C3

A granular material forms a distinct and fascinating phase in physics -- sand acts as a fluid as grains flow through your fingers, the fallen grains form a solid heap on the floor or may suspend in the wind like a gas.

The main challenge of studying granular materials is the development of constitutive models valid across scales, from the micro-scale (collisions between individual particles), via the meso-scale (flow structures inside avalanches) to the macro-scale (dunes, heaps, chute flows).

In this talk, I am highlighting three recent projects from my laboratory, each highlighting physical behavior at a different scale. First, using the property of birefringence, we are quantifying both kinetic and dynamic properties in an avalanche of macroscopic particles and measure rheological properties. Secondly, we explore avalanches on an erodible bed that display an intriguing dynamic intermittency between regimes. Lastly, we take a closer look at aqueous (water-driven) dunes in a novel rotating experiment and resolve an outstanding scaling controversy between migration velocity and dune dimension.

A Surstseyan eruption is a particular kind of volcanic eruption which involves the bulk interaction of water and hot magma. Surtsey Island was born during such an eruption process in the 1940s. I will talk about mathematical modelling of the flashing of water to steam inside a hot erupted lava ball called a Surtseyan bomb. The overall motivation is to understand what determines whether such a bomb will fragment or just quietly fizzle out...
Partial differential equations model transient changes in temperature and pressure in Surtseyan ejecta. We have used a highly simplified approach to the temperature behaviour, to separate temperature from pressure. The resulting pressure diffusion equation was solved numerically and asymptotically to derive a single parametric condition for rupture of ejecta. We found that provided the permeability of the magma ball is relatively large, steam escapes rapidly enough to relieve the high pressure developed at the flashing front, so that rupture does not occur. This rupture criterion is consistent with existing field estimates of the permeability of intact Surtseyan bombs, fizzlers that have survived.
I describe an improvement of this model that allows for the fact that pressure and temperature are in fact coupled, and that the process is not adiabatic. A more systematic reduction of the resulting coupled nonlinear partial differential equations that arise from mass, momentum and energy conservation is described. We adapt an energy equation presented in G.K. Batchelor's book {\em An Introduction to Fluid Dynamics} that allows for pressure-work. This is work in progress.  Work done with Emma Greenbank, Ian Schipper and Andrew Fowler 

In this talk we discuss the recent proof for the existence of $C^{1,1}$ isometric immersions of several classes of negatively curved surfaces into $\R^3$, including the Lobachevsky plane, metrics of helicoid type and a one-parameter family of perturbations of the Enneper surface. Our method, following Chen--Slemrod--Wang and Cao--Huang--Wang, is to transform the Gauss--Codazzi equations into a system of hyperbolic balance laws, and prove the existence of weak solutions by finding the invariant regions. In addition, we provide further characterisation of the $C^{1,1}$ isometrically immersed generalised helicoids/catenoids established in the literature.

The structure of the state of art of scientific research is an important object of study motivated by the understanding of how research evolves and how new fields of study stem from existing research. In the last years complex networks tools contributed to provide insights on the structure of research, through the study of collaboration, citation and co-occurrence networks, in particular keyword co-occurrence networks proved useful to provide maps of knowledge inside a scientific domain. The network approach focuses on pairwise relationships, often compressing multidimensional data structures and inevitably losing information. In this paper we propose to adopt a simplicial complex approach to co-occurrence relations, providing a natural framework for the study of higher-order relations in the space of scientific knowledge. Using topological methods we explore the shape of concepts in mathematical research, focusing on homological cycles, regions with low connectivity in the simplicial structure, and we discuss their role in the understanding of the evolution of scientific research. In addition, we map authors’ contribution to the conceptual space, and explore their role in the formation of homological cycles.

Authors: Daniele Cassese, Vsevolod Salnikov, Renaud Lambiotte
 

We study how the synchronizability of a diffusive network increases (or decreases) when we add some links in its underlying graph. This is of interest in the context of power grids where people want to prevent from having blackouts, or for neural networks where synchronization is responsible of many diseases such as Parkinson. Based on spectral properties for Laplacian matrices, we show some classification results obtained (with Tiago Pereira and Philipp Pade) with respect to the effects of these links.
 

Online discussions are the essence of many social platforms on the Internet. Discussion platforms are receiving increasing interest because of their potential to become deliberative spaces. Although previous studies have proposed approaches to measure online deliberation using the complexity of discussion networks as a proxy, little research has focused on how these networks are affected by changes of platform features.

In this talk, we will focus on how interfaces might influence the network structures of discussions using techniques like interrupted time series analysis and regression discontinuity design. Futhermore, we will review and extend state-of-the-art generative models of discussion threads to explain better the structure and growth of online discussions.
 

Assortative mixing in networks is the tendency for nodes with the same attributes, or metadata, to link to each other. It is a property often found in social networks manifesting as a higher tendency of links occurring between people with the same age, race, or political belief. Quantifying the level of assortativity or disassortativity (the preference of linking to nodes with different attributes) can shed light on the factors involved in the formation of links and contagion processes in complex networks. It is common practice to measure the level of assortativity according to the assortativity coefficient, or modularity in the case of discrete-valued metadata. This global value is the average level of assortativity across the network and may not be a representative statistic when mixing patterns are heterogeneous. For example, a social network spanning the globe may exhibit local differences in mixing patterns as a consequence of differences in cultural norms. Here, we introduce an approach to localise this global measure so that we can describe the assortativity, across multiple scales, at the node level. Consequently we are able to capture and qualitatively evaluate the distribution of mixing patterns in the network. We find that for many real-world networks the distribution of assortativity is skewed, overdispersed and multimodal. Our method provides a clearer lens through which we can more closely examine mixing patterns in networks.

Link to arxiv paper:  https://arxiv.org/abs/1708.01236

Systemic risk arises as a multi-layer network phenomenon. Layers represent direct financial exposures of various types, including interbank liabilities, derivative or foreign exchange exposures. Another network layer of systemic risk emerges through common asset holdings of financial institutions. Strongly overlapping portfolios lead to similar exposures that are caused by price movements of the underlying financial assets. Based on the knowledge of portfolio holdings of financial agents we quantify systemic risk of overlapping portfolios. We present an optimization procedure, where we minimize the systemic risk in a given financial market by optimally rearranging overlapping portfolio networks, under the constraints that the expected returns and risks of the individual portfolios are unchanged. We explicitly demonstrate the power of the method on the overlapping portfolio network of sovereign exposure between major European banks by using data from the European Banking Authority stress test of 2016. We show that systemic-risk-efficient allocations are accessible by the optimization. In the case of sovereign exposure, systemic risk can be reduced by more than a factor of two, without any detrimental effects for the individual banks. These results are confirmed by a simple simulation of fire sales in the government bond market. In particular we show that the contagion probability is reduced dramatically in the optimized network.
 

The story of human progress is often described as a succession of ‘eras’ or ‘ages’ that are characterised by their most dominant technologies (e.g., the bronze age, the industrial revolution or the information age). In modern times, the fast pace of technological progress has accelerated the succession of eras. In addition, the increasing complexity of inventions has made the task of determining when eras begin and end more challenging, as eras are less about the dominance of a single technology and more about the way in which different technologies are combined. We present a data-driven method to determine and uncover technological eras based on networks and patent classification data. We construct temporal networks of technologies that co-appear in patents. By analyzing the evolution of the core-periphery structure and centrality time-series in these networks, we identify periods of time dominated by technological combinations which we identify as distinct ‘eras’. We test the performance of our method using a database of patents in Great Britain spanning a century, and identify five distinct eras.