The behaviour of many large-scale systems can be modelled by a network of pairwise interaction. Examples include spread of epidemics, neural activity in the brain and social media influence. While these models are relatively flexible, their complexity strongly depends both on the structural properties of networks and the precise nature of the process unfolding on them. The complete specification of such models require various amount and quality of information. In the majority of situations, however, such data sets are incomplete and contain errors. Furthermore, pairwise interactions on networks in many instances are hidden from us and are impossible or very difficult to measure directly. Problems of interest in such situations include quantifying the effect of errors and omissions in the data on the predictability of the behaviour of the process unfolding on large-scale networks, and the inference of the underlying structure from partial, erroneous and usually indirect observations. In this symposium we consider different approaches to such problems. This includes approximation of large scale networks through statistical averaging techniques and Bayesian inference of the network structure.
08:30
A network SIR epidemic model with preventive rewiring
David Sirl | University of Nottingham | United Kingdom
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David Sirl | University of Nottingham | United Kingdom
Network epidemic models have developed enormously in the last 20 years or so in response to some of the unrealistic assumptions of homogeneity in most simple epidemic models. A significant feature of most epidemic-on-a-network models is that the epidemic evolves on a static network. We consider an SIR (Susceptible-Infectious-Removed) epidemic spreading on a configuration-model network (a random network with specified degree distribution), with the addition of simple network dynamics whereby susceptible individuals may 'drop' connections to infectious neighbours. A further extension permits such susceptible individuals to then 'rewire' to connect instead with someone else in the population.
For the model with dropping only (i.e. no rewiring), we present limit theorems (in the limit of large population size) for the temporal evolution and for the final size of the epidemic (i.e. the number of initially susceptible individuals that are ultimately recovered), in the event of a large outbreak. For the model with rewiring included too, we present theoretical and numerical results showing that whilst the preventive behaviour of rewiring is always rational at the individual level, it may have negative consequences at the population level. These latter results also cover a variant of the configuration model which incorporates clustering and some empirical social networks.
(Joint work with Frank Ball [Nottingham], Tom Britton [Stockholm] and KaYin Leung [Stockholm, now HAL24K].)
09:00
Comparison of different statistical averaging methods for epidemic processes on networks
Peter Simon | Eötvös Loránd University | Hungary
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Peter Simon | Eötvös Loránd University | Hungary
Epidemic spread on networks can be described by a continuous time Markov chain. The size of its state space blows up exponentially as the number of vertices is increased, hence several statistical averaging methods were derived. The goal of the talk is to compare these models for Susceptible-Infected-Susceptible (SIS) epidemic spread on networks. The simplest mean-field model determines the expected number of infected nodes only and involves simply the average degree of the graph. The degree-based mean-field model is written in terms of the expected number of infected nodes with different degrees and involves the first and second moments of the degree distribution. Both of these models have been extended to include the expected number of susceptible-infected (SI) edges. An even more detailed description can be obtained from the individual-based mean-field model, which is characterized by the adjacency matrix of the network. It will also be presented how the different graph characteristics (e.g. the moments of the degree distribution or the leading eigenvalue of the adjacency matrix) appear in the threshold between the disease-free and the endemic steady state.
09:30
- NEW - Network reconstruction and community detection from dynamics
Tiago Peixoto | Central European University | Hungary
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Tiago Peixoto | Central European University | Hungary
The observed functional behavior of a wide variety of large-scale systems is often the result of a network of pairwise interactions between individual elements. However, in many cases these interactions are hidden from us, either because they are impossible to be measured directly, or because their measurement can be done only at significant experimental cost. In such situations, we are required to infer the network of interactions from the observed functional behavior.
In this talk, I will present a scalable nonparametric Bayesian method to perform network reconstruction from observed functional behavior, that at the same time infers the modular structure (or "communities") present in the network. I will show how the joint reconstruction with community detection has a synergistic effect, where the edge correlations used to inform the existence of communities are also inherently used to improve the accuracy of the reconstruction which, in turn, can better inform the uncovering of communities. I will illustrate the use of the method with observations arising from epidemic models and the Ising model, both on synthetic and empirical networks, as well as on data containing only functional information.
10:00
Non-parametric Bayesian inference for density dependent Networks with SIS discrete data
Jean-Charles Croix | University of Sussex | United Kingdom
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Jean-Charles Croix | University of Sussex | United Kingdom
Inferring contact networks from discrete observations of stochastic processes unfolding on networks is a challenging problem. Indeed, any likelihood-based approach is quickly limited by the size of the state space, exponential in the number of nodes. Within the framework of the Susceptible-Infected-Susceptible (SIS) epidemic model, we alleviate this difficulty by using an approximation inspired from the case of the fully connected network with density dependent infection rates. Indeed, as the number of node becomes large, this stochastic process can be approximated by an Ito stochastic differential equation (SDE) with known coefficients. We extend this idea to different network families, which can all be approximated by a class of degenerate SDEs. The network inference problem is thus framed as identifying the coefficients of the SDE limit model. We deal with this problem using a Bayesian non-parametric approach and provide numerical results for simple network families.