4,012 research outputs found
A Unifying Theory for Graph Transformation
The field of graph transformation studies the rule-based transformation of graphs. An important branch is the algebraic graph transformation tradition, in which approaches are defined and studied using the language of category theory. Most algebraic graph transformation approaches (such as DPO, SPO, SqPO, and AGREE) are opinionated about the local contexts that are allowed around matches for rules, and about how replacement in context should work exactly. The approaches also differ considerably in their underlying formal theories and their general expressiveness (e.g., not all frameworks allow duplication). This dissertation proposes an expressive algebraic graph transformation approach, called PBPO+, which is an adaptation of PBPO by Corradini et al. The central contribution is a proof that PBPO+ subsumes (under mild restrictions) DPO, SqPO, AGREE, and PBPO in the important categorical setting of quasitoposes. This result allows for a more unified study of graph transformation metatheory, methods, and tools. A concrete example of this is found in the second major contribution of this dissertation: a graph transformation termination method for PBPO+, based on decreasing interpretations, and defined for general categories. By applying the proposed encodings into PBPO+, this method can also be applied for DPO, SqPO, AGREE, and PBPO
Covering and Separation for Permutations and Graphs
This is a thesis of two parts, focusing on covering and separation topics of extremal combinatorics and graph theory, two major themes in this area. They entail the existence and properties of collections of combinatorial objects which together either represent all objects (covering) or can be used to distinguish all objects from each other (separation). We will consider a range of problems which come under these areas. The first part will focus on shattering k-sets with permutations. A family of permutations is said to shatter a given k-set if the permutations cover all possible orderings of the k elements. In particular, we investigate the size of permutation families which cover t orders for every possible k-set as well as study the problem of determining the largest number of k-sets that can be shattered by a family with given size. We provide a construction for a small permutation family which shatters every k-set. We also consider constructions of large families which do not shatter any triple. The second part will be concerned with the problem of separating path systems. A separating path system for a graph is a family of paths where, for any two edges, there is a path containing one edge but not the other. The aim is to find the size of the smallest such family. We will study the size of the smallest separating path system for a range of graphs, including complete graphs, complete bipartite graphs, and lattice-type graphs. A key technique we introduce is the use of generator paths - constructed to utilise the symmetric nature of Kn. We continue this symmetric approach for bipartite graphs and study the limitations of the method. We consider lattice-type graphs as an example of the most efficient possible separating systems for any graph
Probabilistic Programming Interfaces for Random Graphs::Markov Categories, Graphons, and Nominal Sets
We study semantic models of probabilistic programming languages over graphs, and establish a connection to graphons from graph theory and combinatorics. We show that every well-behaved equational theory for our graph probabilistic programming language corresponds to a graphon, and conversely, every graphon arises in this way.We provide three constructions for showing that every graphon arises from an equational theory. The first is an abstract construction, using Markov categories and monoidal indeterminates. The second and third are more concrete. The second is in terms of traditional measure theoretic probability, which covers 'black-and-white' graphons. The third is in terms of probability monads on the nominal sets of Gabbay and Pitts. Specifically, we use a variation of nominal sets induced by the theory of graphs, which covers Erdős-Rényi graphons. In this way, we build new models of graph probabilistic programming from graphons
On positive opetopes, positive opetopic cardinals and positive opetopic set
We introduce the notion of a positive opetope and positive opetopic cardinals
as certain finite combinatorial structures. The positive opetopic cardinals to
positive-to-one polygraphs are like simple graphs to free omega-categories over
omega-graphs, c.f. [MZ]. In particular, they allow us to give an explicit
combinatorial description of positive-to-one polygraphs. Using this description
we show, among other things, that positive-to-one polygraphs form a presheaf
category with the exponent category being the category of positive opetopes. We
also show that the category of omega-categories is monadic over the category of
positive-to-one polygraphs with the `free functor' being an inclusion.Comment: 88 page
A BIM - GIS Integrated Information Model Using Semantic Web and RDF Graph Databases
In recent years, 3D virtual indoor and outdoor urban modelling has become an essential geospatial information framework for civil and engineering applications such as emergency response, evacuation planning, and facility management. Building multi-sourced and multi-scale 3D urban models are in high demand among architects, engineers, and construction professionals to achieve these tasks and provide relevant information to decision support systems. Spatial modelling technologies such as Building Information Modelling (BIM) and Geographical Information Systems (GIS) are frequently used to meet such high demands. However, sharing data and information between these two domains is still challenging. At the same time, the semantic or syntactic strategies for inter-communication between BIM and GIS do not fully provide rich semantic and geometric information exchange of BIM into GIS or vice-versa. This research study proposes a novel approach for integrating BIM and GIS using semantic web technologies and Resources Description Framework (RDF) graph databases. The suggested solution's originality and novelty come from combining the advantages of integrating BIM and GIS models into a semantically unified data model using a semantic framework and ontology engineering approaches. The new model will be named Integrated Geospatial Information Model (IGIM). It is constructed through three stages. The first stage requires BIMRDF and GISRDF graphs generation from BIM and GIS datasets. Then graph integration from BIM and GIS semantic models creates IGIMRDF. Lastly, the information from IGIMRDF unified graph is filtered using a graph query language and graph data analytics tools. The linkage between BIMRDF and GISRDF is completed through SPARQL endpoints defined by queries using elements and entity classes with similar or complementary information from properties, relationships, and geometries from an ontology-matching process during model construction. The resulting model (or sub-model) can be managed in a graph database system and used in the backend as a data-tier serving web services feeding a front-tier domain-oriented application. A case study was designed, developed, and tested using the semantic integrated information model for validating the newly proposed solution, architecture, and performance
Factor Graph Neural Networks
In recent years, we have witnessed a surge of Graph Neural Networks (GNNs),
most of which can learn powerful representations in an end-to-end fashion with
great success in many real-world applications. They have resemblance to
Probabilistic Graphical Models (PGMs), but break free from some limitations of
PGMs. By aiming to provide expressive methods for representation learning
instead of computing marginals or most likely configurations, GNNs provide
flexibility in the choice of information flowing rules while maintaining good
performance. Despite their success and inspirations, they lack efficient ways
to represent and learn higher-order relations among variables/nodes. More
expressive higher-order GNNs which operate on k-tuples of nodes need increased
computational resources in order to process higher-order tensors. We propose
Factor Graph Neural Networks (FGNNs) to effectively capture higher-order
relations for inference and learning. To do so, we first derive an efficient
approximate Sum-Product loopy belief propagation inference algorithm for
discrete higher-order PGMs. We then neuralize the novel message passing scheme
into a Factor Graph Neural Network (FGNN) module by allowing richer
representations of the message update rules; this facilitates both efficient
inference and powerful end-to-end learning. We further show that with a
suitable choice of message aggregation operators, our FGNN is also able to
represent Max-Product belief propagation, providing a single family of
architecture that can represent both Max and Sum-Product loopy belief
propagation. Our extensive experimental evaluation on synthetic as well as real
datasets demonstrates the potential of the proposed model.Comment: Accepted by JML
Mixing time for uniform sampling of bipartite graphs with fixed degrees using the trade algorithm
Uniform sampling of bipartite graphs and hypergraphs with given degree
sequences is necessary for building null models to statistically evaluate their
topology. Because these graphs can be represented as binary matrices, the
problem is equivalent to uniformly sampling binary matrices with
fixed row and column sums. The trade algorithm, which includes both the
curveball and fastball implementations, is the state-of-the-art for performing
such sampling. Its mixing time is currently unknown, although is currently
used as a heuristic. In this paper we propose a new distribution-based approach
that not only provides an estimation of the mixing time, but also actually
returns a sample of matrices that are guaranteed (within a user-chosen error
tolerance) to be uniformly randomly sampled. In numerical experiments on
matrices that vary by size, fill, and row and column sum distributions, we find
that the upper bound on mixing time is at least , and that it increases as
a function of both and the fraction of cells containing a 1
Extracting Mergers and Projections of Partitions
We study the problem of extracting randomness from somewhere-random sources,
and related combinatorial phenomena: partition analogues of Shearer's lemma on
projections.
A somewhere-random source is a tuple of (possibly
correlated) -valued random variables where for some unknown , is guaranteed to be uniformly distributed. An
is a seeded device that takes a somewhere-random source as input and
outputs nearly uniform random bits. We study the seed-length needed for
extracting mergers with constant and constant error. We show:
Just like in the case of standard extractors, seedless extracting
mergers with even just one output bit do not exist.
Unlike the case of standard extractors, it possible to have
extracting mergers that output a constant number of bits using only constant
seed. Furthermore, a random choice of merger does not work for this purpose!
Nevertheless, just like in the case of standard extractors, an
extracting merger which gets most of the entropy out (namely, having
output bits) must have seed. This is the main
technical result of our work, and is proved by a second-moment strengthening of
the graph-theoretic approach of Radhakrishnan and Ta-Shma to extractors.
In contrast, seed-length/output-length tradeoffs for condensing mergers
(where the output is only required to have high min-entropy), can be fully
explained by using standard condensers.
Inspired by such considerations, we also formulate a new and basic class of
problems in combinatorics: partition analogues of Shearer's lemma. We show
basic results in this direction; in particular, we prove that in any partition
of the -dimensional cube into two parts, one of the parts has an
axis parallel -dimensional projection of area at least .Comment: Full version of the paper accepted to the International Conference on
Randomization and Computation (RANDOM) 2023. 28 pages, 2 figure
Mining Butterflies in Streaming Graphs
This thesis introduces two main-memory systems sGrapp and sGradd for performing the fundamental analytic tasks of biclique counting and concept drift detection over a streaming graph. A data-driven heuristic is used to architect the systems. To this end, initially, the growth patterns of bipartite streaming graphs are mined and the emergence principles of streaming motifs are discovered. Next, the discovered principles are (a) explained by a graph generator called sGrow; and (b) utilized to establish the requirements for efficient, effective, explainable, and interpretable management and processing of streams. sGrow is used to benchmark stream analytics, particularly in the case of concept drift detection.
sGrow displays robust realization of streaming growth patterns independent of initial conditions, scale and temporal characteristics, and model configurations. Extensive evaluations confirm the simultaneous effectiveness and efficiency of sGrapp and sGradd. sGrapp achieves mean absolute percentage error up to 0.05/0.14 for the cumulative butterfly count in streaming graphs with uniform/non-uniform temporal distribution and a processing throughput of 1.5 million data records per second. The throughput and estimation error of sGrapp are 160x higher and 0.02x lower than baselines. sGradd demonstrates an improving performance over time, achieves zero false detection rates when there is not any drift and when drift is already detected, and detects sequential drifts in zero to a few seconds after their occurrence regardless of drift intervals
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