Branching Distributional Equations and their Applications

Transcription

1 Branching Distributional Equations and their Applications Mariana Olvera-Cravioto UNC Chapel Hill August 22nd, 2018 Bath-UNAM-CMAT, Lecture 3 Branching Distributional Equations and their Applications 1/18

2 Google s PageRank PageRank computes the rank of a webpage as: r i =(1 c)q i + c X j!i r j D j, where, {1, 2,...,n} are the pages under consideration, the sum is taken over all pages pointing to i, D j is the number of outbound links of page j, q =(q 1,...,q n ) is a personalization vector, and c is a damping factor, usually c =0.85. Multiply both sides by n to obtain a scale free rank. n matrix notation, R =(1 c)q + RM, M = matrix of weights q = personalization vector. Bath-UNAM-CMAT, Lecture 3 Branching Distributional Equations and their Applications 2/18

3 The problem to solve We want to analyze the typical behavior of a large class of ranking algorithms on large directed graphs. Bath-UNAM-CMAT, Lecture 3 Branching Distributional Equations and their Applications 3/18

4 The problem to solve We want to analyze the typical behavior of a large class of ranking algorithms on large directed graphs. Can we characterize nodes with very high ranks? Bath-UNAM-CMAT, Lecture 3 Branching Distributional Equations and their Applications 3/18

5 The problem to solve We want to analyze the typical behavior of a large class of ranking algorithms on large directed graphs. Can we characterize nodes with very high ranks? Can we determine the distribution of the ranks? Bath-UNAM-CMAT, Lecture 3 Branching Distributional Equations and their Applications 3/18

6 The problem to solve We want to analyze the typical behavior of a large class of ranking algorithms on large directed graphs. Can we characterize nodes with very high ranks? Can we determine the distribution of the ranks? Can we propose new algorithms that will have a pre specified typical behavior? Bath-UNAM-CMAT, Lecture 3 Branching Distributional Equations and their Applications 3/18

7 The problem to solve We want to analyze the typical behavior of a large class of ranking algorithms on large directed graphs. Can we characterize nodes with very high ranks? Can we determine the distribution of the ranks? Can we propose new algorithms that will have a pre specified typical behavior? Our approach: Step 1: Step 2: Step 3: Start with an appropriate random graph model. Show that we can analyze the rank via a fixed-point equation. Characterize the solutions to this fixed-point equation. Bath-UNAM-CMAT, Lecture 3 Branching Distributional Equations and their Applications 3/18

8 The WWW graph WWW seen as a directed graph (webpages = nodes, links = edges). For ranking purposes we can think of it as being a simple graph. Empirical observations: fraction pages >kin-links / k, =1.1 fraction pages >kout-links / k, =1.72 We want a directed random graph model that matches the degree distributions. Bath-UNAM-CMAT, Lecture 3 Branching Distributional Equations and their Applications 4/18

9 The WWW graph WWW seen as a directed graph (webpages = nodes, links = edges). For ranking purposes we can think of it as being a simple graph. Empirical observations: fraction pages >kin-links / k, =1.1 fraction pages >kout-links / k, =1.72 We want a directed random graph model that matches the degree distributions. nterestingly, fraction pages with PageRank >k/ k The power law hypothesis: This is true in any scale-free graph. Bath-UNAM-CMAT, Lecture 3 Branching Distributional Equations and their Applications 4/18

10 Afirstrandomgraphmodel We consider a directed version of the configuration model. Directed graph on n nodes V = {1, 2,...,n}. n-degree and out-degree: d + i = in-degree of node i = number of edges pointing to i. di = out-degree of node i = number of edges pointing from i. (D +, D )=({d + i }, {d i }) is called a bi-degree-sequence. Target distributions: n-degree: F =(f k : k =0, 1, 2,...), and Out-degree: G =(g k : k =0, 1, 2,...). Assume F has finite 1+ moments and G has finite 2+ moments. Bath-UNAM-CMAT, Lecture 3 Branching Distributional Equations and their Applications 5/18

11 The directed configuration model Assume we have a bi-degree sequence (D +, D ) that is graphical with high probability and whose in-degree and out-degree distributions are F and G, respectively. A method based on i.i.d. samples was given in (Chen-OC, 12). Given the bi-degree sequence, assign to each node i a number of inbound and outbound half edges according to the sequence. We obtain a graph by randomly pairing the inbound half edges with the outbound ones. The result is a multigraph (e.g., with self-loops and multiple edges in the same direction) on nodes {1, 2,...,n}. Theorem: (Chen-OC, 12) By erasing the self loops and multiple edges in the same direction we obtain a simple graph with asymptotic degree distributions F and G as n!1. Bath-UNAM-CMAT, Lecture 3 Branching Distributional Equations and their Applications 6/18

12 Pros and cons of the configuration model Pros: t can be used to fit any degree distribution. Conditionally on the pairing process resulting in a simple graph, the graph is uniformly chosen among all simple graphs having that bi-degree sequence. (Exercise) A uniformly chosen graph is a good null model. Cons: The model is somewhat artificial. t does not provide any explanation as to why some nodes have high degrees whereas others do not. The probability of the pairing process resulting in a simple graph is asymptotically zero if the degrees have infinite variance, and erasing self-loops and multiple-edges destroy the uniformity. Bath-UNAM-CMAT, Lecture 3 Branching Distributional Equations and their Applications 7/18

13 nhomogeneous random digraphs A beautifully simple model: the Erdős-Rényi graph. Bath-UNAM-CMAT, Lecture 3 Branching Distributional Equations and their Applications 8/18

14 nhomogeneous random digraphs A beautifully simple model: the Erdős-Rényi graph. The (undirected) Erdős-Rényi graph on n vertices is constructed by n deciding whether each of the possible edges is present according to 2 an independent coin-flip with probability p = /n. The resulting graph is simple by construction. Problem... it produces degrees that are too homogeneous (Poisson to be precise... Exercise). Goal: generalize the Erdős-Rényi graph to produce directed inhomogeneous graphs with heavy-tailed degrees. Bath-UNAM-CMAT, Lecture 3 Branching Distributional Equations and their Applications 8/18

15 Alargefamilyofmodels Consider a digraph G(V n,e n ) on the set of vertices V n = {1, 2,...,n} having edges in E n. Each vertex i 1 is assigned a type x i 2S. Types are distributed according to some measure µ. Let apple(x, y) :S 2! R + and construct the graph by independently drawing an edge from i to j with probability p (n) ij = P n ((i, j) 2 E n )=apple(x i, x j )(1+' n (x i, x j ))^1, 1 apple i 6= j apple n, where P n ( ) =P ( {x i } n i=1 ) and ' n(x i, x j )!0 su ciently fast. Bath-UNAM-CMAT, Lecture 3 Branching Distributional Equations and their Applications 9/18

16 Some examples included in the family Examples with x =(x +,x ) and apple(x, y) = 1 x y + : Directed Erdős-Rényi model: p (n) ij = n Directed Chung-Lu model: p (n) ij = x i x+ j l n ^ 1, l n = Directed generalized random graph: nx (x + i + x i ) i=1 p (n) ij = x i x+ j l n + x i x + j Directed Poissonian random graph or Norros-Reittu model: p (n) ij =1 e x i x+ j /ln Bath-UNAM-CMAT, Lecture 3 Branching Distributional Equations and their Applications 10/18

17 Degree distributions Let (D + i,d i ) denote the in-degree and out-degree of vertex i. Define Z + (x) = S Z apple(x, y)µ(dy) and (x) = S apple(y, x)µ(dy) Theorem: (Cao-OC, 17) Let be the index of a uniformly chosen vertex in V n. Under some regularity conditions on the kernel apple, wehave (D +,D ) ) (Z+,Z ), n!1, where (Z +,Z ) is a pair of mixed Poisson r.v. s with mixing parameters + (X) and (X), respectively, conditionally independent given X, and X distributed according to µ. Bath-UNAM-CMAT, Lecture 3 Branching Distributional Equations and their Applications 11/18

18 Joint degree distribution Does the model produce scale-free graphs? Bath-UNAM-CMAT, Lecture 3 Branching Distributional Equations and their Applications 12/18

19 Joint degree distribution Does the model produce scale-free graphs? Theorem: (Cao-OC, 17) Suppose that µ is such that if X is distributed according to µ, then( + (X), (X)) has a (non-standard) multivariate regularly varying distribution with scaling functions a(t) 2RV(1/ ) and b(t) 2RV(1/ ). Then(Z +,Z ) is also multivariate regularly varying with the same scaling functions. Remark: in the examples, ( + (x), (x)) = (cx +, (1 c)x ). Bath-UNAM-CMAT, Lecture 3 Branching Distributional Equations and their Applications 12/18

20 Pros and cons of the inhomogeneous random digraph Pros: t always produces simple graphs. t provides an explanation for the inhomogeneity of the degrees based on that of the types. Arcs are independent of each other. Cons: We can only obtain degree distributions that are mixed Poisson (with arbitrary mixing distributions). The graphs produced, although inhomogeneous, are not necessarily realistic. Bath-UNAM-CMAT, Lecture 3 Branching Distributional Equations and their Applications 13/18

21 The local tree-like structure Many random graph models have a local tree-like behavior. Both the configuration model and the inhomogeneous random digraph do. Bath-UNAM-CMAT, Lecture 3 Branching Distributional Equations and their Applications 14/18

22 Analysis through branching processes The analysis of random graphs in general is based in many cases on their local tree-like structure. The key idea is that the exploration of the graph starting from a randomly chosen node can be coupled with a suitable branching process. Examples: the Erdős-Rényi graph can be coupled with a Galton-Watson tree with a Poisson number of o spring, and the inhomogeneous random graph can be coupled with a multi-type branching process. Directed graphs such as the ones described earlier can be coupled with marked branching processes. Bath-UNAM-CMAT, Lecture 3 Branching Distributional Equations and their Applications 15/18

23 The bow-tie structure We say that vertices i and j belong to a strongly connected component if there is a path from i to j and a path from j to i. The largest strongly connected component is said to be giant if it has at least n vertices for some >0. Both the DCM and the RD have a phase transition for the existence of a giant strongly connected component. Fan out Strongly connected component Fan in The phase transition is determined by the survival probabilities of the coupled branching processes. Bath-UNAM-CMAT, Lecture 3 Branching Distributional Equations and their Applications 16/18

24 Coupling for analyzing PageRank We only need to analyze the fan-in of a randomly chosen vertex. We need to keep track of both the in-degrees and out-degrees. Randomly chosen node Tree structure up to distance c log n Bath-UNAM-CMAT, Lecture 3 Branching Distributional Equations and their Applications 17/18

25 More on coupling n general, the in-degree becomes the o spring and the out-degree becomes a mark. Tree is rooted at the randomly chosen vertex. All other nodes have a size bias. For the DCM: The coupling tree is a Galton-Watson process. The root has o spring according to F,allothernodesaccordingto: For the RD: h(m) =P (N = m) = E[1(D+ = m)d ] E[D ] Coupling tree is a multi-type BP with types distributed according to µ. For rank-1 kernels, i.e. x =(x +,x ), apple(x i, x j)= 1 x i x + j,itcanbe reduced to a single-type. The root has o spring according to Z +,amixedpoissonwithmixing distribution + (X), X µ, allothernodesaccordingto: h(m) =P (N = m) = E[1(Z+ = m)x ] E[X ] Bath-UNAM-CMAT, Lecture 3 Branching Distributional Equations and their Applications 18/18