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Jure Leskovec, CMU Lars Backstrom, Cornell Ravi Kumar, Yahoo! Research Andrew Tomkins, Yahoo! Research

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Social networks evolve with additions and deletions of nodes and edges We talk about the evolution but few have actually directly observed atomic events of network evolution (but only via snapshots) Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08 This talk: We observed individual edge and node arrivals in large social networks and so on for millions…

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Test individual edge attachment: Directly observe mechanisms leading to global network properties ▪ E.g., What is really causing power-law degree distributions? Compare models: via model likelihood Compare network models by likelihood (and not by summary network statistics) ▪ E.g., Is Preferential Attachment best model? Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08

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Three processes that govern the evolution P1) Node arrival process: nodes enter the network P2) Edge initiation process: each node decides when to initiate an edge P3) Edge destination process: determines destination after a node decides to initiate Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08 (F) (D) (A) (L)

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Experiments and the complete model of network evolution ProcessOur finding P1) Node arrival P2) Edge initiation P3) Edge destination Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08

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(F) (D) (A) (L) Flickr: Exponential Delicious: Linear Answers: Sub-linear LinkedIn: Quadratic Node arrival process is network dependent

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Lifetime a: time between node’s first and last edge Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08 Node lifetime is exponentially distributed: p(a) = λ exp(-λa) LinkedIn

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What do we know so far? Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08 ProcessOur finding P1) Node arrival Node arrival function is given Node lifetime is exponential P2) Edge initiation P3) Edge destination

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Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08 Edge gap δ(d): time between d th and d+1 st edge of a node Degree d=1 d=3 d=2 Edge time gap (time between 2 consecutive edges of a node) Probability Nodes of higher degree start adding edges faster and faster

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What do we know so far? Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08 ProcessOur finding P1) Node arrival Node arrival function is given Node lifetime is exponential P2) Edge initiation Edge gaps: P3) Edge destination

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Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08 G np PA (D) (F) (L) (A) Networkτ G np 0 PA1 F1 D1 A0.9 L0.6 We unroll the true network edge arrivals Measure node degrees where edges attach First direct proof of preferential attachment!

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u w v Just before the edge (u,v) is placed how many hops is between u and v? Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08 Network% Δ% Δ F66%66% D28% A23% L50% G np PA (D) (F) (L) (A) Fraction of triad closing edges Real edges are local. Most of them close triangles!

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New triad-closing edge (u,w) appears next We model this as: 1. Choose u’s neighbor v 2. Choose v’s neighbor w 3. Connect (u,w) We consider 25 strategies for choosing v and then w Can compute likelihood of each strategy Under Random-Random: p(u,w) = 1/5*1/2+1/5*1 Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08 u w v v’

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Log-likelihood improvement over the baseline Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08 Strategy to select v (1 st node) Select w (2 nd node) Strategies to pick a neighbor: random: uniformly at random deg: proportional to its degree com: prop. to the number of common friends last: prop. to time since last activity comlast: prop. to com*last u w v random-random works well

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The complete network evolution model Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08 ProcessOur finding P1) Node arrival Node arrival function is given Node lifetime is exponential P2) Edge initiation Edge gaps: P3) Edge destination 1 st edge is created preferentially Use random-random to close triangles

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Theorem: node lifetimes and edge gaps lead to power law degree distribution Interesting as temporal behavior predicts structural network property Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08 NetworkTrue γ Predicted γ F1.731.74 D2.382.30 A1.901.75 L2.112.08 Our theorem accurately predicts degree exponents γ as observed data

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We observe network evolution at atomic scale We use log-likelihood of edge placements to compare and infer models Our findings Preferential attachment holds but it is local Triad closure is fundamental mechanism We present a 3 process network evolution model P1) Node lifetimes are exponential P2) Edge interarrival time is power law with exp. cutoff P3) Edge destination is chosen by random-random Gives more realistic evolution that other models Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08

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More details and analyses in the paper Thanks to Yahoo and LinkedIn for providing the data. http://www.cs.cmu.edu/~jure http://www.cs.cmu.edu/~jure Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08

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Edge gap δ(d): time between d th and d+1 st edge Edge interarrivals follow power law with exponential cutoff distribution: Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08 How do α and β change with node degree?

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Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08 This means nodes of higher degree start adding edges faster and faster Degree d=1 d=3 d=2

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