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1. On the effects of memory and topology on the controllability of complex dynamical networks

   https://doi.org/10.1038/s41598-020-74269-5  

   Kyriakis, Panagiotis; Pequito, Sérgio; Bogdan, Paul (December 2020, Scientific Reports)

   null (Ed.)

   Abstract Recent advances in network science, control theory, and fractional calculus provide us with mathematical tools necessary for modeling and controlling complex dynamical networks (CDNs) that exhibit long-term memory. Selecting the minimum number of driven nodes such that the network is steered to a prescribed state is a key problem to guarantee that complex networks have a desirable behavior. Therefore, in this paper, we study the effects of long-term memory and of the topological properties on the minimum number of driven nodes and the required control energy. To this end, we introduce Gramian-based methods for optimal driven node selection for complex dynamical networks with long-term memory and by leveraging the structure of the cost function, we design a greedy algorithm to obtain near-optimal approximations in a computationally efficiently manner. We investigate how the memory and topological properties influence the control effort by considering Erdős–Rényi, Barabási–Albert and Watts–Strogatz networks whose temporal dynamics follow a fractional order state equation. We provide evidence that scale-free and small-world networks are easier to control in terms of both the number of required actuators and the average control energy. Additionally, we show how our method could be applied to control complex networks originating from the human brain and we discover that certain brain cortex regions have a stronger impact on the controllability of network than others. 

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2. The role of long-term power-law memory in controlling large-scale dynamical networks

   https://doi.org/10.1038/s41598-023-46349-9  

   Reed, Emily_A; Ramos, Guilherme; Bogdan, Paul; Pequito, Sérgio (November 2023, Scientific Reports)

   Abstract Controlling large-scale dynamical networks is crucial to understand and, ultimately, craft the evolution of complex behavior. While broadly speaking we understand how to control Markov dynamical networks, where the current state is only a function of its previous state, we lack a general understanding of how to control dynamical networks whose current state depends on states in the distant past (i.e. long-term memory). Therefore, we require a different way to analyze and control the more prevalent long-term memory dynamical networks. Herein, we propose a new approach to control dynamical networks exhibiting long-term power-law memory dependencies. Our newly proposed method enables us to find the minimum number of driven nodes (i.e. the state vertices in the network that are connected to one and only one input) and their placement to control a long-term power-law memory dynamical network given a specific time-horizon, which we define as the ‘time-to-control’. Remarkably, we provide evidence that long-term power-law memory dynamical networks require considerably fewer driven nodes to steer the network’s state to a desired goal for any given time-to-control as compared with Markov dynamical networks. Finally, our method can be used as a tool to determine the existence of long-term memory dynamics in networks. 

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3. Taylor-Lagrange Neural Ordinary Differential Equations: Toward Fast Training and Evaluation of Neural ODEs

   https://doi.org/10.24963/ijcai.2022/405  

   Djeumou, Franck; Neary, Cyrus; Goubault, Eric; Putot, Sylvie; Topcu, Ufuk (July 2022, Thirty-First International Joint Conference on Artificial Intelligence)

   Neural ordinary differential equations (NODEs) -- parametrizations of differential equations using neural networks -- have shown tremendous promise in learning models of unknown continuous-time dynamical systems from data. However, every forward evaluation of a NODE requires numerical integration of the neural network used to capture the system dynamics, making their training prohibitively expensive. Existing works rely on off-the-shelf adaptive step-size numerical integration schemes, which often require an excessive number of evaluations of the underlying dynamics network to obtain sufficient accuracy for training. By contrast, we accelerate the evaluation and the training of NODEs by proposing a data-driven approach to their numerical integration. The proposed Taylor-Lagrange NODEs (TL-NODEs) use a fixed-order Taylor expansion for numerical integration, while also learning to estimate the expansion's approximation error. As a result, the proposed approach achieves the same accuracy as adaptive step-size schemes while employing only low-order Taylor expansions, thus greatly reducing the computational cost necessary to integrate the NODE. A suite of numerical experiments, including modeling dynamical systems, image classification, and density estimation, demonstrate that TL-NODEs can be trained more than an order of magnitude faster than state-of-the-art approaches, without any loss in performance. 

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4. End-to-End Learning Framework for Solving Non-Markovian Optimal Control

   Zhang, Xiaole; Zhang, Peiyu; Xiao, Xiongye; Li, Shixuan; Tzoumas, Vasileios; Gupta, Vijay; Bogdan, Paul (August 2025, International Conference on Machine Learning (ICML))

   Integer-order calculus fails to capture the long-range dependence (LRD) and memory effects found in many complex systems. Fractional calculus addresses these gaps through fractional-order integrals and derivatives, but fractional-order dynamical systems pose substantial challenges in system identification and optimal control tasks. In this paper, we theoretically derive the optimal control via linear quadratic regulator (LQR) for fractional-order linear time-invariant (FOLTI) systems and develop an end-to-end deep learning framework based on this theoretical foundation. Our approach establishes a rigorous mathematical model, derives analytical solutions, and incorporates deep learning to achieve data-driven optimal control of FOLTI systems. Our key contributions include: (i) proposing a novel method for system identification and optimal control strategy in FOLTI systems, (ii) developing the first end-to-end data-driven learning framework, Fractional-Order Learning for Optimal Control (FOLOC), that learns control policies from observed trajectories, and (iii) deriving theoretical bounds on the sample complexity for learning accurate control policies under fractional-order dynamics. Experimental results indicate that our method accurately approximates fractional-order system behaviors without relying on Gaussian noise assumptions, pointing to promising avenues for advanced optimal control. 

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5. Noisy deep networks: chaos, multistationarity, and eternal evolution

   https://doi.org/10.1088/2632-072X/adcdb4  

   Vakulenko, S_A; Sudakow, I. (April 2025, Journal of Physics: Complexity)

   Abstract We study time-recurrent hierarchical networks that model complex systems in biology, economics, and ecology. These networks resemble real-world topologies, with strongly connected hubs (centers) and weakly connected nodes (satellites). Under natural structural assumptions, we develop a mean-field approach that reduces network dynamics to the central nodes alone. Even in the two-layer case, we establish universal dynamical approximation, demonstrating that these networks can replicate virtually any dynamical behavior by tuning center-satellite interactions. In multilayered networks, this property extends further, enabling the approximation of families of structurally stable systems and the emergence of complex bifurcations, such as pitchfork bifurcations under strong inter-satellite interactions. We also show that internal noise within nodes moderates bifurcations, leading to noise-induced phase transitions. A striking effect emerges where central nodes may lose control over satellites, akin to transitions observed in perceptrons studied by E. Gardner-relevant in complex combinatorial problems. Finally, we examine the networks’ responses to stress, demonstrating that increasing complexity during evolution is crucial for long-term viability. 

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