High-dimensional dynamics in low-dimensional networks (2504.13727v1)

Abstract: Many networks that arise in nature and applications are effectively low-dimensional in the sense that their connectivity structure is dominated by a few dimensions. It is natural to expect that dynamics on such networks might also be low-dimensional. Indeed, recent results show that low-rank networks produce low-dimensional dynamics whenever the network is isolated from external perturbations or noise. However, networks in nature are rarely isolated. We show that recurrent networks with low-rank structure often produce high-dimensional dynamics in the presence of high-dimensional perturbations. Counter to intuition, dynamics in these networks are \textit{suppressed} in directions that are aligned with the network's low-rank structure, a phenomenon we term "low-rank suppression." Our results clarify important, but counterintuitive relationships between a network's connectivity structure and the structure of the dynamics it generates.

High-dimensional Dynamics in Low-dimensional Networks: Unveiling Low-rank Suppression

The paper "High-dimensional dynamics in low-dimensional networks" by Wan and Rosenbaum offers a profound investigation into the dynamics of networks characterized by low-rank structures. The authors present an in-depth analysis elucidating the counterintuitive phenomenon that low-rank networks can manifest high-dimensional dynamics in the presence of high-dimensional perturbations, a concept they term "low-rank suppression."

The fundamental premise of the paper is grounded in the observation that real-world networks often exhibit an effectively low-dimensional connectivity; however, when subjected to high-dimensional inputs, these networks can generate surprisingly high-dimensional outputs. Such a finding challenges the conventional intuition that low-rank connectivity should predominantly lead to low-dimensional dynamics. The paper leverages both mathematical structures and robust simulations to demonstrate how dynamics are typically suppressed in directions that align with the network's low-rank structure.

Core Contributions and Numerical Insights

The authors begin by considering a simple linear model of recurrent networks where the connectivity matrix, $W$, is comprised of a rank-one structure $W_0 = c u u^T$ and a full-rank random matrix $W_1$. Through methodical simulations, they reveal that when networks are disturbed by high-dimensional perturbations, the variance in principal components is unexpectedly high-dimensional. This occurs even as the dynamics are suppressed in the directions aligned with the low-rank structure, $u$, a phenomenon demonstrated numerically where the variance along $u$ is over 100 times smaller compared to random directions.

Moreover, the analysis introduces intricate network structures, such as networks with modular or spatial structures. The authors highlight that biases in network weights or modular connectivity can inherently generate low-rank suppressive effects owing to dominant singular values. Networks with strongly low-rank configurations, like those found in certain brain areas or epidemiological models, tend to exhibit a high-dimensional response to high-dimensional stimuli, a property that has significant implications for understanding neural and other complex network systems.

Implications for Theoretical and Applied Network Science

One of the key theoretical implications of this research is the nuanced understanding of dimensions in network dynamics. It challenges existing models that predict low-dimensional dynamics purely on low-rank structures, emphasizing that accounting for external perturbations is crucial. This realization has broad relevance, particularly in computational neuroscience where it may explain varied experimental findings, such as high-dimensional neural responses to complex stimuli.

Practically, the implications extend to any field dealing with networked systems, such as ecological modeling, epidemiology, or engineered network systems. The insight that random and misaligned perturbations relative to low-rank connectivity are most effective at driving network activity suggests strategies for manipulating and controlling network dynamics through external inputs.

Future Directions and Speculative Considerations

The results presented prompt several avenues for future inquiry. One significant direction is the exploration of networks where the alignment of external perturbations can be controlled or predicted. This research could lead to innovative approaches in therapeutic interventions, sensor designs, or information processing systems where the dimensionality of the responses can be optimized.

Further, the paper opens up questions about the interplay between network architecture and intrinsic noise, especially in biological settings where perfect alignments rarely occur. Understanding how these properties scale with network size or integrate with other non-linear dynamics can unlock deeper insights into robust and adaptive network behaviors.

In concluding, the paper by Wan and Rosenbaum provides a critical viewpoint on the dynamics within networks characterized by low-dimensional connectivity. Their findings underscore the necessity of considering perturbation directionality and network architecture cohesively, expanding the conceptual framework within which network functionality and adaptability are understood.