Dynamics on networks

The main focus of this thesis is to study the stability of fix points for a dynamical system. In the first part, we consider two dynamical models whose underlying graph can be represented by a single network. We first consider the Kuramoto model, a canonical model of coupled phase oscillators. We obtain two results on its partial phase-locked state, where a subset of oscillators remain close in phase while others drift away. Firstly, we derive an analytical criterion for the finite-N model to guarantee the existence of partial phase-locking for sufficiently strong coupling, by proving the existence of an attracting ball around a fixed point of a subset of the oscillators. Secondly, we deduce a deterministic condition for the model in the large N limit, giving almost sure existence of a partially entrained cluster of computable size. We then explore a social network model describing the formation of opinions. Two approaches, automorphism reduction method and "nearest-neighbor" mean-field analysis, are proposed to analyze its fix points and their stabilities. Both approaches aim to resolve the problem of the curse of dimensionality. For the first approach, we exploit the graph automorphism of the Petersen graph and find its three stable fix points, a consensus state, a balanced and an unbalanced state. For the second approach, we use the Erdos-Renyi graph to illustrate the idea of approximating a large system by an "averaged" smaller one by considering the distances of neighbors of a given vertex. In the second part, we generalize the single-network case into complex networks to account for real-world problems. In particular, we study a node-aligned multilayer Kuramoto model that encapsulates multiple channels of connectivity among oscillators. Our primary goal is to understand how inter-layer connections affect system stability. We address this question from two aspects: the effect of inter-layer topologies and the effect of a weak inter-layer perturbation. For the first aspect, we discuss two specific topologies: a complete graph and a cycle-tree graph or one containing only no-edge-shared cycles, and explore their effects on the stability of a twisted fixed point of the Kuramoto model. For the second aspect, we focus on a duplex network and provide analytical treatment to measure the effect of an additional weak inter-layer coupling on the system stability using the standard perturbation theory. It is found that under specific conditions our system is always destabilized due to the line addition, conforming to the famously counter-intuitive Braess's Paradox