High-dimensional Graph Research Articles

Locally differentially private graph learning on decentralized social graph

In recent years, decentralized social networks have gained increasing attention, where each client maintains a local view of a social graph. To provide services based on graph learning in such networks, the server commonly needs to collect the local views of the graph structure, which raises privacy issues. In this paper, we focus on learning graph neural networks (GNNs) on decentralized social graphs while satisfying local differential privacy (LDP). Most existing methods collect high-dimensional local views under LDP through Randomized Response, which introduces a large amount of noise and significantly decreases the usability of the collected graph structure for training GNNs. To address this problem, we present Structure Learning-based Locally Private Graph Learning (SL-LPGL). Its main idea is to first collect low-dimensional encoded structural information called cluster degree vectors to reduce the amount of LDP noise, then learn a high-dimensional graph structure from the cluster degree vectors via graph structure learning (GSL) to train GNNs. In SL-LPGL, we propose a Homophily-aware Graph StructurE Initialization (HAGEI) method to provide a low-noise initial graph structure as learning guidance for GSL. We then introduce an Estimated Average Degree Vector Enhanced Graph Structure Learning (EADEGSL) method to further mitigate the negative impact of LDP noise in GSL. We conduct experiments on four real-world graph datasets. The experimental results demonstrate that SL-LPGL outperforms the baselines.

High-dimensional graphs convolution for quantum walks photonic applications

High-dimensional graphs convolution for quantum walks photonic applications

Isoperimetric Inequalities and Supercritical Percolation on High-Dimensional Graphs

It is known that many different types of finite random subgraph models undergo quantitatively similar phase transitions around their percolation thresholds, and the proofs of these results rely on isoperimetric properties of the underlying host graph. Recently, the authors showed that such a phase transition occurs in a large class of regular high-dimensional product graphs, generalising a classic result for the hypercube. In this paper we give new isoperimetric inequalities for such regular high-dimensional product graphs, which generalise the well-known isoperimetric inequality of Harper for the hypercube, and are asymptotically sharp for a wide range of set sizes. We then use these isoperimetric properties to investigate the structure of the giant component L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} in supercritical percolation on these product graphs, that is, when p=1+ϵd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=\\frac{1+\\epsilon }{d}$$\\end{document}, where d is the degree of the product graph and ϵ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon >0$$\\end{document} is a small enough constant. We show that typically L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} has edge-expansion Ω1dlnd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\left( \\frac{1}{d\\ln d}\\right) $$\\end{document}. Furthermore, we show that L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} likely contains a linear-sized subgraph with vertex-expansion Ω1dlnd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\left( \\frac{1}{d\\ln d}\\right) $$\\end{document}. These results are best possible up to the logarithmic factor in d. Using these likely expansion properties, we determine, up to small polylogarithmic factors in d, the likely diameter of L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} as well as the typical mixing time of a lazy random walk on L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document}. Furthermore, we show the likely existence of a cycle of length Ωndlnd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\left( \\frac{n}{d\\ln d}\\right) $$\\end{document}. These results not only generalise, but also improve substantially upon the known bounds in the case of the hypercube, where in particular the likely diameter and typical mixing time of L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} were previously only known to be polynomial in d.

Generating scalable graph states in an atom-nanophotonic interface

Scalable graph states are essential for measurement-based quantum computation and many entanglement-assisted applications in quantum technologies. Generation of these multipartite entangled states requires a controllable and efficient quantum device with delicate design of generation protocol. Here we propose to prepare high-fidelity and scalable graph states in one and two dimensions, which can be tailored in an atom-nanophotonic cavity via state carving technique. We propose a systematic protocol to carve out unwanted state components, which facilitates scalable graph states generations via adiabatic transport of a definite number of atoms in optical tweezers. An analysis of state fidelity is also presented, and the state preparation probability can be optimized via multiqubit state carvings and sequential single-photon probes. Our results showcase the capability of an atom-nanophotonic interface for creating graph states and pave the way toward novel problem-specific applications using scalable high-dimensional graph states with stationary qubits.

Graph Embedding with Similarity Metric Learning

Graph embedding transforms high-dimensional graphs into a lower-dimensional vector space while preserving their structural information and properties. Context-sensitive graph embedding, in particular, performs well in tasks such as link prediction and ranking recommendations. However, existing context-sensitive graph embeddings have limitations: they require additional information, depend on community algorithms to capture multiple contexts, or fail to capture sufficient structural information. In this paper, we propose a novel Graph Embedding with Similarity Metric Learning (GESML). The core of GESML is to learn the optimal graph structure using an attention-based symmetric similarity metric function and establish association relationships between nodes through top-k pooling. Its primary advantage lies in not requiring additional features or multiple contexts, only using the symmetric similarity metric function and pooling operations to encode sufficient topological information for each node. Experimental results on three datasets involving link prediction and node-clustering tasks demonstrate that GESML significantly improves learning for all challenging tasks relative to a state-of-the-art (SOTA) baseline.

Context-sensitive graph representation learning

Graph representation learning, which maps high-dimensional graphs or sparse graphs into a low-dimensional vector space, has shown its superiority in numerous learning tasks. Recently, researchers have identified some advantages of context-sensitive graph representation learning methods in functions such as link predictions and ranking recommendations. However, most existing methods depend on convolutional neural networks or recursive neural networks to obtain additional information outside a node, or require community algorithms to extract multiple contexts of a node, or focus only on the local neighboring nodes without their structural information. In this paper, we propose a novel context-sensitive representation method, Context-Sensitive Graph Representation Learning (CSGRL), which simultaneously combines attention networks and a variant of graph auto-encoder to learn weighty information about various aspects of participating neighboring nodes. The core of CSGRL is to utilize an asymmetric graph encoder to aggregate information about neighboring nodes and local structures to optimize the learning goal. The main benefit of CSGRL is that it does not need additional features and multiple contexts for the node. The message of neighboring nodes and their structures spread through the encoder. Experiments are conducted on three real datasets for both tasks of link prediction and node clustering, and the results demonstrate that CSGRL can significantly improve the effectiveness of all challenging learning tasks compared with 14 state-of-the-art baselines.

Adaptive Reference-Related Graph Embedding for Hyperspectral Anomaly Detection

Graph embedding provides an effective way to reveal intrinsic feature of high-dimensional data on the foundation of preserving topological properties. Under the framework of graph embedding, the hyperspectral image can be represented by a weighted graph, where pixels and similarities among them are treated as vertices and edge weights, respectively. In this article, an adaptive reference-related graph embedding (ARGE) method is proposed to efficaciously obtain the low-dimensional feature and improve computational efficiency. The ARGE method is composed of two primary processes. The key to connecting these two processes is reference vertices set, which is the abstraction of graph topological features. Firstly, the reference vertices are adaptively selected through a three-step adaptive reference set selection (ARSS) algorithm. Secondly, the original high-dimensional graph is embedded as a low-dimensional graph through preserving the reference-related structure. Specifically, the pairwise similarities between vertices and reference vertices are preserved in embedding space. In addition, a new hybrid dissimilarity measure of Rao distance and spectral information divergence (RD-SID) is designed to depict the spectral difference between pixels. To evaluate the effectiveness of the proposed method, the obtained low-dimensional feature is fed into the anomaly detector to detect anomalous pixels. The experimental results on five real and one synthetic hyperspectral datasets demonstrate the superiority of the proposed ARGE method over the compared feature extraction methods.

Self-supervised deep geometric subspace clustering network

Graph mining has been widely studied to analyze real-world graph properties and applied to various applications. In particular, graph subspace clustering performance, defined as partitioning high-dimensional graph data into several clusters by finding minimum weights for the edges, has been consistently improved by exploiting deep learning algorithms with Euclidean features extracted from Euclidean domains (image datasets). Most subspace clustering algorithms tend to extract features from the Euclidean domain to identify graph characteristics and structures, and hence are limited for real-world data applications in non-Euclidean domains. This paper proposes a self-supervised deep geometric subspace clustering algorithm optimized for non-Euclidean high-dimensional graph data by emphasizing spatial features and geometric structures while simultaneously reducing redundant nodes and edges. Quantitative and qualitative experimental results verified the proposed approach is effective for graph clustering compared with previous state-of-the-art algorithms on public datasets.

Novel Projection Schemes for Graph-Based Light Field Coding.

In light field compression, graph-based coding is powerful to exploit signal redundancy along irregular shapes and obtains good energy compaction. However, apart from high time complexity to process high dimensional graphs, their graph construction method is highly sensitive to the accuracy of disparity information between viewpoints. In real-world light field or synthetic light field generated by computer software, the use of disparity information for super-rays projection might suffer from inaccuracy due to vignetting effect and large disparity between views in the two types of light fields, respectively. This paper introduces two novel projection schemes resulting in less error in disparity information, in which one projection scheme can also significantly reduce computation time for both encoder and decoder. Experimental results show projection quality of super-pixels across views can be considerably enhanced using the proposals, along with rate-distortion performance when compared against original projection scheme and HEVC-based or JPEG Pleno-based coding approaches.

Entanglement degree of high-dimensional quantum low-density parity check codes

The important part of quantum information science is quantum error-correction and quantum computation. In thepractical application, some methods of quantum communication, such as the quantum communication based on the atmosphere transmission, require a mix ofmathematics-potentially to solve the problem. The quantum error-correction code is an efficient method to calculate quantum communication. However,since the construction of quantum error-correction codes depends on the degree of quantum entanglement, it is difficult to be widely used in practice.At present, the graph state is a promising solution to reduce the degree of codewords entanglement. But there are still many difficulties in constructingthe high-dimensional graph state. More importantly, the above difficulties can be skillfully solved by the upper and lower bounds of codewords entanglement.Based on the characteristics of the cyclic difference set of stabilizer codes and the combination of U and B of classical low-density parity check (LDPC) codes,a high-dimensional quantum low-density parity check (QLDPC) code is constructed creatively. By calculating the characteristics of the non-$Z$-type generators ofthe new codewords, the minimum number is obtained and the entanglement upper bound of the new code is acquired. And then, the rank of the check matrix of thenew code is calculated as the lower bound of entanglement. When the upper bound and lower bound of entanglement are different, the learning vectorquantization (LVQ) algorithm in machine learning can be used to simultaneously obtain the codewords entanglement and the coding complexity of the code,in which the relationship between them can be derived. In terms of calculating the running speed, compared with the iterative algorithm in the Lagrange multipliermethod, the running speed of LVQ algorithm is increased by $37.68%$. In this paper, a measurement of quantum codewords entanglement is proposed, which supports a helpful role in thedesign of quantum error-correction codes with higher decoding efficiency.

Approximations for the von Neumann and Rényi entropies of graphs with circulant type Laplacians

<abstract><p>In this note, we approximate the von Neumann and Rényi entropies of high-dimensional graphs using the Euler-Maclaurin summation formula. The obtained estimations have a considerable degree of accuracy. The performed experiments suggest some entropy problems concerning graphs whose Laplacians are $ g $-circulant matrices, i.e., circulant matrices with $ g $-periodic diagonals, or quasi-Toeplitz matrices. Quasi means that in a Toeplitz matrix the first two elements in the main diagonal, and the last two, differ from the remaining diagonal entries by a perturbation.</p></abstract>

Graph Signal Compression by Joint Quantization and Sampling

Graph signals arise in various applications, ranging from sensor networks to social media data. The high-dimensional nature of these signals implies that they often need to be compressed in order to be stored and transmitted. The common framework for graph signal compression is based on sampling, resulting in a set of continuous-amplitude samples, which in turn have to be quantized into a finite bit representation. In this work, we study the joint design of graph signal sampling along with quantization, for graph signal compression. We focus on bandlimited graph signals, and show that the compression problem can be represented as a task-based quantization setup, in which the task is to recover the spectrum of the signal. Based on this equivalence, we propose a joint design of the sampling and recovery mechanisms for a fixed quantization mapping, and present an iterative algorithm for dividing the available bit budget among the discretized samples. Furthermore, we show how the proposed approach can be realized using graph filters combining elements corresponding the neighbouring nodes of the graph, thus facilitating distributed implementation at reduced complexity. Our numerical evaluations on both synthetic and real world data shows that the joint sampling and quantization method yields a compact finite bit representation of high-dimensional graph signals, which allows reconstruction of the original signal with accuracy within a small gap of that achievable with infinite resolution quantizers.

Residual finiteness and Abelian subgroup separability of some high dimensional graph manifolds

We generalize $3$-manifolds supporting non-positively curved metric to construct manifolds which have the following properties : (1) They are not locally $\mathrm{CAT}(0)$. (2) Their fundamental groups are residually finite. (3) They have subgroup separability for some abelian subgroups.

Two-dimensional quantum walks of correlated photons

Quantum walks in an elaborately designed graph are a powerful tool for simulating physical and topological phenomena, constructing novel quantum algorithms, and realizing universal quantum computing. Integrated photonics technology has emerged as a versatile platform for implementing a variety of quantum information tasks and as a promising candidate for performing large-scale quantum walks. Both extending physical dimensions and involving more particles will increase the complexity of the evolving systems. Pioneering studies have demonstrated a single particle walking on two-dimensional lattices and multiple walkers interfering on a one-dimensional structure. However, multiple particles evolving in a genuine two-dimensional space in a scalable fashion has remained a vacancy for nearly 10 years. We present a genuine two-dimensional quantum walk with correlated photons on a triangular photonic lattice, which is mapped to a 37 × 37 high-dimensional state space. The genuine two-dimensional quantum walk breaks through the physical restrictions of single-particle evolution, allowing for the encoding of information in large spaces and construction of high-dimensional graphs, which are beneficial for quantum information processing. Between the chip and the two-dimensional fanout interface, site-by-site addressing enables simultaneous detection of over 600 nonclassical interferences and observation of quantum correlations that violate a classical limit by 57 standard deviations. Our implementation provides a paradigm for multi-photon quantum walks in a two-dimensional configuration on a large scale, paving the way for practical quantum simulation and computation beyond the classical regime.

Conjugacy Problem in the Fundamental Groups of High-Dimensional Graph Manifolds

The conjugacy problem for a group G is one of the important algorithmic problems deciding whether or not two elements in G are conjugate to each other. In this paper, we analyze the graph of group structure for the fundamental group of a high-dimensional graph manifold and study the conjugacy problem. We also provide a new proof for the solvable word problem.

Scaling the Growing Neural Gas for Visual Cluster Analysis

• Preventing high-dimensional graph structures in GNG leads to more accurate and meaningful cluster representations. • Using the MapReduce model allows training GNG over distributed infrastructures while preserving or improving accuracy. • Sampling further reduces GNG training times while preserving or improving accuracy. The growing neural gas (GNG) is an unsupervised topology learning algorithm that models a data space through interconnected units that stand on the most populated areas of that space. Its output is a graph that can be visually represented on a two-dimensional plane, disclosing cluster patterns in datasets. It is common, however, for GNG to result in highly connected graphs when trained on high-dimensional data, which in turn leads to highly cluttered 2D representations that may fail to disclose meaningful patterns. Moreover, its sequential learning limits its potential for faster executions on local datasets, and, more importantly, its potential for training on distributed datasets while leveraging from the computational resources of the infrastructures in which they reside. This paper presents two methods that improve GNG for the visualization of cluster patterns in large-scale and high-dimensional datasets. The first one focuses on providing more accurate and meaningful 2D visual representations for cluster patterns of high-dimensional datasets, by avoiding connections that lead to high-dimensional graphs in the modeled topology which may, in turn, result in overplotting and clutter. The second method presented in this paper enables the use of GNG on big and distributed datasets with faster execution times, by modeling and merging separate parts of a dataset using the MapReduce model. Quantitative and qualitative evaluations show that the first method leads to the creation of lower-dimensional graph structures that provide more meaningful (and sometimes more accurate) cluster representations with less overplotting and clutter; and that the second method preserves the accuracy and meaning of the cluster representations while enabling its execution in large-scale and distributed settings.

Algebraic ranks of the fundamental groups of high dimensional graph manifolds

Algebraic ranks of the fundamental groups of high dimensional graph manifolds

How periodic driving stabilizes and destabilizes Anderson localization on random trees

Motivated by the link between Anderson localisation on high-dimensional graphs and many-body localisation, we study the effect of periodic driving on Anderson localisation on random trees. The time dependence is eliminated in favour of an extra dimension, resulting in an extended graph wherein the disorder is correlated along the new dimension. The extra dimension increases the number of paths between any two sites and allows for interference between their amplitudes. We study the localisation problem within the forward scattering approximation (FSA) which we adapt to this extended graph. At low frequency, this favours delocalisation as the availability of a large number of extra paths dominates. By contrast, at high frequency, it stabilises localisation compared to the static system. These lead to a regime of re-entrant localisation in the phase diagram. Analysing the statistics of path amplitudes within the FSA, we provide a detailed theoretical picture of the physical mechanisms governing the phase diagram.

Exploring human-nature interactions in national parks with social media photographs and computer vision.

Understanding the activities and preferences of visitors is crucial for managing protected areas and planning conservation strategies. Conservation culturomics promotes the use of user-generated online content in conservation science. Geotagged social media content is a unique source of in situ information on human presence and activities in nature. Photographs posted on social media platforms are a promising source of information, but analyzing large volumes of photographs manually remains laborious. We examined the application of state-of-the-art computer-vision methods to studying human-nature interactions. We used semantic clustering, scene classification, and object detection to automatically analyze photographs taken in Finnish national parks by domestic and international visitors. Our results showed that human-nature interactions can be extracted from user-generated photographs with computer vision. The different methods complemented each other by revealing broad visual themes related to level of the data set, landscape photogeneity, and human activities. Geotagged photographs revealed distinct regional profiles for national parks (e.g., preferences in landscapes and activities), which are potentially useful in park management. Photographic content differed between domestic and international visitors, which indicates differences in activities and preferences. Information extracted automatically from photographs can help identify preferences among diverse visitor groups, which can be used to create profiles of national parks for conservation marketing and to support conservation strategies that rely on public acceptance. The application of computer-vision methods to automatic content analysis of photographs should be explored further in conservation culturomics, particularly in combination with rich metadata available on social media platforms.

Quantum approximate optimization of non-planar graph problems on a planar superconducting processor

We demonstrate the application of the Google Sycamore superconducting qubit quantum processor to combinatorial optimization problems with the quantum approximate optimization algorithm (QAOA). Like past QAOA experiments, we study performance for problems defined on the (planar) connectivity graph of our hardware; however, we also apply the QAOA to the Sherrington-Kirkpatrick model and MaxCut, both high dimensional graph problems for which the QAOA requires significant compilation. Experimental scans of the QAOA energy landscape show good agreement with theory across even the largest instances studied (23 qubits) and we are able to perform variational optimization successfully. For problems defined on our hardware graph we obtain an approximation ratio that is independent of problem size and observe, for the first time, that performance increases with circuit depth. For problems requiring compilation, performance decreases with problem size but still provides an advantage over random guessing for circuits involving several thousand gates. This behavior highlights the challenge of using near-term quantum computers to optimize problems on graphs differing from hardware connectivity. As these graphs are more representative of real world instances, our results advocate for more emphasis on such problems in the developing tradition of using the QAOA as a holistic, device-level benchmark of quantum processors.