Maximum-entropy moment-closure for stochastic systems on networks

Moment-closure methods are popular tools to simplify the mathematical analysis of stochastic models defined on networks, in which high dimensional joint distributions are approximated (often by some heuristic argument) as functions of lower dimensional distributions. Whilst undoubtedly useful, several such methods suffer from issues of non-uniqueness and inconsistency. These problems are solved by an approach based on the maximisation of entropy, which is motivated, derived and implemented in this article. A series of numerical experiments are also presented, detailing the application of the method to the Susceptible-Infective-Recovered model of epidemics, as well as cautionary examples showing the sensitivity of moment-closure techniques in general.Comment: 20 pages, 7 figure

Assessing node risk and vulnerability in epidemics on networks

Which nodes are most vulnerable to an epidemic spreading through a network, and which carry the highest risk of causing a major outbreak if they are the source of the infection? Here we show how these questions can be answered to good approximation using the cavity method. Several curious properties of node vulnerability and risk are explored: some nodes are more vulnerable than others to weaker infections, yet less vulnerable to stronger ones; a node is always more likely to be caught in an outbreak than it is to start one, except when the disease has a deterministic lifetime; the rank order of node risk depends on the details of the distribution of infectious periods.Comment: Note that Figure 2 does not appear in the final published versio

Universal sum and product rules for random matrices

The spectral density of random matrices is studied through a quaternionic generalisation of the Green's function, which precisely describes the mean spectral density of a given matrix under a particular type of random perturbation. Exact and universal expressions are found in the high-dimension limit for the quaternionic Green's functions of random matrices with independent entries when summed or multiplied with deterministic matrices. From these, the limiting spectral density can be accurately predicted

Modes of competition and the fitness of evolved populations

Competition between individuals drives the evolution of whole species. Although the fittest individuals survive the longest and produce the most offspring, in some circumstances the resulting species may not be optimally fit. Here, using theoretical analysis and stochastic simulations of a simple model ecology, we show how the mode of competition can profoundly affect the fitness of evolved species. When individuals compete directly with one another, the adaptive dynamics framework provides accurate predictions for the number and distribution of species, which occupy positions of maximal fitness. By contrast, if competition is mediated by the consumption of a common resource then demographic noise leads to the stabilization of species with near minimal fitness.Comment: 11 pages, 6 figure

Cavity approach to the spectral density of non-Hermitian sparse matrices

The spectral densities of ensembles of non-Hermitian sparse random matrices are analysed using the cavity method. We present a set of equations from which the spectral density of a given ensemble can be efficiently and exactly calculated. Within this approach, the generalised Girko's law is recovered easily. We compare our results with direct diagonalisation for a number of random matrix ensembles, finding excellent agreement.Comment: 4 pages, 3 figure

How does the EU actually work?

As the UK referendum on EU membership draws closer, final year BSc Government student Tim Rogers outlines the institutions that make up the European Union, and argues that without a basic understanding of how the EU makes laws, it is difficult to make a reasoned judgement about the democratic quality of its practices

Labour’s new leader: what led to Corbyn’s ‘unlikely coup?’

On Wednesday 3 February, Rosa Prince, Assistant Political Editor for the Telegraph, discussed her new book, ‘Comrade Corbyn: a very unlikely coup’ as part of the British Government @ LSE public lecture series. Third year Government student Tim Rogers gives his analysis of the event, and considers how Jeremy Corbyn rose from political outsider to Leader of the Labour Party