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How can complex relationships among molecular or clinico-pathological entities of neurological disorders be represented and analyzed? Graphs seem to be the current answer to the question no matter the type of information: molecular data, brain images or neural signals. We review a wide spectrum of graph representation and graph analysis methods and...
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... detailed review of rs-fMRI is presented by Heuvel et al. [65]. Figure 3, shows a general overview of functional brain network analysis, where the brain image data are first subjected to parcellation that divides the brain into a number of regions or parcels with homogeneous characteristics. The functional connectivity matrix forms a full symmetric matrix between elements (voxels, neurons and recording sites) that provides a simple characterization of functional interactions. ...
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Background: Genomic sequencing, including whole exome sequencing (WES), is enabling a higher resolution for defining diseases, understand mechanisms, and improving the practice of clinical care. However, WES routinely identifies genomic variants with uncertain functional effects. Furthering uncertainty in WES data interpretation is that many genes...
Many real-world relational data can be modeled as graphs that contain vertices and edges representing, respectively, data entities and their relationship. One of the most important tasks is to discover graph clusters or communities, which are interesting subgraphs in the graph data. To find such clusters in graph data, many computational methods ha...
Citations
... The radius of a graph represents the minimum eccentricity of any graph vertex, while the diameter of a graph represents the maximum eccentricity of any graph vertex. (Collantoni et al. 2022;Falsaperla et al. 2021;Farahani et al. 2019;Mijalkov et al. 2017;Thomas et al. 2016;Zenil et al. 2018) The average distance between pairs of vertices is known as the characteristic path length of the graph. The maximum number of edges that connect any two adjacent vertices in a multigraph. ...
... The small-worldness index is quantified using characteristic path length and clustering coefficient, which captues the balance between local clustring and global connectivity. (Collantoni et al. 2022;Falsaperla et al. 2021;Farahani et al. 2019;Mijalkov et al. 2017;Thomas et al. 2016;Zenil et al. 2018) ...
We investigated the differences in functional connectivity based on the source-level electroencephalography (EEG) analysis between stroke patients with and without post-stroke epilepsy (PSE). Thirty stroke patients with PSE and 35 stroke patients without PSE were enrolled. EEG was conducted during a resting state period. We used a Brainstorm program for source estimation and the connectivity matrix. Data were processed according to EEG frequency bands. We used a BRAPH program to apply a graph theoretical analysis. In the beta band, radius and diameter were increased in patients with PSE than in those without PSE (2.699 vs. 2.579, adjusted p = 0.03; 2.261 vs. 2.171, adjusted p = 0.03). In the low gamma band, radius was increased in patients with PSE than in those without PSE (2.808 vs. 2.617, adjusted p = 0.03). In the high gamma band, the radius, diameter, average eccentricity, and characteristic path length were increased (1.828 vs. 1.559, adjusted p < 0.01; 2.653 vs. 2.306, adjusted p = 0.01; 2.212 vs. 1.913, adjusted p < 0.01; 1.425 vs. 1.286, adjusted p = 0.01), whereas average strength, mean clustering coefficient, and transitivity were decreased in patients with PSE than in those without PSE (49.955 vs. 55.055, adjusted p < 0.01; 0.727 vs. 0.810, adjusted p < 0.01; 1.091 vs. 1.215, adjusted p < 0.01). However, in the delta, theta, and alpha bands, none of the functional connectivity measures were different between groups. We demonstrated significant alterations of functional connectivity in patients with PSE, who have decreased segregation and integration in brain network, compared to those without PSE.
... Bioinformatics is another field where the application of graph clustering is prolific. Identifying functional modules of genes or proteins is such an example [5]. Other application areas include image segmentation [6] and personalized recommendations in a recommendation system [7,8] etc. ...
... Cluster 1 (in yellow) showcases an internal degree (int deg (c 1 )) of 8 and an external degree (ext deg (c 1 )) of 10, as determined by Equations (3) and (4), respectively. This yields a relative density (δ r (c 1 )) of 0.44, calculated using Equation (5). In the case of the second cluster (in red), it indicates an internal degree (int deg (c 2 )) of 7 and an external degree (ext deg (c 2 )) of 8. Consequently, the relative density (δ r (c 2 )) for this cluster is 0.46. ...
... This method consistently generates clusters with comparable sizes and topologies across various experiments. As the cluster expands, it continuously monitors its relative density δ r (c i ) using Equation (5) to ensure adherence to the desired density specified by the user U(δ r ). At each expansion step, the algorithm strategically chooses adjacent edges based on their average degree. ...
Graph clustering has received considerable attention recently, and its applications are numerous, ranging from the detection of social communities to the clustering of computer networks. It is classified as an NP-class problem, and several algorithms have been proposed with specific objectives. There also exist various quality metrics for evaluating them. Having clusters with the required density can be beneficial because it permits the effective deployment of resources. This study proposes an approach to partitioning and overlapping clustering of undirected unweighted graphs, allowing users to specify the required density of resultant clusters. This required density is achieved by means of ‘Relative Density’. The proposed algorithm adopts an edge-based approach, commencing with the determination of the edge degree for each edge. The main clustering process is then initiated by an edge with an average degree. A cluster is expanded by considering adjacent edges that can be included while monitoring the relative density of the cluster. Eight empirical networks with diverse characteristics are used to validate the proposed algorithm for both partitioning and overlapping clustering. Their results are assessed using an appropriate metric known as the mean relative density deviation coefficient (MRDDC). This is the first work that attempts to carry out partitioning and overlapping graph clustering, which allows user-specified density.
... Graph-based clustering methods (Thomas et al., 2016) are used to reveal the clusters and communities in network structures. These methods are widely used in network biology (Pavlopoulos et al., 2011;Ali et al., 2023). ...
... The lower small-worldness index in patients with NF1 might be caused by the long mean characteristic path length and low clustering coefficient, despite no statistical difference in the mean clustering coefficient being found in this study. [22] The long characteristic path length means that the number of short-distance connections between the nodes is decreased, with an increased average number of minimum connections between nodes in patients with NF1, which results in a reduced capacity Table 1 Demographic and clinical characteristics of the patients with neurofibromatosis type 1. for information transmission in the brain network. Due to such changes, some information may not be expressed in one node or may be expressed weakly, while the transmission to other nodes may be further strengthened, leading to more than the expression of general functions. ...
... Due to such changes, some information may not be expressed in one node or may be expressed weakly, while the transmission to other nodes may be further strengthened, leading to more than the expression of general functions. [8,22] Similar results have also been confirmed in other neurological disorders, such as epilepsy [23] and Alzheimer disease, [24] suggesting the presence of disrupted topological organization of the brain network in patients with neurological disorders. [22] Furthermore, our study demonstrated differences in the local structural connectivity between patients with NF1 and the healthy control group. ...
... [8,22] Similar results have also been confirmed in other neurological disorders, such as epilepsy [23] and Alzheimer disease, [24] suggesting the presence of disrupted topological organization of the brain network in patients with neurological disorders. [22] Furthermore, our study demonstrated differences in the local structural connectivity between patients with NF1 and the healthy control group. The differences in the local structural connectivity could explain the characteristic behavioral findings of each patient disability. ...
We investigated the changes in structural connectivity (using diffusion tensor imaging [DTI]) and the structural covariance network based on structural volume using graph theory in patients with neurofibromatosis type 1 (NF1) compared to a healthy control group. We included 14 patients with NF1, according to international consensus recommendations, and 16 healthy individuals formed the control group. This was retrospectively observational study followed STROBE guideline. Both groups underwent brain magnetic resonance imaging including DTI and 3-dimensional T1-weighted imaging. We analyzed structural connectivity using DTI and Diffusion Spectrum Imaging Studio software and evaluated the structural covariance network based on the structural volumes using FreeSurfer and Brain Analysis Using Graph Theory software. There were no differences in the global structural connectivity between the 2 groups, but several brain regions showed significant differences in local structural connectivity. Additionally, there were differences between the global structural covariance networks. The characteristic path length was longer and the small-worldness index was lower in patients with NF1. Furthermore, several regions showed significant differences in the local structural covariance networks. We observed changes in structural connectivity and covariance networks in patients with NF1 compared to a healthy control group. We found that global structural efficiency is decreased in the brains of patients with NF1, and widespread changes in the local structural network were found. These results suggest that NF1 is a brain network disease, and our study provides direction for further research to elucidate the biological processes of NF1.
... The mean clustering coefficient is a measure for the tendency of network elements to form local clusters, and the assortative coefficient is the values for degree of connections between nodes with similar degrees. The small-worldness index describes the balance between local connectedness and global integration in the network 38,39 . These are the most commonly used measures to analyze network topology in graph theory 38,39 . ...
... The small-worldness index describes the balance between local connectedness and global integration in the network 38,39 . These are the most commonly used measures to analyze network topology in graph theory 38,39 . We then analyzed the differences in the network measures between patients with RLS and healthy controls, between RLS patients with drug-naïve state and healthy controls, and between RLS patients with drug-treated state and the healthy controls. ...
We aimed to investigate the alterations of thalamic nuclei volumes and intrinsic thalamic network in patients with primary restless legs syndrome (RLS) compared to healthy controls. Seventy-one patients with primary RLS and 55 healthy controls were recruited. They underwent brain MRI using a three-tesla MRI scanner, including three-dimensional T1-weighted images. The intrinsic thalamic network was determined using graph theoretical analysis. The right and left whole thalamic volumes, and the right pulvinar inferior, left ventral posterolateral, left medial ventral, and left pulvinar inferior nuclei volumes in the patients with RLS were lower than those in healthy controls (0.433 vs. 0.447%, p = 0.034; 0.482 vs. 0.502%, p = 0.016; 0.013 vs. 0.015%, p = 0.031; 0.062 vs. 0.065%, p = 0.035; 0.001 vs. 0.001%, p = 0.034; 0.018 vs. 0.020%, p = 0.043; respectively). There was also a difference in the intrinsic thalamic network between the groups. The assortative coefficient in patients with RLS was higher than that in healthy controls (0.0318 vs. − 0.0358, p = 0.048). We demonstrated the alterations of thalamic nuclei volumes and intrinsic thalamic network in patients with RLS compared to healthy controls. These changes might be related to RLS pathophysiology and suggest the pivotal role of the thalamus in RLS symptoms.
... We applied graph theory to determine the differences in the intrinsic limbic network between the groups using network measures such as average degree, average strength, radius, diameter, eccentricity, characteristic path length, global efficiency, local efficiency, mean clustering coefficient, transitivity, modularity, assortativity, and small-worldness index. [35][36][37] These network parameters were compared between patients with OSA and healthy controls. ...
Objectives
We investigated the change in limbic structure volumes and intrinsic limbic network in patients with obstructive sleep apnea (OSA) compared to healthy controls.
Methods
We enrolled 26 patients with OSA and 30 healthy controls. They underwent three-dimensional T1-weighted magnetic resonance imaging (MRI) on a 3 T MRI scanner. The limbic structures were analyzed volumetrically using the FreeSurfer program. We examined the intrinsic limbic network using the Brain Analysis with Graph Theory program and compared the groups' limbic structure volumes and intrinsic limbic network.
Results
There were significant differences in specific limbic structure volumes between the groups. The volumes in the right amygdala, right hippocampus, right hypothalamus, right nucleus accumbens, left amygdala, left basal forebrain, left hippocampus, left hypothalamus, and left nucleus accumbens in patients with OSA were lower than those in healthy controls (right amygdala, 0.102 vs. 0.113%, p = 0.004; right hippocampus, 0.253 vs. 0.281%, p = 0.002; right hypothalamus, 0.028 vs. 0.032%, p = 0.002; right nucleus accumbens, 0.021 vs. 0.024%, p = 0.019; left amygdala, 0.089 vs. 0.098%, p = 0.007; left basal forebrain, 0.020 vs. 0.022%, p = 0.027; left hippocampus, 0.245 vs. 0.265%, p = 0.021; left hypothalamus, 0.028 vs. 0.031%, p = 0.016; left nucleus accumbens, 0.023 vs. 0.027%, p = 0.002). However, there were no significant differences in network measures between the groups.
Conclusion
We demonstrate that the volumes of several limbic structures in patients with OSA are significantly lower than those in healthy controls. However, there are no alterations to the intrinsic limbic network. These findings suggest that OSA is one of the risk factors for cognitive impairments.
... We investigated the difference in the global structural connectivity between the patients with RLS and healthy controls using graph theory, and the transitivity in the patients with RLS was significantly decreased compared to that of healthy controls. A graph's transitivity is determined by the proportion of triangles in the graph to the total number of connected triples of nodes [25,27]. Transitivity is the probability of adjacent nodes being connected in a network, indicating the presence of tightly connected communities, and it is a global measure of overall efficiency of local processing, termed as a segregation, in the brain [25,27]. ...
... A graph's transitivity is determined by the proportion of triangles in the graph to the total number of connected triples of nodes [25,27]. Transitivity is the probability of adjacent nodes being connected in a network, indicating the presence of tightly connected communities, and it is a global measure of overall efficiency of local processing, termed as a segregation, in the brain [25,27]. Thus, our results suggest the presence of decreased global structural connectivity, especially segregation, in patients with RLS compared to healthy controls. ...
... The characteristic path length, radius of graph, and diameter of graph were all positively correlated with RLS severity, whereas mean clustering coefficient, global efficiency, small-worldness index, and transitivity were negatively correlated with RLS severity. The characteristic path length is defined as the average of all path lengths, which is the average distance between any two nodes [25,27,[29][30][31]. The radius of a graph is equal to the minimum eccentricity, the maximum distance between any two nodes, of all nodes, while the diameter of a graph is equal to the maximum eccentricity of all nodes [25,27,[29][30][31]. ...
Study Objectives
To evaluate alterations of global and local structural brain connectivity in patients with restless legs syndrome (RLS).
Methods
Patients with primary RLS and healthy controls were recruited at a sleep center where they underwent diffusion tensor imaging (DTI) of the brain. We calculated the network measures of global and local structural brain connectivity based on the DTI in both groups using DSI studio program and a graph theory.
Results
A total of 69 patients with primary RLS and 51 healthy controls were included in the study. We found a significant difference in the global structural connectivity between the groups. The transitivity in the patients with RLS was lower than that in healthy controls (0.031 vs. 0.033, p=0.035). Additionally, there were significant differences in the local structural connectivity between the groups. The characteristic path length (r=0.283, p=0.018), radius of graph (r=0.260, p=0.030), and diameter of graph (r=0.280, p=0.019) were all positively correlated with RLS severity, whereas the mean clustering coefficient (r=-0.327, p=0.006), global efficiency (r=-0.272, p=0.023), small-worldness index (r=-0.325, p=0.006), and transitivity (r=-0.351, p=0.003) were negatively correlated with RLS severity.
Conclusion
We identified changes in the global structural connectivity of patients with RLS using graph theory based on DTI, which showed decreased segregation in the brain network compared to healthy controls. These changes are well-correlated with RLS severity. We also found changes in local structural connectivity, especially in regions involved in sensorimotor function, which suggests that these areas play a pivotal role in RLS. These findings contribute to a better understanding of the pathophysiology of RLS symptoms.
... Identifying common substructures in networks/graphs underlies many research areas; as ". . . [g]raphs seem to be the current answer to the question no matter the type of information: molecular data, brain images or neural signals" [75]. Thomas, Dongmin and Lee's survey of similarity relations on graphs [75], in particular, shows how they can be used to understand neurological interactions and to identify neurological disorders. ...
... [g]raphs seem to be the current answer to the question no matter the type of information: molecular data, brain images or neural signals" [75]. Thomas, Dongmin and Lee's survey of similarity relations on graphs [75], in particular, shows how they can be used to understand neurological interactions and to identify neurological disorders. Another example [66], uses network topologies based on interactions between and within subnetworks to investigate changes in the brain in people with Alzheimer's disease. ...
Measurement theory is the cornerstone of science, but no equivalent theory underpins the huge volumes of non-numerical data now being generated. In this study, we show that replacing numbers with alternative mathematical models, such as strings and graphs, generalises traditional measurement to provide rigorous, formal systems (`observement') for recording and interpreting non-numerical data. Moreover, we show that these representations are already widely used and identify general classes of interpretive methodologies implicit in representations based on character strings and graphs (networks). This implies that a generalised concept of measurement has the potential to reveal new insights as well as deep connections between different fields of research.
... Americans died of this disease, which is identified as the second common neurological disorder in the United States (48). In a brain network, the correlation network for Parkinson's disease is identified by mining densely-connected regions in PPI, which results in uncovering significant pathways, such as the Parkinson's disease pathway (66). This process, i.e. analyzing the structural and functional of brain networks using dense subgraph mining algorithms, is critical for understanding the causes and mechanisms of disease progression and, thus, promoting better treatments and aiding in drug discovery for Parkinson's disease. ...
... In many application areas such as social networks, communication networks and epidemic spread networks, the corresponding graph models are time-varying in nature [6], [7]. For instance, a company wants to maintain its prominence over a certain user base represented by a graph G(V, E). ...
... The essence of the MEP-based approach lies in successive evaluations of Gibbs distribution in (7). Note that from (9) minimizing L at small values of β is equivalent to maximizing entropy H, which in turn corresponds to uniform distribution. ...
