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Network Games Induced Prior for Graph Topology Learning

Chenyue Zhang, Shangyuan Liu, Hoi-To Wai, Anthony Man-Cho So

TL;DR

This work focuses on social networks with actions modeled by equilibriums of linear quadratic games, and postulates that the social network topologies are optimized with respect to a social welfare function.

Abstract

Learning the graph topology of a complex network is challenging due to limited data availability and imprecise data models. A common remedy in existing works is to incorporate priors such as sparsity or modularity which highlight on the structural property of graph topology. We depart from these approaches to develop priors that are directly inspired by complex network dynamics. Focusing on social networks with actions modeled by equilibriums of linear quadratic games, we postulate that the social network topologies are optimized with respect to a social welfare function. Utilizing this prior knowledge, we propose a network games induced regularizer to assist graph learning. We then formulate the graph topology learning problem as a bilevel program. We develop a two-timescale gradient algorithm to tackle the latter. We draw theoretical insights on the optimal graph structure of the bilevel program and show that they agree with the topology in several man-made networks. Empirically, we demonstrate the proposed formulation gives rise to reliable estimate of graph topology.

Network Games Induced Prior for Graph Topology Learning

TL;DR

This work focuses on social networks with actions modeled by equilibriums of linear quadratic games, and postulates that the social network topologies are optimized with respect to a social welfare function.

Abstract

Learning the graph topology of a complex network is challenging due to limited data availability and imprecise data models. A common remedy in existing works is to incorporate priors such as sparsity or modularity which highlight on the structural property of graph topology. We depart from these approaches to develop priors that are directly inspired by complex network dynamics. Focusing on social networks with actions modeled by equilibriums of linear quadratic games, we postulate that the social network topologies are optimized with respect to a social welfare function. Utilizing this prior knowledge, we propose a network games induced regularizer to assist graph learning. We then formulate the graph topology learning problem as a bilevel program. We develop a two-timescale gradient algorithm to tackle the latter. We draw theoretical insights on the optimal graph structure of the bilevel program and show that they agree with the topology in several man-made networks. Empirically, we demonstrate the proposed formulation gives rise to reliable estimate of graph topology.

Paper Structure

This paper contains 12 sections, 16 theorems, 72 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminary: Graph Learning from Smooth Signals
  3. Graph Learning with Network Games Prior
  4. Structural Interpretation for the Network Games Prior
  5. Efficient Algorithm for Tackling \ref{['eq:glfp']}
  6. Numerical Experiments
  7. Synthetic Data
  8. Case Studies with Real Data
  9. Conclusions
  10. Proof of Proposition \ref{['prop:approx']}
  11. Proof of Proposition \ref{['prop:kkt']}
  12. Proof of Theorem \ref{['thm:ttgd']}

Key Result

Proposition 1

Under Hassu:lipsf, problem eq:net-bilevel-app is equivalent to a relaxed version of problem eq:glfp.

Figures (2)

  • Figure 1: Performance of \ref{['eq:glfp']} via TTGD on learning from PA graphs. (Left) AUC performance. (Right) Social welfare.
  • Figure 2: Comparing the social welfare ${\sf Wel}({\bm W}) = {\bf 1}^\top {\bm y}^{\sf NE}({\bm W})$ against the data fidelity term $J( {\bm W}, {\bm X})$ for GLGP. (Left) Prior with $f(x) = \log(1+x)$. (Right) Prior with $f(x) = x$.

Theorems & Definitions (30)

  • Example 1
  • Proposition 1
  • Proposition 2
  • Theorem 1
  • proof
  • proof
  • Theorem
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 20 more