Recovering Temporally Rewiring Networks: A Model-based Approach

1. Recovering Temporally Rewiring Networks: A Model-based Approach Fan Guo, Steve Hanneke, Wenjie Fu, Eric P. Xing School of Computer Science, Carnegie Mellon University ICML 2007 Presentation

2. Social Networks Physicist Collaborations High School Dating The Internet All the images are from http://www-personal.umich.edu/~mejn/networks/. That page includes original citations.

3. Biological Networks Model for the Yeast cell cycle transcriptional regulatory networkFig. 4 from (T.I. Lee et al., Science 298, 799-804, 25 Oct 2002) Protein-Protein Interaction Network in S. cerevisiaeFig. 1 from (H. Jeong et al., Nature 411, 41-42, 3 May 2001) ICML 2007 Presentation

4. When interactions are hidden… • Infer the hiddennetwork topology from node attribute observations. • Methods: Optimizing a score function; Information-theoretic approaches; Model-based approach … • Most of them pool the data together to infer a static network topology. ICML 2007 Presentation

5. And changing over time • Network topologies and functions are not static: • Social networks can grow as we know more friends • Biological networks rewire under different conditions Fig. 1b fromGenomic analysis of regulatory network dynamics reveals large topological changes N. M. Luscombe, et al. Nature 431, 308-312, 16 September 2004 ICML 2007 Presentation

6. Overview • Network topologies and functions are not always static. • We propose probabilistic models and algorithms for recovering latent network topologies that are changing over time from node attribute observations. ICML 2007 Presentation

7. Rewiring Networks of Genes • Networks rewire over discrete timesteps Part of the image is modified from Fig. 3b (E. Segal et al., Nature Genetics 34, 166-176, June 2003).

8. A Graphical Model Counterpart Transition Model Emission Model ICML 2007 Presentation

9. Technical Challenges • Latent network structures are of higher dimensions than observed node attributes • How to place constraints on the latent space? • Limited evidence per timestep • How to share the information across time? ICML 2007 Presentation

10. Energy Based Conditional Probablities • Energy-based conditional probability model (recall Markov random fields…) • Energy-based model is easier to analysis, but even the design of approximate inference algorithm can be hard. 10/23/2019 ICML 2007 Presentation ICML 2007 Presentation 10

11. Transition Model • Based on our previous work on discrete temporal network models in the ICML’06 SNA-Workshop. • Model network rewiring as a Markov process. • An expressive framework using energy-based local probabilities (based on ERGM): • Features of choice: (Density) (Edge Stability) (Transitivity) 10/23/2019 ICML 2007 Presentation ICML 2007 Presentation 11

12. Emission Model in General • Given the network topology, how to generate the binary node attributes? • Another energy-based conditional model: • All features are pairwise which induces an undirected graph corresponding to the time-specific network topology; • Additional information shared over time is represented by a matrix of parameters Λ; • The design of feature functionΦ is application-specific. 10/23/2019 ICML 2007 Presentation ICML 2007 Presentation 12

13. Design of Features for Gene Expression • The feature function • If no edge between i and j, Φequals 0; • Otherwise the sign of Φ depends on Λijand the empirical correlation of xi, xj at time t. ICML 2007 Presentation

14. Graphical Structure Revisit Hidden rewiring networks Initial network to define the prior on A1 Time-invariant parameters dictating the direction of pairwise correlation in the example ICML 2007 Presentation

15. Inference • A natural approach to infer the hidden networks A1:T is Gibbs sampling: • To evaluate the log-odds • Conditional probabilities in a Markov blanket Tractable transition model; the partition function is the product of per edge terms Computation is straightforward Given the graphical structure, run variable elimination algorithms, works well for small graphs ICML 2007 Presentation

16. Parameter Estimation • Trade-off between the transition model and emission model: • Larger θ : better fit of the rewiring processes; • Larger η : better fit of the observations. • Grid search is very helpful, although Monte Carlo EM can be implemented. ICML 2007 Presentation

17. Results from Simulation • Data generated from the proposed model. • Starting from a network (A0) of 10 nodes and 14 edges. • The length of the time series T = 50. • Compare three approaches using F1 score: • avg: averaged network from “ground truth”(approx. upper bounds the performance of any static network inference algorithm) • htERG: infer timestep-specific networks • sERG: the static counterpart of the proposed algorithm • Study the “edge-switching events” ICML 2007 Presentation

18. Varying Parameter Values • F1 scores on different parameter settings (varying ) ICML 2007 Presentation

19. Varying the Amount of Data • F1 scores on different number of examples ICML 2007 Presentation

20. Capturing Edge Switching • Summary on capturing edge switching in networks • Three cases studied: offset, false positive, missing(false negative) • mean and rms of offset timesteps ICML 2007 Presentation

21. Results on Drosophila Data • The proposed model was applied to infer the muscle development sub-network (Zhao et al., 2006) on Drosophila lifecycle gene expression data (Arbeitman et al., 2002). • 11 genes, 66timesteps over 4 development stages • Further biological experiments are necessary for verification. Network in (Zhao et al. 2006) Embryonic Larval Pupal & Adult ICML 2007 Presentation

22. Summary • A new class of probabilistic models to address the problem of recoving hidden, time-dependent network topologies and an example in a biological context. • An example of employing energy-based model to define meaningful features and simplify parameterization. • Future work • Larger-scale network analysis (100+?) • Developing emission models for richer context ICML 2007 Presentation

23. Acknowledgement • Yanxin Shi CMU • Wentao Zhao Texas A&M University • Hetunandan Kamisetty CMU ICML 2007 Presentation

24. Thank You! ICML 2007 Presentation