Nonlinear dynamics on networks: deterministic and stochastic approaches

Many physical phenomena occurring in nature can be described through complex systems as collective dynamics which emerge at the macroscopic level from the interactions of microscopically constituents. Networks dene the struc- ture of such systems, where the nodes represent the agents interconnected through intricated patterns of interactions. Neurons activity in the brain, species interacting in ecosystems, individuals that constitute social systems are typical examples of systems that can be described as systems hosted on top of networks and whose evolution is strictly related to the organization of the underlying support. Studying dynamical processes on networks represent one of the most inter- esting application in network science. Noise is a key ingredient in the dynamical description of many real-world phenomena where stochastic perturbations can derive from an external noise, due for example to termal uctuations, or can stem from an endogeneous noise, re ecting the inherent discreteness of the scru- tinized medium. In diusion or other linear processes, the dynamics can be described by simple interactions between adjacent nodes, otherwise it could also involve a local reac- tion on the single nodes. A classical example is represented by reaction-diusion processes, where the elements interact depending on specic self-reactions and diuse across the spatial medium dened by the network. Reaction-diusion processes on networks represent the focus of this thesis. The evolution of dynamical systems hosted on the single nodes changes inserting the couplings and unexpected collective behaviors may emerge, like for examples chaotic systems that synchronize to the same solution, or stationary inhomoge- neous patterns emerging from diusing species. The rst dynamical model we present is a simplied version of the cele- brated Wilson-Cowan model for neurons in the brain and consists in a two- species model of the excitatory-inhibitory type where the neurons are supposed to occupy the three nodes located at the vertices of a triangular loop. The phe- nomenon of stochastic quasicycles is investigated as consequence of the interplay between endogeneous noise and non-normality of the system. Non-normality proves a key ingredient to guarantee the amplication eects on noise-assisted oscillations: when the system is constrained to evolve with a constant rate of deterministic damping for the perturbations, the amplication correlates with the degree of non-normal reactivity, a measure that quantify the ability of the system to prompt an initial perturbation. We carry on the study of non-normality considering a stochastic reaction- diusion two-species model on network where the species are assumed to interact on each node following a non-normal reaction scheme. Replicating the interac- tion unit on a directed linear lattice, the combined eect of non-normality and degenerate spectrum of the embedding support guarantee the amplication process of the noise. The same phenomenon takes place when the system is hosted on a quasi-degenerate network, in the sense that the eigenvalues of the Laplacian operator accumulate in a compact region of the complex plane. The coopera- tion of non-normality and quasi-degenerate spectrum may then impact on the perception of stability, as predicted by conventional deterministic methods, and modify the usual denition of resilience, the ability of a system to oppose to ex- ternal perturbations. The concept of resilience is relevant in many elds, from ecology to climate change, via information security and energy development. In this respect, the eigenmodes associated to a quasi-degenerate spectrum rep- resent a route to drive the instability. For this motifs, we perform the rst generative algorithm of this thesis. The idea is to design a network assembling nodes in such a way that the associated Laplacian possesses a quasi-degenerate spectrum. The resulting network is a distorted one-dimensional directed chain and belongs to the class of the so called directed acyclic graphs. This opened the way to the subsequent part of our work: network generation. For reaction-diusion models, dynamics, and the inherent stability, depends on the Laplacian matrix and its spectrum of eigenvalues. As we mentioned above, the concept of stability is of paramount importance as it relates to resilience. It is then crucial to investigate on possible strategies to enforce stability in a desired system. Based on these observations, we aim at provide a novel math- ematical procedure to generate networks with any desired Laplacian spectrum. The obtained networks are weighted and fully connected, so we present two sparsication procedures to remove unessential links preserving the structure of the adjacency matrices. In a special working setting, we write the analytical expressions for the entries of the Laplacian matrix and we consequently derive conditions to have positive non-diagonal entries for such matrix. The built sparsied networks are tested simulating two models of oscillators, the Stuart-Landau and the Kuramoto model. In the nal part of the thesis we focus on high-order interaction structures. In many cases of interest, like for instance brain networks, protein interaction networks, ecological communities and co-authorship networks, the basic inter- action involves more than two nodes at the same time. Such systems necessi- tate to be described by a structure accounting for multi-body interactions, like hypergraphs. In our work, we study dynamical systems dened on top of hyper- graphs. The Master Stability Function proves essential in our applications to reaction-diusion systems and synchronization model, studying the stability of the associated homogeneous equilibria. We show that the role of the localization property of the Laplace operator is crucial in emerging of collective behavior, so we investigate on this property aiming at expand this analysis in a more for- mal way. Localization properties of random networks was already predicted by the perturbation theory, thus, in a very similar way, we apply the perturbation theory to the newly Laplacian matrix introduced for hypergraphs.