Dynamic Network Model from Par5al Observa5ons Elahe Ghalebi 1 , Baharan Mirzasoleiman 2 , Radu Grosu 1 , Jure Leskovec 2 1 TU Wien and 2 Stanford University NeurIPS 2018
Can evolving network be inferred and modeled without directly observing their nodes and edges? •In many applicaGons, the edges of a dynamic network might not be observed •We can only observe the dynamics of stochasGc cascading process e.g. informaGon diffusion, virus propagaGon occurring over the unobserved network ( . ( - ( ) ( + ( * % ( + % % % $ $ $ $ ( - ( * ( , ( , ( - ( + # # # # ' ' ' ' ( , ( . ( * ( ) ( ) ( . ! ! ! ! & & & & " " " "
DYFERENCE framework 1- Extrac5ng Observa5on from Diffusion Data t 3 b t 3 t 1 a t 5 t 2 b t 3 b a t 1 t 4 t 2 t 4 d c t 1 < t 2 < . . . c d d e t 2 f t 1 t 4 e g c 1 t 5 c 2 c 3 t 3 t 3 b b t 2 b t 1 a t 5 t 4 t 2 t 3 a t 1 t 4 c d c d d t 1 e e t 2 g f t 4 E c 3 E c 1 E c 2 t 5 Find the set of possible edges in each cascade as E c i = { e uv | t c i u < t c i c i v < ∞ } Sample a set S c of θ ( | E c | ) edges based on marginal probabiliGes
DYFERENCE framework 1- ExtracGng ObservaGon from Diffusion Data t 3 b t 3 t 1 a t 5 t 2 b t 3 b a Round #1 t 1 t 4 t 2 t 4 d c t 1 < t 2 < . . . c d d e t 2 f t 1 t 4 e g c 1 t 5 c 2 c 3 t 3 t 3 b b t 2 b t 1 a t 5 t 4 t 2 t 3 a t 1 t 4 c d c d d t 1 e e t 2 g f t 4 E c 3 E c 1 E c 2 t 5 t 3 t 3 t 2 b b t 5 t 1 a a c t 4 t 2 t 3 t 1 t 4 d c c d d e t 1 e t 2 S 1 f t 4 S 3 S 2 g t 5 Calculate probability distribuGon over edges consistent with each cascade E c i Calculate marginal probability of every edge in each E c i Sample a set S c of θ ( | E c | ) edges based on marginal probabiliGes Sample a set of edges based on marginal probabiliGes θ ( | E c i | ) S c i
DYFERENCE framework 1- ExtracGng ObservaGon from Diffusion Data 2- Update the model with the extracted observa5on using a collapsed Gibbs sampler X 1 t 3 b t 3 t 1 a t 5 t 2 b t 3 b a t 1 t 4 t 2 t 4 Round #1 d c c d d e t 2 f t 1 t 4 e g c 1 t 5 c 2 c 3 t 3 t 3 b b t 2 b t 1 a t 5 t 4 t 2 t 3 a b t 1 b b t 4 c d a c d a a d c c c t 1 e e d d t 2 g f t 4 E c 3 E c 1 E c 2 t 5 e e f f g f f g t 3 t 3 t 2 b b t 5 t 1 a a c t 4 t 2 t 3 t 1 t 4 X 1 = { S 1 , S 2 , S 3 } d c c d d e t 1 e For the model, we use mixture of Dirichlet t 2 S 1 f t 4 S 3 S 2 g t 5 network distribuGons (MDND) [Williamson’16] Calculate probability distribuGon over edges consistent with each cascade E c i Calculate marginal probability of every edge in each E c i Sample a set S c of θ ( | E c | ) edges based on marginal probabiliGes Sample a set of edges based on marginal probabiliGes θ ( | E c i | ) S c i
DYFERENCE framework 1- ExtracGng ObservaGon from Diffusion Data 2- Update the model with the extracted observaGon using a collapsed Gibbs sampler X 2 t 3 b t 3 t 1 a t 5 t 2 b t 3 b a t 1 t 4 t 2 t 4 Round #2 d c c d d e t 2 f t 1 t 4 e g c 1 t 5 c 2 c 3 t 3 t 3 b b t 2 b t 1 a t 5 t 4 t 2 t 3 a b t 1 b b t 4 c d a c d a a d c c c t 1 e e d d t 2 g f t 4 E c 3 E c 1 E c 2 t 5 e e f f g f f g t 3 t 5 t 1 a t 3 b c t 2 b t 3 X 2 = { S 1 , S 2 , S 3 } t 4 t 2 a d t 1 t 4 c c d d e t 2 t 1 S 3 e t 4 f g S 1 S 2 t 5 Calculate probability distribuGon over each using updated edge probabiliGes from model E c i Calculate marginal probability of every edge in each E c i Sample a set S c of θ ( | E c | ) edges based on marginal probabiliGes Sample a set of edges based on marginal probabiliGes θ ( | E c i | ) S c i
Online Dynamic Network Inference 1. DiscreGze Gme into intervals with length ω 2. Consider only infecGon Gmes in current interval t c ∈ [( i − 1) ω , i ω ], ∀ c ∈ C 3. Update model with the observaGons in current ω
Performance Evalua5on Dynamic Bankruptcy Prediction European country’s financial transacGons: 1,197,116 transacGons;103,497 companies Our algorithm significantly outperforms the baselines
Conclusion ✓ Our algorithm provides a genera&ve probabilis&c model which: ✦ IdenGfies the underlying Gme-varying community structure ✦ Obtains dynamic predicGve distribuGon over the edges ✦ Can be used for diffusion predicGon, predicGng the most influenGal nodes, and bankruptcy predicGon Poster: Today (Wed Dec 5th. @ Room 210 & 230) #7
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