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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis 1 How many samples do we need to converge? How many Random Walk steps to get plot on the right?
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Consider the standard M/M/1 chain Assume with start with K initial customers K ≠ E[N] Q: How long until convergence to π (stationary distribution)? Start on a sunny day Q: How long until P(rainy) = π(rainy)? 2 0 λ 1 2 μ … λ μ λ μ
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Community Detection: Identify (sub)sets of nodes that are better connected to each other than the rest of the network Not easy! visually easy a posteriori, but at first the network on the right is just a large matrix Clustering/Machine Learning/Pattern Recognition 3
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis If a Markov Chain (defined by transition matrix P) is ergodic (irreducible, aperiodic, and positive recurrent) P (n) ik π k and π = [π 1, π 2,…, π n ] Q: But how fast does the chain converge? E.g. how many steps until we are “close enough” to π A: This depends on the eigenvalues of P The convergence time is also called the mixing time 4
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Left Eigenvectors A row vector π is a left eigenvector for eigenvalue λ of matrix P iff πP = λπ Σ k π k p ki = λπ i Right Eigenvectors A column vector v is a right eigenvector for eigenvalue λ of matrix P iff Pv = λv Σ k p ik v k = λv i Q: What eigenvalues and eigenvectors can we guess already? A: λ = 1 is a left eigenvalue with eigenvector π the stationary distr. λ = 1 is a right eigenvalue with eigenvector v=1 (all 1s) 5
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Both sets have non-zero solutions (P - λI) is singular There exists v ≠ 0 such that (P-λI)v = 0 Determinant |P-λI| = 0 (p 11 - λ)(p 22 - λ)-p 12 p 21 = 0 λ 1 =1, λ 2 = 1 – p 12 – p 21 (replace above and confirm using some algebra) |λ 2| < 1 (normalized: π (1) to be a stationary distribution AND v (i) ∙ π (i) = 1, ∀i) 6
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Eigenvalue decomposition: P = U Λ U -1 Q: What is P (n) ? A: => Q: How fast does the chain converge to stationary distrib.? A: It converges exponentially fast in n, as ( λ 2 ) n 7
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis We’ll assume that there are M distinct eigenvalues (see notes for repeated ones) Matrix P is stochastic all eigenvalues |λ i | ≤ 1 Q: Why? A: Q: How fast does an (ergodic) chain converge to stationary distribution? A: Exponentially with rate equal to 2 nd largest eigenvalue 8
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis λ 2 (2 nd largest eigenvalue) related to (balanced) min-cut of the graph The more “partitioned” a graph is into clusters with few links between them the longer the convergence time for the respective MC the slower the random walk search 9 9
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis P7-11 L= D-A= 1 23 4 Diagonal matrix, d ii =d i
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis P7-12 1 23 4 10 0.3 2 4
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis so, zero is an eigenvalue If k connected components, Fiedler (‘73) called “algebraic connectivity of a graph” The further from 0, the more connected.
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis P7-14 G(V,E) L= eig(L)= #zeros = #components 123 6 75 4
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis P7-15 G(V,E) L= eig(L)= #zeros = #components 123 6 75 4 0.01 Indicates a “good cut”
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Device-to-Device Communication (e.g. Bluetooth or WiFi Direct) 19
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Data/Malware Spreading Over Opp. Nets 20 / 38 Contact Process: Due to node mobility Q: How long until X% of nodes “infected”? ACBD D EF D D D D Contact Process: Due to node mobility Q: How long until X% of nodes “infected”?
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis News/Videos on Online Social Networks 21 Contact/Interaction: (random) times when user i posts/writes to user j, or user j checks out i’s page. “transfer” during a contact with probability p Q: How long until a video goes “viral”? interaction (post, share) i j
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Email Network 22 An email with a virus or worm A graph showing which users send emails to whom Pairwise contact process: (random) times of emails between i and j Q: How long for the worm to spread??
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Assumption 1) Underlay Graph Fully meshed Assumption 2) Contact Process Poisson(λ ij ), Indep. Assumption 3) Contact Rate λ ij = λ (homogeneous) 23 Analysis of Epidemics: The Usual Approach
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis 2-hop infection 24
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis A Poisson Graph A Real Contact Graph (ETH Wireless LAN trace) 25
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Bounding the Transition Delay What are we really saying here?? Let a = 3 how can split the graph into a subgraph of 3 and a subgraph of N-3 node, by removing a set of edges whose weight sum is minimum? 27
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Φ is a fundamental property of a graph Related to graph spectrum, community structure 28
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Eurecom, Sophia-Antipolis Thrasyvoulos Spyropoulos / spyropoul@eurecom.fr Navigability: Decentralized Search of Large Networks
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis http://ccl.northwestern.edu/netlogo/models/run.cgi?GiantComponent.884.534
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis http://projects.si.umich.edu/netlearn/NetLogo4/RAndPrefAttachment.html
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Source: http://maps.google.com
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis from Milgram's original article in Psychology Today
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis How to choose among hundreds of acquaintances? Strategy: Simple greedy algorithm - each participant chooses correspondent who is closest to target with respect to the given property Models Geography Kleinberg (2000) Hierarchical groups Watts, Dodds, Newman (2001), Kleinberg(2001) High degree nodes Adamic, Puniyani, Lukose, Huberman (2001), Newman(2003)
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Consider the following simple search algorithm (to find a destination) -- “Decentralized Search” 1. I know all my neighbors and their location 2. At every step I move to my neighbor closest to the destination Q: Does this greedy algorithm find short paths?? (i.e. O(logN) jops)? 36 Erdos-Renyi (Poisson) Random Graphs are small-world! dest
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Cannot find short paths with local, greedy algorithms (even though they exist) Q: What is the problem? A: Even if y is closer than x to dest (in some embedded coordinate system) we have no expectations about y’s neighbors all its neighbors might be further than x Q: How many steps, on average, to reach the destination? 37
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis What about the Watts- Strogatz small-world model? Regular links to k closest Random rewiring with prob p Or this one? Regular 2D lattice Each node has an additional k random links (to any node) 38 dest Decentralized search cannot find short paths either!
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Q: What is wrong this time? A: We have two options 1. Set of “close” neighbors 2. Random shortcut 39 from Milgram's original article in Psychology Today
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Option 1: Close neighbors Traverse up to k hops (constant) Option 2: Shortcuts Traverse ~n ½ hops (constant) Q: What happens when remaining distance < n ½ to destination? A: Small probability that a shortcut will take us closer need to follow lattice links only Q: How many hops for this last part? A: Order of n ½ / k 40
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Intuition: 1. long-range links (shortcuts) not to any node with equal probability 2. The further away the node, the smaller the chance we “know” him Shortcuts: Prob(of link at distance d) ∼ d -q Original model: q = 0 41 small qlarge q
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis How does decentralized search perform for different q values? q = 0 random shortcuts) already saw it doesn’t work Result: best performance for q = 2 Q: Why q = 2? A: number of shortcuts at different “scales” is constant 42
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis 1D lattice (i.e. ring)? long range link at distance d with prob d -q Optimal if q = 1 Result: n-dimensional lattice optimal q = n 43
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis (as before) move to direct neighbor, unless there is a shortcut leading closer Break the sequence of steps into phases Phase j ) 2 j+1 < distance < 2 j 44 a i
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Q:What is the total number of phases? A: Not more than log 2 n (why?) X i = number of steps to finish phase I This is a random variable E[X] = E[X 1 + X 2 + … + X logn ] Goal: Show that E[X i ] ~logn steps 45
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Prob(link d hops far): P(d) ~ 1/d Need to normalize: n nodes 46
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Q: how many nodes at distance d/2 from destination? A: N(d/2) = d/2 + d/2 = d Q: What is the probability of a shortcut to one of them? A: Furthest away at 3d/2 Q: How many steps to leave phase j? A: Xi ~ Geometric(p) with p = (3logn) -1 E[Xi] ≤ 3logn E[X] ~ (logn) 2 47
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis The average user will have ~ 2.5 non-geographic friends The other friends (5.5 on average) are distributed according to an approximate 1/distance relationship But 1/d was proved not to be navigable by Kleinberg, so what gives?
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis = d(u,v) the distance between pairs of people The probability that two people are friends given their distance is equal to P( ) = + f( ), is a constant independent of geography is 5.0 x 10 -6 for LiveJournal users who are very far apart
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Kleinberg assumed a uniformly populated 2D lattice But population is far from uniform population networks and rank-based friendship probability of knowing a person depends not on absolute distance but on relative distance -i.e. how many people live closer Pr[u ->v] ~ 1/rank u (v)
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Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis
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