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Showing papers on "Spectral graph theory published in 2019"


Proceedings ArticleDOI
25 Jul 2019
TL;DR: Wang et al. as discussed by the authors introduced a pooling operator based on graph Fourier transform, which can utilize the node features and local structures during the pooling process, and designed pooling layers based on the pool operator, which are further combined with traditional GCN convolutional layers to form a graph neural network framework for graph classification.
Abstract: Graph neural networks, which generalize deep neural network models to graph structured data, have attracted increasing attention in recent years. They usually learn node representations by transforming, propagating and aggregating node features and have been proven to improve the performance of many graph related tasks such as node classification and link prediction. To apply graph neural networks for the graph classification task, approaches to generate thegraph representation from node representations are demanded. A common way is to globally combine the node representations. However, rich structural information is overlooked. Thus a hierarchical pooling procedure is desired to preserve the graph structure during the graph representation learning. There are some recent works on hierarchically learning graph representation analogous to the pooling step in conventional convolutional neural (CNN) networks. However, the local structural information is still largely neglected during the pooling process. In this paper, we introduce a pooling operator $\pooling$ based on graph Fourier transform, which can utilize the node features and local structures during the pooling process. We then design pooling layers based on the pooling operator, which are further combined with traditional GCN convolutional layers to form a graph neural network framework $\m$ for graph classification. Theoretical analysis is provided to understand $\pooling$ from both local and global perspectives. Experimental results of the graph classification task on $6$ commonly used benchmarks demonstrate the effectiveness of the proposed framework.

257 citations


Journal ArticleDOI
TL;DR: This study indicates that machine learning methods are powerful tools for molecular docking and virtual screening and indicates that spectral geometry or spectral graph theory has the ability to infer geometric properties.
Abstract: Although algebraic graph theory-based models have been widely applied in physical modeling and molecular studies, they are typically incompetent in the analysis and prediction of biomolecular properties, confirming the common belief that "one cannot hear the shape of a drum". A new development in the century-old question about the spectrum-geometry relationship is provided. Novel algebraic graph learning score (AGL-Score) models are proposed to encode high-dimensional physical and biological information into intrinsically low-dimensional representations. The proposed AGL-Score models employ multiscale weighted colored subgraphs to describe crucial molecular and biomolecular interactions in terms of graph invariants derived from graph Laplacian, its pseudo-inverse, and adjacency matrices. Additionally, AGL-Score models are integrated with an advanced machine learning algorithm to predict biomolecular macroscopic properties from the low-dimensional graph representation of biomolecular structures. The proposed AGL-Score models are extensively validated for their scoring power, ranking power, docking power, and screening power via a number of benchmark datasets, namely CASF-2007, CASF-2013, and CASF-2016. Numerical results indicate that the proposed AGL-Score models are able to outperform other state-of-the-art scoring functions in protein-ligand binding scoring, ranking, docking, and screening. This study indicates that machine learning methods are powerful tools for molecular docking and virtual screening. It also indicates that spectral geometry or spectral graph theory has the ability to infer geometric properties.

144 citations


Journal ArticleDOI
TL;DR: This paper leverages recent advances in spectral graph matching to transfer surface data across aligned spectral domains and exploits spectral filters over intrinsic representations of surface neighborhoods in a novel approach to brain parcellation.

47 citations


Journal ArticleDOI
01 Dec 2019
TL;DR: This work gives an exploratory introduction to cellular sheaves and Laplacians, and includes results on eigenvalue interlacing, complex sparsification, effective resistance, synchronization, and sheaf approximation.
Abstract: This paper outlines a program in what one might call spectral sheaf theory—an extension of spectral graph theory to cellular sheaves. By lifting the combinatorial graph Laplacian to the Hodge Laplacian on a cellular sheaf of vector spaces over a regular cell complex, one can relate spectral data to the sheaf cohomology and cell structure in a manner reminiscent of spectral graph theory. This work gives an exploratory introduction, and includes discussion of eigenvalue interlacing, sparsification, effective resistance, synchronization, and sheaf approximation. These results and subsequent applications are prefaced by an introduction to cellular sheaves and Laplacians.

40 citations


Posted Content
TL;DR: This paper introduces a unified graph learning framework lying at the integration of Gaussian graphical models and spectral graph theory, and develops an optimization framework that leverages graph learning with specific structures via spectral constraints on graph matrices.
Abstract: Graph learning from data represents a canonical problem that has received substantial attention in the literature. However, insufficient work has been done in incorporating prior structural knowledge onto the learning of underlying graphical models from data. Learning a graph with a specific structure is essential for interpretability and identification of the relationships among data. Useful structured graphs include the multi-component graph, bipartite graph, connected graph, sparse graph, and regular graph. In general, structured graph learning is an NP-hard combinatorial problem, therefore, designing a general tractable optimization method is extremely challenging. In this paper, we introduce a unified graph learning framework lying at the integration of Gaussian graphical models and spectral graph theory. To impose a particular structure on a graph, we first show how to formulate the combinatorial constraints as an analytical property of the graph matrix. Then we develop an optimization framework that leverages graph learning with specific structures via spectral constraints on graph matrices. The proposed algorithms are provably convergent, computationally efficient, and practically amenable for numerous graph-based tasks. Extensive numerical experiments with both synthetic and real data sets illustrate the effectiveness of the proposed algorithms. The code for all the simulations is made available as an open source repository.

39 citations


Proceedings ArticleDOI
25 Jul 2019
TL;DR: In this article, the spectral density of real-world graphs is computed using tools developed in condensed matter physics and adapted to handle the spectral signatures of common graph motifs, and the resulting methods are highly efficient, as illustrate by computing spectral densities for graphs with over a billion edges on a single compute node.
Abstract: Spectral analysis connects graph structure to the eigenvalues and eigenvectors of associated matrices. Much of spectral graph theory descends directly from spectral geometry, the study of differentiable manifolds through the spectra of associated differential operators. But the translation from spectral geometry to spectral graph theory has largely focused on results involving only a few extreme eigenvalues and their associated eigenvalues. Unlike in geometry, the study of graphs through the overall distribution of eigenvalues --- the \em spectral density --- is largely limited to simple random graph models. The interior of the spectrum of real-world graphs remains largely unexplored, difficult to compute and to interpret. In this paper, we delve into the heart of spectral densities of real-world graphs. We borrow tools developed in condensed matter physics, and add novel adaptations to handle the spectral signatures of common graph motifs. The resulting methods are highly efficient, as we illustrate by computing spectral densities for graphs with over a billion edges on a single compute node. Beyond providing visually compelling fingerprints of graphs, we show how the estimation of spectral densities facilitates the computation of many common centrality measures, and use spectral densities to estimate meaningful information about graph structure that cannot be inferred from the extremal eigenpairs alone.

32 citations


Proceedings ArticleDOI
06 Jan 2019
TL;DR: In this paper, the authors introduced the notion of a submodular transformation F : {0, 1}n → Rm, which applies m sub-modular functions to the n-dimensional input vector.
Abstract: The Cheeger inequality for undirected graphs, which relates the conductance of an undirected graph and the second smallest eigenvalue of its normalized Laplacian, is a cornerstone of spectral graph theory. The Cheeger inequality has been extended to directed graphs and hypergraphs using normalized Laplacians for those, that are no longer linear but piecewise linear transformations.In this paper, we introduce the notion of a submodular transformation F : {0, 1}n → Rm, which applies m submodular functions to the n-dimensional input vector, and then introduce the notions of its Laplacian and normalized Laplacian. With these notions, we unify and generalize the existing Cheeger inequalities by showing a Cheeger inequality for submodular transformations, which relates the conductance of a submodular transformation and the smallest non-trivial eigenvalue of its normalized Laplacian. This result recovers the Cheeger inequalities for undirected graphs, directed graphs, and hypergraphs, and derives novel Cheeger inequalities for mutual information and directed information.Computing the smallest non-trivial eigenvalue of a normalized Laplacian of a submodular transformation is NP-hard under the small set expansion hypothesis. In this paper, we present a polynomial-time O(log n)-approximation algorithm for the symmetric case, which is tight, and a polynomial-time O(log2n + log n · log m)-approximation algorithm for the general case.We expect the algebra concerned with submodular transformations, or submodular algebra, to be useful in the future not only for generalizing spectral graph theory but also for analyzing other problems that involve piecewise linear transformations, e.g., deep learning.

31 citations


Proceedings ArticleDOI
23 Jun 2019
TL;DR: This paper is a continuation of recent work of the authors (FOCS 2018) analyzing random walk algorithms that find forbidden minors, using techniques from spectral graph theory to resolve the open question of property testing P.
Abstract: Let G be a graph with n vertices and maximum degree d. Fix some minor-closed property P (such as planarity). We say that G is e-far from P if one has to remove e dn edges to make it have P. The problem of property testing P was introduced in the seminal work of Benjamini-Schramm-Shapira (STOC 2008) that gave a tester with query complexity triply exponential in e−1. Levi-Ron (TALG 2015) have given the best tester to date, with a quasipolynomial (in e−1) query complexity. It is an open problem to get property testers whose query complexity is (de−1), even for planarity. In this paper, we resolve this open question. For any minor-closed property, we give a tester with query complexity d· (e−1). The previous line of work on (independent of n, two-sided) testers is primarily combinatorial. Our work, on the other hand, employs techniques from spectral graph theory. This paper is a continuation of recent work of the authors (FOCS 2018) analyzing random walk algorithms that find forbidden minors.

27 citations


Journal ArticleDOI
TL;DR: In this article, a new definition of the isoperimetric constant for quantum graphs is introduced and the Cheeger-type estimate of the spectral properties of infinite quantum graphs has been proved.
Abstract: We investigate the bottom of the spectra of infinite quantum graphs, i.e., Laplace operators on metric graphs having infinitely many edges and vertices. We introduce a new definition of the isoperimetric constant for quantum graphs and then prove the Cheeger-type estimate. Our definition of the isoperimetric constant is purely combinatorial and thus it establishes connections with the combinatorial isoperimetric constant, one of the central objects in spectral graph theory and in the theory of simple random walks on graphs. The latter enables us to prove a number of criteria for quantum graphs to be uniformly positive or to have purely discrete spectrum. We demonstrate our findings by considering trees, antitrees and Cayley graphs of finitely generated groups.

24 citations


Proceedings ArticleDOI
13 May 2019
TL;DR: In this article, the problem of selecting a subset S to maximize its current flow closeness centrality (CFCC) with the cardinality constraint |S| = k was investigated. And two greedy algorithms were proposed to minimize the reciprocal of C(S).
Abstract: The problem of selecting a group of vertices under certain constraints that maximize their joint centrality arises in many practical scenarios. In this paper, we extend the notion of current flow closeness centrality (CFCC) to a set of vertices in a graph, and investigate the problem of selecting a subset S to maximizes its CFCC C(S), with the cardinality constraint |S| = k. We show the NP-hardness of the problem, but propose two greedy algorithms to minimize the reciprocal of C(S). We prove the approximation ratios by showing the monotonicity and supermodularity. A proposed deterministic greedy algorithm has an approximation factor and cubic running time. To compare with, a proposed randomized algorithm gives -approximation in nearly-linear time, for any ? > 0. Extensive experiments on model and real networks demonstrate the effectiveness and efficiency of the proposed algorithms, with the randomized algorithm being applied to massive networks with more than a million vertices.

22 citations


Proceedings ArticleDOI
09 Sep 2019
TL;DR: The construction and applications of decimated tight framelets on graphs based on graph clustering algorithms, where a coarse-grained chain of graphs can be constructed where a suitable orthonormal eigenpair can be deduced, are discussed.
Abstract: In this paper, we discuss the construction and applications of decimated tight framelets on graphs. Based on graph clustering algorithms, a coarse-grained chain of graphs can be constructed where a suitable orthonormal eigenpair can be deduced. Decimated tight framelets can then be constructed based on the orthonormal eigen-pair. Moreover, such tight framelets are associated with filter banks with which fast framelet transform algorithms can be realized. An explicit toy example of decimated tight framelets on a graph is provided.

Journal ArticleDOI
TL;DR: A general framework to track, model, and predict the dynamic network structures, including node ranking, community detection, and link prediction, using the spectral graph theory to track the latent feature vectors obtained by a low-rank eigendecomposition of the Laplacian matrices of the networks.
Abstract: Structural network analysis, including node ranking, community detection, and link prediction, has received a lot of attention lately. In the literature, most works focused on the structural analysis of a single network. In this paper, we are particularly interested in how the network structure evolves over time. For this, we propose a general framework to track, model, and predict the dynamic network structures. Unlike some recent works that directly tracks the adjacency matrices of the networks, our framework utilizes the spectral graph theory to track the latent feature vectors obtained by a low-rank eigendecomposition of the Laplacian matrices of the networks. We then use the Finite Impulse Response (FIR) filter to model the evolution of the latent feature vector of each node. By solving a ridge regression problem, the parameters of the FIR filter can be learned and used for predicting the future network structures, including node ranking, community detection, and link prediction. To test the effectiveness of our framework, we perform various experiments based on our synthetic datasets and three real-world datasets. Our experimental results show that our framework is very effective in tracking latent feature vectors and predicting future network structures.

Posted Content
TL;DR: Experimental results show that the proposed GLNN significantly outperforms state-of-the-art approaches over widely adopted social network datasets and citation network datasets for semi-supervised classification.
Abstract: Graph Convolutional Neural Networks (GCNNs) are generalizations of CNNs to graph-structured data, in which convolution is guided by the graph topology. In many cases where graphs are unavailable, existing methods manually construct graphs or learn task-driven adaptive graphs. In this paper, we propose Graph Learning Neural Networks (GLNNs), which exploit the optimization of graphs (the adjacency matrix in particular) from both data and tasks. Leveraging on spectral graph theory, we propose the objective of graph learning from a sparsity constraint, properties of a valid adjacency matrix as well as a graph Laplacian regularizer via maximum a posteriori estimation. The optimization objective is then integrated into the loss function of the GCNN, which adapts the graph topology to not only labels of a specific task but also the input data. Experimental results show that our proposed GLNN outperforms state-of-the-art approaches over widely adopted social network datasets and citation network datasets for semi-supervised classification.

Journal ArticleDOI
TL;DR: It is shown that if det W Q ( G) is odd and square-free, then G is DGQS, which is said to be determined by the generalized Q -spectrum (DGQS for short).

Proceedings ArticleDOI
02 Jun 2019
TL;DR: It is shown that the resultant spectrally-reduced graphs can robustly preserve the first few nontrivial eigenvalues and eigenvectors of the original graph Laplacian and thus allow for developing highly-scalable spectral graph partitioning and circuit simulation algorithms.
Abstract: This paper proposes a scalable algorithmic framework for effective-resistance preserving spectral reduction of large undirected graphs. The proposed method allows computing much smaller graphs while preserving the key spectral (structural) properties of the original graph. Our framework is built upon the following three key components: a spectrum-preserving node aggregation and reduction scheme, a spectral graph sparsification framework with iterative edge weight scaling, as well as effective-resistance preserving post-scaling and iterative solution refinement schemes. By leveraging recent similarity-aware spectral sparsification method and graph-theoretic algebraic multigrid (AMG) Laplacian solver, a novel constrained stochastic gradient descent (SGD) optimization approach has been proposed for achieving truly scalable performance (nearly-linear complexity) for spectral graph reduction. We show that the resultant spectrally-reduced graphs can robustly preserve the first few nontrivial eigenvalues and eigenvectors of the original graph Laplacian and thus allow for developing highly-scalable spectral graph partitioning and circuit simulation algorithms.ACM Reference Format:Zhiqiang Zhao and Zhuo Feng. 2019. Effective-Resistance Preserving Spectral Reduction of Graphs. In The 56th Annual Design Automation Conference 2019 (DAC '19), June 2–6, 2019, Las Vegas, NV, USA. ACM, New York, NY, USA, 6 pages. https://doi.org/10.1145/3316781.3317809

Posted Content
TL;DR: A parameter-free pooling operator is proposed, called iPool, that permits to retain the most informative features in arbitrary graphs and achieves superior or competitive performance in graph classification on a collection of public graph benchmark data sets and superpixel-induced image graph data sets.
Abstract: With the advent of data science, the analysis of network or graph data has become a very timely research problem. A variety of recent works have been proposed to generalize neural networks to graphs, either from a spectral graph theory or a spatial perspective. The majority of these works however focus on adapting the convolution operator to graph representation. At the same time, the pooling operator also plays an important role in distilling multiscale and hierarchical representations but it has been mostly overlooked so far. In this paper, we propose a parameter-free pooling operator, called iPool, that permits to retain the most informative features in arbitrary graphs. With the argument that informative nodes dominantly characterize graph signals, we propose a criterion to evaluate the amount of information of each node given its neighbors, and theoretically demonstrate its relationship to neighborhood conditional entropy. This new criterion determines how nodes are selected and coarsened graphs are constructed in the pooling layer. The resulting hierarchical structure yields an effective isomorphism-invariant representation of networked data in arbitrary topologies. The proposed strategy is evaluated in terms of graph classification on a collection of public graph datasets, including bioinformatics and social networks, and achieves state-of-the-art performance on most of the datasets.

Journal ArticleDOI
TL;DR: In this paper, a new method for symmetrization of directed graphs that constructs an undirected graph with equivalent pairwise effective resistances as a given directed graph is presented.
Abstract: This work presents a new method for symmetrization of directed graphs that constructs an undirected graph with equivalent pairwise effective resistances as a given directed graph. Consequently a gr...

Proceedings ArticleDOI
TL;DR: Beyond providing visually compelling fingerprints of graphs, this paper shows how the estimation of spectral densities facilitates the computation of many common centrality measures, and uses spectral densITIES to estimate meaningful information about graph structure that cannot be inferred from the extremal eigenpairs alone.
Abstract: Spectral analysis connects graph structure to the eigenvalues and eigenvectors of associated matrices. Much of spectral graph theory descends directly from spectral geometry, the study of differentiable manifolds through the spectra of associated differential operators. But the translation from spectral geometry to spectral graph theory has largely focused on results involving only a few extreme eigenvalues and their associated eigenvalues. Unlike in geometry, the study of graphs through the overall distribution of eigenvalues - the spectral density - is largely limited to simple random graph models. The interior of the spectrum of real-world graphs remains largely unexplored, difficult to compute and to interpret. In this paper, we delve into the heart of spectral densities of real-world graphs. We borrow tools developed in condensed matter physics, and add novel adaptations to handle the spectral signatures of common graph motifs. The resulting methods are highly efficient, as we illustrate by computing spectral densities for graphs with over a billion edges on a single compute node. Beyond providing visually compelling fingerprints of graphs, we show how the estimation of spectral densities facilitates the computation of many common centrality measures, and use spectral densities to estimate meaningful information about graph structure that cannot be inferred from the extremal eigenpairs alone.

Journal ArticleDOI
TL;DR: This paper proposes a simple but efficient spectral filter tracking method from the viewpoint of a graph, where each candidate’s image region is modeled as a pixelwise grid graph, and achieves the state-of-the-art performance on OTB-2015 and VOT2016 under the same feature extraction strategy.
Abstract: Visual object tracking is a challenging computer vision task with numerous real-world applications. In this paper, we propose a simple but efficient spectral filter tracking (SFT) method from the viewpoint of a graph, where each candidate’s image region is modeled as a pixelwise grid graph. Instead of the conventional graph matching, we formulate the tracking as a plain least square regression problem of learning spectral filters on graphs to predict an optimal vertex, which indicates the center of the target. To bypass computationally expensive eigenvalue decomposition on graph Laplacian $ \mathcal {L}$ , we parameterize spectral graph filters as a polynomial of $ \mathcal {L}$ to aggregate local graph features according to spectral graph theory, in which $ \mathcal {L}^{k}$ exactly encodes a k-hop local neighborhood of each vertex. Thus, different from the holistic regression in those correlation filter-based methods, SFT can operate on localized regions around a pixel (i.e., a vertex), which can effectively reduce the influence of local variations and cluttered backgrounds. Furthermore, we observe that the correlation filter tracking may be viewed as a specific case of our proposed spectral filtering method. The implementation of SFT can simply boil down to only a few line codes, but surprisingly it beats the correlation filter-based model with the same feature input and achieves the state-of-the-art performance on OTB-2015 and VOT2016 under the same feature extraction strategy.

Journal ArticleDOI
TL;DR: In this paper, the Godsil-McKay algorithm was ported to signed graphs, and it was shown that with suitable adaption, such algorithms can be successfully ported to cospectral switching nonisomorphic signed graphs.
Abstract: A well--known fact in Spectral Graph Theory is the existence of pairs of isospectral nonisomorphic graphs (known as PINGS). The work of A.J. Schwenk (in 1973) and of C. Godsil and B. McKay (in 1982) shed some light on the explanation of the presence of isospectral graphs, and they gave routines to construct PINGS. Here, we consider the Godsil-McKay--type routines developed for graphs, whose adjacency matrices are $(0,1)$-matrices, to the level of signed graphs, whose adjacency matrices allow the presence of $-1$'s. We show that, with suitable adaption, such routines can be successfully ported to signed graphs, and we can build pairs of cospectral switching nonisomorphic signed graphs.

Journal ArticleDOI
TL;DR: Here the majorization theorem is considered to obtain extremal graphs in the class of trees, unicyclic graphs and bicyclic graphs with given order and fixed maximum degree, independence number, matching number, domination number, and/or number of pendant vertices, respectively.

Journal ArticleDOI
TL;DR: This work proposes a method for indirectly determining the Laplacian matrix by estimating its eigenvalues and eigenvectors using the resonance of oscillation dynamics on networks.
Abstract: Spectral graph theory, based on the adjacency matrix or the Laplacian matrix that represents the network topology and link weights, provides a useful approach for analyzing network structure. However, in large scale and complex social networks, since it is difficult to completely know the network topology and link weights, we cannot determine the components of these matrices directly. To solve this problem, we propose a method for indirectly determining the Laplacian matrix by estimating its eigenvalues and eigenvectors using the resonance of oscillation dynamics on networks. key words: Laplacian matrix, spectral graph theory, resonance

Journal ArticleDOI
TL;DR: In this article, a survey on some known properties of the possible Moore graph ϒ with degree 57 and diameter 2 is presented, and some new results about it are given, such as the following: when we consider the distance partition of ϒ induced by a vertex subset C, the following graphs are distance-regular: the induced graph of the vertices at distance 1 from C when C is a set of 400 independent vertices; the induced graphs of the vertex or an edge when c is a vertex or edge, and the line graph of

Posted Content
TL;DR: This thesis develops new algorithmic techniques from both dynamic and sparsification perspective for a multitude of graph-based optimization problems which lie at the core of Spectral Graph Theory, Graph Partitioning, and Metric Embeddings and introduces novel reduction techniques that show unexpected connections between seemingly different areas such as dynamic graph algorithms and graph sparsifiers.
Abstract: Graphs naturally appear in several real-world contexts including social networks, the web network, and telecommunication networks. While the analysis and the understanding of graph structures have been a central area of study in algorithm design, the rapid increase of data sets over the last decades has posed new challenges for designing efficient algorithms that process large-scale graphs. These challenges arise from two usual assumptions in classical algorithm design, namely that graphs are static and that they fit into a single machine. However, in many application domains, graphs are subject to frequent changes over time, and their massive size makes them infeasible to be stored in the memory of a single machine. Driven by the need to devise new tools for overcoming such challenges, this thesis focuses on two areas of modern algorithm design that directly deal with processing massive graphs, namely dynamic graph algorithms and graph sparsification. We develop new algorithmic techniques from both dynamic and sparsification perspective for a multitude of graph-based optimization problems which lie at the core of Spectral Graph Theory, Graph Partitioning, and Metric Embeddings. Our algorithms are faster than any previous one and design smaller sparsifiers with better (approximation) quality. More importantly, this work introduces novel reduction techniques that show unexpected connections between seemingly different areas such as dynamic graph algorithms and graph sparsification.

Posted Content
Wolfgang Erb1
TL;DR: A flexible framework for uncertainty principles in spectral graph theory is presented, which merges several existing uncertainty relations on graphs, among others the Landau-Pollak principle describing the joint admissibility region of two projection operators, and uncertainty relations based on spectral and spatial spreads.
Abstract: We present a flexible framework for uncertainty principles in spectral graph theory. In this framework, general filter functions modeling the spatial and spectral localization of a graph signal can be incorporated. It merges several existing uncertainty relations on graphs, among others the Landau-Pollak principle describing the joint admissibility region of two projection operators, and uncertainty relations based on spectral and spatial spreads. Using theoretical and computational aspects of the numerical range of matrices, we are able to characterize and illustrate the shapes of the uncertainty curves and to study the space-frequency localization of signals inside the admissibility regions.

Journal ArticleDOI
TL;DR: An efficient heuristic for mapping the logical qubits of quantum algorithms to the physical qubitsof connectivity-limited devices is proposed, adding a minimal number of connectivity-compliant SWAP gates.
Abstract: We propose an efficient heuristic for mapping the logical qubits of quantum algorithms to the physical qubits of connectivity-limited devices, adding a minimal number of connectivity-compliant SWAP gates. In particular, given a quantum circuit, we construct an undirected graph with edge weights a function of the two-qubit gates of the quantum circuit. Taking inspiration from spectral graph drawing, we use an eigenvector of the graph Laplacian to place logical qubits at coordinate locations. These placements are then mapped to physical qubits for a given connectivity. We primarily focus on one-dimensional connectivities, and sketch how the general principles of our heuristic can be extended for use in more general connectivities.

Journal ArticleDOI
TL;DR: In this article, the authors consider trees with index in the real interval (2, 2 + 5 ) and their ordering with respect to the index, and obtain the first 8 trees of even order with largest index.

Posted Content
TL;DR: This paper adapts previously developed techniques in spectral graph theory and proposes a novel concept of applying Procrustes techniques to embedded points for vertices in a graph to detect changes in entity behaviour, calling this method CDP (change detection using Pro Crustes analysis).
Abstract: Change detection in dynamic networks is an important problem in many areas, such as fraud detection, cyber intrusion detection and health care monitoring. It is a challenging problem because it involves a time sequence of graphs, each of which is usually very large and sparse with heterogeneous vertex degrees, resulting in a complex, high dimensional mathematical object. Spectral embedding methods provide an effective way to transform a graph to a lower dimensional latent Euclidean space that preserves the underlying structure of the network. Although change detection methods that use spectral embedding are available, they do not address sparsity and degree heterogeneity that usually occur in noisy real-world graphs and a majority of these methods focus on changes in the behaviour of the overall network. In this paper, we adapt previously developed techniques in spectral graph theory and propose a novel concept of applying Procrustes techniques to embedded points for vertices in a graph to detect changes in entity behaviour. Our spectral embedding approach not only addresses sparsity and degree heterogeneity issues, but also obtains an estimate of the appropriate embedding dimension. We call this method CDP (change detection using Procrustes analysis). We demonstrate the performance of CDP through extensive simulation experiments and a real-world application. CDP successfully detects various types of vertex-based changes including (i) changes in vertex degree, (ii) changes in community membership of vertices, and (iii) unusual increase or decrease in edge weight between vertices. The change detection performance of CDP is compared with two other baseline methods that employ alternative spectral embedding approaches. In both cases, CDP generally shows superior performance.

Journal ArticleDOI
TL;DR: In this article, the largest possible order of Kemeny's constant for a graph on n vertices is given for graphs of order n consisting of two large cliques joined by an arbitrary number of parallel paths of equal length.
Abstract: In a graph theory setting, Kemeny’s constant is a graph parameter which measures a weighted average of the mean first passage times in a random walk on the vertices of the graph. In one sense, Kemeny’s constant is a measure of how well the graph is ‘connected’. An explicit computation for this parameter is given for graphs of order n consisting of two large cliques joined by an arbitrary number of parallel paths of equal length, as well as for two cliques joined by two paths of different length. In each case, Kemeny’s constant is shown to be O(n3), which is the largest possible order of Kemeny’s constant for a graph on n vertices. The approach used is based on interesting techniques in spectral graph theory and includes a generalization of using twin subgraphs to find the spectrum of a graph.

Posted ContentDOI
02 Feb 2019-bioRxiv
TL;DR: It was found that graph energy, reciprocal link and cyclomatic complexity can optimally specify network motifs with some degree of degeneracy.
Abstract: Biological processes are based on molecular networks, which exhibit biological functions through interactions among the various genetic elements. This study presents a graph-based method to characterize molecular networks by decomposing them into directed multigraphs: network motifs. Spectral graph theory, reciprocity, and complexity measures were utilized to quantify the network motifs. It was found that graph energy, reciprocity, and cyclomatic complexity can optimally specify network motifs with some degree of degeneracy. A total of 72 molecular networks were analyzed, of three types: cancer networks, signal transduction networks, and cellular processes. It was found that molecular networks are built from a finite number of motif patterns; hence, a graph energy cutoff exists. In addition, it was found that certain motif patterns are absent from the three types of networks; hence, the Shannon entropy of the motif frequency distribution is not maximal. Furthermore, frequently found motifs are irreducible graphs. These are novel findings: they warrant further investigation and may lead to important applications. The present study provides a systematic approach for dissecting biological networks. Our discovery supports the view that there are organizational principles underlying molecular networks.