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Graph theory provides fundamental concepts for many fields of science like statistical physics, network analysis and theoretical computer science. Here we give a pedagogical introduction to graph theory, divided into three sections. In the…

The processes of constructing some graphs from others using binary operations of union with intersection (gluing) are studied. For graph classes closed with respect to gluing operations the elemental and operational bases are introduced.…

Gaussian Processes (GPs) are widely used tools in statistics, machine learning, robotics, computer vision, and scientific computation. However, despite their popularity, they can be difficult to apply; all but the simplest classification or…

Graphs are widely used for describing systems made up of many interacting components and for understanding the structure of their interactions. Various statistical models exist, which describe this structure as the result of a combination…

Prompted models have demonstrated impressive few-shot learning abilities. Repeated interactions at test-time with a single model, or the composition of multiple models together, further expands capabilities. These compositions are…

Much work has been done to give semantics to probabilistic programming languages. In recent years, most of the semantics used to reason about probabilistic programs fall in two categories: semantics based on Markov kernels and semantics…

We introduce a class of random graphs that we argue meets many of the desiderata one would demand of a model to serve as the foundation for a statistical analysis of real-world networks. The class of random graphs is defined by a…

This report outlines an approach to learning generative models from data. We express models as probabilistic programs, which allows us to capture abstract patterns within the examples. By choosing our language for programs to be an…

Generative methods for graphs need to be sufficiently flexible to model complex dependencies between sets of nodes. At the same time, the generated graphs need to satisfy domain-dependent feasibility conditions, that is, they should not…

A natural probabilistic model for motif discovery has been used to experimentally test the quality of motif discovery programs. In this model, there are $k$ background sequences, and each character in a background sequence is a random…

We consider two classes of computations which admit taking linear combinations of execution runs: probabilistic sampling and generalized animation. We argue that the task of program learning should be more tractable for these architectures…

Graph classification aims to categorise graphs based on their structure and node attributes. In this work, we propose to tackle this task using tools from graph signal processing by deriving spectral features, which we then use to design…

A number of novel programming languages and libraries have been proposed that offer simpler-to-use models of concurrency than threads. It is challenging, however, to devise execution models that successfully realise their abstractions…

Graph Generating Dependencies (GGDs) informally express constraints between two (possibly different) graph patterns which enforce relationships on both graph's data (via property value constraints) and its structure (via topological…

Large language models have evolved to process multiple modalities beyond text, such as images and audio, which motivates us to explore how to effectively leverage them for graph reasoning tasks. The key question, therefore, is how to…

Evolution algebras are a new type of non-associative algebras which are inspired from biological phenomena. A special class of such algebras, called Markov evolution algebras, is strongly related to the theory of discrete time Markov…

Small subgraph counts can be used as summary statistics for large random graphs. We use the Stein-Chen method to derive Poisson approximations for the distribution of the number of subgraphs in the stochastic block model which are…

In this paper we will provide an introductory understanding of random graph models, and matchings in the case of Erdos-Renyi random graphs. We will provide a synthesis of background theory to this end. We will further examine pertinent…

Categorical probability has recently seen significant advances through the formalism of Markov categories, within which several classical theorems have been proven in entirely abstract categorical terms. Closely related to Markov categories…

We introduce probabilistic embeddings using Laplacian priors (PELP). The proposed model enables incorporating graph side-information into static word embeddings. We theoretically show that the model unifies several previously proposed…