Disease spreading models such as the ubiquitous SIS compartmental model and its numerous variants are widely used to understand and predict the behavior of a given epidemic or information diffusion process. A common approach to imbue more realism to the spreading process is to constrain simulations to a network structure, where connected nodes update their disease state based on pairwise interactions along the edges of their local neighborhood. Simplicial contagion models (SCM) extend this to hypergraphs such that groups of three nodes are able to interact and propagate the disease along higher-order hyperedges (triangles). Though more flexible, it is not clear the extent to which the inclusion of these higher-order interactions results in dynamics that are characteristically different from those attained from simpler pairwise interactions. Here, we propose an agent-based model that unifies the classical SIS/SIR compartmental model and SCM, and extends it to allow for interactions along hyperedges of arbitrary order. Using this model, we demonstrate how the steady-state dynamics of pairwise interactions can be made to replicate those of simulations that include higher-order topologies by linearly scaling disease parameters based on a proposed measure of network activity. By allowing disease parameters to dynamically vary over time, lower-order pairwise interactions can be made to closely replicate both the transient and steady-state dynamics of higher-order simulations. We demonstrate that this relationship is robust to misspecification in the assumed higher-order interaction model, and applies to non-clique complex hypergraphs with nontrivial heterogeneous topology. For the latter case, it is found that heterogeneities in hypergraph topology result in weakened approximations of higher-order dynamics by pairwise interactions.

The aim of this text is to provide a linguistically accessible, but comprehensive introduction into a variety of topics in dynamical systems and its applications. Whilst preliminary knowledge of dynamical systems is useful, it is not essential and readers are only assumed to have familiarity with foundational undergraduate mathematics topics of calculus, linear algebra and rudimentary statistics. A variety of extended topics on recent publications and research activities in the field have been included in the last four chapters, which the interested reader may use as an introduction into further reading. A collection of exercises and questions both theoretical and computational are also included in this text.

Disease spreading models such as the ubiquitous SIS compartmental model and its numerous variants are widely used to understand and predict the behaviour of a given epidemic or information diffusion process. A common approach to imbue more realism to the spreading process is to constrain simulations to a network structure, where connected nodes update their disease state based on pairwise interactions along the edges of their local neighbourhood. Simplicial contagion models (SCM) extend this to hypergraphs such that groups of three nodes are able to interact and propagate the disease along higher-order hyperedges (triangles). Though more flexible, it is not clear the extent to which the inclusion of these higher-order interactions result in dynamics that are characteristically different to those attained from simpler pairwise interactions. Here, we propose an agent-based model that unifies the classical SIS/SIR compartmental model and SCM, and extends it to allow for interactions along hyperedges of arbitrary order. Using this model, we demonstrate how the steady-state dynamics of pairwise interactions can be made to replicate those of simulations that include higher-order topologies by linearly scaling disease parameters based on a proposed measure network activity. By allowing disease parameters to dynamically vary over time, lower-order pairwise interactions can be made to closely replicate both the transient and steady-state dynamics of higher-order simulations.

Bounded confidence models (BCM) are extensively used to model continuous opinion dynamics in social networks. Typically, these models are analysed on static networks where edges do not vary over time. Following in the footsteps of adaptive voter models, further research has considered BCMs in the setting where agents are able to dynamically adjust their edges, which subsequently feedback into the opinion dynamics of the network. Several methods of updating connections have been proposed ranging from random rewiring to more sophisticated approaches based on concordant edges, homophily and cognitive dissonance. We present a modified form of the bounded confidence model, termed the selfish agent opinion (SAO) model, where connection updates are evaluated using a general cost function. Agents in the SAO model maintain two classes of relationships, friends and acquaintances, based on which they update their opinions and edges to optimise a payoff function that may include multiple social factors. This paper explores the effects, which we describe as "cognitive dissonance"and "introversion"that n" that attempt to mimic postulated behaviours associated with the social concepts of the same name, on social network and opinion dynamics. We find that the SAO model naturally produces echo chambers for social networks with increased sensitivity to cognitive dissonance, whilst introversion produces high levels of fragmentation and low opinion mobility. Additionally, the effect of tolerant agents and inquisitive social encounters is investigated. It is found that both the presence of very small numbers of tolerant agents and inquisitive encounters are able to strongly promote consensus formation.

Dynamical networks are versatile models that describe a variety of behaviours such as synchronisation and feedback in networks of coupled dynamical components.However, applying these models in real systems is difficult as prior information of the connectivity structure or local dynamics is often unknown and must be inferred from node state observations.Additionally, the influence of coupling interactions complicates the isolation of local node dynamics.Given the architectural similarities between dynamical networks and recurrent neural networks (RNNs), we propose a network inference method based on the backpropagation through time (BPTT) algorithm used to train RNNs.This method aims to simultaneously infer both the connectivity structure and isolated local node dynamics from node state observations.An approximation of local node dynamics is first constructed using a neural network.This is alternated with an adapted BPTT algorithm to regress corresponding network weights by minimising prediction errors of the network based on the previously constructed local models until convergence.This method was successful in identifying the connectivity structure for coupled networks of chaotic oscillators.Freerun prediction performance with the resulting local models and weights was comparable to the true system with noisy initial conditions.The method is also extended to asymmetric negative coupling.

A common approach to monitoring the status of physical and biological systems is through the regular measurement of various system parameters. Changes in a system’s underlying dynamics manifest as changes in the behaviour of the observed time series. For example, the transition from healthy cardiac activity to ventricular fibrillation results in erratic dynamics in measured electrocardiogram (ECG) signals. Identifying these transitions—change point detection—can be valuable in preparing responses to mitigate the effects of undesirable system changes. Here, we present a data-driven method of detecting change points using a phase space approach. Delay embedded trajectories are used to construct an ‘attractor network’, a discrete Markov-chain representation of the system’s attractor. Once constructed, the attractor network is used to assess the level of surprise of future observations where unusual movements in phase space are assigned high surprise scores. Persistent high surprise scores indicate deviations from the attractor and are used to infer change points. Using our approach, we find that the attractor network is effective in automatically detecting the onset of ventricular fibrillation (VF) from observed ECG data. We also test the flexibility of our method on artificial data sets and demonstrate its ability to distinguish between normal and surrogate time series.

Delay embedding methods are a staple tool in the field of time series analysis and prediction. However, the selection of embedding parameters can have a big impact on the resulting analysis. This has led to the creation of a large number of methods to optimize the selection of parameters such as embedding lag. This paper aims to provide a comprehensive overview of the fundamentals of embedding theory for readers who are new to the subject. We outline a collection of existing methods for selecting embedding lag in both uniform and non-uniform delay embedding cases. Highlighting the poor dynamical explainability of existing methods of selecting non-uniform lags, we provide an alternative method of selecting embedding lags that includes a mixture of both dynamical and topological arguments. The proposed method, Significant Times on Persistent Strands (SToPS), uses persistent homology to construct a characteristic time spectrum that quantifies the relative dynamical significance of each time lag. We test our method on periodic, chaotic, and fast-slow time series and find that our method performs similar to existing automated non-uniform embedding methods. Additionally, n-step predictors trained on embeddings constructed with SToPS were found to outperform other embedding methods when predicting fast-slow time series.

Dynamical networks are versatile models that can describe a variety of behaviours such as synchronisation and feedback. However, applying these models in real world contexts is difficult as prior information pertaining to the connectivity structure or local dynamics is often unknown and must be inferred from time series observations of network states. Additionally, the influence of coupling interactions between nodes further complicates the isolation of local node dynamics. Given the architectural similarities between dynamical networks and recurrent neural networks (RNN), we propose a network inference method based on the backpropagation through time (BPTT) algorithm commonly used to train recurrent neural networks. This method aims to simultaneously infer both the connectivity structure and local node dynamics purely from observation of node states. An approximation of local node dynamics is first constructed using a neural network. This is alternated with an adapted BPTT algorithm to regress corresponding network weights by minimising prediction errors of the dynamical network based on the previously constructed local models until convergence is achieved. This method was found to be succesful in identifying the connectivity structure for coupled networks of Lorenz, Chua and FitzHugh-Nagumo oscillators. Freerun prediction performance with the resulting local models and weights was found to be comparable to the true system with noisy initial conditions. The method is also extended to non-conventional network couplings such as asymmetric negative coupling.

Dynamical networks are a framework commonly used to model large networks of interacting time-varying components such as power grids and epidemic disease networks. The connectivity structure of dynamical networks play a key role in enabling many interesting behaviours such as synchronisation and chimeras. However, dynamical networks can also be vulnerable to network attack, where the connectivity structure is externally altered. This can cause sudden failure and loss of stability in the network. The ability to detect these network attacks is useful in troubleshooting and preventing system failure. Recently, a back-propagation regression method inspired by RNN training algorithms was proposed to infer both local node dynamics and connectivity structure from measured node signals. This paper explores the application of backpropagation regression for fault detection in dynamical networks. We construct separate models for local dynamics and coupling structure to perform short-term freerun predictions. Due to the separation of models, abnormal increases in prediction error can be attributed to changes in the network structure. Automatic detection is achieved by comparing prediction error statistics across two windows that span a period before and after a network attack. This method is tested on a simulated dynamical network of chaotic Lorenz oscillators undergoing gradual edge corruption via three different processes: edge swapping, moving and deletion. We demonstrate that the correlation between increased prediction error and the occurrence of edge corruption can be used to reliably detect both the onset and approximate location of the attack within the network.

Assessing model accuracy for complex and chaotic systems is a non-trivial task that often relies on the calculation of dynamical invariants, such as Lyapunov exponents and correlation dimensions. Well-performing models are able to replicate the long-term dynamics and ergodic properties of the desired system. We term this phenomenon "dynamics learning." However, existing estimates based on dynamical invariants, such as Lyapunov exponents and correlation dimensions, are not unique to each system, not necessarily robust to noise, and struggle with detecting pathological errors, such as errors in the manifold density distribution. This can make meaningful and accurate model assessment difficult. We explore the use of a topological data analysis technique, persistent homology, applied to uniformly sampled trajectories from constructed reservoir models of the Lorenz system to assess the learning quality of a model. A proposed persistent homology point summary, conformance, was able to identify models with successful dynamics learning and detect discrepancies in the manifold density distribution.

Abstract One of the challenges faced by companies in the oil and gas industry is the difficulty in assessing and quantifying subsurface uncertainties when planning for hydrocarbon exploitation. A commonly employed approach is to use available exploration and appraisal data to produce a range of possible subsurface realisations, through which hydrocarbon production forecasts are generated. Prediction of hydrocarbon production from these simulations are then used to assess the viability of a planned development concept and the associated subsurface uncertainties. However, due to the sparsity of field data and unpredictability of underground geology coupled with the typically large dataset sizes, the ability to rapidly quantify prediction uncertainty and provide an overview of the range of underground geologies leaves much to be desired. Here we show that the application of a network science approach to oil production prediction data provides an intuitive way to visualize and assess reservoir uncertainty. A network transformation utilizing Pearson correlation and mean absolute error as similarity measures were applied to a dataset containing time series predictions of oil production for 10 wells simulated in 50 different subsurface realisations. Realisations were generated using a synthetic reservoir spanning 20 years of production. It was found that the network representation enabled the inference of reservoir uncertainty by simple visual inspection. Additionally, network measures such as the beta index were used with results supporting their viability in quantifying uncertainty. The application of clustering algorithms to the resulting networks was also shown to simplify the time series into component characteristic subsurface realisations. We propose a method to quantify subsurface uncertainties and create a simplified representation of the characteristic modes associated with a range of subsurface realisations, greatly reducing the time required to conduct a cursory analysis. Our results demonstrate how the application of network science ideas may be applied to provide new ways of analyzing production data and performace predictions. These methods provide an opportunity to further refine descriptions of uncertainty by incorporating stochastic elements into the network as well.