arXiv subjects

Amirhossein Nazerian

Publications and source records attributed to Amirhossein Nazerian.

11 recordsLinked to original sources

arXiv2026

Dynamics to decision: A mathematical theory of Lyapunov spectra and decision boundaries in deep classifiers

A deep classifier is defined not only by the decision it produces, but also by the sequence of transformations through which that decision is formed. Treating this evolution as a dynamical system across layers provides a natural framework for asking how decision geometry emerges through depth and how far back we can trace a boundary's dynamical signature. We model a feed-forward classifier as a finite, nonautonomous discrete dynamical system, with layers playing the role of discrete time steps. We study the Finite-Time Maximum Lyapunov Exponent (FTMLE) of the data samples' dynamical trajectory through depths of the classifier. The FTMLE measures the rate of convergence/divergence of nearby trajectories. We move the observation endpoint backward from probabilities to logits and then to hidden representations. For Gaussian classes, we prove that probability-level FTMLE carries a clear geometric signature of the decision boundary, with its dominant direction aligned with the boundary normal. Moving one step backward to the logits, we prove this relationship is no longer universal but depends critically on how the classifier is trained, particularly on the choice of loss function. Moving further backward to the hidden representation, the connection becomes more conditional: boundary-related FTMLE can persist, but only under identifiable structural conditions. We propose geometry-aware fine-tuning for restructuring the classifier's hidden FTMLE, and propose conditions for guaranteed concentration of high hidden FTMLE near the decision boundary. Through our numerical results, we show the generality and validity of our theoretical results. Understanding the evolution of data samples as traveling through the layers of classifier provides a principled foundation for identifying where boundary-relevant sensitivity emerges and for developing layer-aware regularization strategies.

cs.LG↗

arXiv2026

Manifold-Stable Flow Matching

Flow matching (FM) learns generative dynamics through velocity regression. Geometric FM variants commonly assume a prior supported on the data manifold, requiring geometric knowledge that is often unavailable. Without such knowledge, low regression error alone does not guarantee manifold adherence. Adherence keeps generated samples within valid configurations and is empirically associated with better task performance. We introduce manifold-stable flow matching (MSFM), which can start from an arbitrary ambient prior, not necessarily supported on the manifold. Using tools from nonlinear dynamics, namely contraction theory, MSFM combines learned tangential transport with prescribed normal contraction. The construction uses analytical projectors for known manifolds and local affine proxies estimated by principal component analysis for unknown data geometry. By implementing contraction theory in both cases of known and unknown manifolds, we guarantee manifold invariance and transverse convergence to the manifold within a desired time window (e.g., one second). We derive a family of compatible probability paths and decompose the training loss into a learnable tangential term and a normal residual. An ellipse experiment attains a mean terminal off-manifold error of order $10^{-6}$. In Push-T robotic experiments, MSFM raises success from $74\%$ to $82\%$. In the Robomimic Square task, success increases from $60\%$ to $72\%$, while rotation-manifold deviation decreases from order $10^{-2}$ to $10^{-7}$. The MSFM terminal geometric errors are controlled by the chosen numerical tolerance. These results demonstrate stronger geometric adherence and higher observed task performance, supporting prescribed normal contraction as a complement to learned generative transport.

cs.LG↗

arXiv2026

Extreme vulnerability to intruder attacks destabilizes network dynamics

Consensus, synchronization, formation control, and power grid balance are examples of desirable dynamical states that arise in networks. Here we investigate how such states can be destabilized by an intruder agent within an otherwise functioning network. We see that a single adversarial node, coupled through adversarial connections to one or more other nodes, is sufficient to destabilize the entire network. We further show that concentrating the attack on a single low-indegree node induces the greatest instability, challenging the common assumption that hubs are the most critical nodes. This leads to a new characterization of network vulnerability, identifying low-indegree nodes as the most vulnerable components. Although derived for linear systems, our results extend to nonlinear networks, including the Kuramoto model. These findings reveal an intrinsic vulnerability of technological, social, and biological networks.

nlin.AO↗

arXiv2026

The Frequency Response of Networks as Open Systems

Many biological, technological, and social systems can be effectively described as networks of interacting subsystems. Typically, these networks are not isolated objects, but interact with their environment through both signals and information that is received by specific nodes with an input function or released to the environment by other nodes with an output function. An important question is whether the structure of different networks, together with the particular selection of input and output nodes, is such that it favors the passing or blocking of such signals. For a given network and a given choice of the input and output nodes, the H2-norm provides a natural and general quantification of the extent to which input signals, whether deterministic or stochastic, periodic or arbitrary, are amplified. We analyze a diverse set of empirical networks and find that many naturally occurring systems, such as food webs, signaling pathways, and gene regulatory circuits, are structurally organized to enhance the passing of signals; in contrast, the structure of engineered systems like power grids appears to be intentionally designed to suppress signal propagation.

eess.SY↗

arXiv2025

Open networks in discrete time: Passing vs blocking behavior

This paper presents a unified framework for analyzing the input-output behavior of discrete time complex networks viewed as open systems. Importantly, we focus on systems that are inherently modeled in discrete time-such as opinion dynamics, Markov chains, diffusion on networks, and population models-reflecting their natural formulation in many real-world contexts. By an open network, we mean one that is coupled to its environment, through both external signals that are received by designated input nodes and response signals that are released back into the environment via a separate set of output nodes. We develop a general framework for characterizing whether such networks amplify (pass) or suppress (block) the external inputs. Our approach combines the transfer function of the network with the discrete time controllability Gramian, using the H2-norm to quantify signal amplification. We introduce a computationally efficient network index based on the Gramian trace and eigenvalues, enabling scalable comparisons across network topologies. Application of our method to a broad set of empirical networks, spanning biological, technological, and ecological domains, uncovers consistent structural signatures associated with passing or blocking behavior. These findings shed light on how the network architecture and the particular selection of input and output nodes shape information flow in real-world systems, with broad implications for control, signal processing, and network design.

physics.soc-ph↗

arXiv2025

Bridging the Gap between Reactivity, Contraction, and Finite-Time Lyapunov Exponents

Reactivity, contractivity, and Lyapunov exponents are powerful tools for studying the stability properties of dynamical systems and have been extensively investigated in the literature for decades. In this paper, we review and extend the concepts of reactivity, contractivity, and finite-time Lyapunov exponents for discrete-time dynamical systems and establish connections among them. We focus on time-invariant maps, time-varying linear maps, and certain classes of time-varying nonlinear maps. In particular, we show that if the corresponding $p$-iteration systems (with p > 1) are contractive, then the original systems admit stable attractors such as fixed points or limit cycles. We demonstrate the application of these results to the analysis of synchronization stability in coupled networks and discuss how p-iteration systems can serve as a useful framework for studying network synchronization.

math.DS↗

arXiv2024

The Efficiency of Synchronization Dynamics and the Role of Network Syncreactivity

Synchronization of coupled oscillators is a fundamental process in both natural and artificial networks. While much work has investigated the asymptotic stability of the synchronous solution, the fundamental question of the transient behavior toward synchronization has received far less attention. In this work, we present the transverse reactivity as a metric to quantify the instantaneous rate of growth or decay of desynchronizing perturbations. We first use the transverse reactivity to design a coupling-efficient and energy-efficient synchronization strategy that involves varying the coupling strength dynamically according to the current state of the system. We find that our synchronization strategy is able to synchronize networks in both simulation and experiment over a significantly larger (often by orders of magnitude) range of coupling strengths than is possible when the coupling strength is constant. Then, we characterize the effects of network topology on the transient dynamics towards synchronization by introducing the concept of network syncreactivity: A network with a larger syncreactivity has a larger transverse reactivity at every point on the synchronization manifold, independent of the oscillator dynamics. We classify real-world examples of complex networks in terms of their syncreactivity.

nlin.AO↗

arXiv2023

Synchronization in networked systems with large parameter heterogeneity

Systems that synchronize in nature are intrinsically different from one another, with possibly large differences from system to system. While a vast part of the literature has investigated the emergence of network synchronization for the case of small parametric mismatches, we consider the general case that parameter mismatches may be large. We present a unified stability analysis that predicts why the range of stability of the synchronous solution either increases or decreases with parameter heterogeneity for a given network. We introduce a parametric approach, based on the definition of a curvature contribution function, which allows us to estimate the effect of mismatches on the stability of the synchronous solution in terms of contributions of pairs of eigenvalues of the Laplacian. For cases in which synchronization occurs in a bounded interval of a parameter, we study the effects of parameter heterogeneity on both transitions (asynchronous to synchronous and synchronous to asynchronous.)

eess.SY↗

arXiv2023

Single-Integrator Consensus Dynamics over Minimally Reactive Networks

The problem of achieving consensus in a network of connected systems arises in many science and engineering applications. In contrast to previous works, we focus on the system reactivity, i.e., the initial amplification of the norm of the system states. We identify a class of networks that we call minimally reactive, which are such that the indegree and the outdegree of each node of the network are the same. We propose several optimization procedures in which minimum perturbations (links or link weights) are imposed on a given network topology to make it minimally reactive. A new concept of structural reactivity is introduced which measures how much a given network is far from becoming minimally reactive by link perturbations. The structural reactivity of directed random graphs is studied.

eess.SY↗

arXiv2023

Synchronizing Chaos using Reservoir Computing

We attempt to achieve isochronal synchronization between a drive system unidirectionally coupled to a response system, under the assumption that limited knowledge on the states of the drive is available at the response. Machine learning techniques have been previously implemented to estimate the states of a dynamical system from limited measurements. We consider situations in which knowledge of the non-measurable states of the drive system is needed in order for the response system to synchronize with the drive. We use a reservoir computer to estimate the non-measurable states of the drive system from its measured states and then employ these measured states to synchronize the response system with the drive.

nlin.AO↗

arXiv2022

Matryoshka and Disjoint Cluster Synchronization of Networks

The main motivation for this paper is to present a definition of network synchronizability for the case of cluster synchronization (CS), in an analogous fashion to Barahona and Pecora for the case of complete synchronization. We find this problem to be substantially more complex than the original one. We distinguish between the two cases of networks with intertwined clusters and no intertwined clusters and between {the two cases that the master stability function is negative either in a bounded range or in an unbounded range of its argument. We first obtain a definition of synchronizability that applies to each individual cluster within a network and then attempt to generalize this definition to the entire network. For CS, the synchronous solution of each cluster may be stable independent of the stability of the other clusters, which results in possibly different ranges in which each cluster synchronizes (isolated CS.) For each pair of clusters, we distinguish between three different cases: Matryoshka Cluster Synchronization (when the range of the stability of the synchronous solution for one cluster is included in that of the other cluster), Partially Disjoint Cluster Synchronization (when the ranges of stability of the synchronous solutions partially overlap), and Complete Disjoint Cluster Synchronization (when the ranges of stability of the synchronous solutions do not overlap.)

eess.SY↗