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Connection Probabilities Estimation in Multi-layer Networks via Iterative Neighborhood Smoothing

Dingzi Guo, Diqing Li, Jingyi Wang, Wen-Xin Zhou

TL;DR

This work introduces a new Multi-layer Iterative Connection Probability Estimation method that jointly refines inter-layer and intra-layer similarity sets by dynamically updating distance metrics derived from current probability estimates and provides a theoretically grounded and practically scalable framework for probabilistic modeling and inference in multi-layer network systems.

Abstract

Understanding the structural mechanisms of multi-layer networks is essential for analyzing complex systems characterized by multiple interacting layers. This work studies the problem of estimating connection probabilities in multi-layer networks and introduces a new Multi-layer Iterative Connection Probability Estimation (MICE) method. The proposed approach employs an iterative framework that jointly refines inter-layer and intra-layer similarity sets by dynamically updating distance metrics derived from current probability estimates. By leveraging both layer-level and node-level neighborhood information, MICE improves estimation accuracy while preserving computational efficiency. Theoretical analysis establishes the consistency of the estimator and shows that, under mild regularity conditions, the proposed method achieves an optimal convergence rate comparable to that of an oracle estimator. Extensive simulation studies across diverse graphon structures demonstrate the superior performance of MICE relative to existing methods. Empirical evaluations using brain network data from patients with Attention-Deficit/Hyperactivity Disorder (ADHD) and global food and agricultural trade network data further illustrate the robustness and effectiveness of the method in link prediction tasks. Overall, this work provides a theoretically grounded and practically scalable framework for probabilistic modeling and inference in multi-layer network systems.

Connection Probabilities Estimation in Multi-layer Networks via Iterative Neighborhood Smoothing

TL;DR

This work introduces a new Multi-layer Iterative Connection Probability Estimation method that jointly refines inter-layer and intra-layer similarity sets by dynamically updating distance metrics derived from current probability estimates and provides a theoretically grounded and practically scalable framework for probabilistic modeling and inference in multi-layer network systems.

Abstract

Understanding the structural mechanisms of multi-layer networks is essential for analyzing complex systems characterized by multiple interacting layers. This work studies the problem of estimating connection probabilities in multi-layer networks and introduces a new Multi-layer Iterative Connection Probability Estimation (MICE) method. The proposed approach employs an iterative framework that jointly refines inter-layer and intra-layer similarity sets by dynamically updating distance metrics derived from current probability estimates. By leveraging both layer-level and node-level neighborhood information, MICE improves estimation accuracy while preserving computational efficiency. Theoretical analysis establishes the consistency of the estimator and shows that, under mild regularity conditions, the proposed method achieves an optimal convergence rate comparable to that of an oracle estimator. Extensive simulation studies across diverse graphon structures demonstrate the superior performance of MICE relative to existing methods. Empirical evaluations using brain network data from patients with Attention-Deficit/Hyperactivity Disorder (ADHD) and global food and agricultural trade network data further illustrate the robustness and effectiveness of the method in link prediction tasks. Overall, this work provides a theoretically grounded and practically scalable framework for probabilistic modeling and inference in multi-layer network systems.
Paper Structure (12 sections, 3 theorems, 14 equations, 4 figures, 3 tables, 1 algorithm)

This paper contains 12 sections, 3 theorems, 14 equations, 4 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methodology
  3. Notation and model
  4. Iterative connection probability estimation for multi-layer networks
  5. Theoretical Properties
  6. Simulation Studies
  7. Algorithm Effectiveness
  8. Parameter Robustness
  9. Real Data Analysis
  10. Brain Network Data in ADHD Patients
  11. Worldwide Food and Agricultural Trade Network Data
  12. Conclusion

Key Result

Theorem 1

Let $C_l,C_m,C_l',C_m',\tilde{C}_q>0$ be some sufficiently large constants, where $l\in\{2,3\}$, $m\in\{4,5,6,7,8\}$, and $q\in \{1,2,3\}$. Define $C_*=\max\{C_l/3,C_m\}$ and $C_*'=\max\{C_l'/3,C_m'\}$. Then, with probability at least $1-2n^{-C_1/4}-2K^{-C_1'/4}-14n^{-C_*'}K^{-C_*}$, the following e

Figures (4)

  • Figure 1: Connection probability matrices corresponding to Graphons 1--5.
  • Figure 2: Average RMSE with standard deviation error bars under varying choices of $D_i$ and $G_k$. (For the setting $(n=200, K=20)$, the case $\log_2 G_k=2$ is omitted because the resulting neighborhood size exceeds the layer size.)
  • Figure 3: Receiver operating characteristic curves on two real-world datasets.
  • Figure 4: Two examples of validated link predictions. Emerging connections in 2023 are correctly forecasted from 2022 data with estimated probabilities of 0.712 and 0.626.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3