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Inferring Network Structure with Unobservable Nodes from Time Series Data

Mengyuan Chen, Yan Zhang, Zhang Zhang, Lun Du, Jiang Zhang

TL;DR

This work tackles inferring complete network structure when nodes or edges are unobservable by exploiting time-series data from network dynamics. It introduces Gumbel-softmax Inference for Network (GIN), which combines a differentiable network generator, an initial-state generator, and a graph-network dynamics learner to jointly infer unobserved topology, initial conditions, and node dynamics. Through extensive experiments on synthetic and empirical networks with both binary and continuous dynamics, GIN achieves high accuracy in recovering unobserved connections and predicting observable-node trajectories, with AUCs often exceeding 0.9 and up to 0.99 in reconstruction on some networks. The framework demonstrates robustness across network types and scales to large graphs, though performance degrades as the fraction of unobservable nodes grows; its modular design and seed-graph matching evaluation offer a practical path for applications where network data are incomplete or privacy-constrained.

Abstract

Network structures play important roles in social, technological and biological systems. However, the observable nodes and connections in real cases are often incomplete or unavailable due to measurement errors, private protection issues, or other problems. Therefore, inferring the complete network structure is useful for understanding human interactions and complex dynamics. The existing studies have not fully solved the problem of inferring network structure with partial information about connections or nodes. In this paper, we tackle the problem by utilizing time-series data generated by network dynamics. We regard the network inference problem based on dynamical time series data as a problem of minimizing errors for predicting states of observable nodes and proposed a novel data-driven deep learning model called Gumbel-softmax Inference for Network (GIN) to solve the problem under incomplete information. The GIN framework includes three modules: a dynamics learner, a network generator, and an initial state generator to infer the unobservable parts of the network. We implement experiments on artificial and empirical social networks with discrete and continuous dynamics. The experiments show that our method can infer the unknown parts of the structure and the initial states of the observable nodes with up to 90\% accuracy. The accuracy declines linearly with the increase of the fractions of unobservable nodes. Our framework may have wide applications where the network structure is hard to obtain and the time series data is rich.

Inferring Network Structure with Unobservable Nodes from Time Series Data

TL;DR

This work tackles inferring complete network structure when nodes or edges are unobservable by exploiting time-series data from network dynamics. It introduces Gumbel-softmax Inference for Network (GIN), which combines a differentiable network generator, an initial-state generator, and a graph-network dynamics learner to jointly infer unobserved topology, initial conditions, and node dynamics. Through extensive experiments on synthetic and empirical networks with both binary and continuous dynamics, GIN achieves high accuracy in recovering unobserved connections and predicting observable-node trajectories, with AUCs often exceeding 0.9 and up to 0.99 in reconstruction on some networks. The framework demonstrates robustness across network types and scales to large graphs, though performance degrades as the fraction of unobservable nodes grows; its modular design and seed-graph matching evaluation offer a practical path for applications where network data are incomplete or privacy-constrained.

Abstract

Network structures play important roles in social, technological and biological systems. However, the observable nodes and connections in real cases are often incomplete or unavailable due to measurement errors, private protection issues, or other problems. Therefore, inferring the complete network structure is useful for understanding human interactions and complex dynamics. The existing studies have not fully solved the problem of inferring network structure with partial information about connections or nodes. In this paper, we tackle the problem by utilizing time-series data generated by network dynamics. We regard the network inference problem based on dynamical time series data as a problem of minimizing errors for predicting states of observable nodes and proposed a novel data-driven deep learning model called Gumbel-softmax Inference for Network (GIN) to solve the problem under incomplete information. The GIN framework includes three modules: a dynamics learner, a network generator, and an initial state generator to infer the unobservable parts of the network. We implement experiments on artificial and empirical social networks with discrete and continuous dynamics. The experiments show that our method can infer the unknown parts of the structure and the initial states of the observable nodes with up to 90\% accuracy. The accuracy declines linearly with the increase of the fractions of unobservable nodes. Our framework may have wide applications where the network structure is hard to obtain and the time series data is rich.
Paper Structure (24 sections, 9 equations, 8 figures, 7 tables)

This paper contains 24 sections, 9 equations, 8 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Problem and Methods
  3. Problem Definition
  4. Optimization Problem Formulation
  5. Network Inference Framework
  6. Implementation Gumbel-softmax Inference for Network
  7. Graph Matching Problem in the Evaluation Process
  8. Experimental Results
  9. Data Set
  10. Synthetic Network
  11. Empirical Social Networks
  12. Time Series Data of Network Dynamics
  13. Data Preparation
  14. Experimental Setup
  15. Tasks
  16. ...and 9 more sections

Figures (8)

  • Figure 1: The states of some nodes are missing (with dashed box) and only partial network structure can be observed (with bold circles), and the aim is to infer the missing information (the dashed colored circles). In practice, network completion means to infer the “missing” element (with dark grey color) in the adjacency matrix.
  • Figure 2: The modules of the GIN model. At first, the network structure and the initial state can be generated by the network generator and initial state generator modules, respectively. After that, they are input into the dynamics learner to output the predicted value of the node states at the next time step.
  • Figure 3: The dynamics learner consists of four parts: (1) Node to Edge: aggregating the original information of nodes to form representations of edges; (2) Edge to Edge: update the edge representations; (3) Edge to Node: aggregate all information on neighbouring edges of each node to form a new feature vector of the current node; (4) node to node: update the node representations; (5) Output: finally, concatenate the node representations and the input state vectors of node i to feed into a feedforward network, and output the prediction of the next state all nodes.
  • Figure 4: Contrast matrices of the adjacency matrices between the inference and the ground truth for Dolphin network (a), and the Karate network (b). The invert L-shaped part in the figure has four colors, among which blue is the True Positive element, purple is the True Negative element, green is False True element, and red is the False Negative element.
  • Figure 5: AUC of unobservable network decreases with the proportion of observed nodes.
  • ...and 3 more figures