Eigenvector Localization Research Articles

Altered brain connectivity, neuroinflammation and AD‐related markers following 7 days after repetitive mild traumatic brain injury in young wild‐type mice

AbstractBackgroundTraumatic brain injuries (TBI) substantially increase the risk for neurodegenerative disorders including tauopathies and synucleinopathies. Repetitive mild TBI (rmTBI) exacerbate damage in brain impacting cognitive and emotional processes through lifetime. Therefore, there is a critical need to identify the molecular mechanisms underlying TBI‐mediated brain dysfunction. In doing so, we can develop effective therapies as the etiology for dementia remain poorly understood. Our goal is to identify early, unbiased, and novel biomarkers of brain dysfunction following rmTBI. These include defects in the brain connectome and protein expression in the areas most functionally susceptible to damage following rmTBI.MethodTo do this, we exposed young male and female C57BL/6J wild‐type mice to two rmTBI (2 × 0.6 J impacts/24 h apart) using the closed head injury model of engineered rotational acceleration (CHIMERA) or sham procedures. We measured changes in brain functional connectivity (FC) using resting state functional MRI (rsfMRI) and graph theory, examined microstructural differences in white and grey matter regions through diffusion tensor imaging (DTI) analyses, surveyed neuropathological proteins using NanoString‐GeoMx Digital spatial protein profiling spatial profiling (DSP).ResultThe rsfMRI data revealed that rmTBI induces aberrant changes in the graph metrics such as node clustering coefficient, global and local efficiency, participation coefficient, eigenvector centrality, and betweenness centrality in brain regions processing visual, auditory, and somatosensory information. The optic tract and thalamus were the most affected regions showing significant differences in phospho‐tau (S199), neuroinflammatory markers (GFAP, Iba‐1, GPNMB), and other proteolytic (cathepsin‐D) and proliferation (ki‐67) cell markers. We identified extensive white matter damage, defects in brain FC, and exacerbated anatomically distinct protein profiles.ConclusionThese findings align with results reported in individuals with TBI such as diffuse axonal damage, glial activation, and functional network disruption in the visual tract and specific thalamic regions. Upcoming studies will evaluate the potential use of using brain imaging along with molecular markers to predict longitudinal damage after rmTBI.

OS08.7.A DEPRESSIVE SYMPTOMS IN ASSOCIATION WITH TUMOR LOCATIONS AT DIAGNOSIS IN PATIENTS WITH DIFFUSE GLIOMA

Abstract BACKROUND Depressive symptoms are common in glioma patients and occur more frequent than in patients with other cancer types. The presence of tumors in specific brain regions that potentially disturb brain functions could contribute to this difference. These lesions may cause an inadequate emotional response to an incurable and lethal disease with consequences for patient counseling and neurosurgical strategies. In this study, we examine the association between brain tumor location and depressive symptoms at presentation in patients with diffuse glioma using several methods for lesion symptom mapping. MATERIAL AND METHODS The retrospective cohort consisted of patients with diffuse glioma who had a preoperative MR-scan and completed the Center for Epidemiologic Studies Depression scale (CES-D) before treatment. We classified patients based on unexpectedly low ( ≤6), intermediate (7-20) and unexpectedly high (>20) CES-D scores. Tumors were segmented in 3D and corresponding white matter disconnectomes were derived from normative tractography. We first correlated tumor and disconnectome with CES-D using sparse canonical correlation analysis to identify clusters of associated brain voxels. Next, we assessed the association between tumor locations and CES-D category using Bayesian multivariable categorical regression analysis of patient and tumor characteristics, including the overlap of tumors and disconnectomes with specific brain atlas structures, the probability of disconnection of white matter tracts, and functional network changes in local efficiency and eigenvector centrality. RESULTS We included 201 patients with a median age of 45 years (range 18-81), consisting of 62% males and 29% grade 4 gliomas. CES-D was completed a median of three weeks before surgery, was unexpectedly low in 29% and unexpectedly high in 15% of patients. Tumors located in the left (L) pallidum and right (R) thalamus were related to unexpectedly low CES-D, whereas tumors in the L hippocampus, L nucleus accumbens and R putamen were related to unexpectedly high CES-D. Disconnection of the R temporal subsection of the cingulum bundle, L optic radiation, L superior longitudinal fasciculus I and L uncinate fasciculus (UF) were related to unexpectedly low CES-D, whereas disconnection of the R corticospinal tract, R fornix, and L UF were related to unexpectedly high CES-D. Neither voxel clusters nor network measures were associated with CES-D. CONCLUSION Both unexpectedly low and unexpectedly high depressive symptom burden is common in patients with diffuse glioma at presentation. These dysfunctional emotional responses may be associated with the location of the tumor. Awareness of this phenomenon may be important in patient counseling and neurosurgical strategies. * these authors contributed equally

Localization of eigenvectors of nonhermitian banded noisy Toeplitz matrices

Localization of eigenvectors of nonhermitian banded noisy Toeplitz matrices

Eigenvector localization in hypergraphs: Pairwise versus higher-order links.

Localization behaviors of Laplacian eigenvectors of complex networks furnish an explanation to various dynamical phenomena of the corresponding complex systems. We numerically examine roles of higher-order and pairwise links in driving eigenvector localization of hypergraphs Laplacians. We find that pairwise interactions can engender localization of eigenvectors corresponding to small eigenvalues for some cases, whereas higher-order interactions, even being much much less than the pairwise links, keep steering localization of the eigenvectors corresponding to larger eigenvalues for all the cases considered here. These results will be advantageous to comprehend dynamical phenomena, such as diffusion, and random walks on a range of real-world complex systems having higher-order interactions in better manner.

An algorithm for identifying eigenvectors exhibiting strong spatial localization

We introduce an approach for exploring eigenvector localization phenomena for a class of (unbounded) selfadjoint operators. More specifically, given a target region and a tolerance, the algorithm identifies candidate eigenpairs for which the eigenvector is expected to be localized in the target region to within that tolerance. Theoretical results, together with detailed numerical illustrations of them, are provided that support our algorithm. A partial realization of the algorithm is described and tested, providing a proof of concept for the approach.

Resting-state functional connectivity patterns are associated with metal mixture exposure in young adolescents

Background and Aim. Early-life metal mixture exposure may increase the risk for anxiety and depression in adolescence. The neural circuitry subserving internalizing phenotypes begins developing in utero and is vulnerable to early-life exposures. Resting state functional magnetic resonance imaging (rs-fMRI) provides a novel tool to examine underlying mechanisms of metal-associated neurodevelopmental outcomes. We investigate relationships between early-life metal exposure and functional connectivity in pre-adolescents. Methods. In preliminary analysis of 67 children (8-14years; 34 females) enrolled in the Programming Research in Obesity, Growth, Environment and Social Stressors (PROGRESS) study, we estimated weekly exposure (14th week gestation through one year of age) to 15 metals (Ba, Bi, Cd, Co, Cr, Cu, Li, Mg, Mn, Mo, Ni, Pb, Sn, Sr, Zn) in deciduous teeth using laser ablation-inductively coupled plasma-mass spectrometry. Using graph theory analysis of rs-fMRI data, we computed global and local efficiency (GE, LE) and eigenvector centrality (EC) in 111 brain areas (Harvard Oxford Atlas). We used lagged weighted quantile sum (lWQS) regression to examine time-varying associations between metal mixtures and GE, LE or EC in the whole brain and anterior cingulate cortex (ACC), globus pallidus (GP), and insula, adjusting for sex and age. Results. We observed prenatal and postnatal windows between the metal mixture and LE; -13 to 4 weeks, peak βLE (week -2) = -0.23, 95%CI -0.02, -0.45 and 16 to 43 weeks; peak βLE (week 42) = -0.42, 95%CI -0.13, -0.70 and one prenatal window for GE at -18 to -11 weeks, peak βGE (week -18) = -0.35, 95%CI -0.07, -0.64. The metal mixture was significantly associated with decreased EC in the ACC, GP and insula at different critical windows spanning gestation and infancy. Conclusions. Our results combining lWQS modeling with graph theory analysis of rs-fMRI may inform mechanistic understanding of environmentally-associated neurodevelopmental outcomes.

PL01.4.A The relationship between neuronal activity and functional network properties in glioma patients

Abstract Background Glioma is associated with pathologically high neuronal activity around the tumor, which associates with faster tumor progression in patients. Concurrently, patients with glioma have local and widespread disturbances of the functional brain network as measured with magnetoencephalography (MEG) such as higher network clustering (the extent to which regions connected to a particular area are also connected to each other) and locally altered integrative connectivity (for instance assessed with a centrality measure). How local neuronal activity and nodal network properties relate to each other has yet to be investigated. Material and Methods We obtained eyes-closed resting-state MEG in 95 glioma patients before tumor resection and 57 matched healthy controls (HC). Broadband power (BBP, 0.5-48Hz) was used as a proxy for neuronal activity. The local clustering coefficient (CC) and eigenvector centrality (EC) were calculated for the functional network in the lower frequency bands (delta, theta, alpha). We compared averaged nodal network properties and local BBP between peritumoral (regions partially overlapping with the tumor) and contralateral homologue regions in patients and compared these to averaged values in HC. Linear mixed models were used to relate nodal CC and EC to local BBP. Results were corrected for multiple comparisons and deemed significant at alpha 0.05. Results The peritumoral area was significantly more active than the non-peritumoral homologue in patients, and showed pathologically higher activity in comparison to HCs. Pathologically high neuronal activity was restricted to the peritumoral areas and not found in the contralateral homologue. However, patients’ functional network was disturbed throughout the brain in terms of pathologically high clustering, but not with regards to centrality. Whereas high activity related to high centrality in HCs and patients alike, high activity seemed to relate to low clustering in homologue regions in patients but not HCs, potentially indicating that lower neuronal activity relates to pathologically high clustering in these patients. Conclusion Glioma patients show pathologically high brain activity around the tumor and whole-brain disturbances in local network clustering, while integrative connectivity is preserved. We also find that the relationship between neuronal activity and functional network properties is complex in these patients, further underlining the importance of investigating how local activity may impact global network topology in order to understand how neuron-glioma interactions shape brain functioning.

Hierarchical timescales in the neocortex: Mathematical mechanism and biological insights

A cardinal feature of the neocortex is the progressive increase of the spatial receptive fields along the cortical hierarchy. Recently, theoretical and experimental findings have shown that the temporal response windows also gradually enlarge, so that early sensory neural circuits operate on short timescales whereas higher-association areas are capable of integrating information over a long period of time. While an increased receptive field is accounted for by spatial summation of inputs from neurons in an upstream area, the emergence of timescale hierarchy cannot be readily explained, especially given the dense interareal cortical connectivity known in the modern connectome. To uncover the required neurobiological properties, we carried out a rigorous analysis of an anatomically based large-scale cortex model of macaque monkeys. Using a perturbation method, we show that the segregation of disparate timescales is defined in terms of the localization of eigenvectors of the connectivity matrix, which depends on three circuit properties: 1) a macroscopic gradient of synaptic excitation, 2) distinct electrophysiological properties between excitatory and inhibitory neuronal populations, and 3) a detailed balance between long-range excitatory inputs and local inhibitory inputs for each area-to-area pathway. Our work thus provides a quantitative understanding of the mechanism underlying the emergence of timescale hierarchy in large-scale primate cortical networks.

Correlates and predictors of the severity of suicidal ideation in adolescence: an examination of brain connectomics and psychosocial characteristics.

Suicidal ideation (SI) typically emerges during adolescence but is challenging to predict. Given the potentially lethal consequences of SI, it is important to identify neurobiological and psychosocial variables explaining the severity of SI in adolescents. In 106 participants (59 female) recruited from the community, we assessed psychosocial characteristics and obtained resting-state fMRI data in early adolescence (baseline: aged 9-13 years). Across 250 brain regions, we assessed local graph theory-based properties of interconnectedness: local efficiency, eigenvector centrality, nodal degree, within-module z-score, and participation coefficient. Four years later (follow-up: ages 13-19 years), participants self-reported their SI severity. We used least absolute shrinkage and selection operator (LASSO) regressions to identify a linear combination of psychosocial and brain-based variables that best explain the severity of SI symptoms at follow-up. Nested-cross-validation yielded model performance statistics for all LASSO models. A combination of psychosocial and brain-based variables explained subsequent severity of SI (R2 = .55); the strongest was internalizing and externalizing symptom severity at follow-up. Follow-up LASSO regressions of psychosocial-only and brain-based-only variables indicated that psychosocial-only variables explained 55% of the variance in SI severity; in contrast, brain-based-only variables performed worse than the null model. A linear combination of baseline and follow-up psychosocial variables best explained the severity of SI. Follow-up analyses indicated that graph theory resting-state metrics did not increase the prediction of the severity of SI in adolescents. Attending to internalizing and externalizing symptoms is important in early adolescence; resting-state connectivity properties other than local graph theory metrics might yield a stronger prediction of the severity of SI.

The effective potential of an M-matrix

In the presence of a confining potential V, the eigenfunctions of a continuous Schrödinger operator −Δ + V decay exponentially with the rate governed by the part of V, which is above the corresponding eigenvalue; this can be quantified by a method of Agmon. Analogous localization properties can also be established for the eigenvectors of a discrete Schrödinger matrix. This note shows, perhaps surprisingly, that one can replace a discrete Schrödinger matrix by any real symmetric Z-matrix and still obtain eigenvector localization estimates. In the case of a real symmetric non-singular M-matrix A (which is a situation that arises in several contexts, including random matrix theory and statistical physics), the landscape function u = A−11 plays the role of an effective potential of localization. Starting from this potential, one can create an Agmon-type distance function governing the exponential decay of the eigenfunctions away from the “wells” of the potential, a typical eigenfunction being localized to a single such well.

Master equation analysis of mesoscopic localization in contagion dynamics on higher-order networks.

Simple models of infectious diseases tend to assume random mixing of individuals, but real interactions are not random pairwise encounters: they occur within various types of gatherings such as workplaces, households, schools, and concerts, best described by a higher-order network structure. We model contagions on higher-order networks using group-based approximate master equations, in which we track all states and interactions within a group of nodes and assume a mean-field coupling between them. Using the susceptible-infected-susceptible dynamics, our approach reveals the existence of a mesoscopic localization regime, where a disease can concentrate and self-sustain only around large groups in the network overall organization. In this regime, the phase transition is smeared, characterized by an inhomogeneous activation of the groups. At the mesoscopic level, we observe that the distribution of infected nodes within groups of the same size can be very dispersed, even bimodal. When considering heterogeneous networks, both at the level of nodes and at the level of groups, we characterize analytically the region associated with mesoscopic localization in the structural parameter space. We put in perspective this phenomenon with eigenvector localization and discuss how a focus on higher-order structures is needed to discern the more subtle localization at the mesoscopic level. Finally, we discuss how mesoscopic localization affects the response to structural interventions and how this framework could provide important insights for a broad range of dynamics.

Localization and Universality of Eigenvectors in Directed Random Graphs.

Although the spectral properties of random graphs have been a long-standing focus of network theory, the properties of right eigenvectors of directed graphs have so far eluded an exact analytic treatment. We present a general theory for the statistics of the right eigenvector components in directed random graphs with a prescribed degree distribution and with randomly weighted links. We obtain exact analytic expressions for the inverse participation ratio and show that right eigenvectors of directed random graphs with a small average degree are localized. Remarkably, if the fourth moment of the degree distribution is finite, then the critical mean degree of the localization transition is independent of the degree fluctuations, which is different from localization in undirected graphs that is governed by degree fluctuations. We also show that in the high connectivity limit the distribution of the right eigenvector components is solely determined by the degree distribution. For delocalized eigenvectors, we recover in this limit the universal results from standard random matrix theory that are independent of the degree distribution, while for localized eigenvectors the eigenvector distribution depends on the degree distribution.

Achieving spectral localization of network using betweenness-based edge perturbation

Graph is a simple but effective way to represent a complex system, where a node represents a component of the system and edge represents connection between the components. Several insights can be inferred by analyzing such graphs. In this field, optimization of spread and localization are relatively a new research domain. The objective of the spread problem is to maximize the influence, whereas localization controls the diffusion. In this paper, our focus is on the eigenvector localization of the network adjacency matrix using inverse participation ratio (IPR). In this context, we propose betweenness centrality-based perturbation (BP) to localize the network. The results show that the BP approach achieves a better localization than the existing random perturbation (RP) approach. It shows maximum IPR than RP. The performance of the approaches is evaluated using threshold rate of diffusion (τ), number of modifications and IPR. Susceptible–infected–susceptible model is used to investigate the τ value.

Jaccard Guided Perturbation for Eigenvector Localization

Understanding the role of network for epidemic spread is a major concern in the study of complex network and disease spread. Localization of Principal EigenVector (PEV) of the network’s adjacency matrix has been explored in recent past by many researchers. Most of the previous works suggest the localization of PEV through maximization of inverse participation ratio (IPR). The network with maximum IPR leads to maximum PEV localization. Random perturbation (RP) is one of the methods reported recently that performs the random perturbation in the original graph to obtain a perturbed graph that is more localized than the original graph, with same number of edges as the original graph. In this paper, we propose a Jaccard Guided Perturbation (JGP) method that perturbs the graph by using Jaccard’s coefficient. We consider both random and scale-free networks for the experiment. SIS model is used to implement the spread of epidemic. The results show that the proposed JGP approach achieve better PEV localization than the existing RP method. The number of modifications is found to be less in JGP in comparison to RP. The threshold rate of spread is found to be more in JGP than RP. In scale-free networks, JGP shows a better preservation of scale-free property than that of RP.

A Random Matrix Theory Approach to Denoise Single-Cell Data

SummarySingle-cell technologies provide the opportunity to identify new cellular states. However, a major obstacle to the identification of biological signals is noise in single-cell data. In addition, single-cell data are very sparse. We propose a new method based on random matrix theory to analyze and denoise single-cell sequencing data. The method uses the universal distributions predicted by random matrix theory for the eigenvalues and eigenvectors of random covariance/Wishart matrices to distinguish noise from signal. In addition, we explain how sparsity can cause spurious eigenvector localization, falsely identifying meaningful directions in the data. We show that roughly 95% of the information in single-cell data is compatible with the predictions of random matrix theory, about 3% is spurious signal induced by sparsity, and only the last 2% reflects true biological signal. We demonstrate the effectiveness of our approach by comparing with alternative techniques in a variety of examples with marked cell populations.

Susceptible-infected-susceptible model on networks with eigenvector localization.

It is a longstanding debate on the absence of threshold for susceptible-infected-susceptible (SIS) model on networks with finite second order moment of degree distribution. The eigenvector localization of the adjacency matrix for a network gives rise to the inactive Griffiths phase featuring slow decay of the activity localized around highly connected nodes due to the dynamical fluctuation. We show how it dramatically changes our understanding of the SIS model, opening up new possibilities for the debate. We derive the critical condition for Griffiths to active phase transition: on average, an infected node can further infect another one in the characteristic lifespan of the star subgraph composed of the node and its nearest neighbors. The system approaches the critical point of avoiding the irreversible dynamical fluctuation and the trap of absorbing state. As a signature of the phase transition, the infection density of a node is not only proportional to its degree, but also proportional to the exponentially growing lifespan of the star. And the divergence of the average lifespan of the stars is responsible for the vanishing threshold in the thermodynamic limit. The eigenvector localization exponentially reinforces the infection of highly connected nodes, while it inversely suppresses the infection of small-degree nodes.

Wheel graph strategy for PEV localization of networks

Investigation of eigenvector localization properties of complex networks is not only important for gaining insight into fundamental network problems such as network centrality measure, spectral partitioning, development of approximation algorithms, but is also crucial for understanding many real-world phenomena such as disease spreading, criticality in brain network dynamics. For a network, an eigenvector is said to be localized when most of its components take value near to zero, with a few components taking very high values. In this article, we devise a methodology to construct a principal eigenvector (PEV) localized network from a given input network. The methodology relies on adding a small component having a wheel graph to the given input network. By extensive numerical simulation and an analytical formulation based on the largest eigenvalue of the input network, we compute the size of the wheel graph required to localize the PEV of the combined network. Using the susceptible-infected-susceptible model, we demonstrate the success of this method for various models and real-world networks considered as input networks. We show that on such PEV localized networks, the disease gets localized within a small region of the network structure before the outbreaks. The study is relevant in controlling spreading processes on complex systems represented by networks.

Principal eigenvector localization and centrality in networks: Revisited

Complex networks or graphs provide a powerful framework to understand importance of individuals and their interactions in real-world complex systems. Several graph theoretical measures have been introduced to access importance of the individual in systems represented by networks. Particularly, eigenvector centrality (EC) measure has been very popular due to its ability in measuring importance of the nodes based on not only number of interactions they acquire but also particular structural positions they have in the networks. Furthermore, the presence of certain structural features, such as the existence of high degree nodes in a network is recognized to induce localization transition of the principal eigenvector (PEV) of the network’s adjacency matrix. Localization of PEV has been shown to cause difficulties in assigning centrality weights to the nodes based on the EC. We revisit PEV localization and its relation with failure of EC problem, and by using simple model networks demonstrate that in addition to the localization of the PEV, the delocalization of PEV may also create difficulties for using EC as a measure to rank the nodes. Our investigation while providing fundamental insight to the relation between PEV localization and centrality of nodes in networks, suggests that for the networks having delocalized PEVs, it is better to use degree centrality measure to rank the nodes.

Geometrical and spectral study of β-skeleton graphs.

We perform an extensive numerical analysis of β-skeleton graphs, a particular type of proximity graphs. In a β-skeleton graph (BSG) two vertices are connected if a proximity rule, that depends of the parameter β∈(0,∞), is satisfied. Moreover, for β>1 there exist two different proximity rules, leading to lune-based and circle-based BSGs. First, by computing the average degree of large ensembles of BSGs we detect differences, which increase with the increase of β, between lune-based and circle-based BSGs. Then, within a random matrix theory (RMT) approach, we explore spectral and eigenvector properties of random BSGs by the use of the nearest-neighbor energy-level spacing distribution and the entropic eigenvector localization length, respectively. The RMT analysis allows us to conclude that a localization transition occurs at β=1.

Eigenvalue repulsion and eigenvector localization in sparse non-Hermitian random matrices.

Complex networks with directed, local interactions are ubiquitous in nature and often occur with probabilistic connections due to both intrinsic stochasticity and disordered environments. Sparse non-Hermitian random matrices arise naturally in this context and are key to describing statistical properties of the nonequilibrium dynamics that emerges from interactions within the network structure. Here we study one-dimensional (1D) spatial structures and focus on sparse non-Hermitian random matrices in the spirit of tight-binding models in solid state physics. We first investigate two-point eigenvalue correlations in the complex plane for sparse non-Hermitian random matrices using methods developed for the statistical mechanics of inhomogeneous two-dimensional interacting particles. We find that eigenvalue repulsion in the complex plane directly correlates with eigenvector delocalization. In addition, for 1D chains and rings with both disordered nearest-neighbor connections and self-interactions, the self-interaction disorder tends to decorrelate eigenvalues and localize eigenvectors more than simple hopping disorder. However, remarkable resistance to eigenvector localization by disorder is provided by large cycles, such as those embodied in 1D periodic boundary conditions under strong directional bias. The directional bias also spatially separates the left and right eigenvectors, leading to interesting dynamics in excitation and response. These phenomena have important implications for asymmetric random networks and highlight a need for mathematical tools to describe and understand them analytically.