Switching Network System Identification via Convex Optimizations

TL;DR

This work addresses the challenge of identifying switching network systems where both node dynamics and topology switch among a finite set of modes from trajectory data. It develops a bilevel convex optimization framework that models per-mode dynamics as polynomial vector fields $\mathbf{f}_j(x) = C_j\phi_d(x)$ and adjacency matrices via $a_j = \mathrm{vec}(\mathbf{A}_j)$, relaxing binary constraints with simplex and moment-based semidefinite relaxations. An alternating scheme combines mode-search via SDP leveraging order-1 moment relaxations with network-dynamics search via LP, yielding estimates of $\{\mathcal{G}_j, \mathbf{f}_j, \lambda_j(x)\}$. The approach is demonstrated on diffusively coupled oscillators, achieving accurate recovery of switching graphs, mode dynamics, and a learned switching surface that aligns with the ground-truth boundary, underscoring its potential for scalable data-driven discovery of dynamic networks with abrupt structural changes.

Abstract

This paper introduces a convex optimization framework for identifying switched network systems, in which both the node dynamics and the underlying graph topology switch between a finite number of configurations. Building on our recent convex identification method for general switching systems, we extend the formulation to structured network systems where each mode corresponds to a distinct adjacency matrix. We show that both the continuous node dynamics and binary network topologies can be identified from sampled state-velocity data by solving a sequence of convex programs. The proposed framework provides a unified and scalable way to recover piecewise network structures from data without a prior knowledge of mode labels at each state. Numerical results on diffusively coupled oscillators demonstrate accurate recovery of both mode dynamics and switching graphs.

Figures (5)

• Figure 1: Illustrations of graph stricture of $K_3$ and $C_3$. Graphs switch between $\mathcal{G}_1$ and $\mathcal{G}_2$.
• Figure 2: Trajectories of the joint state of the switching network system. Mode 1 and mode 2 are labeled with different colors, and each initial state is marked as red.
• Figure 3: Convergence and identification performance of the proposed bilevel convex optimization framework for switching network systems. Top left: Evolution of the network dynamics reconstruction error per iteration. Top right: Mode classification error showing convergence of mode assignments to the ground truth. Bottom left: Adjacency matrix identification error indicating recovery of the hidden network topology. Bottom right: 3D scatter plot of the sampled state data colored by identified modes.
• Figure 4: Identified switching surface $f_{\text{id}}(x)=0$ obtained via the soft-margin SVM approach in iwasaki2025learning. The surface separates the two identified dynamic regimes in the state space.
• Figure 5: Comparison of recovered switching surface $f_{\text{id}}=0$ and ground truth $f_{\text{true}} = 0$. The identified switching surface well-separates the mode 1 and mode 2 samples, indicating it approximates the true switching rule well.