A closed form for Jacobian reconstruction from time series and its application as an early warning signal in network dynamics.

The Jacobian matrix of a dynamical system describes its response to perturbations. Conversely, one can estimate the Jacobian matrix by carefully monitoring how the system responds to environmental noise. We present a closed-form analytical solution for the calculation of a system's Jacobian from a time series. Being able to access the Jacobian enables a broad range of mathematical analyses by which deeper insights into the system can be gained. Here we consider in particular the computation of the leading Jacobian eigenvalue as an early warning signal for critical transitions. To illustrate this approach, we apply it to ecological meta-foodweb models, which are strongly nonlinear dynamical multi-layer networks. Our analysis shows that accurate results can be obtained, although the data demand of the method is still high.

Wireless localization with diffusion maps.

In the Wireless Localization Matching Problem (WLMP) the challenge is to match pieces of equipment with a set of candidate locations based on wireless signal measurements taken by the pieces of equipment. This challenge is complicated by the noise that is inherent in wireless signal measurements. Here we propose the use of diffusion maps, a manifold learning technique, to obtain an embedding of positions and equipment coordinates in a space that enables coordinate comparison and reliable evaluation of assignment quality at very low computational cost. We show that the mapping is robust to noise and using diffusion maps allows for accurate matching in a realistic setting. This suggests that the diffusion-map-based approach could significantly increase the accuracy of wireless localization in applications.

Manifold cities: social variables of urban areas in the UK.

In the twenty-first century, ongoing rapid urbanization highlights the need to gain deeper insights into the social structure of cities. While work on this challenge can profit from abundant data sources, the complexity of this data itself proves to be a challenge. In this paper, we use diffusion maps, a manifold learning method, to discover hidden manifolds in the UK 2011 census dataset. The census key statistics and quick statistics report 1450 different statistical features for each census output area. Here, we focus primarily on the city of Bristol and the surrounding countryside, comprising 3490 of these output areas. Our analysis finds the main variables that span the census responses, highlighting that university student density and poverty are the most important explanatory variables of variation in census responses.

Impact of cyber-invasive species on a large ecological network.

As impacts of introduced species cascade through trophic levels, they can cause indirect and counter-intuitive effects. To investigate the impact of invasive species at the network scale, we use a generalized food web model, capable of propagating changes through networks with a series of ecologically realistic criteria. Using data from a small British offshore island, we quantify the impacts of four virtual invasive species (an insectivore, a herbivore, a carnivore and an omnivore whose diet is based on a rat) and explore which clusters of species react in similar ways. We find that the predictions for the impacts of invasive species are ecologically plausible, even in large networks. Species in the same taxonomic group are similarly impacted by a virtual invasive species. However, interesting differences within a given taxonomic group can occur. The results suggest that some native species may be at risk from a wider range of invasives than previously believed. The implications of these results for ecologists and land managers are discussed.

Spatial effects in meta-foodwebs.

In ecology it is widely recognised that many landscapes comprise a network of discrete patches of habitat. The species that inhabit the patches interact with each other through a foodweb, the network of feeding interactions. The meta-foodweb model proposed by Pillai et al. combines the feeding relationships at each patch with the dispersal of species between patches, such that the whole system is represented by a network of networks. Previous work on meta-foodwebs has focussed on landscape networks that do not have an explicit spatial embedding, but in real landscapes the patches are usually distributed in space. Here we compare the dispersal of a meta-foodweb on Erdos-Rényi networks, that do not have a spatial embedding, and random geometric networks, that do have a spatial embedding. We found that local structure and large network distances in spatially embedded networks, lead to meso-scale patterns of patch occupation by both specialist and omnivorous species. In particular, we found that spatial separations make the coexistence of competing species more likely. Our results highlight the effects of spatial embeddings for meta-foodweb models, and the need for new analytical approaches to them.

Dynamics of epidemic diseases on a growing adaptive network.

The study of epidemics on static networks has revealed important effects on disease prevalence of network topological features such as the variance of the degree distribution, i.e. the distribution of the number of neighbors of nodes, and the maximum degree. Here, we analyze an adaptive network where the degree distribution is not independent of epidemics but is shaped through disease-induced dynamics and mortality in a complex interplay. We study the dynamics of a network that grows according to a preferential attachment rule, while nodes are simultaneously removed from the network due to disease-induced mortality. We investigate the prevalence of the disease using individual-based simulations and a heterogeneous node approximation. Our results suggest that in this system in the thermodynamic limit no epidemic thresholds exist, while the interplay between network growth and epidemic spreading leads to exponential networks for any finite rate of infectiousness when the disease persists.

Meta-food-chains as a many-layer epidemic process on networks.

Notable recent works have focused on the multilayer properties of coevolving diseases. We point out that very similar systems play an important role in population ecology. Specifically we study a meta-food-web model that was recently proposed by Pillai et al. [Theor. Ecol. 3, 223 (2009)]. This model describes a network of species connected by feeding interactions, which spread over a network of spatial patches. Focusing on the essential case, where the network of feeding interactions is a chain, we develop an analytical approach for the computation of the degree distributions of colonized spatial patches for the different species in the chain. This framework allows us to address ecologically relevant questions. Considering configuration model ensembles of spatial networks, we find that there is an upper bound for the fraction of patches that a given species can occupy, which depends only on the networks mean degree. For a given mean degree there is then an optimal degree distribution that comes closest to the upper bound. Notably scale-free degree distributions perform worse than more homogeneous degree distributions if the mean degree is sufficiently high. Because species experience the underlying network differently the optimal degree distribution for one particular species is generally not the optimal distribution for the other species in the same food web. These results are of interest for conservation ecology, where, for instance, the task of selecting areas of old-growth forest to preserve in an agricultural landscape, amounts to the design of a patch network.