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1 A Random-Surfer Web-Graph Model (Joint work with Avrim Blum & Hubert Chan) Mugizi Rwebangira

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2 The Web as a Graph Consider the World Wide Web as a graph, with web pages as nodes and hyperlinks between pages as edges. links.html resume.html index.html http://cnn.com

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3 Studying the Web Since the Web emerged there has been a lot of interest in: 1.Empirically studying properties of the Web Graph. 2.Modeling the Web Graph mathematically. Benefits of Generative Models: 1.Simulation – When real data is scarce 2.Extrapolation – How will the graph change? 3.Understanding – Inspire further research on real data

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4 Power Law The distribution of a random variable X follows a power law if Prob [X=k] ~ Ck -α f(x) ~ g(x) if Lim x→∞ f(x)/g(x) = 1 e.g (x+1) ~ (x+2) Example: Prob [X=k] = k -2

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5 Power Law: Prob [X=k] = k -2

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6 Power Law log Prob [X=k] ~ log C –α log k Prob [X=k] ~ Ck -α Prob [X=k] = k -2 log Prob [X=k] = -2 log k

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7 Power Law: Log-Log plot

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8 Power Law contd. Prob [X≥k] ~ Ck -α Particularly useful if X takes on real values. More general definition: Sometimes referred to as “heavy tailed” or “scale free.”

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9 Power Laws in Degree distribution Let G be a graph. Let X k be the proportion of nodes with degree k in G. Then if X k ~ Ck -α we say that G has power law degree distribution.

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10 Properties of the Web Graph A Power-law degree distribution has been observed in a wide variety of graphs including citation networks, social networks, protein-protein interaction networks and so on. It has also been observed in the Web Graph. [Barabási & Albert]

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11 Outline Background/Previous Work Motivation Models Theoretical results Experimental results Conclusions

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12 Classic Random Graph Models In the G(n,p) random graph model: 1.There are n nodes. 2.There is an edge between any two nodes with probability p. Was proposed by Erdös and Renyi in 1960s.

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13 Online G(n,p) In this model each new node makes k connections to existing nodes uniformly at random. For this talk we will focus on k = 1, hence the graph will be a tree.

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14 Online G(n,p) T=1 T=2 ½ T=3 ½ T=4 ⅓ ⅓ ⅓

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15 Properties of Online G(n,p) X k = Proportion of nodes with degree k E[X k ] = (½ k ) E[degree of first node] = 1+ 1/2 +1/3+1/4 + … 1/n = (log n) E[max degree] = (log n) NOT POWER LAWED!!

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16 Online G(n,p) (n=100,000, average of 100 runs)

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17 Preferential Attachment In the Preferential Attachment model, each new node connects to the existing nodes with a probability proportional to their degree. [Barabási & Albert]

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18 Preferential Attachment T=2 ¾ T=3 ¼ Deg = 3 Deg = 1 T=4 Deg = 4 Deg = 1 T=1 Degree = in-degree + out-degree

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19 Preferential Attachment Preferential Attachment gives a power-law degree distribution. [Mitzenmacher, Cooper & Frieze 03, KRRSTU00] E[degree of 1st node] = √n

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20 Preferential Attachment

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21 Other Models Kumar et. al. proposed the “copying model.” [KRRSTU00] Leskovec et. al. propose a “forest fire” model which has some similarites to this work. [LKF05]

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22 Outline Background/Previous Work Motivation Models Theoretical results Experimental results Conclusions

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23 Motivating Questions Why would a new node connect to nodes of high degree? -Are high degree nodes more attractive? -Or are there other explanations? How does a new node find out what the high degree nodes are?

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24 Motivating Questions Motivating Observation: If p is small then this is the same as preferential attachment. Suppose a user does a (undirected) random walk until they find an interesting page. What about other processes and directed graphs? Suppose each page has a small probability p of being interesting.

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25 Outline Background/Previous Work Motivation Models Theoretical results Experimental results Conclusions

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26 Directed 1-step Random Surfer, p=.5 ¾ T=3 ¼ (½) (½)+ (½) (½)+ (½) (½) T=1 Start with a single node with a self-loop. T=2 1.Choose a node uniformly at random 2.With probability p connect 3.With probability (1-p) connect to its neighbor

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27 Directed 1-step Random Surfer It turns out this model is a mixture of connecting to nodes uniformly at random and preferential attachment. But taking one step is not very natural. Has a power-law degree distribution. What about doing a real random walk?

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28 NEW NODE RANDOM STARTING NODE 1. COIN TOSS: TAIL (at node A) 2. COIN TOSS: TAIL (at node B) 3. COIN TOSS: HEAD (at node C) 1.Pick a node uniformly at random. 2. Flip a coin of bias pIf HEADS connect to current node, else walk to neighbor A B C D Directed Coin Flipping model

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29 Directed Coin Flipping model 1.At time 1, we start with a single node with a self-loop. 2.At time t, we choose a node u uniformly at random. 3.We then flip a coin of bias p. 4.If the coin comes up heads, we connect to the current node. 5.Else we walk to a random neighbor and go to step 3. “each page has equal probability p of being interesting to us”

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30 Outline Background/Previous Work Motivation Models Theoretical results Experimental results Conclusions

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31 Is Directed Coin-Flipping Power- lawed? We don’t know … but we do have some partial results...

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32 Virtual Degree Definitions: Let l i (u) be the number of level i descendents of node u. l 1 (u) = # of children l 2 (u) = # of grandchildren, e.t.c. Let = (β 1, β 2,..) be a sequence of real numbers with 1 =1. Then v (u) = 1 + β 1 l 1 (u) + β 2 l 2 (u) + β 3 l 3 (u) + … We’ll call v (u) the “Virtual degree of u with respect to .”

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33 u Virtual Degree v(u) = 1 + β 1 (2) + β 2 (4) + β 3 (0) + β 4 (0) +... # of children# of grandchildren

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34 Virtual Degree Easy observation: If we set β i = (1-p) i then the expected increase in deg(u) is proportional to v(u). Expected increase in deg(u) = p/t + (1-p)pl 1 (u)/t + (1-p) 2 pl 2 (u)/t + … = (p/t)v(u) u

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35 Virtual Degree Theorem: There always exist β i such that 1.For i ≥ 1, |β i | · 1. 2.As i → ∞, β i →0 exponentially. 3.The expected increase in v(u) is proportional to v(u). Recurrence: 1 =1, 2 =p, i+1 = i – (1-p) i-1 for p=½, i = 1, 1/2, 0, -1/4, -1/4, -1/8, 0, 1/16, … E.g., for p=¾, i = 1, 3/4, 1/2, 5/16, 3/16, 7/64,...

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36 Virtual Degree, continued Theorem: For any node u and time t ≥ t u, E[v t (u)] = Θ((t/t u ) p ) Let v t (u) be the virtual degree of node u at time t and t u be the time when node u first appears. So, the expected virtual degrees follow a power law.

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37 Actual Degree Theorem: For any node u and time t ≥ t u, E[degree(u)] ≥ Ω((t/t u ) p(1-p) ) We can also obtain lower bounds on the expected values of the actual degrees:

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38 Outline Background/Previous Work Motivation Models Theoretical results Experimental results Conclusions

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39 Experiments Random graphs of n=100,000 nodes Compute statistics averaged over 100 runs. K=1 (Every node has out-degree 1)

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40 Online Erdös-Renyi

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41 Directed 1-Step Random Surfer, p=3/4

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42 Directed 1-Step Random Surfer, p=1/2

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43 Directed 1-Step Random Surfer, p=1/4

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44 Directed Coin Flipping, p=1/2

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45 Directed Coin Flipping, p=1/4

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46 Undirected coin flipping, p=1/2

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47 Undirected Coin Flipping p=0.05

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48 Outline Background/Previous Work Motivation Models Theoretical results Experimental results Conclusions

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49 Conclusions Directed random walk models appear to generate power-laws (and partial theoretical results). Power laws can naturally emerge, even if all nodes have the same intrinsic “attractiveness”.

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50 Open questions Can we prove that the degrees in the directed coin- flipping model do indeed follow a power law? Analyze degree distribution for the undirected coin-flipping model with p=1/2? Suppose page i has “interestingness” p i. Can we analyze the degree as a function of t, i and p i ?

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51 Questions?

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