Finding strongly connected components (SCCs) and the diameter of a directed network play a key role in a variety of machine learning and control theory problems. In this article, we provide for the first time a scalable distributed solution for these two problems by leveraging dynamical consensus-like protocols to find the SCCs. The proposed solution has a time complexity of O(NDd in-degreemax), where N is the number of vertices in the network,D is the (finite) diameter of the network, and din-degreemax is the maximum in-degree of the network. Additionally, we prove that our algorithm terminates in D+2 iterations, which allows us to retrieve the finite diameter of the network. We perform exhaustive simulations that support the outperformance of our algorithm against the state of the art on several random networks, including Erdős-Rényi, Barabási-Albert, and Watts-Strogatz networks.

Fractional-order dynamical networks are increasingly being used to model and describe processes demonstrating long-term memory or complex interlaced dependencies among the spatial and temporal components of a wide variety of dynamical networks. Notable examples include networked control systems or neurophysiological networks which are created using electroencephalographic (EEG) or blood-oxygen-level-dependent data. As a result, the estimation of the states of fractional-order dynamical networks poses an important problem. To this effect, this article addresses the problem of minimum-energy state estimation for discrete-time fractional-order dynamical networks, where the state and output equations are affected by an additive noise that is considered to be deterministic, bounded, and unknown. Specifically, we derive the corresponding estimator and show that the resulting estimation error is exponentially input-to-state stable with respect to the disturbances and to a signal that is decreasing with the increase of the accuracy of the adopted approximation model. An illustrative example shows the effectiveness of the proposed method on real-world neurophysiological networks. Our results may significantly contribute to the development of novel neurotechnologies, particularly in the development of state estimation paradigms for neural signals such as EEG, which are often noisy signals known to be affected by artifacts not having any particular stochastic characterization.

In this paper, we introduce a novel model selection approach to time series forecasting. For linear stationary processes, such as AR processes, the direction of time is independent of the model parameters. By combining theoretical principles of time-reversibility in time series with conventional modeling approaches such as information criteria, we construct a criterion that employs the backwards prediction (backcast) as a proxy for the forecast. Hereby, we aim to adopt a theoretically grounded, data-driven approach to model selection. The novel criterion is named the backwards validated information criterion (BVIC). The BVIC identifies suitable models by trading off a measure of goodness-of-fit and a models ability to predict backwards. We test the performance of the BVIC by conducting experiments on synthetic and real data. In each experiment, the BVIC is examined in contrast to conventionally employed criteria. Our experimental results suggest that the BVIC has comparable performance as conventional information criteria. Specifically, in most of the experiments performed, we did not find statistically significant differences between the forecast error of the BVIC under certain parameterizations and that of the different information criteria. Nonetheless, it is worth emphasizing that the BVIC guarantees are established by design where the model order penalization term depends on strong mathematical properties of time-reversible time series forecasting properties and a finite data assessment. In particular, the penalization term is replaced by a weighted trade-off between functional dimensions pertaining to forecasting.That said, we observed that the BVIC recovered more accurately the real order of the underlying process than the other criteria, which rely on a static penalization of the model order. Lastly, leveraging the latter property we perform the assessment of the order model (or, memory) of time series pertaining to epileptic seizures recorded using electrocorticographic data. Our results provide converging evidence that the order of the model increases during the epileptic events.

A fundamental challenge in neuroscience is to uncover the principles governing how the brain interacts with the external environment. However, assumptions about external stimuli fundamentally constrain current computational models. We show in silico that unknown external stimulation can produce error in the estimated linear time-invariant dynamical system. To address these limitations, we propose an approach to retrieve the external (unknown) input parameters and demonstrate that the estimated system parameters during external input quiescence uncover spatiotemporal profiles of external inputs over external stimulation periods more accurately. Finally, we unveil the expected (and unexpected) sensory and task-related extra-cortical input profiles using functional magnetic resonance imaging data acquired from 96 subjects (Human Connectome Project) during the resting-state and task scans. This dynamical systems model of the brain offers information on the structure and dimensionality of the BOLD signal's external drivers and shines a light on the likely external sources contributing to the BOLD signal's non-stationarity. Our findings show the role of exogenous inputs in the BOLD dynamics and highlight the importance of accounting for external inputs to unravel the brain's time-varying functional dynamics.

Neurotechnology has made great strides in the last 20 years. However, we still have a long way to go to commercialize many of these technologies as we lack a unified framework to study cyber-neural systems (CNS) that bring the hardware, software, and the neural system together. Dynamical systems play a key role in developing these technologies as they capture different aspects of the brain and provide insight into their function. Converging evidence suggests that fractional-order dynamical systems are advantageous in modeling neural systems because of their compact representation and accuracy in capturing the long-range memory exhibited in neural behavior. In this brief survey, we provide an overview of fractional CNS that entails fractional-order systems in the context of CNS. In particular, we introduce basic definitions required for the analysis and synthesis of fractional CNS, encompassing system identification, state estimation, and closed-loop control. Additionally, we provide an illustration of some applications in the context of CNS and draw some possible future research directions. Advancements in these three areas will be critical in developing the next generation of CNS, which will, ultimately, improve people's quality of life.

This paper provides an overview of the research conducted in the context of structural (or structured) systems. These are parametrized models used to assess and design system theoretical properties without considering a specific realization of the parameters (which could be uncertain or unknown). The research in structural systems led to a principled approach to a variety of problems, into what is known as structural systems theory. Hereafter, we perform a systematic overview of the problems and methodologies used in structural systems theory since the latest survey by Dion et al. in 2003. During this period, most of the focus seems to be on structural system's properties related to controllability/observability and decentralized control, in the context of linear time-invariant systems, under the classic assumption that the parameters are independent and belonging to infinite fields. Notwithstanding, it is notable an increase in research in topics that go beyond such scope and underlying assumptions, as well as applications in a variety of domains. Lastly, we provide a compilation of open questions on several settings and we discuss future directions in this field.

Solving optimization problems is a recurrent theme across different fields, including large-scale machine learning systems and deep learning. Often in practical applications, we encounter objective functions where the Hessian is ill-conditioned, which precludes us from using optimization algorithms utilizing second-order information. In this paper, we propose to use fractional time series analysis methods that have successfully been used to model neurophysiological processes in order to circumvent this issue. In particular, the long memory property of fractional time series exhibiting non-exponential power-law decay of trajectories seems to model behavior associated with the local curvature of the objective function at a given point. Specifically, we propose a NEuro-inspired Optimization (NEO) method that leverages this behavior, which contrasts with the short memory characteristics of currently used methods (e.g., gradient descent and heavy-ball). We provide evidence of the efficacy of the proposed method on a wide variety of settings implicitly found in practice.

In this article, we study the target controllability problem of networked dynamical systems,in which we are tasked to steer a subset of network nodes toward a desired objective. More specifically, we derive necessary and sufficient conditions for the structural target controllability of linear time-invariant (LTI) systems with symmetric state matrices, such as those representing undirected dynamical networks with unknown link weights. To achieve our goal, we first characterize the generic rank of symmetrically structured matrices, as well as the modes of any numerical realization. Subsequently, we provide graph-theoretic necessary and sufficient conditions for the structural target controllability of undirected networks with multiple control nodes. In addition, we show that these results can be extended and lead to a necessary and sufficient condition of the structural output controllability. However, different from structural target controllability, we prove that verifying the proposed conditions on structural output controllability in undirected networks is NP-hard.

We provide a necessary and sufficient graph-theoretical characterization of quotient fixes modes occurring in parametric decentralized control systems. Specifically, we introduce the notion of structurally quotient fixed modes (SQFMs) that generically captures the quotient fixed modes and only depends on the system's structure. Additionally, we provide an efficient polynomial-time algorithm for the verification of this graph-theoretical condition. We show that this algorithm can be parallelized, and linear-time computational complexity approximation algorithms can be considered to attain a sub-optimal solution. Lastly, we discuss the implications of the actuation–sensing–communication capabilities and the systems’ interconnections on the existence of SQFM.

Most optimization problems lack closed-form solutions of the argument that minimizes a given function, and even if these were available it might be prohibitive to compute it. As such, we rely on iterative numerical algorithms to find an approximate solution. In this paper, we propose to leverage fractional calculus in the context of time series analysis methods to devise a new iterative algorithm. Specifically, we propose to leverage autoregressive fractional-order integrative moving average time series, whose coefficients encode a proxy for local spatial information. We provide evidence that our algorithm is efficient and particularly suitable for cases where the Hessian is ill-conditioned.

What humans do when exposed to uncertainty, incomplete information, and a dynamic environment influenced by other agents remains an open scientific challenge with important implications in both science and engineering applications. In these contexts, humans handle social situations by employing elaborate cognitive mechanisms such as theory of mind and risk sensitivity. Here we resort to a novel theoretical model, showing that both mechanisms leverage coordinated behaviors among self-regarding individuals. Particularly, we resort to cumulative prospect theory and level-k recursions to show how biases towards optimism and the capacity of planning ahead significantly increase coordinated, cooperative action. These results suggest that the reason why humans are good at coordination may stem from the fact that we are cognitively biased to do so.