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Network Learning with Directional Sign Patterns

Anqi Dong, Can Chen, Tryphon T. Georgiou

TL;DR

The paper tackles learning edge strengths in sign-indefinite networks from nodal marginals by formulating a Schrödinger bridge–type problem with directional sign templates. It derives a Schrödinger-Fortet-Sinkhorn framework, providing closed-form posterior updates and coordinate-ascent-type updates that achieve linear convergence, and extends the approach to general higher-order networks via tensor SBP and virtual-node uniformization. The method is validated on synthetic and ecological datasets, demonstrating accurate magnitude estimation under partial information and sign uncertainty, along with favorable convergence properties. This work offers a scalable, entropically regularized approach for inferring edge magnitudes in complex networks where signs may be uncertain or directional information is embedded in node statistics.

Abstract

Complex systems can be effectively modeled via graphs that encode networked interactions, where relations between entities or nodes are often quantified by signed edge weights, e.g., promotion/inhibition in gene regulatory networks, or encoding political of friendship differences in social networks. However, it is often the case that only an aggregate consequence of such edge weights that characterize relations may be directly observable, as in protein expression of in gene regulatory networks. Thus, learning edge weights poses a significant challenge that is further exacerbated for intricate and large-scale networks. In this article, we address a model problem to determine the strength of sign-indefinite relations that explain marginal distributions that constitute our data. To this end, we develop a paradigm akin to that of the Schrödinger bridge problem and an efficient Sinkhorn type algorithm (more properly, Schrödinger-Fortet-Sinkhorn algorithm) that allows fast convergence to parameters that minimize a relative entropy/likelihood criterion between the sought signed adjacency matrix and a prior. The formalism that we present represents a novel generalization of the earlier Schrödinger formalism in that marginal computations may incorporate weights that model directionality in underlying relations, and further, that it can be extended to high-order networks -- the Schrödinger-Fortet-Sinkhorn algorithm that we derive is applicable all the same and allows geometric convergence to a sought sign-indefinite adjacency matrix or tensor, for high-order networks. We demonstrate our framework with synthetic and real-world examples.

Network Learning with Directional Sign Patterns

TL;DR

The paper tackles learning edge strengths in sign-indefinite networks from nodal marginals by formulating a Schrödinger bridge–type problem with directional sign templates. It derives a Schrödinger-Fortet-Sinkhorn framework, providing closed-form posterior updates and coordinate-ascent-type updates that achieve linear convergence, and extends the approach to general higher-order networks via tensor SBP and virtual-node uniformization. The method is validated on synthetic and ecological datasets, demonstrating accurate magnitude estimation under partial information and sign uncertainty, along with favorable convergence properties. This work offers a scalable, entropically regularized approach for inferring edge magnitudes in complex networks where signs may be uncertain or directional information is embedded in node statistics.

Abstract

Complex systems can be effectively modeled via graphs that encode networked interactions, where relations between entities or nodes are often quantified by signed edge weights, e.g., promotion/inhibition in gene regulatory networks, or encoding political of friendship differences in social networks. However, it is often the case that only an aggregate consequence of such edge weights that characterize relations may be directly observable, as in protein expression of in gene regulatory networks. Thus, learning edge weights poses a significant challenge that is further exacerbated for intricate and large-scale networks. In this article, we address a model problem to determine the strength of sign-indefinite relations that explain marginal distributions that constitute our data. To this end, we develop a paradigm akin to that of the Schrödinger bridge problem and an efficient Sinkhorn type algorithm (more properly, Schrödinger-Fortet-Sinkhorn algorithm) that allows fast convergence to parameters that minimize a relative entropy/likelihood criterion between the sought signed adjacency matrix and a prior. The formalism that we present represents a novel generalization of the earlier Schrödinger formalism in that marginal computations may incorporate weights that model directionality in underlying relations, and further, that it can be extended to high-order networks -- the Schrödinger-Fortet-Sinkhorn algorithm that we derive is applicable all the same and allows geometric convergence to a sought sign-indefinite adjacency matrix or tensor, for high-order networks. We demonstrate our framework with synthetic and real-world examples.
Paper Structure (11 sections, 2 theorems, 38 equations, 4 figures, 2 algorithms)

This paper contains 11 sections, 2 theorems, 38 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Signed Networks
  4. Schrödinger Bridges Problem
  5. Sinkhorn with Directional Sign Templates
  6. General Higher-order Networks
  7. Numerical Examples
  8. Synthetic Example
  9. Ecological Network
  10. Convergence
  11. Conclusion

Key Result

Proposition 1

The optimizer P has a closed-form solution computed as where $\boldsymbol{\mu}\in\mathbb R^{n}$ and $\boldsymbol{\nu}\in\mathbb R^{n}$ are the Lagrangian multipliers of the constraints in eq:constraint2.

Figures (4)

  • Figure 1: Every hyperedge in the example is colored by one color. (A) 3-uniform higher-order network with hyperedges $e_1=\{1,2,3\}$, $e_2=\{3,4,5\}$, $e_3=\{5,6,7\}$ and $e_4=\{8,7,1\}$. (B) Non-uniform higher-order network with hyperedges $e_1=\{1,2,3,4\}$, $e_2=\{4,5\}$, $e_3 = \{5,6,7\}$ and $e_3=\{1,8,7\}$.
  • Figure 2: Uniformity conversion. (A) A non-uniform higher-order network with hyperedges $e_1=\{1,2,3\}$, $e_2=\{1,4\}$, and $e_3=\{3,4\}$ is converted to a $3$-uniform higher-order network by adding virtual node $5$. (B) The resulting 3-uniform higher-order network with hyperedges $e_1=\{1,2,3\}$, $e_2=\{1,4,5\}$, and $e_3=\{3,4,5\}$.
  • Figure 3: The topology of a $10$-node network with sign templates $\textbf{X}$ (left) and $\textbf{Y}$ (right).
  • Figure 4: The convergence of the proposed algorithm and the linear convergence rate in terms of marginal violations.

Theorems & Definitions (4)

  • Proposition 1: Closed-form solution
  • proof
  • Proposition 2: Convergence
  • proof