Peter Grindrod

Evolving graphs: dynamical models, inverse problems and propagation

Applications such as neuroscience, telecommunication, online social networking, transport and retail trading give rise to connectivity patterns that change over time. In this work, we address the resulting need for network models and computational algorithms that deal with dynamic links. We introduce a new class of evolving range-dependent random graphs that gives a tractable framework for modelling and simulation. We develop a spectral algorithm for calibrating a set of edge ranges from a sequence of network snapshots and give a proof of principle illustration on some neuroscience data. We also show how the model can be used computationally and analytically to investigate the scenario where an evolutionary process, such as an epidemic, takes place on an evolving network. This allows us to study the cumulative effect of two distinct types of dynamics.

A matrix iteration for dynamic network summaries

We propose a new algorithm for summarizing properties of large-scale time-evolving networks. This type of data, recording connections that come and go over time, is generated in many modern applications, including telecommunications and online human social behavior. The algorithm computes a dynamic measure of how well pairs of nodes can communicate by taking account of routes through the network that respect the arrow of time. We take the conventional approach of downweighting for length (messages become corrupted as they are passed along) and add the novel feature of downweighting for age (messages go out of date). This allows us to generalize widely used Katz-style centrality measures that have proved popular in network science to the case of dynamic networks sampled at nonuniform points in time. We illustrate the new approach on synthetic and real data.

A roadmap for the computation of persistent homology

Persistent homology (PH) is a method used in topological data analysis (TDA) to study qualitative features of data that persist across multiple scales. It is robust to perturbations of input data, independent of dimensions and coordinates, and provides a compact representation of the qualitative features of the input. The computation of PH is an open area with numerous important and fascinating challenges. The field of PH computation is evolving rapidly, and new algorithms and software implementations are being updated and released at a rapid pace. The purposes of our article are to (1) introduce theory and computational methods for PH to a broad range of computational scientists and (2) provide benchmarks of state-of-the-art implementations for the computation of PH. We give a friendly introduction to PH, navigate the pipeline for the computation of PH with an eye towards applications, and use a range of synthetic and real-world data sets to evaluate currently available open-source implementations for the computation of PH. Based on our benchmarking, we indicate which algorithms and implementations are best suited to different types of data sets. In an accompanying tutorial, we provide guidelines for the computation of PH. We make publicly available all scripts that we wrote for the tutorial, and we make available the processed version of the data sets used in the benchmarking.

Erosion rate and previous extent of interior layered deposits on Mars revealed by obstructed landslides

We describe interior layered deposits on Mars that have obstructed landslides before undergoing retreat by as much as 2 km. These landslides differ from typical Martian examples in that their toe height increases by as much as 500 m in a distinctive frontal scarp that mimics the shape of the layered deposits. By using crater statistics to constrain the formation ages of the individual landslides to between ca. 200 and 400 Ma, we conclude that the retreat of the interior layered deposits was rapid, requiring erosion rates of between 1200 and 2300 nm yr–1. We suggest that the interior layered deposits are either extremely friable, if eroded strictly by wind, or composed of a material whose degradation has been enhanced by ice sublimation. These erosion rates also confirm that the interior layered deposits have been in a state of net degradation over the past 400 m.y., suggesting that the process that caused net deposition in the past has ceased or slowed substantially on Mars relative to erosion. Our results imply that interior layered deposits with a similar morphology across Mars, including the mound in Gale Crater, have probably undergone similar rapid erosion and retreat, suggesting that their total modern volume underrepresents the depositional record and thus sedimentary history of Mars.

The indication of Martian gully formation processes by slope–area analysis

Abstract The formation process of recent gullies on Mars is currently under debate. This study aims to discriminate between the proposed formation processes – pure water flow, debris flow and dry mass wasting – through the application of geomorphological indices commonly used in terrestrial geomorphology. High-resolution digital elevation models (DEMs) of Earth and Mars were used to evaluate the drainage characteristics of small slope sections. Data from Earth were used to validate the hillslope, debris-flow and alluvial process domains previously found for large fluvial catchments on Earth, and these domains were applied to gullied and ungullied slopes on Mars. In accordance with other studies, our results indicate that debris flow is one of the main processes forming the Martian gullies that were being examined. The source of the water is predominantly distributed surface melting, not an underground aquifer. Evidence is also presented indicating that other processes may have shaped Martian crater slopes, such as ice-assisted creep and solifluction, in agreement with the proposed recent Martian glacial and periglacial climate. Our results suggest that, within impact craters, different processes are acting on differently oriented slopes, but further work is needed to investigate the potential link between these observations and changes in Martian climate.

The geometry and motion of reaction-diffusion waves on closed two-dimensional manifolds

Chemical or biological systems modelled by reaction diffusion (R.D.) equations which support simple one-dimensional travelling waves (oscillatory or otherwise) may be expected to produce intricate two or three-dimensional spatial patterns, either stationary or subject to certain motion. Such structures have been observed experimentally. Asymptotic considerations applied to a general class of such systems lead to fundamental restrictions on the existence and geometrical form of possible structures. As a consequence of the geometrical setting, it is a straightforward matter to consider the propagation of waves on closed two-dimensional manifolds. We derive a fundamental equation for R.D. wave propagation on surfaces and discuss its significance. We consider the existence and propagation of rotationally symmetric and double spiral waves on the sphere and on the torus.

Three-dimensional waves in excitable reaction-diffusion systems

The eikonal equation [5] for excitable media is generalised to three dimensional systems. The main result of the investigation is the demonstration of the existence of toroidal and twisted toroidal scroll waves in the limit of large values of the major radius of the torus. The existence of a helical wave near the z-axis follows from the eidonal equation but its connection with the twisted toroidal scroll remains to be demonstrated. The eikonal equation also predicts a non-uniform rate of rotation of the cross-sectional spiral wave near the toroidal axis. The notion of geometrical stability is introduced for the case of an expanding sphere; in particular it is shown that a discussion of stability of solutions of the eikonal equation must take into account the possible shift in the origin of the coordinate systems with respect to which patterns are defined.

A dynamical systems view of network centrality

To gain insights about dynamic networks, the dominant paradigm is to study discrete snapshots, or timeslices, as the interactions evolve. Here, we develop and test a new mathematical framework where network evolution is handled over continuous time, giving an elegant dynamical systems representation for the important concept of node centrality. The resulting system allows us to track the relative influence of each individual. This new setting is natural in many digital applications, offering both conceptual and computational advantages. The novel differential equations approach is convenient for modelling and analysis of network evolution and gives rise to an interesting application of the matrix logarithm function. From a computational perspective, it avoids the awkward up-front compromises between accuracy, efficiency and redundancy required in the prevalent discrete-time setting. Instead, we can rely on state-of-the-art ODE software, where discretization takes place adaptively in response to the prevailing system dynamics. The new centrality system generalizes the widely used Katz measure, and allows us to identify and track, at any resolution, the most influential nodes in terms of broadcasting and receiving information through time-dependent links. In addition to the classical static network notion of attenuation across edges, the new ODE also allows for attenuation over time, as information becomes stale. This allows ‘running measures’ to be computed, so that networks can be monitored in real time over arbitrarily long intervals. With regard to computational efficiency, we explain why it is cheaper to track good receivers of information than good broadcasters. An important consequence is that the overall broadcast activity in the network can also be monitored efficiently. We use two synthetic examples to validate the relevance of the new measures. We then illustrate the ideas on a large-scale voice call network, where key features are discovered that are not evident from snapshots or aggregates.

Experimental investigation of the mechanical properties of synthetic magnesium sulfate hydrates: Implications for the strength of hydrated deposits on Mars

[1] We have carried out uniaxial compression experiments to determine the mechanical properties of three crystalline magnesium sulfate hydrates that may be present in the near‐surface environment of Mars. Our synthetic samples of kieserite (MgSO4 ·H 2O), epsomite (MgSO4 ·7 H2O), and meridianiite (MgSO4 · 11H2O) have mean values of unconfined compressive strength of 6.3 ± 0.7, 12.9 ± 1.8, and 30.1 ± 4.5 MPa, respectively, Young’s modulus of 0.8 ± 0.1, 2.9 ± 0.4, and 5.9 ± 0.8 GPa, respectively, and mean porosity values of 47.8% ± 0.5%, 11.1% ± 0.6%, and 2.9% ± 0.2%, respectively. Although our tests cannot quantify a systematic relationship between hydration state and mechanical properties, the different porosities produced by consistent sample preparation methods suggest that the addition of non‐cation‐coordinated water molecules likely reduces the strength of individual sulfate hydrate phases. However, the bulk mechanical properties of our synthetic specimens are instead controlled predominantly by the sample porosity; generally, the strength increases as the porosity decreases. We expect the mechanical properties of sulfate hydrate deposits on Mars to be governed by the bulk porosity rather than the strength of the pure solid phase. We have performed cyclic stressing tests, replicating possible periodic depositional and erosional periods on Mars resulting from obliquity changes. A gradual compaction and reduction in sample porosity, rather than an increase in crack damage, is observed with each loading cycle, suggesting that the evolution of mechanical properties will depend on local factors such as bulk density, in addition to the overall stress history.

Groundwater processes in Hebes Chasma, Mars

We describe a conceptual model of groundwater processes at Hebes Chasma, Mars, which can account for the distribution of hydrated minerals and their subsequent evolution. At Hebes Chasma, pressure gradients set up by the large central mound, Hebes Mensa, could cause groundwater to be sourced predominantly from beneath the central region, if such water were present. Evaporation of upwelling groundwater would cause monohydrates to form at or near the surface through efflorescence, and polyhydrates to form inside the central mound through subflorescence. This crystallization could lead to an excess pore pressure, causing large-scale weakening and subsequent collapse that can reveal the interior polyhydrated deposits. If evaporation is high compared to groundwater inflow, then increased crystallization would promote the formation of collapse zones. If evaporation is low compared to groundwater inflow then there would be a greater chance for water reaching the surface and the possible formation of karst landforms.