Microscopic Evolution of Social Networks

1. Jure Leskovec, CMU Lars Backstrom, Cornell Ravi Kumar, Yahoo! Research Andrew Tomkins, Yahoo! Research Microscopic Evolution of Social Networks

2. Introduction This talk:We observed individual edge and node arrivals in largesocial networks and so on for millions… 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

3. Questions we ask • 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

4. The setting: Edge-by-edge evolution (F) (D) (A) (L) • Threeprocesses 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

5. The rest of the talk Experiments and the complete model of network evolution Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08

6. P1) How fast are nodes arriving? (F) (D) Flickr: Exponential Delicious: Linear (A) (L) Node arrival process is network dependent Answers: Sub-linear LinkedIn: Quadratic Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08

7. P1) What is node lifetime? LinkedIn Node lifetime is exponentially distributed: p(a) = λ exp(-λa) Lifetime a: time between node’s first and last edge Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08

8. The model so far … What do we know so far? Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08

9. P2) How do α & β evolve with degree? d=3 d=2 Degree d=1 Probability Nodes of higher degree start adding edges faster and faster Edge time gap (time between 2 consecutive edges of a node) Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08 Edge gap δ(d): time between dth and d+1stedge of a node

10. The model so far … What do we know so far? Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08

11. Preferential attachment: Does it hold? PA (L) Gnp (F) (A) (D) First direct proof of preferential attachment! Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08 We unroll the true network edge arrivals Measure node degrees where edges attach

12. PA? Not really. Edges are local! w Fraction of triad closing edges u (D) PA v Gnp (F) Real edges are local. Most of them close triangles! (L) (A) 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

13. How to close triangles? v’ w u v • New triad-closing edge (u,w) appears next • We model this as: • Choose u’s neighbor v • Choose v’s neighbor w • 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

14. Triad closing strategies Strategy to select v (1st node) w u v Select w (2nd 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 random-random works well Log-likelihood improvement over the baseline Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08

15. The complete model The complete network evolution model Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08

16. Analysis of our model Our theorem accurately predicts degree exponents γ as observed data 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

17. Summary and conclusion • 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 processnetwork evolution model • P1) Node lifetimes are exponential • P2) Edge interarrivaltime 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

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

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

20. 2) How do α & β evolve with degree? d=3 d=2 Degree d=1 This means nodes of higher degree start adding edges faster and faster Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08