How directed is a directed network?
Speaker: R.S.MacKay (University of Warwick)
Date: Wednesday 29 April 2020, 3:00 p.m.
Abstract: Many systems can be represented by directed graphs, e.g. food webs, supply networks, social networks, metabolic networks, language networks, financial networks. The nodes represent the objects and a directed edge indicates a flow from one node to another or influence of one node on another.
In some networks, the edges line up in an overall direction; in others, they do not. The old notion of “trophic level” from ecology, and its more recent analogue “upstreamness” in economics, provide one way to quantify this, but they require basal or top nodes and have various other shortcomings.
In joint work with Samuel Johnson and Bazil Sansom, we present an improved notion of trophic level and a resulting notion of trophic coherence. We illustrate their application to a wide variety of real-world networks and we derive some nice mathematical relationships of trophic coherence with other significant network properties like non-normality, stability of contagion processes, and cyclicality.
The work was supported by the Economic and Social Research Council via the Instability hub of the Rebuilding Macroeconomics programme of the National Institute for Economic and Social Research.
Is dispersion a stabilising or destabilising mechanism? Landau-damping induced by fast background flows
Speaker: Edriss Titi (Cambridge, Texas A&M Univ, Weizmann Institute of Science)
Date: Wednesday 4 March 2020
Abstract: In this talk Edriss Titi will present a unified approach for the effect of fast rotation and dispersion as an averaging mechanism for, on the one hand, regularising and stabilising certain evolution equations, such as the Navier-Stokes and Burgers equations. On the other hand, Edriss will present some results in which large dispersion acts as a destabilising mechanism for the long-time dynamics of certain dissipative evolution equations, such as the Kuramoto-Sivashinsky equation. In addition, they will present some new results concerning two- and three-dimensional turbulent flows with high Reynolds numbers in periodic domains, which exhibit ``Landau-damping" mechanism due to large spatial average in the initial data.
Weighing the Fog of War: A physicist's adventures in operations research and history
Speaker: Niall MacKay (York)
Date: Wednesday 11 December 2019
Abstract: A tour through ten years' work with operations researchers and historians on combat modelling, military history and their interaction. I'll begin with some elementary ideas from Lanchester theory and present some neat results for multilateral models. Then I shall describe some of our historical work, including on the First World War at sea and the Second World War in the air.
Global stability properties of the climate: Melancholia states, invariant measures, and phase transitions
Speaker: Valerio Lucarini (Reading)
Date: Tuesday 3 December 2019
Abstract: For a wide range of values of the incoming solar radiation, the Earth features at least two attracting states, which correspond to competing climates. The warm climate is analogous to the present one; the snowball climate features global glaciation and conditions that can hardly support life forms. Paleoclimatic evidences suggest that in past our planet flipped between these two states. The main physical mechanism responsible for such instability is the ice-albedo feedback. Following an idea developed by Eckhardt and co. for the investigation of multistable turbulent flows, we study the global instability giving rise to the snowball/warm multistability in the climate system by identifying the climatic edge state, a saddle embedded in the boundary between the two basins of attraction of the stable climates. We refer to these states as Melancholia States. We then introduce random perturbations as modulations to the intensity of the incoming solar radiation. We observe noise-induced transitions between the competing basins of attractions. In the weak noise limit, large deviation laws define the invariant measure and the statistics of escape times. By empirically constructing the instantons, we show that the Melancholia states are the gateways for the noise-induced transitions. In the region of multistability, in the zero-noise limit, the measure is supported only on one of the competing attractors. For low (high) values of the solar irradiance, the limit measure is the snowball (warm) climate. The changeover between the two regimes corresponds to a first order phase transition in the system. The framework we propose seems of general relevance for the study of complex multistable systems. Finally, we propose a new method for constructing Melancholia states from direct numerical simulations, thus bypassing the need to use the edge-tracking algorithm.
Refs.
On extreme and record events in dynamical systems
Speaker: Mark Holland (Exeter)
Date: Wednesday 6 November 2019
Abstract: Understanding extreme events, such as severe weather, climatic or financial events is a major challenge. In this talk I will explain some of the mathematical approaches used in understanding extreme events, such as finding their probability distribution. We will assume that the underlying time series process is modelled by a deterministic dynamical system, such as a discrete time map or differential equation. These latter processes have some dependency, and so any results known about extremes for independent, identically distributed (i.i.d) random processes cannot immediately be transferred to dynamical systems. As part of the talk, I will review relevant extreme value theory associated to the i.i.d case. I will then discuss this theory for dynamical systems, illustrating with examples from low dimensional chaotic maps.
