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- 1. © Pieter Musterd/ flickr Competitive Diffusion A Scalable Framework for Modeling Competitive Diffusion in Social Networks Matthias Broecheler, Paulo Shakarian & V.S. Subrahmanian
- 2. Competitive Diffusion Diffusion in Social Networks Diffusion is a widely studied dynamic of social networks - Epidemiology • SIR Disease Model - Marketing • Viral Marketing - Health • Obesity Study - Campaign Management © Christakis, Fowler • Opinion Leaders 2
- 3. Competitive Diffusion Diffusion in Social Networks 3
- 4. Competitive Diffusion Competitive Diffusion 4
- 5. Competitive Diffusion Competitive Product Diffusion pictures are copyright of the Apple Inc, Google Inc and Samsung Inc. Used under fair use. 5
- 6. Competitive Diffusion Competitive Opinion Diffusion Pictures copyright of the respective party. Used under fair use. 6
- 7. Competitive Diffusion Key Points Competitive Diffusion occurs often Expressive Model for Competitive Diffusion Scale to large Social Networks
- 8. Competitive Diffusion General Diffusion Models One diffusion model for each problem class - Time consuming - Complex Wouldn’t it be nice to have a general, easy to use language to express such models? - Like SQL for data access Language Model SN Data Diffusion Process 8
- 9. Competitive Diffusion General Diffusion Models Approach: Develop an expressive framework to represent diffusion models using logical rules - Logic rules easy to read and write by humans • Shakarian & Subrahmanian [2010] - General idea: “If some condition holds in the network, then some diffusion is likely to occur with some confidence” - Based on General Annotated Programming (GAP) • More expressive than “standard” boolean rules • Can represent non-linear diffusion processes (arbitrary functions) • Kifer & Subrahmanian [92] 9
- 10. Competitive Diffusion Modeling in Practice I Think of your network as a database of facts - represented as (ground) atoms - Can represent un-/directed, multi-labeled data knows Mary Bob coworker best friends wife knows siblings Jeff John Jane wife(John,Mary), knows(John, Jane), siblings(Bob, Jane), etc 10
- 11. Competitive Diffusion Modeling in Practice II Write a set of rules that describe the diffusion process you are interested in - using annotated rules with variables which are grounded against your network database knows Mary Bob coworker best friends wife knows siblings Jeff John Jane vote(B,Dem):X wife(A,B):1 vote(A,Dem):X knows(A,Bi):1 vote(Bi,Dem):Xi vote(A,Dem):ΣXi÷n 11
- 12. Competitive Diffusion Diffusion Rules Ground Rule B1:X1 .. Bn:Xn H:f(Xi) Given Interpretation I: Satisfaction: I(H) ≥ f(I(Bi)) Distance from Satisfaction: max(0, f(I(Bi))-I(H) ) 12
- 13. Competitive Diffusion Competition Hard competition expressed as constraints - Example: A person has only one vote vote(A,Dem) + vote(A,Republican) ≤ 1 Soft competition expressed by rule weights which represent the relative probability that the described diffusion will happen - Example: If person B votes democratic, then B’s husband is likely to vote democrats as well (but not necessarily): vote(B,Dem):X wife(A,B):1 vote(A,Dem):X | 0.8 13
- 14. Competitive Diffusion Diffusion Rules II Ground Rule B1:X1 .. Bn:Xn H:f(Xi) | w Given Interpretation I: Satisfaction: I(H) ≥ f(I(Bi)) Weighted Distance from Satisfaction: w * max(0, f(I(Bi))-I(H) ) 14
- 15. Competitive Diffusion Probabilistic Model Semantics We use the rules and their weights to define a probability distribution over the space of “possible unfoldings of the diffusion process” - i.e. interpretations or confidence assignments - Exponential family distribution (as used e.g. in p* models) d(R1,I) All ground rules d(P,I) = d(R,I) x = d(Rn,I) P = set of rules x Ri = ground rule ( I | P) = 1/Z exp (- d(P,I)) 15
- 16. Competitive Diffusion Most Probable Interpretation Finding the most probable interpretation (MPI) is an optimization problem - Reminiscent of determining least energy state argmaxI ( I | P) = argminI d(P,I) Restricting the GAP annotations to be convex makes the problem tractable - We currently focus on conic annotations which give O(n3.5) complexity (i.e. SOCP) - n=number of ground rules 16
- 17. Competitive Diffusion 500 million users 50M tweets / day Scalability © Ludwig Gatzke
- 18. Competitive Diffusion Minimal Grounding Use fixpoint operator to determine the minimal non-ground interpretation - Keep the number of ground atoms small - Intuition: If there is no evidence for it, we don’t consider it • If John and Jane aren’t married, don’t need to consider rules with wife(John,Jane) - Implementation: Ground out rules iteratively until no further ground atoms are added to the interpretation. 18
- 19. Competitive Diffusion Dependency Graph vote(Mary,Dem) vote(Jane, Dem) vote(Mary,Dem) wife(John,Mary) vote(Jane,Dem) friend(John,Jane) vote(John,Dem) | 0.8 vote(John,Dem) | 0.3 vote(John,Dem) 19
- 20. Competitive Diffusion Dependency Graph vote(Mary,Dem) vote(Jane, Dem) vote(Mary,Dem) wife(John,Mary) vote(Jane,Dem) friend(John,Jane) vote(John,Dem) | 0.8 vote(John,Dem) | 0.3 vote(John,Dem) Idea: Partition Dependency graph into strongly connected components and solve MPI on each independently 20
- 21. Competitive Diffusion Approximate Algorithm 1. Ground out dependency graph with fixpoint operator 2. Partition dependency graph using a modularity maximizing clustering alg - Inspired by Blondel et al [06] - Aggregate rule weights 3. Compute MPI on each cluster fixing confidence values of outside atoms 4. Go to 1 until change in I < Θ 21
- 22. USA dean author Competitive Diffusion member Prof Prof Jones Baneri Italy in Paper “ABC” comment author UC CS UMD CS in faculty friends faculty Prof Calero department in member faculty presented Prof Dooley attended Social Science department University MD Universita Calabria department in dean ASONAM 09 attended faculty submitted Prof Roma author UMD Physics author member visited organized accepted friends author KPLLC Paper 09 “UVW” S3 Prof Smith Paper “HIJ” submitted Paper “XYZ” comment attended comment student of author S2 student of Prof Olsen collaborates Prof Lund member dean Prof Larsen faculty Jamie Lock member Karl Oede Social Science visited Odense SDU Physics Odense colleagues John Doe department Denmark
- 23. Competitive Diffusion Experiments Synthetically generated scale free, labeled social networks 6 edge types, 7 rules Used different parameter settings for convergence condition Executed on single 16 core machine with 256 GB of memory. 23
- 24. Competitive Diffusion Scalability 16000 Exact vs Approximate Algorithm Running Times 14000 12000 Time in Seconds Exact Algorithm 10000 Approximate Algorithm with Parameters A 8000 6000 4000 2000 0 0 10000 20000 30000 40000 50000 60000 70000 80000 # Edges in Graph 24
- 25. Competitive Diffusion Accuracy 7% Relative Error compared to Exact Inference 6% Percentage Relative Error 5% Parameters B 4% Parameters A 3% Parameters C Parameters D 2% Parameters E 1% 0% 0 10000 20000 30000 40000 50000 60000 70000 80000 Number of Edges 25
- 26. Competitive Diffusion Runtime Running Time Comparison of Approximate Algorithm 500 450 Parameters B 400 Parameters A 350 Time in Seconds Parameters C 300 Parameters D 250 Parameters E 200 150 100 50 0 0 10000 20000 30000 40000 50000 60000 70000 80000 Number of Edges 26
- 27. Competitive Diffusion Accuracy on large SN Relative Error on Large Networks 6% Percentage Relative Error 5% Parameters B Parameters C 4% Parameters D 3% Parameters E 2% 1% 0% 3.5E+05 7.0E+05 1.4E+06 2.8E+06 5.6E+06 Number of Edges Log-scale 27
- 28. Competitive Diffusion Runtime on large SN Runtime Comparison on Large Networks 40000 Time in Seconds 4000 Parameters B Parameters A 2M edges Parameters C in 48 min Parameters D Parameters E 400 3.5E+05 7.0E+05 1.4E+06 2.8E+06 5.6E+06 Number of Edges Log-log-scale 28
- 29. Competitive Diffusion Conclusion Expressive language for competitive diffusion models Scalable algorithm to compute such models on large social networks Verify model on real diffusion data 29
- 30. ? Competitive Diffusion Questions? Comments?

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