Graph-theoretical Study Research Articles

Identifying regions in wide-angle scattering via graph-theoretical approaches

The method of regions, which provides a systematic approach for computing Feynman integrals involving multiple kinematic scales, proposes that a Feynman integral can be approximated and even reproduced by summing over integrals expanded in certain regions. A modern perspective of the method of regions considers any given Feynman integral as a specific Newton polytope, defined as the convex hull of the points associated with Symanzik polynomials. The regions then correspond one-to-one with the lower facets of this polytope.As Symanzik polynomials correspond to the spanning trees and spanning 2-trees of the Feynman graph, a graph-theoretical study of these polynomials may allow us to identify the complete set of regions for a given expansion. In this work, our primary focus is on three specific expansions: the on-shell expansion of generic wide-angle scattering, the soft expansion of generic wide-angle scattering, and the mass expansion of heavy-to-light decay. For each of these expansions, we employ graph-theoretical approaches to derive the generic forms of the regions involved in the method of regions. The results, applicable to all orders, offer insights that can be leveraged to investigate various aspects of scattering amplitudes.

A graph theoretical study of porphyrin-cored dendrimers by means of Sombor indices:a computational approach

Topological indices are numerical parameters calculated on the molecular graph of chemical structures which do not depend on the method the molecular graph is labelled or depicted and have no need of energy minimisation of the molecular structure. The Sombor index is a recently established topological index possessing the ability to predict the enthalpy of vaporisation and entropy of octane isomers. Here, we apply the edge-partition technique to study the Sombor index and some of its variants namely delta Sombor index, reduced Sombor index, increased Sombor index, modified Sombor index, reverse Sombor index and Kulli-Gutman Sombor index for four types of dendrimers with porphyrin core. Our results could be helpful in better understanding the physico-chemical characteristics of this family of macromolecules.

On the metric-based resolving parameter of the line graph of certain structures

Let G be a graph and R = {r1, r2, …, rk} be an ordered subset of vertices of G, if every two vertices of G have different representation r (v|R) = (d (v, r1) , d (v, r2) , …, d (v, rk)) with respect to R, then R is said to be a metric-based resolving parameter or resolving set of G and its minimum cardinality is called the metric dimension of graph G . Metric dimension is considered as an important applied concept of graph theory especially in the localization of a network and also in the chemical graph theoretical study of molecular compounds. Therefore, it is hot topic to study for different families of graphs as well. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. In this paper, we determine the metric-based resolving parameter of line graph of a convex polytope Sn, and conclude that it has constant metric dimension but vary with the parity of n . This article presents a measurement of the line graph of a convex polytope, denoted as ( S n ) . The subsequent section provides the metric dimension of the resulting graph. There are two scenarios pertaining to the metric dimension of a selected graph with respect to the metric dimension. The metric dimension of even cycle-based convex polytopes is three, whereas for other values, the metric dimension is four.

Circadian clock crosstalks with autism.

The mechanism underlying autism spectrum disorder (ASD) remains incompletely understood, but researchers have identified over a thousand genes involved in complex interactions within the brain, nervous, and immune systems, particularly during the mechanism of brain development. Various contributory environmental effectsincluding circadian rhythm have also been studied in ASD. Thus, capturing the global picture of the ASD-clock network in combined form is critical. We reconstructed the protein-protein interaction network of ASD and circadian rhythm to understand the connection between autism and the circadian clock. A graph theoretical study is undertaken to evaluate whether the network attributes are biologically realistic. The gene ontology enrichment analyses provide information about the most important biological processes. This study takes a fresh look at metabolic mechanisms and the identification of potential key proteins/pathways (ribosome biogenesis, oxidative stress, insulin/IGF pathway, Wnt pathway, and mTOR pathway), as well as the effects of specific conditions (such as maternal stress or disruption of circadian rhythm) on the development of ASD due to environmental factors. Understanding the relationship between circadian rhythm and ASD provides insight into the involvement of these essential pathways in the pathogenesis/etiology of ASD, as well as potential early intervention options and chronotherapeutic strategies for treating or preventing the neurodevelopmental disorder.

Functional disruptions of the brain network in low back pain: a graph-theoretical study.

The aim of this study was to investigate alterations in the topological organization of whole-brain functional networks in patients with chronic low back pain (CLBP) and characterize the relationship of these alterations with pain characteristics. Thirty-three CLBP patients and 34 matched healthy controls (HCs) underwent fMRI scans. A graph-theoretical approach was applied to identify brain network changes in patients suffering from chronic low back pain given its nonspecific etiology and complexity. Graph theory-based analysis was used to construct functional connectivity matrices and extract the features of small-world networks of the brain in both groups. Then, the whole-brain functional connectivity differences were characterized by network-based statistics (NBS) analysis, and the relationship between the altered brain features and clinical measures was explored. At the global level, patients with CLBP showed significantly decreased gamma, sigma, global efficiency, and local efficiency and increased lambda and shortest path length compared with HCs. At the regional level, there were deficits in nodal efficiency within the default mode network and salience network. NBS analysis demonstrated that decreased functional connectivity was present in the CLBP patients, mainly in the frontolimbic circuit and temporal regions. Furthermore, aspects of topological dysfunctions in CLBP were correlated with pain severity. This study highlighted the aberrant topological organization of functional brain networks in CLBP, which may shed light on the pathophysiology of CLBP and support the development of pain management approaches.

Evidence of neuroplasticity with brain-computer interface in a randomized trial for post-stroke rehabilitation: a graph-theoretic study of subnetwork analysis.

Brain-computer interface (BCI) has been widely used for functional recovery after stroke. Understanding the brain mechanisms following BCI intervention to optimize BCI strategies is crucial for the benefit of stroke patients. Forty-six patients with upper limb motor dysfunction after stroke were recruited and randomly divided into the control group or the BCI group. The primary outcome was measured by the assessment of Fugl-Meyer Assessment of Upper Extremity (FMA-UE). Meanwhile, we performed resting-state functional magnetic resonance imaging (rs-fMRI) in all patients, followed by independent component analysis (ICA) to identify functionally connected brain networks. Finally, we assessed the topological efficiency of both groups using graph-theoretic analysis in these brain subnetworks. The FMA-UE score of the BCI group was significantly higher than that of the control group after treatment (p = 0.035). From the network topology analysis, we first identified seven subnetworks from the rs-fMRI data. In the following analysis of subnetwork properties, small-world properties including γ (p = 0.035) and σ (p = 0.031) within the visual network (VN) decreased in the BCI group. For the analysis of the dorsal attention network (DAN), significant differences were found in assortativity (p = 0.045) between the groups. Additionally, the improvement in FMA-UE was positively correlated with the assortativity of the dorsal attention network (R = 0.498, p = 0.011). Brain-computer interface can promote the recovery of upper limbs after stroke by regulating VN and DAN. The correlation trend of weak intensity proves that functional recovery in stroke patients is likely to be related to the brain's visuospatial processing ability, which can be used to optimize BCI strategies. The trial is registered in the Chinese Clinical Trial Registry, number ChiCTR2000034848. Registered 21 July 2020.

Network analysis of the human structural connectome including the brainstem.

The underlying anatomical structure is fundamental to the study of brain networks, but the role of brainstem from a structural perspective is not very well understood. We conduct a computational and graph-theoretical study of the human structural connectome incorporating a variety of subcortical structures including the brainstem. Our computational scheme involves the use of Python DIPY and Nibabel libraries to develop structural connectomes using 100 healthy adult subjects. We then compute degree, eigenvector, and betweenness centralities to identify several highly connected structures and find that the brainstem ranks highest across all examined metrics, a result that holds even when the connectivity matrix is normalized by volume. We also investigated some global topological features in the connectomes, such as the balance of integration and segregation, and found that the domination of the brainstem generally causes networks to become less integrated and segregated. Our results highlight the importance of including the brainstem in structural network analyses.

Global Alterations of Whole Brain Structural Connectome in Parkinson's Disease: A Meta-analysis.

Recent graph-theoretical studies of Parkinson's disease (PD) have examined alterations in the global properties of the brain structural connectome; however, reported alterations are not consistent. The present study aimed to identify the most robust global metric alterations in PD via a meta-analysis. A comprehensive literature search was conducted for all available diffusion MRI structural connectome studies that compared global graph metrics between PD patients and healthy controls (HC). Hedges' g effect sizes were calculated for each study and then pooled using a random-effects model in Comprehensive Meta-Analysis software, and the effects of potential moderator variables were tested. A total of 22 studies met the inclusion criteria for review. Of these, 16 studies reporting 10 global graph metrics (916 PD patients; 560 HC) were included in the meta-analysis. In the structural connectome of PD patients compared with HC, we found a significant decrease in clustering coefficient (g = -0.357, P = 0.005) and global efficiency (g = -0.359, P < 0.001), and a significant increase in characteristic path length (g = 0.250, P = 0.006). Dopaminergic medication, sex and age of patients were potential moderators of global brain network changes in PD. These findings provide evidence of decreased global segregation and integration of the structural connectome in PD, indicating a shift from a balanced small-world network to 'weaker small-worldization', which may provide useful markers of the pathophysiological mechanisms underlying PD.

Design, synthesis, graph theoretical analysis and molecular modelling studies of novel substituted quinoline analogues as promising anti-breast cancer agents.

The most promising class of heterocyclic compounds in medicinal chemistry are those with the quinolin-2-one nucleus. It is a versatile heterocyclic molecule that has been put together with numerous pharmaceutical substances and is crucial in the creation of anticancer medications. In this view, the present research work deals with design, synthesis, and characterization of various analogous of quinolin-2-one nucleus and evaluation of their anticancer activity against MCF-7 cells (adenoma breast cancer cell line). Fourteen new compounds have been synthesised using suitable synthetic route and are characterized by FTIR, 1H NMR, 13C NMR and Mass spectral data. Molecular docking studies of the title compounds were carried out using PyRx 0.8 tool in AutoDock Vina program. All the synthesised compounds were exhibited well conserved hydrogen bonding with one or more amino acid residues in the active pocket of EGFR tyrosine kinase (PDB ID: 1m17). The docking score of the derivatives ranged from -6.7 to -9.5kcalmol-1, standard drug Imatinib with -9.6kcalmol-1 and standard active ligand 4-anilinoquinazoline with -7.7kcalmol-1. The designed compound IV-A1 showed least binding energy (-9.5kcalmol-1) against EGFR tyrosine kinase receptor. Further, top scored compound, IV-A1 found to be most significant against MCF-7 cells with IC50 value of 0.0870µMmL-1, TGI of 0.0958µMmL-1, GI50 of 0.00499µMmL-1, LC50 of 1.670µMmL-1.

Computing Edge Version of Resolvability and Double Resolvability of a Graph

The field of graph theory is extensively used to investigate structure models in biology, computer programming, chemistry, and combinatorial optimization. In order to work with the chemical structure, chemists require a mathematical form of the compound. The chemical structure can be depicted using nodes (which represent the atom) and links (which represent the many types of bonds). As a result, a graph theoretic explanation of this problem is to give representations for the nodes of a graph such that different nodes have unique representations. This graph theoretic study is referred to as the metric dimension. In this article, we have computed the edge version of the metric dimension and doubly resolving sets for the family of cycle with chord C n t for n ≥ 6 and 2 ≤ t ≤ ⌊ n / 2 ⌋ .

Altered Connectedness of the Brain Chronnectome During the Progression to Alzheimer's Disease.

Graph theory has been extensively used to investigate brain network topology and its changes in disease cohorts. However, many graph theoretic analysis-based brain network studies focused on the shortest paths or, more generally, cost-efficiency. In this work, we use two new concepts, connectedness and 2-connectedness, to measure different global properties compared to the previously widely adopted ones. We apply them to unravel interesting characteristics in the brain, such as redundancy design and further conduct a time-varying brain functional network analysis for characterizing the progression of Alzheimer's disease (AD). Specifically, we define different connectedness and 2-connectedness states and evaluate their dynamics in AD and its preclinical stage, mild cognitive impairment (MCI), compared to the normal controls (NC). Results indicate that, compared to MCI and NC, brain networks of AD tend to be more frequently connected at a sparse level. For MCI, we found that their brains are more likely to be 2-connected in the minimal connected state as well indicating increasing redundancy in brain connectivity. Such a redundant design could ensure maintained connectedness of the MCI's brain network in the case that pathological damages break down any link or silenced any node, making it possible to preserve cognitive abilities. Our study suggests that the redundancy in the brain functional chronnectome could be altered in the preclinical stage of AD. The findings can be successfully replicated in a retest study and with an independent MCI dataset. Characterizing redundancy design in the brain chronnectome using connectedness and 2-connectedness analysis provides a unique viewpoint for understanding disease affected brain networks.

Metabolic Brain Network and Surgical Outcome in Temporal Lobe Epilepsy: A Graph Theoretical Study Based on 18F-fluorodeoxyglucose PET

Drug-resistant temporal lobe epilepsy (TLE) is a potential candidate for surgery; however, nearly one-third subjects had a poor surgical prognosis. We studied the underlying neuromechanism related to the surgical prognosis using graph theory based on metabolic brain network. Sixty-four unilateral TLE subjects with preoperative 18F-fluorodeoxyglucose (FDG) PET scanning were retrospectively enrolled and divided into Ia (Engel class Ia, n = 32) and non-Ia (Engel class Ib-IV, n = 32) groups according to more than 3-year follow-up after unilateral anterior temporal lobectomy (ATL). The metabolic brain network was constructed and the changed metabolic connectivity of Ia and non-Ia was detected compared with 15 matched healthy controls (HCs). Further, the network properties, including small-worldness and global efficiency, were calculated and hub nodes were also identified for the 3 groups respectively. Non-Ia group exhibited increased connectivity between contralateral fusiform gyrus and contralateral lingual gyrus; while Ia showed decreased connectivity mainly among bilateral frontal, temporal and parietal cortex. Graph theoretical analysis revealed that non-Ia group showed increased small-worldness (35%<s < 55%, P ≤ 0.05) compared to HCs; and elevated global efficiency (P = 0.05) and decreased Lp (P = 0.05) compared to Ia group. Ia group showed reduced Cp (55%<s < 63%, P < 0.05) and increased small-worldness (35%<s < 37%, P < 0.05) compared to HCs; Furthermore, disrupted hub nodes distribution pattern with the midcingulate gyrus disappeared, was also found in non-Ia group compared with the Ia group. All those results revealed that elevated network integration and metabolic connectivity, redistributed hub nodes pattern is associated with ongoing postoperative seizures in subjects with intractable TLE.

Functional network topology in drug resistant and well-controlled idiopathic generalized epilepsy: a resting state functional MRI study.

Despite an increasing number of drug treatment options for people with idiopathic generalized epilepsy (IGE), drug resistance remains a significant issue and the mechanisms underlying it remain poorly understood. Previous studies have largely focused on potential cellular or genetic explanations for drug resistance. However, epilepsy is understood to be a network disorder and there is a growing body of literature suggesting altered topology of large-scale resting networks in people with epilepsy compared with controls. We hypothesize that network alterations may also play a role in seizure control. The aim of this study was to compare resting state functional network structure between well-controlled IGE (WC-IGE), drug resistant IGE (DR-IGE) and healthy controls. Thirty-three participants with IGE (10 with WC-IGE and 23 with DR-IGE) and 34 controls were included. Resting state functional MRI networks were constructed using the Functional Connectivity Toolbox (CONN). Global graph theoretic network measures of average node strength (an equivalent measure to mean degree in a network that is fully connected), node strength distribution variance, characteristic path length, average clustering coefficient, small-world index and average betweenness centrality were computed. Graphs were constructed separately for positively weighted connections and for absolute values. Individual nodal values of strength and betweenness centrality were also measured and ‘hub nodes’ were compared between groups. Outcome measures were assessed across the three groups and between both groups with IGE and controls. The IGE group as a whole had a higher average node strength, characteristic path length and average betweenness centrality. There were no clear differences between groups according to seizure control. Outcome metrics were sensitive to whether negatively correlated connections were included in network construction. There were no clear differences in the location of ‘hub nodes’ between groups. The results suggest that, irrespective of seizure control, IGE interictal network topology is more regular and has a higher global connectivity compared to controls, with no alteration in hub node locations. These alterations may produce a resting state network that is more vulnerable to transitioning to the seizure state. It is possible that the lack of apparent influence of seizure control on network topology is limited by challenges in classifying drug response. It is also demonstrated that network topological features are influenced by the sign of connectivity weights and therefore future methodological work is warranted to account for anticorrelations in graph theoretic studies.

Disentangling age- and disease-related alterations in schizophrenia brain network using structural equation modeling: A graph theoretical study based on minimum spanning tree.

Functional brain networks have been shown to undergo fundamental changes associated with aging or schizophrenia. However, the mechanism of how these factors exert influences jointly or interactively on brain networks remains elusive. A unified recognition of connectomic alteration patterns was also hampered by heterogeneities in network construction and thresholding methods. Recently, an unbiased network representation method regardless of network thresholding, so called minimal spanning tree algorithm, has been applied to study the critical skeleton of the brain network. In this study, we aimed to use minimum spanning tree (MST) as an unbiased network reconstruction and employed structural equation modeling (SEM) to unravel intertwined relationships among multiple phenotypic and connectomic variables in schizophrenia. First, we examined global and local brain network properties in 40 healthy subjects and 40 schizophrenic patients aged 21–55 using resting‐state functional magnetic resonance imaging (rs‐fMRI). Global network alterations are measured by graph theoretical metrics of MSTs and a connectivity‐transitivity two‐dimensional approach was proposed to characterize nodal roles. We found that networks of schizophrenic patients exhibited a more star‐like global structure compared to controls, indicating excessive integration, and a loss of regional transitivity in the dorsal frontal cortex (corrected p <.05). Regional analysis of MST network topology revealed that schizophrenia patients had more network hubs in frontal regions, which may be linked to the “overloading” hypothesis. Furthermore, using SEM, we found that the level of MST integration mediated the influence of age on negative symptom severity (indirect effect 95% CI [0.026, 0.449]). These findings highlighted an altered network skeleton in schizophrenia and suggested that aging‐related enhancement of network integration may undermine functional specialization of distinct neural systems and result in aggravated schizophrenic symptoms.

The Effects of 10 Hz and 20 Hz tACS in Network Integration and Segregation in Chronic Stroke: A Graph Theoretical fMRI Study.

Transcranial alternating current stimulation (tACS) has emerged as a promising technique to non-invasively modulate the endogenous oscillations in the human brain. Despite its clinical potential to be applied in routine rehabilitation therapies, the underlying modulation mechanism has not been thoroughly understood, especially for patients with neurological disorders, including stroke. In this study, we aimed to investigate the frequency-specific stimulation effect of tACS in chronic stroke. Thirteen chronic stroke patients underwent tACS intervention, while resting-state functional magnetic resonance imaging (fMRI) data were collected under various frequencies (sham, 10 Hz and 20 Hz). The graph theoretical analysis indicated that 20 Hz tACS might facilitate local segregation in motor-related regions and global integration at the whole-brain level. However, 10 Hz was only observed to increase the segregation from whole-brain level. Additionally, it is also observed that, for the network in motor-related regions, the nodal clustering characteristic was decreased after 10 Hz tACS, but increased after 20 Hz tACS. Taken together, our results suggested that tACS in various frequencies might induce heterogeneous modulation effects in lesioned brains. Specifically, 20 Hz tACS might induce more modulation effects, especially in motor-related regions, and they have the potential to be applied in rehabilitation therapies to facilitate neuromodulation. Our findings might shed light on the mechanism of neural responses to tACS and facilitate effectively designing stimulation protocols with tACS in stroke in the future.

Graph theory in paediatric epilepsy: A systematic review.

Graph theoretical studies have been designed to investigate network topologies during life. Network science and graph theory methods may contribute to a better understanding of brain function, both normal and abnormal, throughout developmental stages. The degree to which childhood epilepsies exert a significant effect on brain network organisation and cognition remains unclear. The hypothesis suggests that the formation of abnormal networks associated with epileptogenesis early in life causes a disruption in normal brain network development and cognition, reflecting abnormalities in later life. Neurological diseases with onset during critical stages of brain maturation, including childhood epilepsy, may threaten this orderly neurodevelopmental process. According to the hypothesis that the formation of abnormal networks associated with epileptogenesis in early life causes a disruption in normal brain network development, it is then mandatory to perform a proper examination of children with new-onset epilepsy early in the disease course and a deep study of their brain network organisation over time. In regards, graph theoretical analysis could add more information. In order to facilitate further development of graph theory in childhood, we performed a systematic review to describe its application in functional dynamic connectivity using electroencephalographic (EEG) analysis, focussing on paediatric epilepsy.

On Efficiently Solvable Cases of Quantum k-SAT

Estimating ground state energies of local Hamiltonian models is a central problem in quantum physics. The question of whether a given local Hamiltonian is frustration-free, meaning the ground state is the simultaneous ground state of all local interaction terms, is known as the Quantum k-SAT (k-QSAT) problem. In analogy to its classical Boolean constraint satisfaction counterpart, the NP-complete problem k-SAT, Quantum k-SAT is hbox {QMA}_1-complete (for kge 3, and where hbox {QMA}_1 is a quantum generalization of NP with one-sided error), and thus likely intractable. But whereas k-SAT has been well-studied for special tractable cases, as well as from a “parameterized complexity” perspective, much less is known in similar settings for k-QSAT. Here, we study the open problem of computing satisfying assignments to k-QSAT instances which have a “dimer covering” or “matching”; such systems are known to be frustration-free, but it remains open whether one can efficiently compute a ground state. Our results fall into three directions, all of which relate to the “dimer covering” setting: (1) We give a polynomial-time classical algorithm for k-QSAT when all qubits occur in at most two interaction terms or clauses. (2) We give a “parameterized algorithm” for k-QSAT instances from a certain non-trivial class, which allows us to obtain exponential speedups over brute force methods in some cases. This is achieved by reducing the problem to solving for a single root of a single univariate polynomial. An explicit family of hypergraphs, denoted Crash, for which such a speedup is obtained is introduced. (3) We conduct a structural graph theoretic study of 3-QSAT interaction graphs which have a “dimer covering”. We remark that the results of (2), in particular, introduce a number of new tools to the study of Quantum SAT, including graph theoretic concepts such as transfer filtrations and blow-ups from algebraic geometry.

Altered White Matter Structural Network in Frontal and Temporal Lobe Epilepsy: A Graph-Theoretical Study.

Temporal lobe epilepsy (TLE) and frontal lobe epilepsy (FLE) are the largest subgroup of partial epilepsy, and focal cortical dysplasias (FCDs) are highly epileptogenic brain lesions and are a frequent cause for antiepileptic drug (AED)-resistant focal epilepsies that mostly occur in the temporal and frontal lobes. We performed a graph-theoretical study based on the diffusion tensor imaging (DTI) data of patients with FLE or TLE caused by FCDs or lesions with high suspicion of FCDs and evaluated their cognitive function by the Chinese version of the Montreal Cognitive Assessment-Basic (MoCA-BC). The construction of the white matter structural network and graph-theoretical analysis was performed by Pipeline for Analysing Brain Diffusion Images (PANDA) and Graph-theoretical Network Analysis (GRETNA). We used the nonparametric analysis of covariance to compare the differences in diffusion metrics, network attributes and nodal attributes among FLE, TLE, and healthy control (HC) groups and then performed post hoc pairwise comparisons. Nonparametric Spearman partial correlation analysis was performed to analyse the correlation of network attributes with the age of onset, duration of disease, and MoCA-BC scores in patients with FLE and TLE. The results showed that the white matter structural network in patients with FLE and TLE was impaired in a more extensive set of regions than the FCD location. The similarities in white matter alterations between FLE and TLE suggested that their epileptogenic network might affect the fronto-temporal white matter tracts and thalamo-occipital connections, which might be responsible for the overlapping cognitive deficits in FLE and TLE. The white matter impairments in patients with FLE were more severe than those in patients with TLE, which might be explained by more affected nodes in the areas of DMN in patients with FLE.

Altered Functional and Structural Connectomes in Female High School Soccer Athletes After a Season of Head Impact Exposure and the Effect of a Novel Collar.

Background: Characterization of, and evaluation of strategies to mitigate, the effects of sub-concussive impacts (SCI) on brain structure and function are crucial to understanding potential long-term neurological risks associated with sports participation. Objectives: To evaluate the efficacy of a jugular vein compression collar for preserving functional and structural measures of brain network organization in a cohort of female high school soccer players throughout a season of competitive play. Methods: Athletes were assigned to a collar (N = 72) or non-collar (N = 56) group before engaging in a season of play, during which head impact data were recorded via accelerometer for every practice and competition. Participants completed neuroimaging sessions before and following the season. A graph theoretical framework was applied to the functional and structural connectivity measures computed from resting state functional magnetic resonance imaging (rs-fMRI) and diffusion tensor imaging (DTI) data. Results: Non-collar-wearing athletes exhibited significantly increased rs-fMRI-derived global clustering coefficients (p = 0.032) and DTI-derived modularity (p = 0.042), compared to collar-wearing athletes. No longitudinal changes in any graph measures were observed for the collar group (p > 0.05). Conclusion: The observed increase in graph measures in the non-collar group is congruent with previous studies of SCI and is similar to graph theoretical studies of traumatic brain injury. The absence of alterations in graph metrics in the collar group indicates a potential ameliorating effect of the collar device against network reorganization, in line with previous literature.

Associate Submersions and Qualitative Properties of Nonlinear Circuits with Implicit Characteristics

We introduce in this paper an equivalence notion for submersions [Formula: see text], [Formula: see text] open in [Formula: see text], which makes it possible to identify a smooth planar curve with a unique class of submersions. This idea, which extends to the nonlinear setting the construction of a dual projective space, provides a systematic way to handle global implicit descriptions of smooth planar curves. We then apply this framework to model nonlinear electrical devices as classes of equivalent functions. In this setting, linearization naturally accommodates incremental resistances (and other analogous notions) in homogeneous terms. This approach, combined with a projectively-weighted version of the matrix-tree theorem, makes it possible to formulate and address in great generality several problems in nonlinear circuit theory. In particular, we tackle unique solvability problems in resistive circuits, and discuss a general expression for the characteristic polynomial of dynamic circuits at equilibria. Previously known results, which were derived in the literature under unnecessarily restrictive working assumptions, are simply obtained here by using dehomogenization. Our results are shown to apply also to circuits with memristors. We finally present a detailed, graph-theoretic study of certain stationary bifurcations in nonlinear circuits using the formalism here introduced.