vector: r 1 r +1 r 2 = +1 r n We will now prove tht A G r = r Note tht y definition of G, the trix A G is A G = If we ultiply, we get: for i {2, n}: ,2 1,n +1 2,1 2,2 2,n +1 n,1 n,2 n,n [A G r ] 1 = = [A G r ] i = r 1 + j=2 1,j r i,1r 1 + j=2 + 1 r j = r i,j 1,j r j = j=2 1,j r j = r r 1 = r r j = + 1 i,j r j = + 1 r i So, we get A G r = r s required Also r 1 = r 1 = 1, so P R G (v j ) = r j for ll j {1,, n 1} Assue v 2 P G R required Now ssue v 2 P G R required v 1 Then, P R G (v 2 ) = r 2 < r 2 = P R G (v 2 ) P R G (v 1 ) = r 1 = r 1 = P R G (v 1), s v 3 Then, P R G (v 2 ) = r 2 = r 2 = P R G (v 2 ) P R G (v 3 ) = r 3 = r 3 = P R G (v 3), s iii

15 A2 Proof of Le 42 Proof Let V e vertex set, let v V ; v 1, v 2 V \ {v} e vertices nd let G = (V, E) G V e grph st S(v) = {s}, P (v) = {p}, nd (s, p) / E Assue v 1 F G v 2 Let s 0 = v nd S(p) = {s 0, s 1, s 2,, s } Let G 1 = (V 1, E 1 ), where V 1 = V {p } E 1 = E \ {(p, s i ) i = 0,, } {p, p } {(p, s i ) i = 0,, } By the vote y coittee xio with preter 1, v 1 F G 1 v 2 Let G 2 = (V 2, E 2 ), where V 2 = V 1 {u i i = 0,, } E 2 = E 1 \ {(p, p )} {(p, u i ), (u i, p ) i = 0,, } By the vote y coittee xio with preter + 1, v 1 F G 2 v 2 Let G 3 = (V 3, E 3 ), where V 3 = V 2 \ {p } E 3 = E 2 \ {(u i, p ), (p, s i ) i = 0,, } {(u i, s i ) i = 0,, } By the isoorphis xio, u i G2 u j for ll i, j {0,, } By the proxy xio, v 1 F G 3 v 2 Let G 4 = (V 4, E 4 ), where V 4 = V 3 \ {v} E 4 = E 3 \ {(u 0, v), (v, s)} {(u 0, s)} By the vote y coittee xio with preter 1, v 1 F G 4 v 2 Let G = Del(G, v) By the vote y coittee, isoorphis, nd proxy xios, s etween G nd G 3 ove, v 1 F G v 2 v 1 F G 4 v 2 Thus, v 1 F G v 2 s required A3 Proof of Le 44 Proof Let V e vertex set, let v V ; v 1, v 2 V \ {v} e vertices nd let G = (V, E) G V e grph st S(v) = {s 1, s 2,, s t }, P (v) = {p i j j = 1,, t; i = 0,, }, S(pi j ) = {v} for ll j {1, t} nd i {0,, }, nd p i j = pk j for ll j {1, t} nd i, k {0,, } Assue v 1 F G v 2 Denote u 0 = v Let G 1 = (V 1, E 1 ), where V 1 = V {u i i = 1,, } E 1 = E \ {(p i j, v) i = 1,, ; j = 1,, t} {(p i j, u i ), (u i, s j ) i = 1,, ; j = 1,, t} By the collpsing xio pplied in the reverse direction totl of ties for {(u i 1, u i ) i = 1,, }, v 1 F G 1 v 2 iv

16 Let G 2 = (V 2, E 2 ), where V 2 = V 1 \ {u i i = 0,, } E 2 = E 1 \ {(p i j, ui ), (u i, s j ) i = 0,, ; j = 1,, t} {(p i j, s j) i = 0,, ; j = 1,, t} By the proxy xio pplied totl of + 1 ties for {u i i = 0,, }, v 1 F G 2 v 2 Note tht G 2 is exctly G = Delete(G, v, {(s 1, {p i 1 i = 0, }), (s t, {p i t i = 0, })}), so v 1 F G v 2 s required A4 Proof of Le 46 Proof Let V e vertex set, let v, v 1, v 2 V e vertices, nd let N e nturl nuer Assue > 1 (otherwise G = G), nd let = 1 Let G = (V, E) G V e grph Assue v 1 F G v 2, nd let S(v) = {s 0 1, s0 2,, s0 t } Let G 1 = (V 1, E 1 ), where V 1 = V {u i j i = 0,, ; j = 1, t} E 1 = E \ {(v, x) x S G (v)} {(v, u i j ) i = 0,, ; j = 1, t} {(u i j, x) x S G (v), i = 0,, ; j = 1, t} By the vote y coittee xio with preter ( + 1)t, v 1 F G 1 v 2 Let G 2 = (V 2, E 2 ), where V 2 = V 1 {wj i i = 0,, ; j = 1, t} E 2 = E 1 \ {(v, u i j) i = 0,, ; j = 1, t} {(v, w i j), (w i j, u i j) i = 0,, ; j = 1, t} By the vote y coittee xio (pplied ( 1)t ties) with preter 1, v 1 F G 2 v 2 Let G 3 = (V 3, E 3 ), where V 3 = V 2 \ {u i j i = 0,, ; j = 2,, t} E 3 = E 2 \ {(u i j, x x S G (v); i = 0,, ; j = 2,, t} \ \{(wj i, ui j ) i = 0,, ; j = 2,, t} {(wj i, ui 1 ) i = 0,, ; j = 2,, t} By the collpsing xio pplied totl of (+1)(t 1) ties for {(u i j 1, ui j ) j = 2,, t; i = 0, }, v 1 F G 3 v 2 Let G 4 = (V 4, E 4 ), where V 4 = V 3 \ {u i 1 i = 0,, } E 4 = E 3 \ {(u i 1, x) i = 0,, ; x S G (v)} \ \{(wj i, ui 1 ) i = 0,, ; j = 1,, t} {(wj i, s0 j ) i = 0,, ; j = 1,, t} v

17 By the isoorphis xio, wj i wi k for ll i {0,, } nd j, k {1,, t} By the proxy xio (pplied totl of + 1 ties for {u i 1 i = 0, }), v 1 F G 4 v 2 Let G 5 = (V 5, E 5 ), where V 5 = V 4 {s i j i = 1,, ; j = 1,, t} E 5 = E 4 \ {(w i j, s 0 j ) i = 1,, ; j = 1,, t} {(wj i, si j ) i = 1,, ; j = 1,, t} {(s i j, x) x S(s0 j ); i = 1,, } By the collpsing xio pplied in the reverse direction totl of t ties for {(s i 1 j, s i j ) i = 1,, ; j = 1,, t}, v 1 F G 5 v 2 Let G 6 = (V 6, E 6 ), where V 6 = V 5 \ {wj i i = 0,, ; j = 1, t} E 6 = E 5 \ {(v, wj i ), (wi j, si j ) i = 0,, ; j = 1, t} {(v, s i j ) i = 0,, ; j = 1, t} By the vote y coittee xio pplied in the reverse direction totl of ( + 1) t ties for {w i j i = 0,, ; j = 1, t}, v 1 F G 6 v 2 Note tht G 6 is exctly Duplicte(G, v, + 1) = Duplicte(G, v, ) = G, so v 1 F G v 2 s required A5 Proof of Proposition 52 Proof Let V e vertex set nd let G = (V, E) G V e soe grph If V = 1, then there exists only one ordering on V, so trivilly F1 G F2 G Assue V = {v 1, v 2,, v n } We will show tht v 1 F1 G v 2 v 1 F2 G v 2 Without loss of generlity we cn show only one direction Let F {F 1, F 2 } Let G 2 = (V 2, E 2 ) e the following grph (G with vertex dded on every edge): Note tht V 2 = V {u i,j (v i, v j ) E} E 2 = {(v i, u i,j ), (u i,j, v j ) (v i, v j ) E} G = Del(Del( Del(G 2, u 1 ), u E 1 ), u E ) where {u 1,, u E } = {u i,j (v i, v j ) E} nd tht G 2 stisfies the conditions of wek deletion property for the vertices {u i,j (v i, v j ) E}, thus v 1 F G v 2 v 1 F G 2 v 2 For ll strongly connected directed grphs G such tht for ll v V nd for ll v P G (v) S G (v) st S G (v ) = P G (v ) = 1, let us denote for ll v V : S 2 G (v) = {v V : x S G (v), S G (x) = {v }} nd P 2 G (v) = {v V : x P G (v), P G (x) = {v }} G i 1 For, i = 3,, n, we recursively define G i s follows: Let {q 1, q 2,, q } = S Gi 1 (v i ) P Gi 1 (v i ) Let e the grph G i 1 = SelfEdge 1 (Del( SelfEdge 1 (Del(G i 1, q 1 ), v i ), q ), v i ) vi

18 Now, let PG 2 = {p i 1(v) 1,, p k } nd let S G i 1 (v i ) = {s 1, s 2,, s l } Let G i 1 e defined s: G i 1 = Duplicte( Duplicte(G i 1, p 1, l), p k, l) Let {p i j i = 1,, l} = S G i 1 (p j) e the duplicted successors of p j for j = 1 k Now let G i = (V i, E i ) e defined s: G i 1 = Delete(G i 1, v i, {(s 1, {p 1 j j = 1,, k}),, (s l, {p l j j = 1,, k})}) G i = Delete( Delete(Delete(G i 1, s 1), s 2 ), s l ) By the edge dupliction nd strong deletion properties nd the self edge xio, v 1 F G i v 2 for ll i {2,, n} We will now prove tht for ll i {2,, n} nd for ll v V i \ V : P Gi (v) = S Gi (v) = 1 nd P Gi (v) S Gi (v) V nd for ll v V : (P Gi (v) S Gi (v)) V = Proof y induction: G 2 trivilly stisfies oth requireents Now ssue tht for ll v V i \ V : P Gi (v) = S Gi (v) = 1 nd P Gi (v) S Gi (v) V nd for ll v V : (P Gi (v) S Gi (v)) V = Clerly, G i stisfies the conditions, ecuse we only reoved eleents fro V i, nd not chnged the predecessors or successors of ny v V \ V i Also, ll edges dded etween vertices in V were reoved The Duplicte(,, ) opertion dds vertices with in-degree 1 nd out-degree equl to the out degree of the successors of v, which is lso 1 So, the new vertices dded in G i stisfy the conditions Furtherore, no edges were dded etween eleents of V Thus, G i stisfies the keep their out-degree 1, nd point to eleents of SG 2 (v), nd thus still eet the requireents Other eleents of V i \ V hve i not chnged their edges, nd thus still eet the requireents Still, no edges were dded etween eleents of V Therefore, for ll v V i+1 \ V : P Gi+1 (v) = S Gi+1 (v) = 1 nd P Gi+1 (v) S Gi+1 (v) V nd for ll v V : (P Gi+1 (v) S Gi+1 (v)) V = conditions In G i+1, we reoved v nd ll its successors The predecessors of v in G i Specificlly, this is true for G n = (V n, E n ) Furtherore, V n V = {v 1, v 2 } Thus, G n could e descried s: V n = {v 1, v 2 } {v i jk j, k {1, 2}; i = 1,, n jk } E n = {(v j, v i jk ), (vi jk, v k) j, k {1, 2}; i = 1,, n jk } The only preters which ffect the structure of G n re n jk (j, k {1, 2}), so we cn denote G n = G[n 11, n 12, n 21, n 22 ] Now, let G n = Duplicte(Duplicte(G n, v 1, n 21 ), v 2, n 12 ) = G[n 21 n 11, n 21 n 12, n 12 n 21, n 12 n 22 ] By the edge dupliction property, v 1 F G v 2 v 1 F G n v 2 Consider the following 3 cses: If n 21 n 11 = n 12 n 22, then the grph is isoorphic to itself, replcing v 1 with v 2 nd vjk i with vi kj In this cse, y the isoorphis xio, v 1 F G v 2 nd thus v n 1 F G v 2, nd therefore v 1 F G v 2 for F {F 1, F 2 } If n 21 n 11 > n 12 n 22, let δ = n 21 n 11 n 12 n 22 > 0 Now we define for i = n + 1, n + δ: G i = SelfEdge(G i 1, v 2 ) G i = G[n 21 n 11, n 21 n 12, n 12 n 21, n 12 n 22 + i n] Note tht G i = Del(G i, v n12n22+i n 22 ) Thus, y the self-edge xio nd the wek deletion property, v 1 F G v 2 v 2 F G n+δ v 1 Now, note tht G n+δ = G[n 21 n 11, n 12 n 21, n 12 n 21, n 21 n 11 ], thus s efore, y isoorphis, v 1 F G n+δ v 2 Therefore we conclude tht v 1 F G v 2 for F {F 1, F 2 } vii

19 If n 21 n 11 < n 12 n 22, we cn siilrly conclude tht v 2 F G v 1, nd therefore v 1 F G v 2 for F {F 1, F 2 } We hve shown tht for every vertex set V, for ll G = (V, E) G V, nd for every v 1, v 2 V : v 1 F1 G v 1 F2 G v 2 Thus, F 1 F 2, concluding the proof of the proposition v 2 viii