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1. 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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2. THE ACTUATION SPECTRUM OF SPATIOTEMPORAL NETWORKS WITH POWER-LAW TIME DEPENDENCIES

   https://doi.org/10.1142/S0219525919500231  

   CAO, QI; RAMOS, GUILHERME; BOGDAN, PAUL; PEQUITO, SÉRGIO (November 2019, Advances in Complex Systems)

   null (Ed.)

   The ability to steer the state of a dynamical network towards a desired state within a time horizon is intrinsically dependent on the number of driven nodes considered, as well as the network’s topology. The trade-off between time-to-control and the minimum number of driven nodes is captured by the notion of the actuation spectrum (AS). We study the actuation spectra of a variety of artificial and real-world networked systems, modeled by fractional-order dynamics that are capable of capturing non-Markovian time properties with power-law dependencies. We find evidence that, in both types of networks, the actuation spectra are similar when the time-to-control is less or equal to about 1/5 of the size of the network. Nonetheless, for a time-to-control larger than the network size, the minimum number of driven nodes required to attain controllability in networks with fractional-order dynamics may still decrease in comparison with other networks with Markovian properties. These differences suggest that the minimum number of driven nodes can be used to determine the true dynamical nature of the network. Furthermore, such differences also suggest that new generative models are required to reproduce the actuation spectra of real fractional-order dynamical networks. 

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3. 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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4. Structure-based approach to identifying small sets of driver nodes in biological networks

   https://doi.org/10.1063/5.0080843  

   Newby, Eli; Tejeda Zañudo, Jorge Gómez; Albert, Réka (June 2022, Chaos: An Interdisciplinary Journal of Nonlinear Science)

   In network control theory, driving all the nodes in the Feedback Vertex Set (FVS) by node-state override forces the network into one of its attractors (long-term dynamic behaviors). The FVS is often composed of more nodes than can be realistically manipulated in a system; for example, only up to three nodes can be controlled in intracellular networks, while their FVS may contain more than 10 nodes. Thus, we developed an approach to rank subsets of the FVS on Boolean models of intracellular networks using topological, dynamics-independent measures. We investigated the use of seven topological prediction measures sorted into three categories—centrality measures, propagation measures, and cycle-based measures. Using each measure, every subset was ranked and then evaluated against two dynamics-based metrics that measure the ability of interventions to drive the system toward or away from its attractors: To Control and Away Control. After examining an array of biological networks, we found that the FVS subsets that ranked in the top according to the propagation metrics can most effectively control the network. This result was independently corroborated on a second array of different Boolean models of biological networks. Consequently, overriding the entire FVS is not required to drive a biological network to one of its attractors, and this method provides a way to reliably identify effective FVS subsets without the knowledge of the network dynamics. 

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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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