Normalized Graph Laplacian Research Articles

Bi-stochastically normalized graph Laplacian: convergence to manifold Laplacian and robustness to outlier noise.

Bi-stochastic normalization provides an alternative normalization of graph Laplacians in graph-based data analysis and can be computed efficiently by Sinkhorn-Knopp (SK) iterations. This paper proves the convergence of bi-stochastically normalized graph Laplacian to manifold (weighted-)Laplacian with rates, when [Formula: see text] data points are i.i.d. sampled from a general [Formula: see text]-dimensional manifold embedded in a possibly high-dimensional space. Under certain joint limit of [Formula: see text] and kernel bandwidth [Formula: see text], the point-wise convergence rate of the graph Laplacian operator (under 2-norm) is proved to be [Formula: see text] at finite large [Formula: see text] up to log factors, achieved at the scaling of [Formula: see text]. When the manifold data are corrupted by outlier noise, we theoretically prove the graph Laplacian point-wise consistency which matches the rate for clean manifold data plus an additional term proportional to the boundedness of the inner-products of the noise vectors among themselves and with data vectors. Motivated by our analysis, which suggests that not exact bi-stochastic normalization but an approximate one will achieve the same consistency rate, we propose an approximate and constrained matrix scaling problem that can be solved by SK iterations with early termination. Numerical experiments support our theoretical results and show the robustness of bi-stochastically normalized graph Laplacian to high-dimensional outlier noise.

Laplacian Change Point Detection for Single and Multi-view Dynamic Graphs

Dynamic graphs are rich data structures that are used to model complex relationships between entities over time. In particular, anomaly detection in temporal graphs is crucial for many real-world applications such as intrusion identification in network systems, detection of ecosystem disturbances, and detection of epidemic outbreaks. In this article, we focus on change point detection in dynamic graphs and address three main challenges associated with this problem: (i) how to compare graph snapshots across time, (ii) how to capture temporal dependencies, and (iii) how to combine different views of a temporal graph. To solve the above challenges, we first propose Laplacian Anomaly Detection (LAD) which uses the spectrum of graph Laplacian as the low dimensional embedding of the graph structure at each snapshot. LAD explicitly models short-term and long-term dependencies by applying two sliding windows. Next, we propose MultiLAD, a simple and effective generalization of LAD to multi-view graphs. MultiLAD provides the first change point detection method for multi-view dynamic graphs. It aggregates the singular values of the normalized graph Laplacian from different views through the scalar power mean operation. Through extensive synthetic experiments, we show that (i) LAD and MultiLAD are accurate and outperforms state-of-the-art baselines and their multi-view extensions by a large margin, (ii) MultiLAD’s advantage over contenders significantly increases when additional views are available, and (iii) MultiLAD is highly robust to noise from individual views. In five real-world dynamic graphs, we demonstrate that LAD and MultiLAD identify significant events as top anomalies such as the implementation of government COVID-19 interventions which impacted the population mobility in multi-view traffic networks.

Random matrices with row constraints and eigenvalue distributions of graph Laplacians

Symmetric matrices with zero row sums occur in many theoretical settings and in real-life applications. When the offdiagonal elements of such matrices are i.i.d. random variables and the matrices are large, the eigenvalue distributions converge to a peculiar universal curve that looks like a cross between the Wigner semicircle and a Gaussian distribution. An analytic theory for this curve, originally due to Fyodorov, can be developed using supersymmetry-based techniques. We extend these derivations to the case of sparse matrices, including the important case of graph Laplacians for large random graphs with N vertices of mean degree c. In the regime , the eigenvalue distribution of the ordinary graph Laplacian (diffusion with a fixed transition rate per edge) tends to a shifted and scaled version of , centered at c with width . At smaller c, this curve receives corrections in powers of accurately captured by our theory. For the normalized graph Laplacian (diffusion with a fixed transition rate per vertex), the large c limit is a shifted and scaled Wigner semicircle, again with corrections captured by our analysis.

On the spectrum of dense random geometric graphs

In this paper we study the spectrum of the random geometric graph G(n,r), in a regime where the graph is dense and highly connected. In the Erdős–Rényi G(n,p) random graph it is well known that upon connectivity the spectrum of the normalized graph Laplacian is concentrated around 1. We show that such concentration does not occur in the G(n,r) case, even when the graph is dense and almost a complete graph. In particular, we show that the limiting spectral gap is strictly smaller than 1. In the special case where the vertices are distributed uniformly in the unit cube and r=1, we show that for every 0≤k≤d there are at least ( d k) eigenvalues near 1−2−k, and the limiting spectral gap is exactly 1/2. We also show that the corresponding eigenfunctions in this case are tightly related to the geometric configuration of the points.

Energy Levels Based Graph Neural Networks for Heterophily

The representation power of graph neural networks (GNNs) under heterophily has drawn much attention recently, in which some connected nodes do not have common labels or similar features. Previous approaches typically incorporate high-order neighbourhoods’ information so as to reinforce aggregation operation. However, they do not directly face with heterophily in first-order neighbours. To deal with this problem, we propose the energy levels guided GNNs in which energy levels of classes are incorporated for better expressiveness. We prove that the label with higher energy level would imply higher energy in the spectrum of the normalized graph Laplacian. Experiments on common benchmark datasets show the effectiveness of the proposed approach.

Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

We offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least frac{n+1}{n-1} provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size frac{n-1}{2}. With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most frac{n-1}{2}.

A note on the degree conjecture for separability of multipartite quantum states

The degree conjecture for bipartite quantum states which are normalized graph Laplacians was first put forward by Braunstein et al. [Phys. Rev. A 73 (2006) 012320]. The degree criterion, which is equivalent to PPT criterion, is simpler and more efficient to detect the separability of quantum states associated with graphs. Hassan et al. settled the degree conjecture for the separability of multipartite quantum states in [J. Math. Phys. 49 (2008) 0121105]. It is proved that the conjecture is true for pure multipartite quantum states. However, the degree condition is only necessary for separability of a class of quantum mixed states. It does not apply to all mixed states. In this paper, we show that the degree conjecture holds for the mixed quantum states of nearest point graph. As a byproduct, the degree criterion is necessary and sufficient for multipartite separability of [Formula: see text]-qubit quantum states associated with graphs.

Single image super-resolution via non-local normalized graph Laplacian regularization: A self-similarity tribute

The process of producing a high-resolution image given a single low-resolution noisy measurement is called single-frame image super-resolution (SISR). Historically, many fractal-based schemes have been proposed in the literature to address the SISR problem. Many conventional interpolation schemes fail to preserve important edge information of natural images and cannot be used blindly for resolution enhancement. Generally, a-priori constraints are required in the process of high resolution image approximation. We model the SISR problem as an energy minimization procedure which balances data fidelity and a regularization term. The regularization term will implicitly incorporate natural image redundancy via a normalized graph Laplacian operator, as a self-similarity based prior. This operator applies a non-local kernel similarity measure due to the choice of a non-local operator for the weight assignment. The data fidelity term is modelled as a likelihood estimator that is scaled using a sharpening term composed from the normalized graph Laplacian operator. Finally, a conjugate gradient scheme is used to minimize the objective functional. Promising results on resolution enhancement for a variety of digital images will be presented.

Design of affinity-aware encoding by embedding graph centrality for graph classification

Deep learning methods for graph classification are critical for graph data mining. Recently, graph convolutional networks (GCNs) have been able to achieve state-of-the-art node classification. A typical process for GCNs includes two iterative steps: node feature encoding and message passing. While the former encodes each graph node independently via the uniform encoding function, the latter updates the features of each node by weighted aggregation of the features of neighboring nodes, where the weights are generated by predefined or learned graph Laplacian. However, their accuracy deteriorates for graph classification tasks because the uniform encoding function encodes all the node features involved. In this study, we propose a novel affinity-aware encoding for graph classification. In our model, we implement a separate encoding function for the neighboring nodes of each node for updating the node features, where the nodes are arranged in the order of affinity values, such as graph centrality, in order to determine the correspondence between an encoding function and a specific neighboring node. Our separate encoding function performs non-Euclidean neighboring encoding for each node by weight sharing, which enables message passing. We also develop two variants based on our separate encoding function: the graph centrality-convolutional neural network (C-CNN) and the graph centrality-graph convolutional network (C-GCN). The former performs the separate encoding function on graph data directly by the function of message passing. The latter combines the separate encoding function with the normalized graph Laplacian implemented on the graph data. Experiments demonstrate that the results obtained by our models are consistent with those obtained by classical convolutional neural networks (CNNs) on the MNIST dataset, and they outperform existing GCNs on the 20NEWS, Reuters8, and Reuters52 datasets. We also apply our two variants to online car-hailing service data for traffic congestion recognition. Our methods show state-of-the-art results compared with GCNs.

Random Walks on Simplicial Complexes and the Normalized Hodge 1-Laplacian

Focusing on coupling between edges, we generalize the relationship between the normalized graph Laplacian and random walks on graphs by devising an appropriate normalization for the Hodge Laplacian -- the generalization of the graph Laplacian for simplicial complexes -- and relate this to a random walk on edges. Importantly, these random walks are intimately connected to the topology of the simplicial complex, just as random walks on graphs are related to the topology of the graph. This serves as a foundational step towards incorporating Laplacian-based analytics for higher-order interactions. We demonstrate how to use these dynamics for data analytics that extract information about the edge-space of a simplicial complex that complements and extends graph-based analysis. Specifically, we use our normalized Hodge Laplacian to derive spectral embeddings for examining trajectory data of ocean drifters near Madagascar and also develop a generalization of personalized PageRank for the edge-space of simplicial complexes to analyze a book co-purchasing dataset.

On Laplacian and Normalized Laplacian of a Social Network

In the task of structural identification of a network a vital tool is the underlying spectrum associated with the normalized graph Laplacian. To comprehend such spectrum, we need to determine the eigenvalues. In this paper we have found certain useful bounds involving the eigenvalues of both combinatorial Laplacian and normalized Laplacian and applied the same on a collaboration graph obtained from a social network.

Characterizing and Comparing Phylogenetic Trait Data from Their Normalized Laplacian Spectrum

The dissection of the mode and tempo of phenotypic evolution is integral to our understanding of global biodiversity. Our ability to infer patterns of phenotypes across phylogenetic clades is essential to how we infer the macroevolutionary processes governing those patterns. Many methods are already available for fitting models of phenotypic evolution to data. However, there is currently no comprehensive nonparametric framework for characterizing and comparing patterns of phenotypic evolution. Here, we build on a recently introduced approach for using the phylogenetic spectral density profile (SDP) to compare and characterize patterns of phylogenetic diversification, in order to provide a framework for nonparametric analysis of phylogenetic trait data. We show how to construct the SDP of trait data on a phylogenetic tree from the normalized graph Laplacian. We demonstrate on simulated data the utility of the SDP to successfully cluster phylogenetic trait data into meaningful groups and to characterize the phenotypic patterning within those groups. We furthermore demonstrate how the SDP is a powerful tool for visualizing phenotypic space across traits and for assessing whether distinct trait evolution models are distinguishable on a given empirical phylogeny. We illustrate the approach in two empirical data sets: a comprehensive data set of traits involved in song, plumage, and resource-use in tanagers, and a high-dimensional data set of endocranial landmarks in New World monkeys. Considering the proliferation of morphometric and molecular data collected across the tree of life, we expect this approach will benefit big data analyses requiring a comprehensive and intuitive framework.

To Detect Irregular Trade Behaviors In Stock Market By Using Graph Based Ranking Methods

: To detect the irregular trade behaviors in the stock market is the important problem in machine learning field. These irregular trade behaviors are obviously illegal. To detect these irregular trade behaviors in the stock market, data scientists normally employ the supervised learning techniques. In this paper, we employ the three graph Laplacian based semi-supervised ranking methods to solve the irregular trade behavior detection problem. Experimental results show that that the un-normalized and symmetric normalized graph Laplacian based semisupervised ranking methods outperform the random walk Laplacian based semi-supervised ranking method.

Spectral Gaps of Random Graphs and Applications

AbstractWe study the spectral gap of the Erdős–Rényi random graph through the connectivity threshold. In particular, we show that for any fixed $\delta> 0$ if $$\begin{equation*} p \geq \frac{(1/2 + \delta) \log n}{n}, \end{equation*}$$then the normalized graph Laplacian of an Erdős–Rényi graph has all of its nonzero eigenvalues tightly concentrated around $1$. This is a strong expander property. We estimate both the decay rate of the spectral gap to $1$ and the failure probability, up to a constant factor. We also show that the $1/2$ in the above is optimal, and that if $p = \frac{c \log n}{n}$ for $c < 1/2,$ then there are eigenvalues of the Laplacian restricted to the giant component that are separated from $1.$ We then describe several applications of our spectral gap results to stochastic topology and geometric group theory. These all depend on Garland’s method [24], a kind of spectral geometry for simplicial complexes. The following can all be considered to be higher-dimensional expander properties. First, we exhibit a sharp threshold for the fundamental group of the Bernoulli random $2$-complex to have Kazhdan’s property (T). We also obtain slightly more information and can describe the large-scale structure of the group just before the (T) threshold. In this regime, the random fundamental group is with high probability the free product of a (T) group with a free group, where the free group has one generator for every isolated edge. The (T) group plays a role analogous to that of a “giant component” in percolation theory. Next we give a new, short, self-contained proof of the Linial–Meshulam–Wallach theorem [35, 39], identifying the cohomology-vanishing threshold of Bernoulli random $d$-complexes. Since we use spectral techniques, it only holds for $\mathbb{Q}$ or $\mathbb{R}$ coefficients rather than finite field coefficients, as in [35] and [39]. However, it is sharp from a probabilistic point of view, providing for example, hitting time type results and limiting Poisson distributions inside the critical window. It is also a new method of proof, circumventing the combinatorial complications of cocycle counting. Similarly, results in an earlier preprint version of this article were already applied in [33] to obtain sharp cohomology-vanishing thresholds in every dimension for the random flag complex model.

A longitudinal model for tau aggregation in Alzheimer's disease based on structural connectivity.

Tau tangles are a pathological hallmark of Alzheimer?s disease (AD) with strong correlations existing between tau aggregation and cognitive decline. Studies in mouse models have shown that the characteristic patterns of tau spatial spread associated with AD progression are determined by neural connectivity rather than physical proximity between different brain regions. We present here a network diffusion model for tau aggregation based on longitudinal tau measures from positron emission tomography (PET) and structural connectivity graphs from diffusion tensor imaging (DTI). White matter fiber bundles reconstructed via tractography from the DTI data were used to compute normalized graph Laplacians which served as graph diffusion kernels for tau spread. By linearizing this model and using sparse source localization, we were able to identify distinct patterns of propagative and generative buildup of tau at a population level. A gradient descent approach was used to solve the sparsity-constrained optimization problem. Model fitting was performed on subjects from the Harvard Aging Brain Study cohort. The fitted model parameters include a scalar factor controlling the network-based tau spread and a network-independent seed vector representing seeding in different regions-of-interest. This parametric model was validated on an independent group of subjects from the same cohort. We were able to predict with reasonably high accuracy the tau buildup at a future time-point. The network diffusion model, therefore, successfully identifies two distinct mechanisms for tau buildup in the aging brain and offers a macroscopic perspective on tau spread.

On the von Neumann entropy of graphs

AbstractThe von Neumann entropy of a graph is a spectral complexity measure that has recently found applications in complex networks analysis and pattern recognition. Two variants of the von Neumann entropy exist based on the graph Laplacian and normalized graph Laplacian, respectively. Due to its computational complexity, previous works have proposed to approximate the von Neumann entropy, effectively reducing it to the computation of simple node degree statistics. Unfortunately, a number of issues surrounding the von Neumann entropy remain unsolved to date, including the interpretation of this spectral measure in terms of structural patterns, understanding the relation between its two variants and evaluating the quality of the corresponding approximations. In this article, we aim to answer these questions by first analysing and comparing the quadratic approximations of the two variants and then performing an extensive set of experiments on both synthetic and real-world graphs. We find that (1) the two entropies lead to the emergence of similar structures, but with some significant differences; (2) the correlation between them ranges from weakly positive to strongly negative, depending on the topology of the underlying graph; (3) the quadratic approximations fail to capture the presence of non-trivial structural patterns that seem to influence the value of the exact entropies; and (4) the quality of the approximations, as well as which variant of the von Neumann entropy is better approximated, depends on the topology of the underlying graph.

Constructing brain connectivity group graphs from EEG time series

ABSTRACTGraphical analysis of complex brain networks is a fundamental area of modern neuroscience. Functional connectivity is important since many neurological and psychiatric disorders, including schizophrenia, are described as ‘dys-connectivity’ syndromes. Using electroencephalogram time series collected on each of a group of 15 individuals with a common medical diagnosis of positive syndrome schizophrenia we seek to build a single, representative, brain functional connectivity group graph. Disparity/distance measures between spectral matrices are identified and used to define the normalized graph Laplacian enabling clustering of the spectral matrices for detecting ‘outlying’ individuals. Two such individuals are identified. For each remaining individual, we derive a test for each edge in the connectivity graph based on average estimated partial coherence over frequencies, and associated p-values are found. For each edge these are used in a multiple hypothesis test across individuals and the proportion rejecting the hypothesis of no edge is used to construct a connectivity group graph. This study provides a framework for integrating results on multiple individuals into a single overall connectivity structure.

Network Partitioning Algorithms for Solving the Traffic Assignment Problem using a Decomposition Approach

Recent methods in the literature to parallelize the traffic assignment problem consider partitioning a network into subnetworks to reduce the computation time. In this article, a partitioning method is sought that generates subnetworks minimizing the computation time of a decomposition approach for solving the traffic assignment (DSTAP). The aim is to minimize the number of boundary nodes, the interflow between subnetworks, and the computation time when the traffic assignment problem is solved in parallel on the subnetworks. Two different methods for partitioning are tested. The first is an agglomerative clustering heuristic that reduces the subnetwork boundary nodes. The second is a flow weighted spectral partitioning algorithm that uses the normalized graph Laplacian to partition the network. The performance of both algorithms is assessed on different test networks. The results indicate that the agglomerative heuristic generates subnetworks with a lower number of boundary nodes, which reduces the per iteration computation time of DSTAP. However, the partitions generated may be heavily imbalanced leading to a higher computation time when the subnetworks are solved in parallel separately at a particular DSTAP iteration. For the Austin network partitioned into four subnetworks, the agglomerative heuristic requires 3.5 times more computation time to solve the subnetworks in parallel. The results also show that the spectral partitioning method is superior for minimizing the interflow between subnetworks. This leads to a faster convergence rate of the DSTAP algorithm.

Efficient large scale commute time embedding

Commute time embedding involves computing eigenfunctions of the graph Laplacian matrix. Spectral decomposition requires computational burden proportional to O(n3), which may not be suitable for large scale dataset. This paper proposes computationally efficient commute time embedding by applying Nyström method to the normalized graph Laplacian. The performance of the proposed algorithms is analysed by checking the embedding results on a patch graph.

A variational approach to the consistency of spectral clustering

Abstract This paper establishes the consistency of spectral approaches to data clustering. We consider clustering of point clouds obtained as samples of a ground-truth measure. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. We investigate the spectral convergence of both unnormalized and normalized graph Laplacians towards the appropriate operators in the continuum domain. We obtain sharp conditions on how the connectivity radius can be scaled with respect to the number of sample points for the spectral convergence to hold. We also show that the discrete clusters obtained via spectral clustering converge towards a continuum partition of the ground truth measure. Such continuum partition minimizes a functional describing the continuum analogue of the graph-based spectral partitioning. Our approach, based on variational convergence, is general and flexible.