Eigenvector Localization Research Articles

Local law and complete eigenvector delocalization for supercritical Erdős–Rényi graphs

We prove a local law for the adjacency matrix of the Erdős–Rényi graph $G(N,p)$ in the supercritical regime $pN\geq C\log N$ where $G(N,p)$ has with high probability no isolated vertices. In the same regime, we also prove the complete delocalization of the eigenvectors. Both results are false in the complementary subcritical regime. Our result improves the corresponding results from (Ann. Probab. 41 (2013) 2279–2375) by extending them all the way down to the critical scale $pN=O(\log N)$. A key ingredient of our proof is a new family of multilinear large deviation estimates for sparse random vectors, which carefully balance mixed $\ell^{2}$ and $\ell^{\infty }$ norms of the coefficients with combinatorial factors, allowing us to prove strong enough concentration down to the critical scale $pN=O(\log N)$. These estimates are of independent interest and we expect them to be more generally useful in the analysis of very sparse random matrices.

Graph theoretical approaches for the characterization of damage in hierarchical materials

We discuss the relevance of methods of graph theory for the study of damage in simple model materials described by the random fuse model. While such methods are not commonly used when dealing with regular random lattices, which mimic disordered but statistically homogeneous materials, they become relevant in materials with microstructures that exhibit complex multi-scale patterns. We specifically address the case of hierarchical materials, whose failure, due to an uncommon fracture mode, is not well described in terms of either damage percolation or crack nucleation-and-growth. We show that in these systems, incipient failure is accompanied by an increase in eigenvector localization and a drop in topological dimension. We propose these two novel indicators as possible candidates to monitor a system in the approach to failure. As such, they provide alternatives to monitoring changes in the precursory avalanche activity, which is often invoked as a candidate for failure prediction in materials which exhibit critical-like behavior at failure, but may not work in the context of hierarchical materials which exhibit scale-free avalanche statistics even very far from the critical load.

Sparse general Wigner-type matrices: Local law and eigenvector delocalization

We prove a local law and eigenvector delocalization for general Wigner-type matrices. Our methods allow us to get the best possible interval length and optimal eigenvector delocalization in the dense case, and the first results of such kind for the sparse case down to p=g(n)log⁡nn with g(n) → ∞. We specialize our results to the case of the stochastic block model, and we also obtain a local law for the case when the number of classes is unbounded.

Locally-biased spectral approximation for community detection

We propose a Locally-Biased Spectral Approximation (LBSA) approach for identifying all latent members of a local community from very few seed members. To reduce the computation complexity, we first apply a fast random walk, personalized PageRank and heat kernel diffusion to sample a comparatively small subgraph covering almost all potential community members around the seeds. Then starting from a normalized indicator vector of the seeds and by a few steps of either Lanczos iteration or power iteration on the sampled subgraph, a local eigenvector is gained for approximating the eigenvector of the transition matrix with the largest eigenvalue. Elements of this local eigenvector is a relaxed indicator for the affiliation probability of the corresponding nodes to the target community. We conduct extensive experiments on real-world datasets in various domains as well as synthetic datasets. Results show that the proposed method outperforms state-of-the-art local community detection algorithms. To the best of our knowledge, this is the first work to adapt the Lanczos method for local community detection, which is natural and potentially effective. Also, we did the first attempt of using heat kernel as a sampling method instead of detecting communities directly, which is proved empirically to be very efficient and effective.

Engineering chimera patterns in networks using heterogeneous delays.

Symmetry breaking spatial patterns, referred to as chimera states, have recently been catapulted into the limelight due to their coexisting coherent and incoherent hybrid dynamics. Here, we present a method to engineer a chimera state by using an appropriate distribution of heterogeneous time delays on the edges of a network. The time delays in interactions, intrinsic to natural or artificial complex systems, are known to induce various modifications in spatiotemporal behaviors of the coupled dynamics on networks. Using a coupled chaotic map with the identical coupling environment, we demonstrate that control over the spatial location of the incoherent region of a chimera state in a network can be achieved by appropriately introducing time delays. This method allows for the engineering of tailor-made one cluster or multi-cluster chimera patterns. Furthermore, borrowing a measure of eigenvector localization from the spectral graph theory, we introduce a spatial inverse participation ratio, which provides a robust way for the identification of the chimera state. This report highlights the necessity to consider the heterogeneous time delays to develop applications for the chimera states in particular and understand coupled dynamical systems in general.

A spectral decomposition approach for the mechanical statistical characterization of distributed fiber-reinforced tissues

We discuss a spectral decomposition formulation for the mechanical statistical characterization of the anisotropic strain energy density of soft hyperelastic materials embedded with distributed fibers. We consider a generalized angular probability density function (PDF) of the reinforcement built upon the local eigenvalue and eigenvector system of the Cauchy–Green deformation tensor. We focus our analysis on material models dependent on the fourth pseudo-invariant of the deformation, I4, and on exponential forms of the fiber strain energy function. Within such a spectral reference system, we derive the closed-form expression of the PDF for I4 by generalizing the multi-value random variable transformation procedure recently developed in Gizzi et al. 2016. Our formulation bypasses the cumbersome extension–contraction switch, commonly adopted for shutting down the contribution of contracted fibers in models based on generalized structure tensors. Accordingly, we identify analytically the support of the fibers in pure extension for significant loading conditions. We can readily compute any statistics of the fourth pseudo-invariant and we can derive the direct definition of the average second Piola–Kirchhoff stress tensor according to the second order approximation.

Eigenvector Localization in Real Networks and Its Implications for Epidemic Spreading

The spectral properties of the adjacency matrix, in particular its largest eigenvalue and the associated principal eigenvector, dominate many structural and dynamical properties of complex networks. Here we focus on the localization properties of the principal eigenvector in real networks. We show that in most cases it is either localized on the star defined by the node with largest degree (hub) and its nearest neighbors, or on the densely connected subgraph defined by the maximum $K$-core in a $K$-core decomposition. The localization of the principal eigenvector is often strongly correlated with the value of the largest eigenvalue, which is given by the local eigenvalue of the corresponding localization subgraph, but different scenarios sometimes occur. We additionally show that simple targeted immunization strategies for epidemic spreading are extremely sensitive to the actual localization set.

Optimized evolution of networks for principal eigenvector localization

Network science is increasingly being developed to get new insights about behavior and properties of complex systems represented in terms of nodes and interactions. One useful approach is investigating the localization properties of eigenvectors having diverse applications including disease-spreading phenomena in underlying networks. In this work, we evolve an initial random network with an edge rewiring optimization technique considering the inverse participation ratio as a fitness function. The evolution process yields a network having a localized principal eigenvector. We analyze various properties of the optimized networks and those obtained at the intermediate stage. Our investigations reveal the existence of a few special structural features of such optimized networks, for instance, the presence of a set of edges which are necessary for localization, and rewiring only one of them leads to complete delocalization of the principal eigenvector. Furthermore, we report that principal eigenvector localization is not a consequence of changes in a single network property and, preferably, requires the collective influence of various distinct structural as well as spectral features.

Charge transfer and anderson localization in one-dimensional finite-size disordered systems

The metal–insulator transition in finite one-dimensional lattices with diagonal disorder is considered. Although the true phase transition does not occur in such systems, a conditional criterion can be introduced to distinguish the regions of localization and delocalization of eigenvectors and wave functions, depending on the degree of disorder W. A relationship between the degree of disorder W and the characteristic length separating the localized and delocalized states is obtained: with high accuracy, ξcr ~ W –2. The calculations are performed by integrating the evolutional Schrodinger equations and through analyzing the spectral properties of the corresponding random matrices. The probability of charge transfer is estimated within the framework of a simple model of DNA.

Localization of Laplacian eigenvectors on random networks

In large random networks, each eigenvector of the Laplacian matrix tends to localize on a subset of network nodes having similar numbers of edges, namely, the components of each Laplacian eigenvector take relatively large values only on a particular subset of nodes whose degrees are close. Although this localization property has significant consequences for dynamical processes on random networks, a clear theoretical explanation has not yet been established. Here we analyze the origin of localization of Laplacian eigenvectors on random networks by using a perturbation theory. We clarify how heterogeneity in the node degrees leads to the eigenvector localization and that there exists a clear degree-eigenvalue correspondence, that is, the characteristic degrees of the localized nodes essentially determine the eigenvalues. We show that this theory can account for the localization properties of Laplacian eigenvectors on several classes of random networks, and argue that this localization should occur generally in networks with degree heterogeneity.

Bioinformatic Analysis of Coronary Disease Associated SNPs and Genes to Identify Proteins Potentially Involved in the Pathogenesis of Atherosclerosis.

Factors that contribute to the onset of atherosclerosis may be elucidated by bioinformatic techniques applied to multiple sources of genomic and proteomic data. The results of genome wide association studies, such as the CardioGramPlusC4D study, expression data, such as that available from expression quantitative trait loci (eQTL) databases, along with protein interaction and pathway data available in Ingenuity Pathway Analysis (IPA), constitute a substantial set of data amenable to bioinformatics analysis. This study used bioinformatic analyses of recent genome wide association data to identify a seed set of genes likely associated with atherosclerosis. The set was expanded to include protein interaction candidates to create a network of proteins possibly influencing the onset and progression of atherosclerosis. Local average connectivity (LAC), eigenvector centrality, and betweenness metrics were calculated for the interaction network to identify top gene and protein candidates for a better understanding of the atherosclerotic disease process. The top ranking genes included some known to be involved with cardiovascular disease (APOA1, APOA5, APOB, APOC1, APOC2, APOE, CDKN1A, CXCL12, SCARB1, SMARCA4 and TERT), and others that are less obvious and require further investigation (TP53, MYC, PPARG, YWHAQ, RB1, AR, ESR1, EGFR, UBC and YWHAZ). Collectively these data help define a more focused set of genes that likely play a pivotal role in the pathogenesis of atherosclerosis and are therefore natural targets for novel therapeutic interventions.

Localized eigenvectors of the non-backtracking matrix

In the case of graph partitioning, the emergence of localized eigenvectors can cause the standard spectral method to fail. To overcome this problem, the spectral method using a non-backtracking matrix was proposed. Based on numerical experiments on several examples of real networks, it is clear that the non-backtracking matrix does not exhibit localization of eigenvectors. However, we show that localized eigenvectors of the non-backtracking matrix can exist outside the spectral band, which may lead to deterioration in the performance of graph partitioning.

Distinct types of eigenvector localization in networks.

The spectral properties of the adjacency matrix provide a trove of information about the structure and function of complex networks. In particular, the largest eigenvalue and its associated principal eigenvector are crucial in the understanding of nodes’ centrality and the unfolding of dynamical processes. Here we show that two distinct types of localization of the principal eigenvector may occur in heterogeneous networks. For synthetic networks with degree distribution P(q) ~ q−γ, localization occurs on the largest hub if γ > 5/2; for γ < 5/2 a new type of localization arises on a mesoscopic subgraph associated with the shell with the largest index in the K-core decomposition. Similar evidence for the existence of distinct localization modes is found in the analysis of real-world networks. Our results open a new perspective on dynamical processes on networks and on a recently proposed alternative measure of node centrality based on the non-backtracking matrix.

Loss of brain graph network efficiency in alcohol dependence.

Alcohol dependence (AD) is characterized by corticostriatal impairments in individual brain areas such as the striatum. As yet however, complex brain network topology in AD and its association with disease progression are unknown. We applied graph theory to resting-state functional magnetic resonance imaging (RS-fMRI) to examine weighted global efficiency and local (clustering coefficient, degree and eigenvector centrality) network topology and the functional role of the striatum in 24 AD patients compared with 20 matched healthy controls (HCs), and their association with dependence characteristics. Graph analyses were performed based on Pearson's correlations between RS-fMRI time series, while correcting for age, gender and head motion. We found no significant group differences between AD patients and HCs in network topology. Notably, within the patient group, but not in HCs, the whole-brain network showed reduced average cluster coefficient with more severe alcohol use, whereas longer AD duration within the patient group was associated with a global decrease in efficiency, degree and clustering coefficient. Additionally, within four a-priori chosen bilateral striatal nodes, alcohol use severity was associated with lower clustering coefficient in the left caudate. Longer AD duration was associated with reduced clustering coefficient in caudate and putamen, and reduced degree in bilateral caudate, but with increased eigenvector centrality in left posterior putamen. Especially changes in global network topology and clustering coefficient in anterior striatum remained strikingly robust after exploratory variations in network weight. Our results show adverse effects of AD on overall network integration and possibly on striatal efficiency, putatively contributing to the increasing behavioral impairments seen in chronically addicted patients.

Limitations in the spectral method for graph partitioning: Detectability threshold and localization of eigenvectors.

Investigating the performance of different methods is a fundamental problem in graph partitioning. In this paper, we estimate the so-called detectability threshold for the spectral method with both un-normalized and normalized Laplacians in sparse graphs. The detectability threshold is the critical point at which the result of the spectral method is completely uncorrelated to the planted partition. We also analyze whether the localization of eigenvectors affects the partitioning performance in the detectable region. We use the replica method, which is often used in the field of spin-glass theory, and focus on the case of bisection. We show that the gap between the estimated threshold for the spectral method and the threshold obtained from Bayesian inference is considerable in sparse graphs, even without eigenvector localization. This gap closes in a dense limit.

Localization of epileptogenic zones in Lennox–Gastaut syndrome (LGS) using graph theoretical analysis of ictal intracranial EEG: A preliminary investigation

Precise localization of epileptogenic zones is essential for the successful surgical treatment of refractory epilepsy including Lennox-Gastaut syndrome (LGS). The surgical resection areas are generally determined by epileptologists based on diverse neuroimaging modalities; however, exact epileptogenic zones cannot be accurately localized in many patients with LGS using the conventional methods. Therefore, new reliable algorithms are still required for enhancing the success rate of the resective epilepsy surgery. In the present study, we introduce an approach to localize epileptogenic zones in LGS based on the graph theoretical analysis of ical intracranial EEG (iEEG). Four patients with LGS who became seizure-free after the resective epilepsy surgery were selected. Before the surgery, their epileptogenic zones were delineated using EEG, iEEG, and several conventional imaging modalities. Phase locking value (PLV) analysis was applied to construct functional connectivity networks during ictal events, and then several graph theoretical indices including betweenness centrality (BC) were evaluated for each iEEG sensor to find the primary hubs of the ictal epileptic network. The graph theoretical index values were then overlaid on 3D individual cortical surface. The iEEG channels with high BC values coincided well with the surgical resection areas. Among various graph theoretical measures such as local efficiency, participation coefficient, and eigenvector centrality, only BC showed fair correspondence with the surgical resection areas. The primary hubs in the ictal epileptic networks coincided well with areas of surgical resection in LGS patients with successful surgical outcomes. This observation warrants further studies to determine if the graph theoretical network analysis of ictal iEEG recordings can serve as a new auxiliary tool to localize epileptogenic zones in LGS.

On Enhanced Binding and Related Effectsin the Non- and Semi-Relativistic Pauli-Fierz Models

We prove enhanced binding and increase of binding energies in the non- and semi-relativistic Pauli-Fierz models, for arbitrary values of the fine-structure constant and the ultra-violet cut-off, and discuss the resulting improvement of exponential localization of ground state eigenvectors. For the semi-relativistic model we also discuss the increase of the renormalized electron mass and determine the linear leading order term in the asymptotics of the self-energy, as the ultra-violet cut-off goes to infinity.

Hand gesture recognition based on bag of features and support vector machine

According to the influence of approximate skin color information or complex background,it is hard to get precise gesture contour by hand gesture segmentation,which will have effect on later gesture recognition rate and real-time interaction.Therefore,this paper proposed a gesture recognition method based on the BOF-SVM(Bag Of Features-Support Vector Machine).At first,local invariant features of the gesture images were extracted by the Scale Invariant Feature Transformation(SIFT) algorithm.Then the visual code book was generated by gesture local eigenvector(SIFT descriptors) through K-means clustering.And visual code set of every image got quantized by visual code book.As a result,the characterized vector of gesture images with fixed dimensional was obtained to train multi-class SVM classifier.This method only needed to frame the gesture area instead of segmenting gesture accurately.The experimental results indicate that the average recognition rate of the nine interactive hand gestures based on this method can reach 92.1%.Besides,it has good robustness and efficiency,and can adapt to the changes of environment.

Localization and delocalization of eigenvectors for heavy-tailed random matrices

Consider an \(n \times n\) Hermitian random matrix with, above the diagonal, independent entries with \(\alpha \)-stable symmetric distribution and \(0 < \alpha < 2\). We establish new bounds on the rate of convergence of the empirical spectral distribution of this random matrix as \(n\) goes to infinity. When \(1 < \alpha < 2\) and \( p > 2\), we give vanishing bounds on the \(L^p\)-norm of the eigenvectors normalized to have unit \(L^2\)-norm. On the contrary, when \(0 < \alpha < 2/3\), we prove that these eigenvectors are localized.

Localization of eigenvectors in random graphs

Using exact numerical diagonalization, we investigate localization in two classes of random matrices corresponding to random graphs. The first class comprises the adjacency matrices of Erdős-Rényi (ER) random graphs. The second one corresponds to random cubic graphs, with Gaussian random variables on the diagonal. We establish the position of the mobility edge, applying the finite-size analysis of the inverse participation ratio. The fraction of localized states is rather small on the ER graphs and decreases when the average degree increases. On the contrary, on cubic graphs the fraction of localized states is large and tends to 1 when the strength of the disorder increases, implying that for sufficiently strong disorder all states are localized. The distribution of the inverse participation ratio in localized phase has finite width when the system size tends to infinity and exhibits complicated multi-peak structure. We also confirm that the statistics of level spacings is Poissonian in the localized regime, while for extended states it corresponds to the Gaussian orthogonal ensemble.