Tensor Graph Research Articles

Structure Tensor Graph Searches Based Fully Automated Grading and 3D Profiling of Maculopathy From Retinal OCT Images

Maculopathy is a collective group of diseases that damages the central region of a retina known as macula. The major two forms of maculopathy are macular edema (ME) and central serous chorioretinopathy (CSCR). Different researchers have worked on the identification of these macular disorders using optical coherence tomography (OCT) images. However, to the best of our knowledge, no research is reported until now that can automatically extract retinal information for the grading of ME and CSCR as per clinically significant ME (CSME), non-clinically significant ME (non-CSME), Type-I CSCR, and Type-II CSCR clinical standards from OCT images. Therefore, this paper presents a novel structure tensor graph searches (ST-GS)-based segmentation framework that combines a structure tensor and graph theory to automatically extract retinal and choroidal layers along with fluid pores followed by automated reconstruction of 3-D retinal surfaces. The ST-GS can extract retinal information even from highly degraded OCT scans. Furthermore, the proposed system automatically grades ME and CSCR pathologies as CSME and non-CSME and Type-I CSCR and Type-II CSCR, respectively. The proposed system extracts seven distinct features, and it is trained on 30 (10 healthy, 10 ME, and 10 CSCR) labeled OCT volumes containing 3840 brightness scans (B-scans). After that, 90 (30 healthy, 30 CSCR, and 30 ME) unlabeled OCT volumes containing 11520 B-scans were used for testing the proposed system, where it correctly classified 88 out of 90 cases with the sensitivity, specificity, and accuracy ratings of 96.77%, 100%, and 97.78%, respectively. Furthermore, the proposed system has achieved a mean dice coefficient of 0.875±0.0342 and 0.92±0.0258 for extracting cyst and serous fluids, respectively.

The 1/N Expansion of Tensor Models with Two Symmetric Tensors

It is well known that tensor models for a tensor with no symmetry admit a $1/N$ expansion dominated by melonic graphs. This result relies crucially on identifying \emph{jackets} which are globally defined ribbon graphs embedded in the tensor graph. In contrast, no result of this kind has so far been established for symmetric tensors because global jackets do not exist. In this paper we introduce a new approach to the $1/N$ expansion in tensor models adapted to symmetric tensors. In particular we do not use any global structure like the jackets. We prove that, for any rank $D$, a tensor model with two symmetric tensors and interactions the complete graph $K_{D+1}$ admits a $1/N$ expansion dominated by melonic graphs.

Graph operations and upper bounds on graph homomorphism counts

AbstractWe construct a family of countexamples to a conjecture of Galvin [5], which stated that for any n‐vertex, d‐regular graph G and any graph H (possibly with loops), urn:x-wiley:03649024:media:jgt22148:jgt22148-math-0001where is the number of homomorphisms from G to H. By exploiting properties of the graph tensor product and graph exponentiation, we also find new infinite families of H for which the bound stated above on holds for all n‐vertex, d‐regular G. In particular, we show that if HWR is the complete looped path on three vertices, also known as the Widom–Rowlinson graph, then urn:x-wiley:03649024:media:jgt22148:jgt22148-math-0004for all n‐vertex, d‐regular G. This verifies a conjecture of Galvin.

Tensor network states with three-site correlators

We present a detailed analysis of various tensor network parameterizations within the complete graph tensor network states (CGTNS) approach. We extend our 2-site CGTNS scheme by introducing 3-site correlators. For this we devise three different strategies. The first relies solely on 3-site correlators and the second on 3-site correlators added on top of the 2-site correlator ansatz. To avoid an inflation of the variational space introduced by higher-order correlators, we limit the number of higher-order correlators to the most significant ones in the third strategy. Approaches for the selection of these most significant correlators are discussed. The sextet and doublet spin states of the spin-crossover complex manganocene serve as a numerical test case. In general, the CGTNS scheme achieves a remarkable accuracy for a significantly reduced size of the variational space. The advantages, drawbacks, and limitations of all CGTNS parameterizations investigated are rigorously discussed.

Structural Connectivity Changes in The Brain After Aerobic Training in Adolescents with Persistent Symptoms After Mild TBI

To combine diffusion tensor imaging (DTI) and graph theoretical analysis to characterize abnormalities in structural connectivity in the brain network of adolescents with persistent symptoms after mild traumatic brain injury (mTBI).

On the dynamics of endomorphisms of finite groups

Aiming at a better understanding of finite groups as finite dynamical systems, we show that by a version of Fitting’s Lemma for groups, each state space of an endomorphism of a finite group is a graph tensor product of a finite directed 1-tree whose cycle is a loop with a disjoint union of cycles, generalizing results of Hernández-Toledo on linear finite dynamical systems, and we fully characterize the possible forms of state spaces of nilpotent endomorphisms via their “ramification behavior”. Finally, as an application, we will count the isomorphism types of state spaces of endomorphisms of finite cyclic groups in general, extending results of Hernández-Toledo on primary cyclic groups of odd order.

O(N) Random Tensor Models

We define in this paper a class of three indices tensor models, endowed with $O(N)^{\otimes 3}$ invariance ($N$ being the size of the tensor). This allows to generate, via the usual QFT perturbative expansion, a class of Feynman tensor graphs which is strictly larger than the class of Feynman graphs of both the multi-orientable model (and hence of the colored model) and the $U(N)$ invariant models. We first exhibit the existence of a large $N$ expansion for such a model with general interactions. We then focus on the quartic model and we identify the leading and next-to-leading order (NLO) graphs of the large $N$ expansion. Finally, we prove the existence of a critical regime and we compute the critical exponents, both at leading order and at NLO. This is achieved through the use of various analytic combinatorics techniques.

Characterizing structural neural networks in de novo Parkinson disease patients using diffusion tensor imaging.

Parkinson disease (PD) can be considered as a brain multisystemic disease arising from dysfunction in several neural networks. The principal aim of this study was to assess whether large-scale structural topological network changes are detectable in PD patients who have not been exposed yet to dopaminergic therapy (de novo patients). Twenty-one drug-naïve PD patients and thirty healthy controls underwent a 3T structural MRI. Next, Diffusion Tensor Imaging (DTI) and graph theoretic analyses to compute individual structural white-matter (WM) networks were combined. Centrality (degree, eigenvector centrality), segregation (clustering coefficient), and integration measures (efficiency, path length) were assessed in subject-specific structural networks. Moreover, Network-based statistic (NBS) was used to identify whether and which subnetworks were significantly different between PD and control participants. De novo PD patients showed decreased clustering coefficient and strength in specific brain regions such as putamen, pallidum, amygdala, and olfactory cortex compared with healthy controls. Moreover, NBS analyses demonstrated that two specific subnetworks of reduced connectivity characterized the WM structural organization of PD patients. In particular, several key pathways in the limbic system, basal ganglia, and sensorimotor circuits showed reduced patterns of communications when comparing PD patients to controls. This study shows that PD is characterized by a disruption in the structural connectivity of several motor and non-motor regions. These findings provide support to the presence of disconnectivity mechanisms in motor (basal ganglia) as well as in non-motor (e.g., limbic, olfactory) circuits at an early disease stage of PD. Hum Brain Mapp 37:4500-4510, 2016. © 2016 Wiley Periodicals, Inc.

Square-free graphs are multiplicative

A graph K is square-free if it contains no four-cycle as a subgraph (i.e., for every quadruple of vertices, if v0v1,v1v2,v2v3,v3v0∈E(K), then v0=v2 or v1=v3). A graph K is multiplicative if G×H→K implies G→K or H→K, for all graphs G, H. Here G×H is the tensor (or categorical) graph product and G→K denotes the existence of a graph homomorphism from G to K. Hedetniemi's conjecture, which states that χ(G×H)=min⁡(χ(G),χ(H)), is equivalent to the statement that all cliques Kn are multiplicative. However, the only non-trivial graphs known to be multiplicative are K3, odd cycles, and still more generally, circular cliques Kp/q with 2≤pq<4. We make no progress for cliques, but show that all square-free graphs are multiplicative. In particular, this gives the first multiplicative graphs of chromatic number higher than 4. Generalizing, in terms of the box complex, the topological insight behind existing proofs for odd cycles, we also give a different proof for circular cliques with 2≤pq<4.

Abnormal organization of white matter networks in patients with subjective cognitive decline and mild cognitive impairment

Network analysis has been widely used in studying Alzheimer's disease (AD). However, how the white matter network changes in cognitive impaired patients with subjective cognitive decline (SCD) (a symptom emerging during early stage of AD) and amnestic mild cognitive impairment (aMCI) (a pre-dementia stage of AD) is still unclear. Here, structural networks were constructed respectively based on FA and FN for 36 normal controls, 21 SCD patients, and 33 aMCI patients by diffusion tensor imaging and graph theory. Significantly lower efficiency was found in aMCI patients than normal controls (NC). Though not significant, the values in those with SCD were intermediate between aMCI and NC. In addition, our results showed significantly altered betweenness centrality located in right precuneus, calcarine, putamen, and left anterior cingulate in aMCI patients. Furthermore, association was found between network metrics and cognitive impairment. Our study suggests that the structural network properties might be preserved in SCD stage and disrupted in aMCI stage, which may provide novel insights into pathological mechanisms of AD.

Abnormal topological organization of the white matter network in Mandarin speakers with congenital amusia.

Congenital amusia is a neurogenetic disorder that mainly affects the processing of musical pitch. Brain imaging evidence indicates that it is associated with abnormal structural and functional connections in the fronto-temporal region. However, a holistic understanding of the anatomical topology underlying amusia is still lacking. Here, we used probabilistic diffusion tensor imaging tractography and graph theory to examine whole brain white matter structural connectivity in 31 Mandarin-speaking amusics and 24 age- and IQ-matched controls. Amusics showed significantly reduced global connectivity, as indicated by the abnormally decreased clustering coefficient (Cp) and increased normalized shortest path length (λ) compared to the controls. Moreover, amusics exhibited enhanced nodal strength in the right inferior parietal lobule relative to controls. The co-existence of the lexical tone deficits was associated with even more deteriorated global network efficiency in amusics, as suggested by the significant correlation between the increments in normalized shortest path length (λ) and the insensitivity in lexical tone perception. Our study is the first to reveal reduced global connectivity efficiency in amusics as well as an increase in the global connectivity cost due to the co-existed lexical tone deficits. Taken together these results provide a holistic perspective on the anatomical substrates underlying congenital amusia.

Left hemisphere structural connectivity abnormality in pediatric hydrocephalus patients following surgery

Neuroimaging research in surgically treated pediatric hydrocephalus patients remains challenging due to the artifact caused by programmable shunt. Our previous study has demonstrated significant alterations in the whole brain white matter structural connectivity based on diffusion tensor imaging (DTI) and graph theoretical analysis in children with hydrocephalus prior to surgery or in surgically treated children without programmable shunts. This study seeks to investigate the impact of brain injury on the topological features in the left hemisphere, contratelateral to the shunt placement, which will avoid the influence of shunt artifacts and makes further group comparisons feasible for children with programmable shunt valves. Three groups of children (34 in the control group, 12 in the 3-month post-surgery group, and 24 in the 12-month post-surgery group, age between 1 and 18 years) were included in the study. The structural connectivity data processing and analysis were performed based on DTI and graph theoretical analysis. Specific procedures were revised to include only left brain imaging data in normalization, parcellation, and fiber counting from DTI tractography. Our results showed that, when compared to controls, children with hydrocephalus in both the 3-month and 12-month post-surgery groups had significantly lower normalized clustering coefficient, lower small-worldness, and higher global efficiency (all p < 0.05, corrected). At a regional level, both patient groups showed significant alteration in one or more regional connectivity measures in a series of brain regions in the left hemisphere (8 and 10 regions in the 3-month post-surgery and the 12-month post-surgery group, respectively, all p < 0.05, corrected). No significant correlation was found between any of the global or regional measures and the contemporaneous neuropsychological outcomes [the General Adaptive Composite (GAC) from the Adaptive Behavior Assessment System, Second Edition (ABAS-II)]. However, one global network measure (global efficiency) and two regional network measures in the insula (local efficiency and between centrality) tested at 3-month post-surgery were found to correlate with GAC score tested at 12-month post-surgery with statistical significance (all p < 0.05, corrected). Our data showed that the structural connectivity analysis based on DTI and graph theory was sensitive in detecting both global and regional network abnormality when the analysis was conducted in the left hemisphere only. This approach provides a new avenue enabling the application of advanced neuroimaging analysis methods in quantifying brain damage in children with hydrocephalus surgically treated with programmable shunts.

Renormalization and Hopf algebraic structure of the five-dimensional quartic tensor field theory

This paper is devoted to the study of renormalization of the quartic melonic tensor model in dimension (=rank) five. We review the perturbative renormalization and the computation of the one loop beta function, confirming the asymptotic freedom of the model. We then define the Connes–Kreimer-like Hopf algebra describing the combinatorics of the renormalization of this model and we analyze in detail, at one- and two-loop levels, the Hochschild cohomology allowing to write the combinatorial Dyson–Schwinger equations. Feynman tensor graph Hopf subalgebras are also exhibited.

Brain network alterations and vulnerability to simulated neurodegeneration in breast cancer

Breast cancer and its treatments are associated with mild cognitive impairment and brain changes that could indicate an altered or accelerated brain aging process. We applied diffusion tensor imaging and graph theory to measure white matter organization and connectivity in 34 breast cancer survivors compared with 36 matched healthy female controls. We also investigated how brain networks (connectomes) in each group responded to simulated neurodegeneration based on network attack analysis. Compared with controls, the breast cancer group demonstrated significantly lower fractional anisotropy, altered small-world connectome properties, lower brain network tolerance to systematic region (node), and connection (edge) attacks and significant cognitive impairment. Lower tolerance to network attack was associated with cognitive impairment in the breast cancer group. These findings provide further evidence of diffuse white matter pathology after breast cancer and extend the literature in this area with unique data demonstrating increased vulnerability of the post-breast cancer brain network to future neurodegenerative processes.

Aberrant white matter networks mediate cognitive impairment in patients with silent lacunar infarcts in basal ganglia territory.

Silent lacunar infarcts, which are present in over 20% of healthy elderly individuals, are associated with subtle deficits in cognitive functions. However, it remains largely unclear how these silent brain infarcts lead to cognitive deficits and even dementia. Here, we used diffusion tensor imaging tractography and graph theory to examine the topological organization of white matter networks in 27 patients with silent lacunar infarcts in the basal ganglia territory and 30 healthy controls. A whole-brain white matter network was constructed for each subject, where the graph nodes represented brain regions and the edges represented interregional white matter tracts. Compared with the controls, the patients exhibited a significant reduction in local efficiency and global efficiency. In addition, a total of eighteen brain regions showed significantly reduced nodal efficiency in patients. Intriguingly, nodal efficiency-behavior associations were significantly different between the two groups. The present findings provide new aspects into our understanding of silent infarcts that even small lesions in subcortical brain regions may affect large-scale cortical white matter network, as such may be the link between subcortical silent infarcts and the associated cognitive impairments. Our findings highlight the need for network-level neuroimaging assessment and more medical care for individuals with silent subcortical infarcts.

Asymptotic Expansion of the Multi-Orientable Random Tensor Model

Three-dimensional random tensor models are a natural generalization of the celebrated matrix models. The associated tensor graphs, or 3D maps, can be classified with respect to a particular integer or half-integer, the degree of the respective graph. In this paper we analyze the general term of the asymptotic expanion in $N$, the size of the tensor, of a particular random tensor model, the multi-orientable tensor model. We perform their enumeration and we establish which are the dominant configurations of a given degree.

Neural signatures of autism spectrum disorders: insights into brain network dynamics.

Neuroimaging investigations of autism spectrum disorders (ASDs) have advanced our understanding of atypical brain function and structure, and have recently converged on a model of altered network-level connectivity. Traditional task-based functional magnetic resonance imaging (MRI) and volume-based structural MRI studies have identified widespread atypicalities in brain regions involved in social behavior and other core ASD-related behavioral deficits. More recent advances in MR-neuroimaging methods allow for quantification of brain connectivity using diffusion tensor imaging, functional connectivity, and graph theoretic methods. These newer techniques have moved the field toward a systems-level understanding of ASD etiology, integrating functional and structural measures across distal brain regions. Neuroimaging findings in ASD as a whole have been mixed and at times contradictory, likely due to the vast genetic and phenotypic heterogeneity characteristic of the disorder. Future longitudinal studies of brain development will be crucial to yield insights into mechanisms of disease etiology in ASD sub-populations. Advances in neuroimaging methods and large-scale collaborations will also allow for an integrated approach linking neuroimaging, genetics, and phenotypic data.

Next-to-Leading Order in the Large N Expansion of the Multi-Orientable Random Tensor Model

In this paper we analyze in detail the next-to-leading order (NLO) of the recently obtained large N expansion for the multi-orientable (MO) tensor model. From a combinatorial point of view, we find the class of Feynman tensor graphs contributing to this order in the expansion. Each such NLO graph is characterized by the property that it contains a certain non-orientable ribbon subgraph (a non-orientable jacket). Furthermore, we find the radius of convergence and the susceptibility exponent of the NLO series for this model. These results represent a first step towards the larger goal of defining an appropriate double-scaling limit for the MO tensor model.

The 1/N Expansion of Multi-Orientable Random Tensor Models

Multi-orientable group field theory (GFT) has been introduced in A. Tanasa, J. Phys. A 45 (2012) 165401, arXiv:1109.0694, as a quantum field theoretical simplification of GFT, which retains a larger class of tensor graphs than the colored one. In this paper we define the associated multi-orientable identically independent distributed multi-orientable tensor model and we derive its 1/N expansion. In order to obtain this result, a partial classification of general tensor graphs is performed and the combinatorial notion of jacket is extended to the multi-orientable graphs. We prove that the leading sector is given, as in the case of colored models, by the so-called melon graphs.

The trade-off between wiring cost and network topology in white matter structural networks in health and migraine

The human brain organization of cortical networks has optimized trade-off architecture for the economical minimization of connection distance and maximizing valuable topological properties; however, whether this network configuration is disrupted in chronic migraine remains unknown. Here, employing the diffusion tensor imaging and graph theory approaches to construct white matter networks in 26 patients with migraine (PM) and 26 gender-matched healthy controls (HC), we investigated relationships between structural connectivity, cortical network architecture and anatomical distance in the two groups separately. Compared with the HC group, the patients showed longer global distance connection in PM, with proportionally less short-distance and more medium-distance; correspondingly, the patients showed abnormal global topology in their structural networks, mainly presented as a higher clustering coefficient. Moreover, the abnormal association between these two network features was also found. Intriguingly, the network measure that combined the nodal anatomical distance and network topology could distinguish PM from HC with high accuracy of 90.4%. We also demonstrated a high reproducibility of our findings across different parcellation schemes. Our results demonstrated that long-term migraine may result in a abnormal optimization of a trade-off between wiring cost and network topology in white matter structural networks and highlights the potential for combining spatial and topological aspects as a network marker, which may provide valuable insights into the understanding of brain network reorganization that could be attributed to the underlying pathophysiology resulting from migraine.