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Representation learning of dynamic networks

Haixu Wang, Jiguo Cao, Jian Pei

TL;DR

This work addresses learning representations for dynamic networks by reframing the problem in functional data analysis. It introduces time-continuous, asymmetric node embeddings $(\bm{\alpha}_j(t), \bm{\beta}_j)$ in a shared $R$-dimensional functional space $\mathcal{F}$, with $p_{jk}(t) = \mathrm{logit}^{-1}(\langle \bm{\alpha}_j(t), \bm{\beta}_k\rangle)$ and basis expansions $\alpha_{ir}(t) = \sum_{d=1}^D \gamma_{ird} \phi_d(t)$. The model estimation combines a penalized likelihood with clustering penalties to preserve community structure and enforce smooth, time-consistent embeddings; theoretical results guarantee consistency and asymptotic normality of the embeddings. Empirically, the approach outperforms existing network-embedding methods in link prediction under missing data and yields interpretable, dynamic community structure in real ant-colony networks. Overall, the framework enables continuous-time network inference, prediction, and community analysis using low-dimensional, time-evolving representations.

Abstract

This study presents a novel representation learning model tailored for dynamic networks, which describes the continuously evolving relationships among individuals within a population. The problem is encapsulated in the dimension reduction topic of functional data analysis. With dynamic networks represented as matrix-valued functions, our objective is to map this functional data into a set of vector-valued functions in a lower-dimensional learning space. This space, defined as a metric functional space, allows for the calculation of norms and inner products. By constructing this learning space, we address (i) attribute learning, (ii) community detection, and (iii) link prediction and recovery of individual nodes in the dynamic network. Our model also accommodates asymmetric low-dimensional representations, enabling the separate study of nodes' regulatory and receiving roles. Crucially, the learning method accounts for the time-dependency of networks, ensuring that representations are continuous over time. The functional learning space we define naturally spans the time frame of the dynamic networks, facilitating both the inference of network links at specific time points and the reconstruction of the entire network structure without direct observation. We validated our approach through simulation studies and real-world applications. In simulations, we compared our methods link prediction performance to existing approaches under various data corruption scenarios. For real-world applications, we examined a dynamic social network replicated across six ant populations, demonstrating that our low-dimensional learning space effectively captures interactions, roles of individual ants, and the social evolution of the network. Our findings align with existing knowledge of ant colony behavior.

Representation learning of dynamic networks

TL;DR

This work addresses learning representations for dynamic networks by reframing the problem in functional data analysis. It introduces time-continuous, asymmetric node embeddings in a shared -dimensional functional space , with and basis expansions . The model estimation combines a penalized likelihood with clustering penalties to preserve community structure and enforce smooth, time-consistent embeddings; theoretical results guarantee consistency and asymptotic normality of the embeddings. Empirically, the approach outperforms existing network-embedding methods in link prediction under missing data and yields interpretable, dynamic community structure in real ant-colony networks. Overall, the framework enables continuous-time network inference, prediction, and community analysis using low-dimensional, time-evolving representations.

Abstract

This study presents a novel representation learning model tailored for dynamic networks, which describes the continuously evolving relationships among individuals within a population. The problem is encapsulated in the dimension reduction topic of functional data analysis. With dynamic networks represented as matrix-valued functions, our objective is to map this functional data into a set of vector-valued functions in a lower-dimensional learning space. This space, defined as a metric functional space, allows for the calculation of norms and inner products. By constructing this learning space, we address (i) attribute learning, (ii) community detection, and (iii) link prediction and recovery of individual nodes in the dynamic network. Our model also accommodates asymmetric low-dimensional representations, enabling the separate study of nodes' regulatory and receiving roles. Crucially, the learning method accounts for the time-dependency of networks, ensuring that representations are continuous over time. The functional learning space we define naturally spans the time frame of the dynamic networks, facilitating both the inference of network links at specific time points and the reconstruction of the entire network structure without direct observation. We validated our approach through simulation studies and real-world applications. In simulations, we compared our methods link prediction performance to existing approaches under various data corruption scenarios. For real-world applications, we examined a dynamic social network replicated across six ant populations, demonstrating that our low-dimensional learning space effectively captures interactions, roles of individual ants, and the social evolution of the network. Our findings align with existing knowledge of ant colony behavior.

Paper Structure

This paper contains 8 sections, 1 theorem, 18 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Method
  3. Model Specification
  4. Model Estimation
  5. Theoretical Properties
  6. Simulation Studies
  7. Analysis on the Dynamic Network of Ant Colonies
  8. Conclusions and Discussion

Key Result

Theorem 1

Given that the regularity assumptions A1-5 are met, then there exists a maximizer $\hat{\mathbf{\gamma}}$ to the penalized likelihood (eq:finalobj) such that $||\hat{\mathbf{\gamma}} - \mathbf{\gamma}_{0}|| = O_{p}(n^{-\frac{1}{2}} + \xi_{n})$ in an open neighborhood of the true coefficients $\mathb

Figures (7)

  • Figure 1: Three common changes that may happen in dynamic networks. The first row illustrates the insertion (blue) and deletion (red) of links from time point $t_{1}$ to $t_{2}$. The second row depicts the addition (blue) and deletion (red) of nodes in the dynamic network from time point $t_{1}$ to $t_{2}$. The third row presents how communities are constructed through linking individual nodes of the dynamic network.
  • Figure 2: The percentages represent the portion of time points that are skipped for estimating the time-dependent node embeddings. For example, $2\%$ represents that $2\%$ of $n$ time points have been skipped between a pair of actually observed adjacency matrices.
  • Figure 3: On the left: The total degree of the dynamic network structures ants in each colony over time. On the right: the sum of estimated probabilities of connections at each time point. The color indicates the ant colony.
  • Figure 4: Static clustering of an ant colony. On the left: hierarchical clustering of the network summed over time. In the middle: clustering on the $\bm{\alpha}_{j}(t)$'s. On the left: clustering of the $\bm{\beta}_{j}$'s.
  • Figure 5: Display of the out-going embedding function $\bm{\alpha}_{j}(t)$'s. Different colors indicate different clusters of $\bm{\alpha}_{j}(t)$'s. The solid line represents the pointwise mean and the shaded area demonstrates the values of the embedding functions.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Remark 1