Ranking of nodes in graphs

Transcription

1 Ranking of nodes in graphs Alejandro Ribeiro & Cameron Finucane Dept. of Electrical and Systems Engineering University of Pennsylvania October 3, 200 Stoch. Systems Analysis Ranking of nodes in graphs

2 Ranking of nodes in graphs: Random walk Ranking of nodes in graphs: Random walk Ranking of nodes in graphs: Markov chain Stoch. Systems Analysis Ranking of nodes in graphs 2

3 Graphs Graph A set of J nodes j =,..., J Connected by a set of edges E defined as ordered pairs (i, j) In figure nodes are j =, 2, 3, 4, 5, edges E = {(, 2), (, 5),(2, 3), (2, 5), (3, 4),... (3, 6), (4, 5), (4, 6), (5, 4)} Websites and links The web. People and friendship Social network Stoch. Systems Analysis Ranking of nodes in graphs 3

4 How well connected nodes are? Q: Which node is the most connected? A: Define most connected Can define most connected in different ways Node rankings to measure website quality, social influence There are two important connectivity indicators How many nodes point to a link (outgoing links irrelevant) How Important are the links that point to a node Stoch. Systems Analysis Ranking of nodes in graphs 4

5 Connectivity ranking Insight There is information in the structure of the network Knowledge is distributed through the network The network (not the nodes) know the rankings Idea exploited by Google s PageRank c to rank webpages... by social scientists to study trust & reputation in social networks... by ISI to rank scientific papers, transactions & magazines... No one points to Only points to 2 Only 2 points to 3, but 2 more important than 4 as high as 5 with less links 2 3 Links to 5 have lower rank Same for 6 Stoch. Systems Analysis Ranking of nodes in graphs 5

6 Preliminary definitions Graph G = (V, E) sets of vertices V = {, 2,..., J} and edges E Edges (elements of E) are ordered pairs (i, j) We say there is a connection from i to j Outgoing neighborhood of i is the set of nodes j to which i points n(i) := {j : (i, j) E} Incoming neighborhood, n (i) is the set of nodes that point to i: n (i) := {j : (j, i) E} Connected graph There is a path from any node to any other node Stoch. Systems Analysis Ranking of nodes in graphs 6

7 Definition of rank Agent A chooses node i, e.g., web page, at random for initial visit Next visit randomly chosen between links in the neighborhood n(i) All neighbors chosen with equal probability If reach a dead end because node i has no neighbors Chose next visit at random equiprobably among all nodes Redefine graph G = (V, E) adding edges from dead ends to all nodes Restrict attention to connected (modified) graphs Rank of node i is the average number of visits of A to i Stoch. Systems Analysis Ranking of nodes in graphs 7

8 Equiprobable random walk Formally, let A n be the node visited at time n Define transition probability P ij from node i into node j P ij := P [ A n+ = j A n = i ] Next visit equiprobable among neighbors P ij = #[n(i)] = N i, Defined number of neighbors N i = #[n(i)] for all j n(i) /2 /2 5 4 /2 /2 /2 /2 2 3 /2 /2 6 /5 /5 /5 /5 /5 to to 2 to 3 to 4 to 5 Still have a graph But also a MC Red (not blue) circles Stoch. Systems Analysis Ranking of nodes in graphs 8

9 Formal definition of rank Consider variable I {A m = i} to indicate visit to state i at time m. Rank r i of i-th node defined as time average of number of visits, i.e., r i := lim n n n I {A m = i} m= Define vector of ranks r := [r, r 2,..., r J ] T Rank r i can be approximated by average r ni at time n r ni := n n I {A m = i} m= Since lim n r ni = r i, it holds r ni r i for n sufficiently large Define vector of approximate ranks r n := [r n, r n2,..., r nj ] T If modified graph is connected, rank independent of initial visit Stoch. Systems Analysis Ranking of nodes in graphs 9

10 Algorithm Output : Vector r(i) with ranking of node i Input : Vector N(i) containing number of neighbors of i Input : Matrix N(i, k) containing indices j of neighbors of i m = ; r=zeros(j,); % Initialize time and ranks A 0 = random( unid,j); % Draw first visit uniformly at random while m < n do jump = random( unid,n Am ); % Neighbor uniformly at random A m = N(A m, jump); % Jump to selected neighbor r(a m ) = r(a m ) + ; % Update ranking for A m m = m + ; end r = r/n; % Normalize by number of iterations n Stoch. Systems Analysis Ranking of nodes in graphs 0

11 Example: Social graph Asked students taking ESE303 about homework collaboration Created (crude) graph of the social network of students in this class Used ranking algorithm to understand connectedness E.g., If I want to know how well students are coping with the class it is best to ask people with higher connectivity ranking 2009 data Students in 200 don t show up in class in enough numbers Stoch. Systems Analysis Ranking of nodes in graphs

12 Ranked class graph Harish Venkatesan Jacci Jeffries Pallavi Yerramilli Xiang-Li Lim Thomas Cassel Owen Tian Eric Lamb Daniela Savoia Ceren Dumaz Priya Takiar Lindsey Eatough Sugyan Lohiaa Ankit Aggarwal Lisa Zheng Madhur Agarwal Anthony Dutcher Robert Feigenberg Carolina Lee Aarti Kochhar Ciara Kennedy Amanda Smith Saksham Karwal Michael Harker Amanda Zwarenstein Ranga Ramachandran Varun Balan Katie Joo Shahid Bosan Pia Ramchandani Jesse Beyroutey Rebecca Gittler Ivan Levcovitz Jane Kim Paul Deren Alexandra Malikova Jihyoung Ahn Chris Setian Charles Jeon Ella Kim Aditya Kaji Stoch. Systems Analysis Ranking of nodes in graphs 2

13 Convergence metrics Recall r is vector of ranks and r n of rank iterates By definition lim r n = r. How fast r n converges to r (r given)? n Can measure by distance between r and r n ( J ) /2 ζ n := r r n 2 = (r ni r i ) 2 If interest is only on largest ranked nodes, e.g., a web search Denote r (i) as the index of the i-th highest ranked node Similarly, r (i) n i= is the index of the i-th highest ranked node at time n First element wrongly ranked at time n ξ n := min r (i) r n (i) i Stoch. Systems Analysis Ranking of nodes in graphs 3

14 Evaluation of convergence metrics 0 Distance correctly ranked nodes Distance gets close to 0 2 in approx iterations time (n) 4 2 First element wrongly ranked Bad: Two largest ranks in iterations Awful: Six best ranks in iterations correctly ranked nodes Convergence appears (very) slow time (n) Stoch. Systems Analysis Ranking of nodes in graphs 4

15 When does this algorithm converge? Can confidently claim convergence not until 0 5 iterations True for particular case. Slow convergence inherent to algorithm Example has 40 nodes, want to use in network with 0 9 nodes correctly ranked nodes time (n) x 0 5 Use fact that this process is a MC to obtain faster algorithm Stoch. Systems Analysis Ranking of nodes in graphs 5

16 Ranking of nodes in graphs: Markov chain Ranking of nodes in graphs: Random walk Ranking of nodes in graphs: Markov chain Stoch. Systems Analysis Ranking of nodes in graphs 6

17 Limit probabilities Recall definition of rank r i := lim t t t I{A(u) = i} u= Rank is time average of number of state visits in a MC Can be equally obtained from limiting probabilities Recall transition probabilities P ij = N i, for all j n(i) Stationary distribution π = [π, π,..., π J ] T solution of π i = j n (i) P ji π j = π j N j j n (i) Plus normalization equation J i= π i = As per ergodicity r = π for all i Stoch. Systems Analysis Ranking of nodes in graphs 7

18 Matrix notation, eigenvalue problem As always, can define matrix P with elements P ij π i = J P ji π j = P ji π j j n (i) j= for all i Right hand side is just definition of a matrix product leading to π = P T π, π T = Also added normalization equation Can solve as system of linear equations or eigenvalue problem on P T Non-iterative method Convergence not an issue But requires matrix P available at a central location Computationally costly (matrix P with 0 9 rows and columns) All methods are costly to compute exact solution This one is costly to find even approximate solution Stoch. Systems Analysis Ranking of nodes in graphs 8

19 What are limit probabilities? Let p i (n) denote probability of agent A visiting node i at time t p i (n) := P [A n = i] Probabilities at time n + and n can be related P [A n+ = i] = P [ A n = i An = i ] P [A n = j] j n (i) Which is, of course, probability propagation in a MC p i (n + ) = P ji p j (n) j n (i) By definition limit probabilities are (let p(n) = [p (n),..., p J (n)] T ) lim p(n) = π = r n Compute ranks from limit of probability propagation Stoch. Systems Analysis Ranking of nodes in graphs 9

20 Probability propagation Can also write probability propagation in matrix form p i (n + ) = J P ji p j (n) = P ji p j (n) j n (i) j= for all i Right hand side is just definition of a matrix product leading to p(n + ) = P T p(n) Can approximate rank by probability distribution r = lim n p(n) p(n) for n sufficiently large Stoch. Systems Analysis Ranking of nodes in graphs 20

21 Algorithm Algorithm is just a recursive matrix product Output : Vector r(i) with ranking of node i Input : Matrix P containing transition probabilities m = ; % Initialize time r=(/j)ones(j,); % Initial distribution uniform across all nodes while m < n do r = P T r; % Probability propagation m = m + ; end Stoch. Systems Analysis Ranking of nodes in graphs 2

22 Interpretation of probability propagation Why does the random walk converges so slow? What does it take to obtain a time average r ni close to r i? Need to register a large number of agent visits to every state Back of hand: 40 nodes, some 00 visits to each iters. Idea: Unleash a large number of agents K r i = lim n n n m= K K I {A km = i} k= Visits are now spread over time and space Converges K times faster (depends agents initial distribution) But haven t changed computational cost Stoch. Systems Analysis Ranking of nodes in graphs 22

23 Interpretation of prob. propagation (continued) What happens if we unleash infinite number of agents K? n K r i = lim lim I {A km = i} n n K K m= Using law of large numbers and expected value of indicator function n n r i = lim E [I {A m = i}] = lim P [A m = i] n n n n m= Graph walk is a MC, then r i = lim n n k= m= lim P [A m = i] = lim p i(m) exists, and m m n m= p i (m) = lim n p i(n) Probability propagation Unleashing infinite number of agents Interpretation true for any MC Stoch. Systems Analysis Ranking of nodes in graphs 23

24 Distance to rank Initialize with uniform probability distribution p(0) = (/J) Distance between p(n) and r Distance time (n) Distance is 0 2 in approximately 30 iterations, 0 4 in 40 iterations Convergence is two orders of magnitude faster than random walk Stoch. Systems Analysis Ranking of nodes in graphs 24

25 Number of nodes correctly ranked Rank of highest ranked node that is wrongly ranked by time n correctly ranked nodes time (n) Not bad: All nodes correctly ranked in 20 iterations Good: Ten best ranks in 80 iterations Great: Four best ranks in 20 iterations Stoch. Systems Analysis Ranking of nodes in graphs 25

26 Distributed algorithm to compute ranks Nodes want to compute their rank r i Can communicate with neighbors only (incoming + outgoing) Access to neighborhood information only Recall probability update p i (n + ) = Uses local information only j n (i) P ji p j (n) = j n (i) Algorithm. Nodes keep local rank estimates p i (n) N j p j (n) Receive rank (probability) estimates p j (n) from neighbors j n (i) Update local rank estimate p i (n + ) = j n (i) p j(n)/n j Communicate rank estimate p i (n + ) to outgoing neighbors j n(i) Need only know number of neighbors of my neighbors Stoch. Systems Analysis Ranking of nodes in graphs 26

27 Distributed implementation of random walk Can communicate with neighbors only (incoming + outgoing) But cannot access neighborhood information Pass agent around Local rank estimates r i (n) and counter with number of visits V i Algorithm run by node i at time n if Agent received from neighbor then V i = V i + Choose random neighbor Send agent to chosen neighbor end n = n + ; r i (n) = V i /n; Speed up convergence by generating many agents to pass around Stoch. Systems Analysis Ranking of nodes in graphs 27

28 Comparison of different algoithms Random walk implementation Most secure & robust. No information shared with other nodes Implementation can be distributed Convergence exceedingly slow System of linear equations Least security and robustness. Graph in central server Distributed implementation not clear Non-iterative method, convergence not a problem But computationally costly to obtain approximate solutions Probability propagation, matrix powers Somewhat secure/robust. Info. shared with neighbors only Implementation can be distributed Convergence rate acceptable (orders of magnitude faster than RW) Stoch. Systems Analysis Ranking of nodes in graphs 28