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 • 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) We observe individual edge and node arrivals in largesocial networks

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?

4. The setting: Edge-by-edge evolution • 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

5. Structure of our contributions • Experiments and the complete model of network evolution

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

7. Node lifetime is exponential LinkedIn • Lifetime a: time between node’s first and last edge Node lifetime is exponential: p(a) = λ exp(-λa)

8. 2) How are edges initiated? Edge gap δ(d): inter-arrival time between dth and d+1st edge LinkedIn

9. 2) How do α & β evolve with degree? d=3 Degree d=1 d=2 Probability Edge time gap (time between 2 consecutive edges of a node)

10. Degree of Preferential Attachment

11. Edge Locality: closing triangles

12. Degree of PA & Edge locality Fraction of triad closing edges

13. How to close triangles? • We consider 25 strategies for choosing node v and then w • Compute likelihood of each strategy

14. Triad closing strategies • Log-likelihood improvement over the baseline 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

15. The complete model

16. Analysis of our model • Theorem: node lifetimes and edge gaps lead to power law degree distribution • Interesting as temporal behavior predicts structural network property

17. Evolving the networks • Given our model one can take an existing network continue its evolution

18. Comparison with other models • Take Flickr at time T/2 and then further evolve it continue evolving it using PA and our model.

19. 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 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

20. Choosing edge source and destination