1,284 research outputs found

    A Unifying Theory for Graph Transformation

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    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

    Robustness, Heterogeneity and Structure Capturing for Graph Representation Learning and its Application

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    Graph neural networks (GNNs) are potent methods for graph representation learn- ing (GRL), which extract knowledge from complicated (graph) structured data in various real-world scenarios. However, GRL still faces many challenges. Firstly GNN-based node classification may deteriorate substantially by overlooking the pos- sibility of noisy data in graph structures, as models wrongly process the relation among nodes in the input graphs as the ground truth. Secondly, nodes and edges have different types in the real-world and it is essential to capture this heterogeneity in graph representation learning. Next, relations among nodes are not restricted to pairwise relations and it is necessary to capture the complex relations accordingly. Finally, the absence of structural encodings, such as positional information, deterio- rates the performance of GNNs. This thesis proposes novel methods to address the aforementioned problems: 1. Bayesian Graph Attention Network (BGAT): Developed for situations with scarce data, this method addresses the influence of spurious edges. Incor- porating Bayesian principles into the graph attention mechanism enhances robustness, leading to competitive performance against benchmarks (Chapter 3). 2. Neighbour Contrastive Heterogeneous Graph Attention Network (NC-HGAT): By enhancing a cutting-edge self-supervised heterogeneous graph neural net- work model (HGAT) with neighbour contrastive learning, this method ad- dresses heterogeneity and uncertainty simultaneously. Extra attention to edge relations in heterogeneous graphs also aids in subsequent classification tasks (Chapter 4). 3. A novel ensemble learning framework is introduced for predicting stock price movements. It adeptly captures both group-level and pairwise relations, lead- ing to notable advancements over the existing state-of-the-art. The integration of hypergraph and graph models, coupled with the utilisation of auxiliary data via GNNs before recurrent neural network (RNN), provides a deeper under- standing of long-term dependencies between similar entities in multivariate time series analysis (Chapter 5). 4. A novel framework for graph structure learning is introduced, segmenting graphs into distinct patches. By harnessing the capabilities of transformers and integrating other position encoding techniques, this approach robustly capture intricate structural information within a graph. This results in a more comprehensive understanding of its underlying patterns (Chapter 6)

    Classical and quantum algorithms for scaling problems

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    This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size

    Algorithms and complexity for approximately counting hypergraph colourings and related problems

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    The past decade has witnessed advancements in designing efficient algorithms for approximating the number of solutions to constraint satisfaction problems (CSPs), especially in the local lemma regime. However, the phase transition for the computational tractability is not known. This thesis is dedicated to the prototypical problem of this kind of CSPs, the hypergraph colouring. Parameterised by the number of colours q, the arity of each hyperedge k, and the vertex maximum degree Δ, this problem falls into the regime of Lovász local lemma when Δ ≲ qᵏ. In prior, however, fast approximate counting algorithms exist when Δ ≲ qᵏ/³, and there is no known inapproximability result. In pursuit of this, our contribution is two-folded, stated as follows. • When q, k ≥ 4 are evens and Δ ≥ 5·qᵏ/², approximating the number of hypergraph colourings is NP-hard. • When the input hypergraph is linear and Δ ≲ qᵏ/², a fast approximate counting algorithm does exist

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Mining Butterflies in Streaming Graphs

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    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

    A proof of the Ryser-Brualdi-Stein conjecture for large even nn

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    A Latin square of order nn is an nn by nn grid filled using nn symbols so that each symbol appears exactly once in each row and column. A transversal in a Latin square is a collection of cells which share no symbol, row or column. The Ryser-Brualdi-Stein conjecture, with origins from 1967, states that every Latin square of order nn contains a transversal with n1n-1 cells, and a transversal with nn cells if nn is odd. Keevash, Pokrovskiy, Sudakov and Yepremyan recently improved the long-standing best known bounds towards this conjecture by showing that every Latin square of order nn has a transversal with nO(logn/loglogn)n-O(\log n/\log\log n) cells. Here, we show, for sufficiently large nn, that every Latin square of order nn has a transversal with n1n-1 cells. We also apply our methods to show that, for sufficiently large nn, every Steiner triple system of order nn has a matching containing at least (n4)/3(n-4)/3 edges. This improves a recent result of Keevash, Pokrovskiy, Sudakov and Yepremyan, who found such matchings with n/3O(logn/loglogn)n/3-O(\log n/\log\log n) edges, and proves a conjecture of Brouwer from 1981 for large nn.Comment: 71 pages, 13 figure
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