Differential Calculus on Graphon Space

Recently, the theory of dense graph limits has received attention from multiple disciplines including graph theory, computer science, statistical physics, probability, statistics, and group theory. In this paper we initiate the study of the general structure of differentiable graphon parameters $F$. We derive consistency conditions among the higher G\^ateaux derivatives of $F$ when restricted to the subspace of edge weighted graphs $\mathcal{W}_{\bf p}$. Surprisingly, these constraints are rigid enough to imply that the multilinear functionals $\Lambda: \mathcal{W}_{\bf p}^n \to \mathbb{R}$ satisfying the constraints are determined by a finite set of constants indexed by isomorphism classes of multigraphs with $n$ edges and no isolated vertices. Using this structure theory, we explain the central role that homomorphism densities play in the analysis of graphons, by way of a new combinatorial interpretation of their derivatives. In particular, homomorphism densities serve as the monomials in a polynomial algebra that can be used to approximate differential graphon parameters as Taylor polynomials. These ideas are summarized by our main theorem, which asserts that homomorphism densities $t(H,-)$ where $H$ has at most $N$ edges form a basis for the space of smooth graphon parameters whose $(N+1)$st derivatives vanish. As a consequence of this theory, we also extend and derive new proofs of linear independence of multigraph homomorphism densities, and characterize homomorphism densities. In addition, we develop a theory of series expansions, including Taylor's theorem for graph parameters and a uniqueness principle for series. We use this theory to analyze questions raised by Lov\'asz, including studying infinite quantum algebras and the connection between right- and left-homomorphism densities.Comment: Final version (36 pages), accepted for publication in Journal of Combinatorial Theory, Series

Are there any good digraph width measures?

Many width measures for directed graphs have been proposed in the last few years in pursuit of generalizing (the notion of) treewidth to directed graphs. However, none of these measures possesses, at the same time, the major properties of treewidth, namely, 1. being algorithmically useful , that is, admitting polynomial-time algorithms for a large class of problems on digraphs of bounded width (e.g. the problems definable in MSO1MSO1); 2. having nice structural properties such as being (at least nearly) monotone under taking subdigraphs and some form of arc contractions (property closely related to characterizability by particular cops-and-robber games). We investigate the question whether the search for directed treewidth counterparts has been unsuccessful by accident, or whether it has been doomed to fail from the beginning. Our main result states that any reasonable width measure for directed graphs which satisfies the two properties above must necessarily be similar to treewidth of the underlying undirected graph

Full abstraction for fair testing in CCS (expanded version)

In previous work with Pous, we defined a semantics for CCS which may both be viewed as an innocent form of presheaf semantics and as a concurrent form of game semantics. We define in this setting an analogue of fair testing equivalence, which we prove fully abstract w.r.t. standard fair testing equivalence. The proof relies on a new algebraic notion called playground, which represents the `rule of the game'. From any playground, we derive two languages equipped with labelled transition systems, as well as a strong, functional bisimulation between them.Comment: 80 page

Semigroup-valued metric spaces

The structural Ramsey theory is a field on the boundary of combinatorics and model theory with deep connections to topological dynamics. Most of the known Ramsey classes in finite binary symmetric relational language can be shown to be Ramsey by utilizing a variant of the shortest path completion (e.g. Sauer's $S$-metric spaces, Conant's generalised metric spaces, Braunfeld's $\Lambda$-ultrametric spaces or Cherlin's metrically homogeneous graphs). In this thesis we explore the limits of the shortest path completion. We offer a unifying framework --- semigroup-valued metric spaces --- for all the aforementioned Ramsey classes and study their Ramsey expansions and EPPA (the extension property for partial automorphisms). Our results can be seen as evidence for the importance of studying the completion problem for amalgamation classes and have some further applications (such as the stationary independence relation). As a corollary of our general theorems, we reprove results of Hubi\v{c}ka and Ne\v{s}et\v{r}il on Sauer's $S$-metric spaces, results of Hub\v{c}ka, Ne\v{s}et\v{r}il and the author on Conant's generalised metric spaces, Braunfeld's results on $\Lambda$-ultrametric spaces and the results of Aranda et al. on Cherlin's primitive 3-constrained metrically homogeneous graphs. We also solve several open problems such as EPPA for $\Lambda$-ultrametric spaces, $S$-metric spaces or Conant's generalised metric spaces. Our framework seems to be universal enough that we conjecture that every primitive strong amalgamation class of complete edge-labelled graphs with finitely many labels is in fact a class of semigroup-valued metric spaces.Comment: Master thesis, defended in June 201

Network modularity and local environment similarity as descriptors of protein structure

As the number of solved protein structures increases, the opportunities for meta-analysis of this dataset increase too. Here we explore two approaches for analysing protein structure, both starting from the three-dimensional co-ordinates of each atom within the structure, which are then abstracted into a more useful form. The first method transforms the protein into a network in which its amino acids are the nodes, and where the edges are generated using a simple proximity test. By applying the Infomap community detection algorithm, we can fragment the protein into highly intra-connected subregions - these subregions are compact and globular, and can be compared with known structural and functional subunits of the protein (also known as domains). By performing this fragmentation process systematically across a large set of proteins, and checking for structurally conserved fragments, we can search for novel candidate domains. This method for automatically decomposing a protein into compact substructures may also be useful in coarse-graining molecular dynamics, analysing the protein’s topology, in de novo protein design, or in fitting electron density maps derived from single particle electron microscopy. The second method calculates a descriptor for each atom of the protein based on its local environment, known as a Smooth Overlap of Atomic Positions (SOAP) descriptor. Using these descriptors we can perform overall comparisons of the subregions identified above. In addition, by comparing the descriptors of a set of proteins known to share common structural or functional features (such as binding of a particular ligand), we can automatically identify the most highly conserved atoms of the set. These atoms may line ligand binding pockets or correspond to allosteric sites, which could inform drug design

Graph layout using subgraph isomorphisms

Today, graphs are used for many things. In engineering, graphs are used to design circuits in very large scale integration. In computer science, graphs are used in the representation of the structure of software. They show information such as the flow of data through the program (known as the data flow graph [1]) or the information about the calling sequence of programs (known as the call graph [145]). These graphs consist of many classes of graphs and may occupy a large area and involve a large number of vertices and edges. The manual layout of graphs is a tedious and error prone task. Algorithms for graph layout exist but tend to only produce a 'good' layout when they are applied to specific classes of small graphs. In this thesis, research is presented into a new automatic graph layout technique. Within many graphs, common structures exist. These are structures that produce 'good' layouts that are instantly recognisable and, when combined, can be used to improve the layout of the graphs. In this thesis common structures are given that are present in call graphs. A method of using subgraph isomorphism to detect these common structures is also presented. The method is known as the ANHOF method. This method is implemented in the ANHOF system, and is used to improve the layout of call graphs. The resulting layouts are an improvement over layouts from other algorithms because these common structures are evident and the number of edge crossings, clusters and aspect ratio are improved

Local weak convergence and propagation of ergodicity for sparse networks of interacting processes

We study the limiting behavior of interacting particle systems indexed by large sparse graphs, which evolve either according to a discrete time Markov chain or a diffusion, in which particles interact directly only with their nearest neighbors in the graph. To encode sparsity we work in the framework of local weak convergence of marked (random) graphs. We show that the joint law of the particle system varies continuously with respect to local weak convergence of the underlying graph. In addition, we show that the global empirical measure converges to a non-random limit, whereas for a large class of graph sequences including sparse Erd\"{o}s-R\'{e}nyi graphs and configuration models, the empirical measure of the connected component of a uniformly random vertex converges to a random limit. Finally, on a lattice (or more generally an amenable Cayley graph), we show that if the initial configuration of the particle system is a stationary ergodic random field, then so is the configuration of particle trajectories up to any fixed time, a phenomenon we refer to as "propagation of ergodicity". Along the way, we develop some general results on local weak convergence of Gibbs measures in the uniqueness regime which appear to be new.Comment: 46 pages, 1 figure. This version of the paper significantly extends the convergence results for diffusions in v1, and includes new results on propagation of ergodicity and discrete-time models. The complementary results in v1 on autonomous characterization of marginal dynamics of diffusions on trees and generalizations thereof are now presented in a separate paper arXiv:2009.1166