Manuel Gomez Rodriguez 1,2 Jure Leskovec 1 Andreas Krause 3

1. 1 Stanford University2 MPI for Biological Cybernetics3 California Institute of Technology Inferring Networks of Diffusion and Influence Manuel Gomez Rodriguez1,2Jure Leskovec1Andreas Krause3

2. Networks and Processes Many times hard to directly observe a social or information network: Hidden/hard-to-reach populations: Drug injection users Implicit connections: Network of information sharing in online media Often easier to observe results of the processes taking place on such (invisible) networks: Virus propagation: People get sick, they see the doctor Information networks: Blogs mention information

3. Information Diffusion Network Information diffuses through the network • We only see who mentions but not where they got the information from • Can we reconstruct the (hidden) diffusion network?

4. More Examples Virus propagation Word of mouth & Viral marketing Viruses propagate through the network We only observe when people get sick But NOT who infected them • Recommendations and influence propagate • We only observe when people buy products • But NOT who influenced them Process We observe It’s hidden Can we infer the underlying social or information network?

5. Inferring the Network • There is ahiddendirected network: • We only see times when nodes get infected: • Contagion c1: (a, 1), (c, 2), (b, 3), (e, 4) • Contagion c2: (c, 1), (a, 4), (b, 5), (d, 6) • Want to infer the who-infects-whom network a a a b b b d d c c c e e

6. Our Problem Formulation Plan for the talk: Define a continuous time model of diffusion Define the likelihood of the observed propagation data given a graph Show how to efficiently compute the likelihood Show how to efficiently optimize the likelihood Find a graph G that maximizes the likelihood There is a super-exponential number of graphs G Our method finds a near-optimal solution in O(N2)!

7. Cascade generation model: Cascade reaches u at tu, and spreads to u’s neighbors v: with probability β cascade propagates along (u, v) and tv = tu + Δ, with Δ~ f() Cascade Generation Model ta tb tc te tf Δ1 Δ2 Δ3 Δ4 a a a b b b c c c d We assume each node v has only one parent! e e f f

8. Likelihood of a Cascade • If u infected v in a cascade c, its transmission probability is: • Pc(u, v)  f(tv - tu)with tv > tuand (u, v) are neighbors a a b b • To model that in reality any node v in a cascade can have been infected by an external influence m:Pc(m, j) =ε d c c m • Prob. that cascade c propagates in a tree T: ε e e ε ε Tree pattern T on cascade c: (a, 1), (b, 2), (c, 4), (e, 8)

9. Finding the Diffusion Network There are many possible propagation trees: c: (a, 1), (c, 2), (b, 3), (e, 4) Need to consider all possible propagation trees T supported by G: Good news Computing P(c|G) is tractable: • Bad news • We actually want to find: • We have a super-exponential number of graphs! a a a a a a b b b b b b For each c, consider all O(nn) possible transmission trees of G. Matrix Tree Theorem can compute this sum in O(n3)! d d d c c c c c c e e e e e e • Likelihood of a set of cascades C on G: • Want to find:

10. An Alternative Formulation We consider only the most likely tree Maximum log-likelihood for a cascade c under a graph G: Log-likelihood of G given a set of cascades C: The problem is NP-hard: MAX-k-COVER Our algorithm can do it near-optimally in O(N2)

11. Max Directed Spanning Tree Given a cascade c, What is the most likely propagation tree? where • A maximum directed spanning tree (MDST): • Just need to compute the MDST of a the sub-graph of G induced by c (i.e., a DAG) • For each node, just picks an in-edge ofmax-weight: Subgraph of G induced by c doesn’t have loops (DAG) a a 3 5 b b 2 c c 4 6 1 d d Local greedy selection gives optimal tree!

12. Great News: Submodularity Theorem: Log-likelihood Fc(G) is monotonic, and submodular in the edges of the graph G Fc(A  {e}) – Fc (A) ≥ Fc (B  {e}) – Fc (B) Gain of adding an edge to a “small” graph Gain of adding an edge to a “large“ graph A B  VxV • Proof: • Single cascade c, edge e with weight x A r i i w x • Let w be max weight in-edge of s in A s k k • Let w’ be max weight in-edge of s in B B • We know: w ≤ w’ j w’ • Now: Fc(A  {e}) – Fc(A) = max (w, x) – w ≥ max (w’, x) – w’ = Fc(B {e}) – Fc(B) o a Then, log-likelihood FC(G) is monotonic, and submodular too

13. Finding the Diffusion Graph Use the greedy hill-climbing to maximize FC(G): At every step, pick the edge that maximizes the marginal improvement Benefits • 1. Approximation guarantee (≈ 0.63 of OPT) • 2. Tight on-line bounds on the solution quality • 3. Speed-ups: • Lazy evaluation (by submodularity) • Localized update (by the structure of the problem) Marginal gains b a : 12 a c a : 3 b d a : 6 a b : 20 : 17 c b : 18 Localized update d Localized update : 2 : 1 d b : 4 c : 1 : 3 e b : 5 a c : 15 Localized update b c : 6 : 8 e b d : 16 Localized update c d : 7 : 8 e d : 8 : 10 b e : 7 d e : 13

14. Experimental Setup We validate our method on: Synthetic data Generate a graph G on k edges Generate cascades Record node infection times Reconstruct G Real data MemeTracker: 172m news articles Aug ’08 – Sept ‘09 343m textual phrases (quotes) • How well do we optimize Fc(G)? • How many edges of G can we find? • Precision-Recall • Break-even point • How fast are we? • How many cascades do we need?

15. Small Synthetic Example True network Baseline network Our method • Pick kstrongest edges: • Small synthetic network: 15

16. Synthetic Networks 1024 node hierarchical Kronecker exponential transmission model 1000 node Forest Fire (α = 1.1) power law transmission model • Our performance does not depend on the network structure: • Synthetic Networks: Forest Fire, Kronecker, etc. • Prob. of transmission: Exponential, Power Law • Break-even points of > 90% when the baseline gets 30-50%!

17. How good is our graph? We achieve ≈ 90 % of the best possible network!

18. How many cascades do we need? With 2x as many infections as edges, the break-even point is already 0.8 - 0.9!

19. Running Time Lazy evaluation and localized updates speed up 2 orders of magnitude!

20. Real Data MemeTracker dataset: 172m news articles Aug ’08 – Sept ‘09 343m textual phrases (quotes) Times tc(w) when site w mentions phrase (quote) c http://memetracker.org • Given times when sites mention phrases • We infer the network of information diffusion: • Who tends to copy (repeat after) whom

21. Real Network • We use the hyperlinks in the MemeTracker dataset to generate the edges of a ground truth G • From the MemeTracker dataset, we have the timestamps of: • 1. cascades of hyperlinks: • sites link other sites • 2.cascades of (MemeTracker) quotes • sites copy quotes from other sites e a c f Are they correlated? e a c f Can we infer the hyperlinks network from… …cascades of hyperlinks? …cascades of MemeTracker quotes?

22. Real Network 500 node hyperlink network using hyperlinks cascades 500 node hyperlink network using MemeTracker cascades • Break-even points of 50% for hyperlinks cascades and 30% for MemeTracker cascades!

23. Diffusion Network Blogs Mainstream media • 5,000 news sites:

24. Diffusion Network (small part) Blogs Mainstream media

25. Networks and Processes We infer hidden networks based on diffusion data (timestamps) Problem formulation in a maximum likelihood framework NP-hard problem to solve exactly We develop an approximation algorithm that: It is efficient -> It runs in O(N2) It is invariant to the structure of the underlying network It gives a sub-optimal network with tight bound Future work: Learn both the network and the diffusion model Extensions to other processes taking place on networks Applications to other domains: biology, neuroscience, etc.

26. Thanks! For more (Code & Data):http://snap.stanford.edu/netinf