VIEWS: 16 PAGES: 41 CATEGORY: Internet / Online POSTED ON: 11/22/2010
PageRank is the Google ranking algorithm (ranking formula) is part of Google used to identify web pages for rank / importance of a method used to measure a website, Google the only criterion for good or bad. In a blend of identification, such as Title Keywords such as logo and all other relevant factors, Google PageRank to adjust results through, so that more "level / importance" of pages in search results get promoted in another site's ranking to improve search results relevance and quality.
The PageRank Axioms Alon Altman and Moshe Tennennholtz Technion, IIT The PageRank Axioms – p. Main Contribution A representation theorem for PageRank, bridging the gap between two well-known disciplines: Social choice – The theory of combining individual preferences into a social preference over alternatives. Ranking systems – Systems designated to rank agents based on mutual reports or links between the agents. The PageRank Axioms – p. Social Choice The classical social choice setting is comprised of: A set of agents. A set of alternatives. Each agent has a preference relation over the set of alternatives. A social welfare function is a mapping from the agents’ individual preferences into a social ranking over the alternatives. The goal of social choice theory is to produce “good” social welfare functions. The PageRank Axioms – p. Ranking Systems Ranking systems are widely used in practical implementations: Reputation systems such as E-bay’s Journal inﬂuence measurement Internet page ranking: The PageRank Axioms – p. Page Ranking The page ranking problem can be described as: Given a graph G = (V, E) representing the Internet, produce a ranking over V which represents the “power” or “relevance” of the vertices in G. Several solutions were suggested to the page ranking problem: Google’s PageRank — The most widely implemented technology behind the most popular search engine. Hubs&Authorities. The PageRank Axioms – p. Page Ranking as Social Choice Such problem can be viewed as a social choice problem: The set of agents is V . The set of alternatives coincides with the set of agents, and is also V . Each agent v ∈ V prefers the agents V = {v |(v, v ) ∈ E} it links to over the other agents. These individual preferences are to be combined to a social preference over V , which “represents” the individual preferences. The PageRank Axioms – p. The Axiomatic Approach In the axiomatic approach, we try to ﬁnd basic properties satisﬁed by page ranking systems. The axiomatic approach encompasses two distinct approaches: The normative approach, in which we study sets of axioms that intuitively should be satisﬁed by a ranking system; and The descriptive approach, in which we devise a set of axioms that are uniquely satisﬁed by a known ranking system. In this presentation, we apply the descriptive approach to PageRank. The PageRank Axioms – p. Previous Results The only work we know of suggesting an axiomatization of the PageRank is the one recently published by Palacois-Huerta and Volij(2004), dealing with relative inﬂuence of scientiﬁc journals. Our axiomatization differs in the fact that it captures the relative ranking using purely ordinal axioms, in the sense that there is no underlying numeric measure of power. This is opposed to the axioms suggested by Palacois-Huerta and Volij, which include arithmetic manipulation of numerical weights. The PageRank Axioms – p. PageRank The simpliﬁed version of PageRank we axiomatize here could be deﬁned as ranking the vertices according to the stationary probabilities of a random walk on the graph. We assume the graph G = (V, E) to be strongly connected. Let AG be the following matrix: 1 |S(vj )| (vj , vi ) ∈ E [AG ]i,j = 0 Otherwise where S(v) is the successor set of v. The PageRank Axioms – p. PageRank (cont.) The PageRank of a graph G is deﬁned as the principal eigenvector of the matrix A. That is, the PageRank of G is the vector x satisfying AG x = x The PageRank ranking system P R can now be deﬁned as the ordering on V according to x: v1 PR v2 ⇔ x1 ≤ x2 The PageRank Axioms – p. 1 The Axioms Our representation theorem for PageRank requires the following ﬁve axioms: 1. Isomorphism; 2. Self-Edge; 3. Vote by Committee; 4. Collapsing; and 5. Proxy The PageRank Axioms – p. 1 Axiom 1: Isomorphism This axiom states that the ranking must not rely on the names of the vertices, but only on the voting structure between the agents. Formally: A ranking system F satisﬁes isomorphism if for every isomorphism function ϕ : V1 → V2 , and two isomorphic graphs G ∈ GV1 , ϕ(G) ∈ GV2 : F ϕ(G) = ϕ( F ). G The PageRank Axioms – p. 1 Axiom 2: Self-Edge This axiom states that adding a self edge on v strengthens v, but does not change the ranking of other vertices. Formally: SelfEdge(G, v) – The graph G with the self edge (v, v) added. Let F be a ranking system. F satisﬁes the self edge axiom if for every vertex set V and for every vertex v ∈ V and for every graph G = (V, E) ∈ GV s.t. (v, v) ∈ E, and for every v1 , v2 ∈ V \ {v}: Let / G = SelfEdge(G, v). If v1 F v then v F v1 ; G G and v1 F v2 iff v1 F v2 . G G The PageRank Axioms – p. 1 Axiom 3: Vote by Committee b b a a c c The Vote by Committee axiom captures the fact that an agent may vote indirectly via any number of intermediate agents, each of which vote to the agent’s original preferences. The PageRank Axioms – p. 1 Axiom 3: Vote by Committee (cont.) Formal Deﬁnition: Let F be a ranking system. F satisﬁes vote by committee if for every vertex set V , for every vertex v ∈ V , for every graph G = (V, E) ∈ GV , for every v1 , v2 ∈ V , and for every m ∈ N: Let G = (V ∪ {u1 , u2 , . . . , um }, E \ {(v, x)|x ∈ SG (v)} ∪ {(v, ui )|i = 1, . . . , m} ∪ {(ui , x)|x ∈ SG (v), i = 1, . . . , m}), where {u1 , u2 , . . . , um } ∩ V = ∅. Then, v1 F v2 iff v1 F v2 . G G The PageRank Axioms – p. 1 Axiom 4: Collapsing a a b The collapsing axiom captures the fact that voters which have the same preferences may be collapsed to a single voter with the same preferences, voted by all the voters for both. We assume the voter sets for the collapsed vertices are disjoint and do not include a or b. This collapsing only change the rank of a. The PageRank Axioms – p. 1 Axiom 4: Collapsing (cont.) Formal Deﬁnition: Let F be a ranking system. F satisﬁes collapsing if for every vertex set V , for every v, v ∈ V , for every v1 , v2 ∈ V \ {v, v }, and for every graph G = (V, E) ∈ GV for which SG (v) = SG (v ), PG (v) ∩ PG (v ) = ∅, and [PG (v) ∪ PG (v )] ∩ {v, v } = ∅: Let G = (V \ {v }, E \ {(v , x)|x ∈ SG (v )} \ {(x, v )|x ∈ PG (v )} ∪ {(x, v)|x ∈ PG (v )}). Then, v1 F v2 iff v1 F v2 . G G The PageRank Axioms – p. 1 Axiom 5: Proxy = x = The proxy axiom captures the fact that n voters of equal rank who have voted via a proxy (another agent) for n alternatives, can achieve the same result by directly voting for one alternative each. The PageRank Axioms – p. 1 Axiom 5: Proxy (cont.) Formal Deﬁnition: Let F be a ranking system. F satisﬁes proxy if for every vertex set V , for every vertex v ∈ V , for every v1 , v2 ∈ V \ {v}, and for every graph G = (V, E) ∈ GV for which |PG (v)| = |SG (v)|, for all p ∈ PG (v): SG (p) = {v}, and for all p, p ∈ PG (v): p p : Assume PG (v) = {p1 , p2 , . . . , pm } and SG (v) = {s1 , s2 , . . . , sm }. Let G = (V \ {v}, E \ {(x, v), (v, x)|x ∈ V } ∪ {(pi , si )|i ∈ {1, . . . , m}}). Then, v1 F v2 iff G v1 F v2 . G The PageRank Axioms – p. 1 Soundness Proposition 1 The PageRank ranking system P R satisﬁes isomorphism, self edge, vote by committee, collapsing, and proxy. This proposition is proven by a simple application of linear algebra. The PageRank Axioms – p. 2 Completeness In order to prove completeness, we will ﬁrst show two strong properties that are entailed by our ﬁve axioms: Weak Deletion; Strong Deletion; and Duplication The PageRank Axioms – p. 2 Weak Deletion The Weak Deletion property allows us to remove a vertex that has both an in-degree and an out-degree of 1. Formally, Let V be a vertex set and let v ∈ V be a vertex. Let G = (V, E) ∈ GV be a graph where S(v) = {s}, P (v) = {p}, and (s, p) ∈ E. We will use Del(G, v) to denote the graph / G = (V , E ) deﬁned by: V = V \ {v} E = E \ {(p, v), (v, s)} ∪ {(p, s)}. The PageRank Axioms – p. 2 Weak Deletion (cont.) Now we can state the weak deletion property: Let F be a ranking system. F has the weak deletion property if for every vertex set V , for every vertex v ∈ V and for all vertices v1 , v2 ∈ V \ {v}, and for every graph G = (V, E) ∈ GV s.t. S(v) = {s}, P (v) = {p}, and (s, p) ∈ E: Let G = Del(G, v). / Then, v1 F v2 iff v1 F v2 . G G The PageRank Axioms – p. 2 Strong Deletion = = x = = The Strong Deletion property is a generalization of the proxy axiom, allowing removal of a vertex with m sets of t equal predecessors, and t successors. One element of each equal sets is set to point to each of the original successors. This change does not affect the relative rank of the remaining vertices. The PageRank Axioms – p. 2 Strong Deletion (cont.) Formally, the strong deletion operator is deﬁned: Let V be a vertex set and let v ∈ V be a vertex. Let G = (V, E) ∈ GV be a graph where S(v) = {s1 , s2 , . . . , st } and P (v) = {pi |j = 1, . . . , t; i = 0, . . . , m}, and S(pi ) = {v} for all j j j ∈ {1, . . . t} and i ∈ {0, . . . , m}. We will use Delete(G, v, {(s1 , {pi |i = 0, . . . m}), . . . , (st , {pi |i = 0, . . . m})}) to 1 t denote the graph G = (V , E ) deﬁned by: V = V \ {v} E = E \ {(pi , v), (v, sj )|i = 0, . . . , m; j = 1, . . . , t} ∪ j ∪{(pi , sj )|i = 0, . . . , m; j = 1, . . . , t}. j The PageRank Axioms – p. 2 Strong Deletion (cont.) Now we can state the strong deletion property: Let F be a ranking system. F has the strong deletion property if for every vertex set V , for every vertex v ∈ V , for all v1 , v2 ∈ V \ {v}, and for every graph G = (V, E) ∈ GV s.t. S(v) = {s1 , s2 , . . . , st }, P (v) = {pi |j = 1, . . . , t; i = 0, . . . , m}, S(pi ) = {v} for all j ∈ {1, . . . t} j j and i ∈ {0, . . . , m}, and pi F pi for all i ∈ {0, . . . , m} and j G k j, k ∈ {1, . . . t}: Let G = Delete(G, v, {(s1 , {pi |i = 0, . . . m}), . . . (st , {pi |i = 0, . . . m})}). 1 t Then, v1 F v2 iff v1 F v2 . G G The PageRank Axioms – p. 2 Duplication b b a c a c d d The duplication property allows duplication of an agent’s successors by any factor. The new vertices have the same successors as the old. The relative ranking of all vertices except the duplicated successors does not change. The PageRank Axioms – p. 2 Duplication (cont.) Formally, the duplication operator is deﬁned: Let V be a vertex set and let G = (V, E) ∈ GV be a graph. Let S(v) = {s0 , s0 , . . . , s0 }. We will use Duplicate(G, v, m) to denote 1 2 t the graph G = (V , E ) deﬁned by: V = V ∪ {si |i = 1, . . . , m − 1; j = 1, . . . t} j E = E ∪ {(v, si )|i = 1, . . . , m − 1; j = 1, . . . t} ∪ j ∪{(si , u)|i = 1, . . . , m − 1; j = 1, . . . t; u ∈ SG (s0 )}. j j The PageRank Axioms – p. 2 Duplication (cont.) Now we can state the duplication property: Let F be a ranking system. F has the edge duplication property if for every vertex set V , for all vertices v, v1 , v2 ∈ V , for every m ∈ N, and for every graph G = (V, E) ∈ GV : Let S(v) = {s0 , s0 , . . . , s0 }, and let G = Duplicate(G, v, m). 1 2 t Then, v1 F v2 iff v1 F v2 . G G The PageRank Axioms – p. 2 Satisﬁcation These three properties are entailed by our axioms: Lemma 1 Let F be a ranking system that satisﬁes isomorphism, vote by committee, and proxy. Then, F has the edge duplication property. Lemma 2 Let F be a ranking system that satisﬁes collapsing and proxy. Then, F has the strong deletion property. Lemma 3 Let F be a ranking system that satisﬁes isomorphism, vote by committee, collapsing, and proxy. Then, F has the edge duplication property. The PageRank Axioms – p. 3 Idea of Completeness Proof The completeness proof is a constructive one. The idea is to ﬁx to vertices a and b, and then to manipulate the graph by applying the axioms and properties, preserving their relative ranking. Further manipulation is then applied to the graph using the self edge axiom to modify the relative ranking of a and b in one direction, until a and b can be proved of equal rank using the isomorphism axiom. As the relative ranking was manipulated in only one direction, we show that a and b must have a relative ranking opposite to the manipulation in any ranking system satisfying the axioms. The PageRank Axioms – p. 3 Demonstration of Proof a a c Start with the input graph and c two vertices a and b to be b compared. b Add a vertex on each edge. The relative ranking of a and d b does not change because of d the weak deletion property. The PageRank Axioms – p. 3 Demonstration of Proof (cont.) Select an original vertex except a and b (c in our a example), and delete all its self edges with a vertex on them. c This does not change the relative ranking of a and b due to the self-edge and weak deletion axioms. b Next, we use the duplication property to duplicate the pre- decessors of c by c’s out de- d gree, without changing the relative ranking of a and b. The PageRank Axioms – p. 3 Demonstration of Proof (cont.) a The isomorphism axiom guarantees that c satisﬁes the conditions of the strong b deletion property. Thus, we can apply Delete(G, c). d Due to the strong deletion property, this does not change the relative ranking of a and b. The PageRank Axioms – p. 3 Demonstration of Proof (cont.) We apply the strong deletion property again to delete the new vertices that were a successors of c. Again, this does not change the relative ranking of a and b. b Note that now again all suc- cessors and predecessors of the original vertices are new d vertices, and all successors and predecessors of the new vertices are original vertices. The PageRank Axioms – p. 3 Demonstration of Proof (cont.) Repeat the previous steps, selecting a different vertex each time, until the only remaining original vertices are a and b: a b The PageRank Axioms – p. 3 Demonstration of Proof (cont.) Equalize the number of edges with vertices from a to b to the number of edges with vertices from b to a by duplicating a by the number edges with vertices from b to a and vice versa. In our example b is duplicated by 3. a b The PageRank Axioms – p. 3 Demonstration of Proof (cont.) Assume without loss of generality that b has fewer self edges with vertices than a. Add self edges with vertices to b, until a and b have the same number of self edges (with vertices). a b The PageRank Axioms – p. 3 Completeness Proof – Conclusion Now, a b according to the isomorphism axiom. But, according to the self edge axiom we increased the relative rank of b compared to a, so we conclude that in the original graph, b a. This unique outcome is general, and thus the axioms guarantee a unique ranking, and thus exactly represent PageRank. The PageRank Axioms – p. 3 Concluding Remarks We have shown a ﬁrst (graph-theoretic, ordinal) axiomatization of PageRank. This axiomatization bridges the gap between classical social choice theory, and the most widely used algorithm for ranking Internet pages. The completeness proof is constructive, and suggest a (grossly inefﬁcient) algorithm for computing the relative PageRank of two nodes. The PageRank Axioms – p. 4 Future and Additional Work Descriptive Approach Hubs & Authorities PageRank with damping factor Normative Approach Impossibility results Incentive compatibility Impossibility result Negative votes The PageRank Axioms – p. 4