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Recovering Temporally Rewiring Networks: A Model-based Approach

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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

Physicist Collaborations

High School Dating

The Internet

All the images are from http://www-personal.umich.edu/~mejn/networks/. That page includes original citations.

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)

The small image is from http://www.raiks.de/img/dyna_title_zoom.jpg

- 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

- 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

- 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

- 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).

Transition Model

Emission Model

ICML 2007 Presentation

- 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

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.

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ICML 2007 Presentation

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- 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)

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ICML 2007 Presentation

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- 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.

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ICML 2007 Presentation

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- 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

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

- 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

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

ICML 2007 Presentation

- 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

- F1 scores on different parameter settings (varying )

ICML 2007 Presentation

- F1 scores on different number of examples

ICML 2007 Presentation

- 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

- 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

- 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

- Yanxin Shi CMU
- Wentao Zhao Texas A&M University
- Hetunandan Kamisetty CMU

ICML 2007 Presentation

ICML 2007 Presentation