arXiv subjects

Matthew Sheerin

Publications and source records attributed to Matthew Sheerin.

3 recordsLinked to original sources

arXiv2017

Random Neighborhood Graphs as Models of Fracture Networks on Rocks: Structural and Dynamical Analysis

We propose a new model to account for the main structural characteristics of rock fracture networks (RFNs). The model is based on a generalization of the random neighborhood graphs to consider fractures embedded into rectangular spaces. We study a series of 29 real-world RFNs and find the best fit with the random rectangular neighborhood graphs (RRNGs) proposed here. We show that this model captures most of the structural characteristics of the RFNs and allows a distinction between small and more spherical rocks and large and more elongated ones. We use a diffusion equation on the graphs in order to model diffusive processes taking place through the channels of the RFNs. We find a small set of structural parameters that highly correlates with the average diffusion time in the RFNs. We found analytically some bounds for the diameter and the algebraic connectivity of these graphs that allow to bound the diffusion time in these networks. We also show that the RRNGs can be used as a suitable model to replace the RFNs in the study of diffusion-like processes. Indeed, the diffusion time in RFNs can be predicted by using structural and dynamical parameters of the RRNGs. Finally, we also explore some potential extensions of our model to include variable fracture apertures, the possibility of long-range hops of the diffusive particles as a way to account for heterogeneities in the medium and possible superdiffusive processes, and the extension of the model to 3-dimensional space.

physics.geo-ph↗

arXiv2016

Epidemic Spreading in Random Rectangular Networks

The use of network theory to model disease propagation on populations introduces important elements of reality to the classical epidemiological models. The use of random geometric graphs (RGG) is one of such network models that allows for the consideration of spatial properties on disease propagation. In certain real-world scenarios -like in the analysis of a disease propagating through plants- the shape of the plots and fields where the host of the disease is located may play a fundamental role on the propagation dynamics. Here we consider a generalization of the RGG to account for the variation of the shape of the plots/fields where the hosts of a disease are allocated. We consider a disease propagation taking place on the nodes of a random rectangular graph (RRG) and we consider a lower bound for the epidemic threshold of a Susceptible-Infected-Susceptible (SIS) or Susceptible-Infected-Recovered (SIR) model on these networks. Using extensive numerical simulations and based on our analytical results we conclude that (ceteris paribus) the elongation of the plot/field in which the nodes are distributed makes the network more resilient to the propagation of a disease due to the fact that the epidemic threshold increases with the elongation of the rectangle. These results agree with accumulated empirical evidence and simulation results about the propagation of diseases on plants in plots/fields of the same area and different shapes.

physics.soc-ph↗

arXiv2015

Random Rectangular Graphs

A generalization of the random geometric graph (RGG) model is proposed by considering a set of points uniformly and independently distributed on a rectangle of unit area instead of on a unit square [0,1]^2. The topological properties of the random rectangular graphs (RRGs) generated by this model are then studied as a function of the rectangle sides lengths a and b=1/a, and the radius r used to connect the nodes. When a=1 we recover the RGG, and when a-->infinity the very elongated rectangle generated resembles a one-dimensional RGG. We obtain here analytical expressions for the average degree, degree distribution, connectivity, average path length and clustering coefficient for RRG. These results provide evidence that show that most of these properties depend on the connection radius and the side length of the rectangle, usually in a monotonic way. The clustering coefficient, however, increases when the square is transformed into a slightly elongated rectangle, and after this maximum it decays with the increase of the elongation of the rectangle. We support all our findings by computational simulations that show the goodness of the theoretical models proposed for RRGs.

physics.soc-ph↗