In an increasingly connected world, the resilience of networked dynamical systems is important in the fields of ecology, economics, critical infrastructures, and organizational behaviour. Whilst we understand small-scale resilience well, our understanding of large-scale networked resilience is limited. Recent research in predicting the effective network-level resilience pattern has advanced our understanding of the coupling relationship between topology and dynamics. However, a method to estimate the resilience of an individual node within an arbitrarily large complex network governed by non-linear dynamics is still lacking. Here, we develop a sequential mean-field approach and show that after 1-3 steps of estimation, the node-level resilience function can be represented with up to 98% accuracy. This new understanding compresses the higher dimensional relationship into a onedimensional dynamic for tractable understanding, mapping the relationship between local dynamics and the statistical properties of network topology. By applying this framework to case studies in ecology and biology, we are able to not only understand the general resilience pattern of the network, but also identify the nodes at the greatest risk of failure and predict the impact of perturbations. These findings not only shed new light on the causes of resilience loss from cascade effects in networked systems, but the identification capability could also be used to prioritize protection, quantify risk, and inform the design of new system architectures. SIGNIFICANCE STATEMENTA gap in understanding exists between individual dynamics and the coupled dynamics of a large-scale networked complex system. Here, we present a framework for tractably analyzing the resilience of individual nodes as a function of the individual dynamics and the network property. Quantifying connected resilience as a function of the dynamics enables us to prioritize actions more effectively and predict resilience loss more accurately. Conversely, we may also discover hidden cascade effects, whereby disconnecting a weakly connected node can lead to failure in other nodes. In general, the node-level precision methods developed here will enable practitioners in ecology, infrastructure, and other application areas to prioritize protection and intervention resources, such as maintenance, preservation, rewiring, and upgrades.
Load balancing between adjacent base stations (BSs) is important for balancing load distributions and improving service provisioning. Whilst load balancing between any given pair of BSs is beneficial, cascade load sharing can cause network level instability that is hard to predict. The relationship between each BS's load balancing dynamics and the network topology is not understood. In this seminal work on stability analysis, we consider a frequency re-use network with no interference, whereby load balancing dynamics doesn't perturb the individual cells' capacity.Our novelty is to show an exact analytical and also a probabilistic relationship for stability, relating generalized local load balancing dynamics with generalized network topology, as well as the uncertainty we have in load balancing parameters. We prove that the stability analysis given is valid for any generalized load balancing dynamics and topological cell deployment and we believe this general relationship can inform the joint design of both the load balancing dynamics and the neighbour list of the network.
Trophic coherence, a measure of a graph’s hierarchical organisation, has been shown to be linked to a graph’s structural and dynamical aspects such as cyclicity, stability and normality. Trophic levels of vertices can reveal their functional properties, partition and rank the vertices accordingly. Trophic levels and hence trophic coherence can only be defined on graphs with basal vertices, i.e. vertices with zero in-degree. Consequently, trophic analysis of graphs had been restricted until now. In this paper we introduce a hierarchical framework which can be defined on any simple graph. Within this general framework, we develop several metrics: hierarchical levels, a generalisation of the notion of trophic levels, influence centrality, a measure of a vertex’s ability to influence dynamics, and democracy coefficient, a measure of overall feedback in the system. We discuss how our generalisation relates to previous attempts and what new insights are illuminated on the topological and dynamical aspects of graphs. Finally, we show how the hierarchical structure of a network relates to the incidence rate in a SIS epidemic model and the economic insights we can gain through it.
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