Self-exciting point processes have been proposed as models for the location of criminal events in space and time. Here we consider the case where the triggering function is isotropic and takes a non-parametric form that is determined from data. We pay special attention to normalisation issues and to the choice of spatial distance measure, thereby extending the current methodology. After validating these ideas on synthetic data, we perform inference and prediction tests on public domain burglary data from Chicago. We show that the algorithmic advances that we propose lead to improved predictive accuracy.

Within the online media universe, there are many underlying communities. These may be defined, for example, through politics, location, health, occupation, extracurricular interests or retail habits. Government departments, charities and commercial organisations can benefit greatly from insights about the structure of these communities; the move to customer-centred practices requires knowledge of the customer base. Motivated by this issue, we address the fundamental question of whether a sub-network looks like a collection of individuals who have effectively been picked at random from the whole, or instead forms a distinctive community with a new, discernible structure. In the former case, to spread a message to the intended user base it may be best to use traditional broadcast media (TV, billboard), whereas in the latter case a more targeted approach could be more effective. In this work, we therefore formalise a concept of testing for sub-structure and apply it to social interaction data. First, we develop a statistical test to determine whether a given sub-network (induced sub-graph) is likely to have been generated by sampling nodes from the full network uniformly at random. This tackles an interesting inverse alternative to the more widely studied “forward” problem. We then apply the test to a Twitter reciprocated mentions network where a range of brand name based sub-networks are created via tweet content. We correlate the computed results against the independent views of 16 digital marketing professionals. We conclude that there is great potential for social media based analytics to quantify, compare and interpret online brand allegiances systematically, in real time and at large scale.

We develop and test an intuitively simple dynamic network model to describe the type of time-varying connectivity structure present in many technological settings. The model assumes that nodes have an inherent hierarchy governing the emergence of new connections. This idea draws on newly established concepts in online human behaviour concerning the existence of discussion catalysts, who initiate long threads, and online leaders, who trigger feedback. We show that the model captures an important property found in e-mail and voice call data – ‘dynamic communicators’ with sufficient foresight or impact to generate effective links and having an influence that is grossly underestimated by static measures based on snaphots or aggregated data.

Bipartivity is an important network concept that can be applied to nodes, edges and communities. Here we focus on directed networks and look for subnetworks made up of two distinct groups of nodes, connected by ‘one-way’ links. We show that a spectral approach can be used to find hidden substructures of this form. Theoretical support is given for the idealized case where there is limited overlap between subnetworks. Numerical experiments show that the approach is robust to spurious and missing edges. A key application of this work is in the analysis of high-throughput gene expression data, and we give an example where a biologically meaningful directed bipartite subnetwork is found from a cancer microarray dataset.

Long-term solutions of the theta method applied to scalar nonlinear differential equations are studied in this paper. In the case where the equation has a stable steady state, lower bounds on the basin of non-oscillatory, monotonic attraction for the theta method are derived. Spurious period two solutions are then analysed. Under mild assumptions, precise results are obtained concerning the generic nature and stability of these solutions for small timesteps. Particular problem classes are studied, and direct connections are made between the existence and stability of period two solutions and the dynamics of the theta method. The analysis is extended to a wide class of semi-discretized partial differential equations. Numerical examples are given.

Positive results are proved here about the ability of numerical simulations to reproduce the exponential mean-square stability of stochastic differential equations (SDEs). The first set of results applies under finite-time convergence conditions on the numerical method. Under these conditions, the exponential mean-square stability of the SDE and that of the method (for sufficiently small step sizes) are shown to be equivalent, and the corresponding second-moment Lyapunov exponent bounds can be taken to be arbitrarily close. The required finite-time convergence conditions hold for the class of stochastic theta methods on globally Lipschitz problems. It is then shown that exponential mean-square stability for non-globally Lipschitz SDEs is not inherited, in general, by numerical methods. However, for a class of SDEs that satisfy a one-sided Lipschitz condition, positive results are obtained for two implicit methods. These results highlight the fact that for long-time simulation on nonlinear SDEs, the choice of numerical method can be crucial.