This paper studies the multi-agent average consensus problem under the requirement of differential privacy of the agents' initial states against an adversary that has access to all the messages. We first establish that a differentially private consensus algorithm cannot guarantee convergence of the agents' states to the exact average in distribution, which in turn implies the same impossibility for other stronger notions of convergence. This result motivates our design of a novel differentially private Laplacian consensus algorithm in which agents linearly perturb their state-transition and message-generating functions with exponentially decaying Laplace noise. We prove that our algorithm converges almost surely to an unbiased estimate of the average of agents' initial states, compute the exponential mean-square rate of convergence, and formally characterize its differential privacy properties. We show that the optimal choice of our design parameters (with respect to the variance of the convergence point around the exact average) corresponds to a one-shot perturbation of initial states and compare our design with various counterparts from the literature. Simulations illustrate our results.

We study a class of distributed convex constrained optimization problems where a group of agents aim to minimize the sum of individual objective functions while each desires that any information about its objective function is kept private. We prove the impossibility of achieving differential privacy using strategies based on perturbing the inter-agent messages with noise when the underlying noise-free dynamics are asymptotically stable. This justifies our algorithmic solution based on the perturbation of individual functions with Laplace noise. To this end, we establish a general framework for differentially private handling of functional data. We further design postprocessing steps that ensure the perturbed functions regain the smoothness and convexity properties of the original functions while preserving the differentially private guarantees of the functional perturbation step. This methodology allows us to use any distributed coordination algorithm to solve the optimization problem on the noisy functions. Finally, we explicitly bound the magnitude of the expected distance between the perturbed and true optimizers which leads to an upper bound on the privacyaccuracy trade-off curve. Simulations illustrate our results.

Recent advances in computational models of signal propagation and routing in the human brain have underscored the critical role of white matter structure. A complementary approach has utilized the framework of network control theory to better understand how white matter constrains the manner in which a region or set of regions can direct or control the activity of other regions. Despite the potential for both of these approaches to enhance our understanding of the role of network structure in brain function, little work has sought to understand the relations between them. Here, we seek to explicitly bridge computational models of communication and principles of network control in a conceptual review of the current literature. By drawing comparisons between communication and control models in terms of the level of abstraction, the dynamical complexity, the dependence on network attributes, and the interplay of multiple spatiotemporal scales, we highlight the convergence of and distinctions between the two frameworks. Based on the understanding of the intertwined nature of communication and control in human brain networks, this work provides an integrative perspective for the field and outlines exciting directions for future work.

A central challenge in the computational modeling of neural dynamics is the trade-off between accuracy and simplicity. At the level of individual neurons, nonlinear dynamics are both experimentally established and essential for neuronal functioning. One may therefore expect the collective dynamics of massive networks of such neurons to only increase in their complexity, thereby supporting an expanded repertoire of nonlinear behaviors. An implicit assumption has thus formed that an “accurate” computational model of whole-brain dynamics must inevitably be nonlinear whereas linear models may provide a first-order approximation. To what extent this assumption holds, however, has remained an open question. Here, we provide a rigorous and data-driven answer at the level of whole-brain blood-oxygen-level-dependent (BOLD) and macroscopic field potential dynamics by leveraging the theory of system identification. Using functional magnetic resonance imaging (fMRI) and intracranial electroencephalography (iEEG), we model the spontaneous, resting state activity of 700 subjects in the Human Connectome Project (HCP) and 122 subjects from the Restoring Active Memory (RAM) project using state-of-the-art linear and nonlinear model families. We assess relative model fit using predictive power, computational complexity, and the extent of residual dynamics unexplained by the model. Contrary to our expectations, linear auto-regressive models achieve the best measures across all three metrics, eliminating the trade-off between accuracy and simplicity. To understand and explain this linearity, we highlight four properties of macroscopic neurodynamics which can counteract or mask microscopic nonlinear dynamics: averaging over space, averaging over time, observation noise, and limited data samples. Whereas the latter two are technological limitations and can improve in the future, the former two are inherent to aggregated macroscopic brain activity. Our results demonstrate the discounted potential of linear models in accurately capturing macroscopic brain dynamics. This, together with the unparalleled interpretability of linear models, can greatly facilitate our understanding of macroscopic neural dynamics, which in turn may facilitate the principled design of model-based interventions for the treatment of neuropsychiatric disorders.

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