Orthographies in Early Modern Europe

This volume provides, for the first time, a pan-European view of the development of written languages at a key time in their history: that of the 16th century. The major cultural and intellectual upheavals that affected Europe at the time - Humanism, the Reformation and the emergence of modern nation-states - were not isolated phenomena, and the evolution of the orthographical systems of European languages shows a large number of convergences, due to the mobility of scholars, ideas and technological innovations throughout the period

Supermodularity in Unweighted Graph Optimization I: Branchings and Matchings

The main result of this paper is motivated by the following two apparently unrelated graph optimization problems: (A) As an extension of Edmonds' disjoint branchings theorem, characterize digraphs comprising k disjoint branchings B-i each having a specified number mu(i) of arcs. (B) As an extension of Ryser's maximum term rank formula, determine the largest possible matching number of simple bipartite graphs complying with degree-constraints. The solutions to these problems and to their generalizations will be obtained from a new min-max theorem on covering a supermodular function by a simple degree-constrained bipartite graph. A specific feature of the result is that its minimum cost extension is already NP-hard. Therefore classic polyhedral tools themselves definitely cannot be sufficient for solving the problem, even though they make some good service in our approach

Orthographies in Early Modern Europe

This volume provides, for the first time, a pan-European view of the development of written languages at a key time in their history: that of the 16th century. The major cultural and intellectual upheavals that affected Europe at the time - Humanism, the Reformation and the emergence of modern nation-states - were not isolated phenomena, and the evolution of the orthographical systems of European languages shows a large number of convergences, due to the mobility of scholars, ideas and technological innovations throughout the period

CHALLENGES IN RANDOM GRAPH MODELS WITH DEGREE HETEROGENEITY: EXISTENCE, ENUMERATION AND ASYMPTOTICS OF THE SPECTRAL RADIUS

In order to understand how the network structure impacts the underlying dynamics, we seek an assortment of methods for efficiently constructing graphs of interest that resemble their empirically observed counterparts. Since many real world networks obey degree heterogeneity, where different nodes have varying numbers of connections, we consider some challenges in constructing random graphs that emulate the property. Initially we focus on the Uniform Model, where we would like to uniformly sample from all graphs that realize a given bi-degree sequence. We provide easy to implement, sufficient criteria to guarantee that a bi-degree sequence corresponds to a graph. Consequently, we construct novel results regarding asymptotics of the number of graphs that realize a given degree sequence, where knowledge of the aforementioned enumeration result will assist us in constructing realizations from the Uni- form Model. Finally, we consider another random directed graph model that exhibits degree heterogeneity, the Chung-Lu random graph model and prove concentration results regarding the dominating eigenvalue of the corresponding adjacency matrix. We extend our analysis to a more generalized model that allows for intricate community structure and demonstrate the impact of the community structure in networks with Kuramoto and SIS epidemiological dynamics

Sampling uniform hypergraphs with given degrees

Graphs are combinatorial objects commonly used to model relationships between pairs of entities. Hypergraphs are a generalization of graphs in which edges connect an arbitrary number of vertices. We consider hypergraphs in which each edge has size k, each vertex has a degree specified by a degree sequence d, and all edges are unique. These are known as simple k-uniform hypergraphs with degree sequence d. We focus on algorithms for sampling these hypergraphs, particularly when the degree sequence is approximately regular and sufficiently sparse. The goal is an algorithm which produces approximately uniform output with expected running time that is polynomial in the number of vertices. We first discuss an algorithm for this problem which used a rejection sampling approach and a black-box bipartite graph sampler. This algorithm was presented in a paper by myself and co-authors: my specific contributions to the publication are described. As a new contribution (not contained in the paper), the rejection sampling approach is extended to give an algorithm for sampling linear hypergraphs, which are hypergraphs in which no two distinct edges share more than one common vertex. We also define and analyse an algorithm for sampling simple k-uniform hypergraphs with degree sequence d. Our algorithm uses a black-box sampler A for producing (possibly non-simple) hypergraphs and a ‘switchings’ process to remove any repeated edges from the hypergraph. This analysis additionally produces explicit tail bounds for the number and multiplicity of repeated edges in uniformly distributed random hypergraphs, under certain conditions for d and k. We show that our algorithm is asymptotically approximately uniform and has an expected running time that is polynomial in the number of vertices for a large range of degree sequences d, provided d is near-regular. This extends the range of degree sequences for which efficient sampling schemes are known

The Nature of Writing – A Theory of Grapholinguistics [book cover]

Cover illustration: Purgatory: Canto VII – The Rule of the Mountain from A Typographic Dante (2008) by Barrie Tullett (also displayed in Barrie Tullett, Typewriter Art: A Modern Anthology, London: Laurence King Publishing, 2014, p. 167). With kind permission by Barrie Tullett. The text is taken from Dante. The Divine Comedy, translated by Dorothy L. Sayers, Harmondsworth­Middlesex: The Penguin Classics, 1949. On the lower part of the illustration, one can read the concluding verses of the Canto: But now the poet was going on before; “Forward!” said he; “look how the sun doth stand Meridian­high, while on the Western shore Night sets her foot upon Morocco’s strand.