Showing papers in "Understanding complex systems in 2022"

COVID-19 Epidemiology and Virus Dynamics

Abstract: This book addresses the COVID-19 pandemic from a quantitative perspective based on mathematical models and methods used in nonlinear physics

Signal Processing on Simplicial Complexes

Abstract: Higher-order networks have so far been considered primarily in the context of studying the structure of complex systems, i.e., the higher-order or multi-way relations connecting the constituent entities. More recently, a number of studies have considered dynamical processes that explicitly account for such higher-order dependencies, e.g., in the context of epidemic spreading processes or opinion formation. In this chapter, we focus on a closely related, but distinct third perspective: how can we use higher-order relationships to process signals and data supported on higher-order network structures. In particular, we survey how ideas from signal processing of data supported on regular domains, such as time series or images, can be extended to graphs and simplicial complexes. We discuss Fourier analysis, signal denoising, signal interpolation, and nonlinear processing through neural networks based on simplicial complexes. Key to our developments is the Hodge Laplacian matrix, a multi-relational operator that leverages the special structure of simplicial complexes and generalizes desirable properties of the Laplacian matrix in graph signal processing.

Consensus Dynamics and Opinion Formation on Hypergraphs

Abstract: In this chapter, we derive and analyse models for consensus dynamics on hypergraphs. As we discuss, unless there are nonlinear node interaction functions, it is always possible to rewrite the system in terms of a new network of effective pairwise node interactions, regardless of the initially underlying multi-way interaction structure. We thus focus on dynamics based on a certain class of non-linear interaction functions, which can model different sociological phenomena such as peer pressure and stubbornness. Unlike for linear consensus dynamics on networks, we show how our nonlinear model dynamics can cause shifts away from the average system state. We examine how these shifts are influenced by the distribution of the initial states, the underlying hypergraph structure and different forms of non-linear scaling of the node interaction function.

Topological Data Analysis of Spatial Systems

Abstract: In this chapter, we discuss applications of topological data analysis (TDA) to spatial systems. We briefly review a recently proposed level-set construction of filtered simplicial complexes, and we then examine persistent homology in two cases studies: street networks in Shanghai and anomalies in the spread of COVID-19 infections. We then summarize our results and provide an outlook on TDA in spatial systems.

Non-pairwise Interaction in Oscillatory Ensembles: from Theory to Data Analysis

Abstract: In this chapter, we briefly review several cases when non-pairwise interaction naturally appears in oscillatory networks. First, we analyze globally coupled ensembles of phase oscillators. We demonstrate that nonlinear high-order mean-field coupling is equivalent to a hypernetwork with non-pairwise interactions of the population elements. Next, we consider small networks of limit-cycle oscillators. We show that pairwise interaction in the state variables description results in non-pairwise interaction on the level of phase dynamics, if one goes beyond the first order in the weak-coupling phase reduction. Finally, we discuss the implications for recovery of the network connectivity in terms of the phase dynamics from observations.

Flow-Based Community Detection in Hypergraphs

Abstract: To connect structure, dynamics and function in systems with multibody interactions, network scientists model random walks on hypergraphs and identify communities that confine the walks for a long time. The two flow-based community-detection methods Markov stability and the map equation identify such communities based on different principles and search algorithms. But how similar are the resulting communities? We explain both methods’ machinery applied to hypergraphs and compare them on synthetic and real-world hypergraphs using various hyperedge-size biased random walks and time scales. We find that the map equation is more sensitive to time-scale changes and that Markov stability is more sensitive to hyperedge-size biases.

Graphs, Simplicial Complexes and Hypergraphs: Spectral Theory and Topology

Abstract: In this chapter we discuss the spectral theory of discrete structures such as graphs, simplicial complexes and hypergraphs. We focus, in particular, on the corresponding Laplace operators. We present the theoretical foundations, but we also discuss the motivation to model and study real data with these tools.

Sustainable Outcomes Through the Structured Forward Supply Chain: A System Dynamic Approach

Abstract: The concepts of sustainability and supply chains are critical components for modern businesses as they face enormous competition and manage economic, social, and environmental sustainability. However, the poultry livestock sub-sector has received insufficient attention from academics, according to current literature. As a result, this particular industry is suffering from unstructured supply chain processes, a lack of awareness of the implications of the sustainability concept, and a failure to recycle poultry wastes. Furthermore, the Covid 19 pandemic puts additional strain on this industry and its supply chain. As a result, the current study uses a case study to develop an integrated poultry forward supply chain model. The integration process model is an expanded version derived from real-world scattered processes performed by various supply chain members. With the help of ‘system dynamics’ and case study method, this quantitative study used the positivist paradigm and ‘design science’ methodology. The findings revealed that supply chain integration could provide economic and social sustainability and a structured manufacturing process to support the research objectives and questions. At the end of the chapter, the pandemic effects will discuss briefly to determine the future direction.

The Master Stability Function for Synchronization in Simplicial Complexes

Abstract: The collective behaviors of a complex system are determined by the intricate way in which its components interact. In this chapter we discuss a novel and general analytical framework to study synchronized states in systems of many dynamical units with many-body interactions, which allows to account for the microscopic structure of the interactions at any possible order. In such a framework, the N dynamical units of a system are associated to the N nodes of a D dimensional ( $$D \ge 1$$ ) simplicial complex, whose simplices represent the structure of the different types of coupling. Namely, 1-simplices (links) describe pairwise interactions, 2-simplices (triangles) describe three-body interactions, 3-simpliced (tetrahedra) four-body interactions, and so on. Such a description generalizes that of a complex network of dynamical units, and reduces to it in the particular case of $$D=1$$ simplicial complexes. Within this framework, we study the onset of full synchronization and the conditions for the stability of a synchronized state in systems of identical dynamical units. We show that, under certain assumptions on the network topology or on the form of the coupling, these conditions can be written in terms of a Master Stability Function that generalizes the existing results valid for pairwise interactions (i.e. networks) to the case of complex systems with the most general possible architecture. As an example of the potential utility of the proposed method we study the dynamics of $$D=3$$ simplicial complexes of chaotic systems (Rössler oscillators) and we investigate how the stability of synchronized states depends on the interplay between the control parameters of the chaotic units and the structural properties of the simplicial complex.

Explosive Synchronization and Multistability in Large Systems of Kuramoto Oscillators with Higher-Order Interactions

Abstract: Since its original formulation, the Kuramoto model and its many variants have served as critical tools for uncovering and understanding the emergence of nonlinear collective behavior. However, recent evidence suggests that in such phase-reduced systems, interactions beyond the typical pair-wise angle differences need to be considered to develop a full picture of the dynamics. In particular, higher-order interactions, namely non-additive, nonlinear interactions that take place between three or more oscillators are required. Here we explore these interactions and their effect on the macroscopic dynamics of coupled phase oscillator systems. The analysis for these systems begins with all-to-all coupled systems where a range of techniques including dimensionality reductions and self-consistency analyses may be employed. The effects of the various higher-order coupling terms on the macroscopic dynamics may then be explored, revealing a natural mechanism for nonlinear phenomena that includes abrupt (i.e., explosive) synchronization transitions and extensive multistability. These dynamics are qualitatively preserved under more heterogeneous network topologies. Moreover, the high degree of multistability in such networks allows for the system to store information and memory.

Random Simplicial Complexes: Models and Phenomena

Abstract: We review a collection of models of random simplicial complexes together with some of the most exciting phenomena related to them. We do not attempt to cover all existing models, but try to focus on those for which many important results have been recently established rigorously in mathematics, especially in the context of algebraic topology. In application to real-world systems, the reviewed models are typically used as null models, so that we take a statistical stance, emphasizing, where applicable, the entropic properties of the reviewed models. We also review a collection of phenomena and features observed in these models, and split the presented results into two classes: phase transitions and distributional limits. We conclude with an outline of interesting future research directions.

Multiorder Laplacian for Kuramoto Dynamics with Higher-Order Interactions

Abstract: Many real-world systems are characterised by higher-order interactions, where influences among units involve more than two nodes at a time, and which can significantly affect the emergence of collective behaviors. A paradigmatic case is that of synchronization, occuring when oscillators reach coherent dynamics through their mutual couplings, and which is known to display richer collective phenomena when connections are not limited to simple dyads. Here, we consider an extension of the Kuramoto model with higher-order interactions, where oscillators can interact in groups of any size, arranged in any arbitrary complex topology. We present a new operator, the multiorder Laplacian, which allows us to treat the system analytically and that can be used to assess the stability of synchronization in general higher-order networks. Our spectral approach, originally devised for Kuramoto dynamics, can be extended to a wider class of dynamical processes beyond pairwise interactions, advancing our quantitative understanding of how higher-order interactions impact network dynamics.

From Symmetric Networks to Heteroclinic Dynamics and Chaos in Coupled Phase Oscillators with Higher-Order Interactions

Abstract: We highlight some results from normal form theory for symmetric bifurcations that give a rational way to organize higher-order interactions between phase oscillators in networks with fully symmetric coupling. For systems near Hopf bifurcation the lowest order (pairwise) interactions correspond to the system of Kuramoto and Sakaguchi. At next asymptotic order one must generically include higher-order interactions of up to four oscillators. We discuss some dynamical consequences of these interactions in terms of heteroclinic attractors, chaos, and chimeras for related systems.

Persistent Homology: A Topological Tool for Higher-Interaction Systems

Abstract: The aim of this chapter is to give a handy but thorough introduction to persistent homology and its applications. The chapter’s path is made by the following steps. First, we deal with the constructions from data to simplicial complexes according to the kind of data: filtrations of data, point clouds, networks, and topological spaces. For each construction, we underline the possible dependence on a fixed scale parameter. Secondly, we introduce the necessary algebraic structures capturing topological informations out of a simplicial complex at a fixed scale, namely the simplicial homology groups and the Hodge Laplacian operator. The so-obtained linear structures are then integrated into the multiscale framework of persistent homology where the entire persistence information is encoded in algebraic terms and the most advantageous persistence summaries available in the literature are discussed. Finally, we introduce the necessary metrics in order to state properties of stability of the introduced multiscale summaries under perturbations of input data. At the end, we give an overview of applications of persistent homology as well as a review of the existing tools in the broader area of Topological Data Analysis (TDA).