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Change-Point Detection in Dynamic Networks with Missing Links

Farida Enikeeva, Olga Klopp

TL;DR

The paper develops a Matrix CUSUM framework for detecting and localizing change-points in dynamic, high-dimensional networks observed with missing links. It establishes minimax detection boundaries in spectral norm, introduces a flexible sparsity notion κ_n, and extends the methodology to graphon-based settings with changing node sets, deriving upper and lower bounds for both step-function and Hölder graphons. The authors also provide a consistent change-point estimator and validate the approach through extensive simulations on SBMs and graphons, along with a real-data TfL dataset, demonstrating robustness to missing data and practical applicability. Overall, the work delivers a theoretically grounded, adaptable procedure for reliable change-point detection in partially observed dynamic networks, with clear implications for fraud detection, cybersecurity, and network monitoring.

Abstract

Structural changes occur in dynamic networks quite frequently and its detection is an important question in many situations such as fraud detection or cybersecurity. Real-life networks are often incompletely observed due to individual non-response or network size. In the present paper we consider the problem of change-point detection at a temporal sequence of partially observed networks. The goal is to test whether there is a change in the network parameters. Our approach is based on the Matrix CUSUM test statistic and allows growing size of networks. We show that the proposed test is minimax optimal and robust to missing links. We also demonstrate the good behavior of our approach in practice through simulation study and a real-data application.

Change-Point Detection in Dynamic Networks with Missing Links

TL;DR

The paper develops a Matrix CUSUM framework for detecting and localizing change-points in dynamic, high-dimensional networks observed with missing links. It establishes minimax detection boundaries in spectral norm, introduces a flexible sparsity notion κ_n, and extends the methodology to graphon-based settings with changing node sets, deriving upper and lower bounds for both step-function and Hölder graphons. The authors also provide a consistent change-point estimator and validate the approach through extensive simulations on SBMs and graphons, along with a real-data TfL dataset, demonstrating robustness to missing data and practical applicability. Overall, the work delivers a theoretically grounded, adaptable procedure for reliable change-point detection in partially observed dynamic networks, with clear implications for fraud detection, cybersecurity, and network monitoring.

Abstract

Structural changes occur in dynamic networks quite frequently and its detection is an important question in many situations such as fraud detection or cybersecurity. Real-life networks are often incompletely observed due to individual non-response or network size. In the present paper we consider the problem of change-point detection at a temporal sequence of partially observed networks. The goal is to test whether there is a change in the network parameters. Our approach is based on the Matrix CUSUM test statistic and allows growing size of networks. We show that the proposed test is minimax optimal and robust to missing links. We also demonstrate the good behavior of our approach in practice through simulation study and a real-data application.

Paper Structure

This paper contains 40 sections, 20 theorems, 166 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Contributions and comparison with previous results
  3. Notation
  4. Modeling dynamic networks with missing links
  5. Modeling sparse networks
  6. Change-point detection problem
  7. Matrix CUSUM statistic
  8. Definitions from the minimax testing theory
  9. Change-point detection at a given location
  10. Testing for change-point at an unknown location with missing links
  11. Change-point localization
  12. Testing the change in the sparse graphon model
  13. Numerical experiments
  14. Results for Stochastic Block Models
  15. Scenario 1: SBM with 2 communities, change in connection probability between communities.
  16. ...and 25 more sections

Key Result

Theorem 1

Let $\alpha,\beta \in(0,1)$ be given significance levels. Assume that for some universal constant $C$ depending only on $\epsilon\in(0,1/2]$, Then, for the test $\psi_{n,T}^\tau$ defined in (eq:test:P1_wo_log) with threshold (eq:H_P1_wo_log) we have

Figures (7)

  • Figure 1: The test powers with known (tests $\psi$) and estimated sparsity parameters (tests $\widehat{\psi}$) for $n=100$, $\tau/T=0.5$ and $T=100$ for Scenario 1 (on the left) and $T=250$ for Scenario 4 (on the right).
  • Figure 2: The risk of the change-point estimator under Scenario 1 for $T=100$, $n=100$ and the change-points $\tau\in\{5,25,50\}$ (left to right). The links are observed at the constant sampling rate $p_n\in(0,1]$. The estimation is easier when the change-point is located in the middle (the graph to the right).
  • Figure 3: The risk of the change-point estimator under three different patterns of missing links for $T=100$, $n=100$ and $\tau=50$ (Setting A--C, from left to right).
  • Figure 4: The test power for the Markov dependent networks with $n=100$ vertexes and $\lambda =0.6$ following Scenario 1 (on the left) for $T=100$, and Scenario 4 (on the right) for $T=250$.
  • Figure 5: The risk of the change-point estimator for $T=100$, $n=100$ and the change-point in the middle, $\tau=50$ for Scenarios 1, 4, and 5.
  • ...and 2 more figures

Theorems & Definitions (48)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • proof
  • Theorem 2
  • Remark 1
  • Remark 2
  • Theorem 3
  • proof
  • ...and 38 more