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Identifying disturbances in network-coupled dynamical systems without knowledge of the disturbances or underlying dynamics is a problem with a wide range of applications. For example, one might want to know which nodes in the network are being disturbed and identify the type of disturbance. Here, we present a model-free method based on machine learning to identify such unknown disturbances based only on prior observations of the system when forced by a known training function. We find that this method is able to identify the locations and properties of many different types of unknown disturbances using a variety of known forcing functions. We illustrate our results with both linear and nonlinear disturbances using food web and neuronal activity models. Finally, we discuss how to scale our method to large networks. 

Single-cell measurements routinely demonstrate high levels of variation between cells, but fewer studies provide insight into the analytical and biological sources of this variation. This is particularly true of chemical cytometry, in which individual cells are lysed and their contents separated, compared to more established single-cell measurements of the genome and transcriptome. To characterize population level variation and its sources, we analyzed oxidative stress levels in 1278 individual Dictyostelium discoideum cells as a function of exogenous stress level and cell cycle position. Cells were exposed to varying levels of oxidative stress via singlet oxygen generation using the photosensitizer Rose Bengal. Single-cell data reproduced the dose-response observed in ensemble measurements by CE-LIF, superimposed with high levels of heterogeneity. Through experiments and data analysis, we explored possible biological sources of this heterogeneity. No trend was observed between population variation and oxidative stress level, but cell cycle position was a major contributor to heterogeneity in oxidative stress. Cells synchronized to the same stage of cell division were less heterogeneous than unsynchronized cells (RSD of 37-51% vs 93%), and mitotic cells had higher levels of reactive oxygen species than interphase cells. While past research has proposed changes in cell size during the cell cycle as a source of biological noise, the measurements presented here use an internal standard to normalize for effects of cell volume, suggesting that a more complex contribution of cell cycle in heterogeneity of oxidative stress. 

We study the effect of structured higher-order interactions on the collective behavior of coupled phase oscillators. By combining a hypergraph generative model with dimensionality reduction techniques, we obtain a reduced system of differential equations for the system’s order parameters. We illustrate our framework with the example of a hypergraph with hyperedges of sizes 2 (links) and 3 (triangles). For this case, we obtain a set of two coupled nonlinear algebraic equations for the order parameters. For strong values of coupling via triangles, the system exhibits bistability and explosive synchronization transitions. We find conditions that lead to bistability in terms of hypergraph properties and validate our predictions with numerical simulations. Our results provide a general framework to study the synchronization of phase oscillators in hypergraphs, and they can be extended to hypergraphs with hyperedges of arbitrary sizes, dynamic-structural correlations, and other features. 

We study synchronization dynamics in populations of coupled phase oscillators with higher-order interactions and community structure. We find that the combination of these two properties gives rise to a number of states unsupported by either higher-order interactions or community structure alone, including synchronized states with communities organized into clusters in-phase, anti-phase, and a novel skew-phase, as well as an incoherent-synchronized state. Moreover, the system displays strong multistability with many of these states stable at the same time. We demonstrate our findings by deriving the low dimensional dynamics of the system and examining the system’s bifurcations using stability analysis and perturbation theory. 

Despite the vast literature on network dynamics, we still lack basic insights into dynamics on higher-order structures (e.g., edges, triangles, and more generally, [Formula: see text]-dimensional “simplices”) and how they are influenced through higher-order interactions. A prime example lies in neuroscience where groups of neurons (not individual ones) may provide building blocks for neurocomputation. Here, we study consensus dynamics on edges in simplicial complexes using a type of Laplacian matrix called a Hodge Laplacian, which we generalize to allow higher- and lower-order interactions to have different strengths. Using techniques from algebraic topology, we study how collective dynamics converge to a low-dimensional subspace that corresponds to the homology space of the simplicial complex. We use the Hodge decomposition to show that higher- and lower-order interactions can be optimally balanced to maximally accelerate convergence and that this optimum coincides with a balancing of dynamics on the curl and gradient subspaces. We additionally explore the effects of network topology, finding that consensus over edges is accelerated when two-simplices are well dispersed, as opposed to clustered together.