Unique Reconstruction From Mean-Field Measurements

TL;DR

This work tackles the problem of inferring both the topology and the dynamics of a high-dimensional network from under-determined mean-field measurements obtained via pinching initial conditions. It introduces a two-tier approach: first, topological reconstruction under sparsity via wTRC and sTRC guarantees, and second, a two-stage full-network reconstruction leveraging an oracle dictionary to identify local dynamics. Theoretical results establish uniqueness and stability under RIP-type conditions, with Gaussian and ER-graph settings providing concrete scaling laws for the required number of measurements $P$ (e.g., $P = \Theta(\log N)$ in certain regimes and $P = \Theta(\log^2 N)$ in others). Numerical experiments on Erdős–Rényi graphs validate the theory, demonstrating accurate topology recovery and robust dynamic identification across regimes, thereby outlining practical foundations for inferring high-dimensional networks from low-dimensional observations.

Abstract

We address the inverse problem of reconstructing both the structure and dynamics of a network from mean-field measurements, which are linear combinations of node states. This setting arises in applications where only a few aggregated observations are available, making network inference challenging. We focus on the case when the number of mean-field measurements is smaller than the number of nodes. To tackle this ill-posed recovery problem, we propose a framework that combines localized initial perturbations with sparse optimization techniques. We derive sufficient conditions that guarantee the unique reconstruction of the network's adjacency matrix from mean-field data and enable recovery of node states and local governing dynamics. Numerical experiments demonstrate the robustness of our approach across a range of sparsity and connectivity regimes. These results provide theoretical and computational foundations for inferring high-dimensional networked systems from low-dimensional observations.

Figures (5)

• Figure 1: The light-colored ellipses represent the mean-field measurements. A set of mean-field measurements, generated through $q$-pinching initial conditions, is provided, while the state vectors remain unknown.
• Figure 2: Critical number of mean-fields for the supercritical ER graphs. ER graphs are generated with $p_i = \frac{\log N}{N}(1 - \varepsilon_i)$, where $\varepsilon_1 = 0.5$ (in black) and $\varepsilon_2 = 0.2$ (in orange). The right insets depic the network topologies. In the left inset, we show the MCC-score and define $P_c$ as the critical number of mean fields that excedees $99\%$ of the score. For each case, the empirical critical number of mean-field measurements $P_c$ is approximately 7% and 10% of the network size $N$, respectively.
• Figure 3: Critical number of mean-fields across ER regimes. We plot the empirical critical number of mean-field measurements $P_c$ required for exact reconstruction, as a function of the sparsity parameter $\varepsilon$, where $p = \frac{\log N}{N}(1 + \varepsilon)$ for a support recovery threshold of $10^{-9}$ and $10^{-10}$. This suggests that operating beyond the stricter $P_c$ thresholds ensures robustness: if recovery is successful under a tighter tolerance (e.g., $10^{-10}$), it remains successful under looser ones (e.g., $10^{-9}$).
• Figure 4: Cumulative MCC Score. The ER graph on the right is generated with the given edge probability $p$. The number of mean-field measurements is set to 60% of the network size $N$. The cumulative MCC score, computed from the reconstructed trajectories $\{\{\hat{x}^q(t)\}_{q=1}^N\}_{t=1}^T$ up to each time step $T$, is shown on the left.
• Figure 5: Heatmap for coefficient reconstruction. The number of mean-field measurements is set to 60% of the network size $N$. The reconstructed trajectories $\{\{\hat{x}^q(t)\}_{q=1}^N\}_{t=1}^T$, accumulated up to each time step $T$, are used to generate the coefficient recovery. Each subplot corresponds to a different $T$, with the respective MSE values shown above. The results demonstrate that recovery remains stable up to $T = 5$, consistent with Theorem \ref{['two-stage']}.

Theorems & Definitions (36)

• Definition 1
• Definition 2
• Definition 3
• Remark 1
• Theorem 1
• Remark 2
• Corollary 1
• Corollary 2
• Definition 4
• Theorem 2