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Application of Graph Theory for Identifying Connectivity Patterns in Human Brain Networks: A Systematic Review

Application of Graph Theory for Identifying Connectivity Patterns in Human Brain Networks: A... SYSTEMATIC REVIEW published: 06 June 2019 doi: 10.3389/fnins.2019.00585 Application of Graph Theory for Identifying Connectivity Patterns in Human Brain Networks: A Systematic Review 1 1 2 Farzad V. Farahani , Waldemar Karwowski and Nichole R. Lighthall Computational Neuroergonomics Laboratory, Department of Industrial Engineering and Management Systems, University of Central Florida, Orlando, FL, United States, Department of Psychology, University of Central Florida, Orlando, FL, United States Background: Analysis of the human connectome using functional magnetic resonance imaging (fMRI) started in the mid-1990s and attracted increasing attention in attempts to discover the neural underpinnings of human cognition and neurological disorders. In general, brain connectivity patterns from fMRI data are classified as statistical dependencies (functional connectivity) or causal interactions (effective connectivity) Edited by: among various neural units. Computational methods, especially graph theory-based Gabriel A. Silva, University of California, San Diego, methods, have recently played a significant role in understanding brain connectivity United States architecture. Reviewed by: Giuseppe D’Avenio, Objectives: Thanks to the emergence of graph theoretical analysis, the main purpose of Istituto Superiore di Sanità (ISS), Italy the current paper is to systematically review how brain properties can emerge through the Vishnu Suppiramaniam, interactions of distinct neuronal units in various cognitive and neurological applications Auburn University, United States using fMRI. Moreover, this article provides an overview of the existing functional and *Correspondence: Waldemar Karwowski effective connectivity methods used to construct the brain network, along with their advantages and pitfalls. Specialty section: Methods: In this systematic review, the databases Science Direct, Scopus, arXiv, This article was submitted to Google Scholar, IEEE Xplore, PsycINFO, PubMed, and SpringerLink are employed for Neural Technology, exploring the evolution of computational methods in human brain connectivity from 1990 a section of the journal Frontiers in Neuroscience to the present, focusing on graph theory. The Cochrane Collaboration’s tool was used to Received: 28 November 2018 assess the risk of bias in individual studies. Accepted: 23 May 2019 Published: 06 June 2019 Results: Our results show that graph theory and its implications in cognitive Citation: neuroscience have attracted the attention of researchers since 2009 (as the Human Farahani FV, Karwowski W and Connectome Project launched), because of their prominent capability in characterizing Lighthall NR (2019) Application of the behavior of complex brain systems. Although graph theoretical approach can be Graph Theory for Identifying Connectivity Patterns in Human Brain generally applied to either functional or effective connectivity patterns during rest or Networks: A Systematic Review. task performance, to date, most articles have focused on the resting-state functional Front. Neurosci. 13:585. doi: 10.3389/fnins.2019.00585 connectivity. Frontiers in Neuroscience | 1 June 2019 | Volume 13 | Article 585 Farahani et al. Graph Theory and Brain Networks Conclusions: This review provides an insight into how to utilize graph theoretical measures to make neurobiological inferences regarding the mechanisms underlying human cognition and behavior as well as different brain disorders. Keywords: brain connectivity, functional connectivity, effective connectivity, fMRI, brain networks, graph theory, small-world, connectome INTRODUCTION Graph-based network analysis reveals meaningful information about the topological architecture of human The human brain comprises ∼86 billion neurons brain networks, such as small-worldness, modular organization, connected through ∼150 trillion synapses that allow and highly connected or centralized hubs (Bullmore and Sporns, neurons to transmit electrical or chemical signals to other 2009, 2012; He and Evans, 2010; Meunier et al., 2010; Bullmore neurons (Pakkenberg et al., 2003; Azevedo et al., 2009). and Bassett, 2011; van den Heuvel and Sporns, 2013). Small- Studies on modeling the human brain as a complex system worldness is a property of some networks in which most nodes have grown remarkably as neuroscientists seek to understand are not neighbors of each other but can be reached from every the comprehensive information underlying cognition, behavior, other node by a small number of steps. This characteristic is well and perception (Bassett and Bullmore, 2006; Reijneveld et al., suited to the study of complex brain dynamics, and it confirms 2007; Bullmore and Sporns, 2009, 2012; He and Evans, 2010; efficient information segregation and integration in the human Friston, 2011; Craddock et al., 2013; Park and Friston, 2013). brain networks with low energy and wiring costs (Watts and Exploring the human brain from the viewpoint of connectivity Strogatz, 1998). Recent studies demonstrate that the small-world patterns reveals important information regarding the structural, property of brain networks experiences topological alterations functional, and causal organization of the brain. Among the under different cognitive loads and during development (Bassett connectivity techniques, functional, and effective connectivity et al., 2011; Braun et al., 2015; Cao et al., 2016; Liang et al., 2016), have been the focus of the computational studies in recent years as well as in neurological and mental disorders (Xia and He, (Friston, 1994, 2011; Farahani and Karwowski, 2018). Functional 2011; Fornito et al., 2012; Filippi et al., 2013; Dai and He, 2014; connectivity refers to the temporal correlations among spatially Stam, 2014; Fornito and Bullmore, 2015; Gong and He, 2015; remote neurophysiological events, whereas effective connectivity Abós et al., 2017; Fleischer et al., 2017; Hojjati et al., 2017; Jalili, refers to the causal interactions between neuronal units of 2017; Miri Ashtiani et al., 2018). These alterations may provide the brain network (Friston, 1994). Computational methods novel insights into the biological mechanisms underlying human for functional brain connectivity are generally divided into cognition, as well as health and disease. model-based and model-free (Li et al., 2009a). For the analysis Recent advances in neuroimaging have enabled mapping of of effective brain connectivity, methods such as Granger the human connectome in different applications (Van Essen casualty, dynamic causal modeling, and Bayesian networks et al., 2012; Fornito et al., 2015). Brain function can be have been of interest to researchers (Friston, 2009; Zhang localized through neuroimaging techniques that assess changes in et al., 2015). Further, the human connectome (i.e., mapping metabolism via positron emission tomography (PET) or changes the connectivity patterns of the human brain) has become an in blood oxygenation level-dependent (BOLD) responses via increasing topic of interest in the area of human neuroscience fMRI. Structural pathways can be captured using diffusion and can be studied using network science and graph theory tensor imaging (DTI), in which MRI is applied to trace white (Sporns et al., 2005; Kelly et al., 2012; Van Essen et al., 2012; matter tracts. Finally, the timing of brain activity and its Sporns, 2013c). locus can be determined from electroencephalogram (EEG) or The human brain is one of the most complex networks in magnetoencephalogram, which respectively, measure electrical the world, and studies on its static and dynamic properties and magnetic signals outside the skull. Used separately or have undergone explosive growth in recent years (Bullmore together, these techniques constitute the neuroimaging toolkit of and Sporns, 2012; Sporns, 2013b; Kriegeskorte and Douglas, scientists investigating the physiology of human brain networks 2018). The advances in graph theory and network neuroscience (Chugani et al., 1987; Ogawa et al., 1990; Pfurtscheller and Lopes, (i.e., the study of the structure or function of the nervous 1999; Le Bihan et al., 2001). Among them, fMRI and PET offer a system) offer an opportunity to understand the details of this relatively low temporal resolution but have a significant spatial complex phenomenon and its modeling (Vecchio et al., 2017; resolution, making them particularly useful for determining Sporns, 2018). Graph theoretical approaches have set up a where neural signals are generated (Mehta and Parasuraman, mathematical framework to model the pairwise communications 2013). However, PET scanning can measure the blood flow between elements of a network. In human neuroscience, changes in an area of ∼5–10 cubic millimeters while fMRI can graph theory is generally applied to either functional or resolve down to 3 cubic millimeters and even lower. Moreover, effective connectivity. However, most studies have been PET scanning is much more expensive than fMRI and requires devoted to functional connectivity (Bullmore and Sporns, 2009; radioactive isotopes to work (Friston et al., 1996). During the Goldenberg and Galván, 2015). last two decades, there has been an explosion of fMRI studies Frontiers in Neuroscience | 2 June 2019 | Volume 13 | Article 585 Farahani et al. Graph Theory and Brain Networks mapping neural functions to distinct parts of the brain at rest or • RQ5: What can be learned from current graph-based during task performance (Greicius et al., 2003), however, more research in human connectome that will lead to topics for attention has been directed toward resting-state fMRI (rs-fMRI) further investigation? data (Lee et al., 2013). Search Strategy The main purpose of this paper is to review the recent studies utilizing graph-based methods to analyze connectivity The search strategy was able to first explore the search space patterns in the human brain network using fMRI data. We properly, and secondly, exploit the relevant material with a expect to see whether the recognition of brain connectivity rigorous evaluation process. Current and seminal research properties by graph theory (as measured by fMRI) has been literature in the realm of fMRI brain connectivity focusing effective in understanding the mechanisms underlying human on graph-based methods including peer-reviewed journal cognition compared to the traditional approaches. The remaining articles, textbooks, reference books, proceedings, and conference sections are organized as follows. Section Methodology presents presentations were considered key sources for this systematic the methodology and criteria used for selecting papers to be review. During the exploration phase, the bibliographic search studied in the current paper, as well as data synthesis and validity was carried out using a list of academic databases and search risk assessment. Section Theoretical Background: Connectivity engines such as Science Direct, Scopus, arXiv, Google Scholar, Patterns Using fMRI first summarizes existing methods for IEEE Xplore, PsycINFO, PubMed, and SpringerLink. To meet examining the brain network connectivity, which are categorized the eligibility criteria for creating search space, articles must into functional and effective patterns (3.1 and 3.2), then, focuses have been published after 1990, the time when fMRI technique on the graph-theoretic concepts required for analyzing the was invented, with the following keyword combinations in the brain connectivity architecture (3.3). Section Results provides title, keywords or abstract: (“graph theory” or “graph analysis” the results of literature search, study characteristics, validity or “network analysis” or “connectome” or “connectomics” assessment of the considered studies, as well as a general overview or “small-world” or “modularity” or “topological change” or of the selected articles. Then, section Discussion discusses “topological pattern” or “functional connectivity” or “effective the potential implications and applications of graph theory connectivity” or “brain connectivity” or “connectivity analysis” in human cognition (5.1), as well as common neurological or “brain network” or “network connectivity” or “functional illnesses (5.2). Finally, section Challenges and Future Directions network”) and (“fMRI” or “functional MRI” or “functional highlights challenging issues and future perspectives in this magnetic resonance imaging”). These criteria resulted in a rapidly growing field. narrowing of the focus to identify the population addressing the research questions. Eligibility Criteria METHODOLOGY Published original articles with the following features were included in the current study: (a) be written in English; (b) be This systematic review was conducted based on the PRISMA peer reviewed; (c) identify, describe, or use empirical and/or (Preferred Reporting Items for Systematic Reviews and Meta- modeled graph-based methods to quantify and/or compare Analyses) guidelines (Moher et al., 2010). The starting point connectivity patterns in the human brain network; (d) be for this systematic review was a protocol where the research applied to fMRI data. Other exclusion criteria were: (a) book questions and the search strategy were specified to reduce the chapters; (b) papers which upon review were not related to the effect of research expectations on the review. Furthermore, research questions; (c) opinions, viewpoints, anecdotes, letters, the literature searches and systematic review adhered to the and editorials. Two authors (FVF and WK) independently Cochrane Collaboration guidance (Higgins et al., 2011), to screened the titles and abstracts to find the relevant papers based minimize the risk of bias and error. on the inclusion and exclusion criteria and any discrepancies were resolved through discussion. Research Questions Quality Assessment Based on the objectives of this systematic review described in the Risk of bias in individual studies was assessed by two abstract, the following research questions were derived and form authors independently (FVF and WK) using the Cochrane the basis of this literature review: Collaboration’s tool (Higgins et al., 2011). The following • RQ1: How has the computational methods for modeling the domains were evaluated: random sequence generation, brain connectivity patterns using fMRI evolved? allocation concealment, blinding of participants, blinding • RQ2: How can research of mapping the human connectome of outcome assessment, incomplete outcome data, selective using fMRI be classified? outcome reporting. To evaluate the quality of evidence across • RQ3: What is the significance of graph-based approaches studies, we examined for lack of completeness (publication bias) among the identified toolkit for brain connectivity analysis? and missing data from the included studies (selective reporting • RQ4: With the advent of graph theory in cognitive bias). The risk of missing studies is heavily dependent on the neuroscience, what applications have been studied in selected keywords and the limitations of the applied search modeling human cognition and psychiatric disorders? engines. To mitigate this risk, a well-known and heavily cited Frontiers in Neuroscience | 3 June 2019 | Volume 13 | Article 585 Farahani et al. Graph Theory and Brain Networks set of papers was employed to construct the keyword search selection, and the inability to detect non-linear forms of list in an iterative process. Accordingly, a Pareto analysis of the interaction, restrict the discovery of all plausible functional top keywords was conducted to assess the quality of selected architectures (Farahani and Karwowski, 2018). keywords in search strategy. Cross-correlation and coherence An important concern to the validity of evidence across Cross-correlation analysis is the most traditional method for studies is the issue of limited attention span (i.e., the length testing functional connectivity, which is defined by measuring the of time a person can concentrate on a task without becoming correlation between the BOLD signals of any two brain regions distracted) for reviewing the sheer volume of identified scientific (Cao and Worsley, 1999). The computational complexity of this articles. To put it another way, the likelihood of erroneously method is extremely high when calculating the correlation of two omitting relevant articles as well as information from the series at all lags (Cecchi et al., 2007). Fortunately, a large number included studies increases due to the repetitive and monotonous of fMRI studies have overcome this drawback by computing only nature of reviewing a large number of papers for content under the correlation with zero lag due to the short duration of the perceived and/or real-time constraints. Reduction of this risk was hemodynamic response of blood (Friston et al., 1994b; Saad et al., achieved by breaking up the articles into controllable, discrete 2001). Moreover, correlations are sensitive to the shape of the quantities of 20–40 articles depending on article length, and hemodynamic response function (HRF), which causes variations providing sufficient time separation between reviews. Moreover, across different individuals and different brain areas (Miezin to prevent the formation of taxonomy with insufficient breadth et al., 2000; Lee et al., 2001). Furthermore, a high correlation when categorizing selected articles, an iterative content analysis may be observed among regions that practically have no blood method was employed to assure adequate classes for every new flow fluctuations. Uncontrolled physiological noise in the brain concept encountered in the literature review. (e.g., from cardiac and respiratory variations) can also result in high correlations between brain regions (Friston et al., 1994a). THEORETICAL BACKGROUND: To address these problems, Sun et al. (2004) suggested a new CONNECTIVITY PATTERNS USING FMRI measure, termed coherence, which is the spectral representation of correlation in the frequency domain. Brain connectivity investigations using fMRI time-series were initiated in the mid-1990s and provided a new tool for Statistical parametric mapping (SPM) researchers, especially neuroscientists, to study the human brain SPM is another model-based approach used to detect region- network with high precision. Computational methods available specific effects (e.g., brain activation patterns) in neuroimaging for brain connectivity are divided into two general categories: data, such as fMRI and PET, using a combination of the general functional connectivity and effective connectivity (Friston, 1994, linear model (GLM) and Gaussian random field (GRF) (Friston 2011). Briefly, functional connectivity provides information et al., 1991). The GLM helps estimate the parameters describing about the statistical dependencies or temporal correlations the spatially continuous data by performing a univariate test between spatially remote neurophysiological events, whereas statistic on each voxel. GRF theory is applied to address the effective connectivity is concerned with the directed influence multiple comparisons problem for continuous data (i.e., images) of brain regions on each other (Friston, 2011). In the following, when making statistical inferences over a volume of the brain, an we will review the computational methods that are presented approach similar to the Bonferroni correction for the analysis of in the literature for investigating both types of connectivity discrete data (Worsley et al., 1992). with a greater focus on graph theoretical approaches in separate Model-Free Methods sections (Figure 1). In contrast to seeds-based methods, model-free methods need Functional Connectivity no seeds selection. Also, model-free methods may be beneficial Functional connectivity refers to the temporal correlations in studies where there are no temporal or spatial patterns, between BOLD signals from spatially remote brain regions as well as in quantifying non-linear neuronal interactions (Friston et al., 1993; Lee et al., 2003). Functional connectivity (Farahani and Karwowski, 2018). methods in fMRI studies are broadly divided into model- Decomposition-based analysis based (e.g., cross-correlation, coherence analysis, and statistical PCA can express the fMRI data with a linear combination parametric mapping) and model-free (e.g., decomposition-based of orthogonal contributors that have the greatest impact on analysis, clustering, and mutual information) groups. the data variance. Each contributor contains a pattern of time Model-Based Methods variability (or a principal component) multiplied by a pattern Model-based methods typically identify brain connectivity of spatial variability (or an eigen map). The created eigen maps networks by selecting one or more “seed” regions and then reflect the connectivity architecture of the brain (Baumgartner determining whether there is a linear link between seed regions et al., 2000; Worsley et al., 2005). Despite the ability to explore and other regions using predefined criteria (Li et al., 2009a). the whole-brain connectivity, PCA fails to detect activations Despite their widespread use and simple interpretation in when the contrast-to-noise ratio is low (Baumgartner et al., identifying functional connectivity, the requirement for prior 2000). Also, how to select the optimal number of components knowledge (particularly in rs-fMRI), dependency on the seed has become an open question. Thus, PCA commonly serves Frontiers in Neuroscience | 4 June 2019 | Volume 13 | Article 585 Farahani et al. Graph Theory and Brain Networks FIGURE 1 | Taxonomy of existing methods for modeling functional and effective connectivity patterns using fMRI. Each of the identified methods can be represented in terms of a graph, where the nodes correspond to cortical or subcortical regions and the edges represent (directed or undirected) connections (Bullmore and Sporns, 2012); thereby all of them can be further examined with graph-theoretic measures. as a preprocessing step in fMRI studies through dimension Clustering reduction (Li et al., 2009a). Another decomposition-based The primary goal of clustering algorithms is to group voxels method, called independent component analysis (ICA), attracted or regions of interest into different clusters based on the the attention of researchers in rs-fMRI studies. The major similarity between their BOLD time courses (Golay et al., difference between ICA and PCA is that the components in 1998). Hierarchical clustering, k-means, fuzzy clustering (fuzzy ICA should be as independent as possible (Comon, 1994; c-means), self-organizing maps, graph-based, and bootstrap Hyvärinen and Oja, 2000). Note that a violation of component analysis are the most well-known algorithms used in fMRI studies independence would reduce the efficiency of ICA (Calhoun et al., (Chuang et al., 1999; Ngan and Hu, 1999; Cordes et al., 2002; 2001). Furthermore, finding the optimal number of independent Golland et al., 2008; van den Heuvel et al., 2008a; Bellec et al., components is controversial because choosing a small number 2010; Lee et al., 2012). Among these methods, the largest volume of components can have a significant effect on ICA results of studies utilizes hierarchical and fuzzy clustering. Hierarchical (Ma et al., 2007), particularly when used for decoding purposes clustering seeks to construct a hierarchy of clusters based on an (Douglas et al., 2011, 2013). Finally, ICA cannot discriminate agglomerative or divisive strategy (Rokach and Maimon, 2005). between signals of interest and signals of no interest (e.g., Although this method exhibits good efficacy in the presence of physiological noise, unexplained signal variations), leading to respiratory or cardiac noise, its high computational complexity is overfitting and invalid assessment of statistical significance. To a serious limitation when examining the whole brain connectivity address this pitfall, Beckmann and Smith (2004) proposed a (Cordes et al., 2002). Fuzzy c-means (FCM) is a method in probabilistic ICA that allows for non-square mixing when there which each data point has a membership value to each cluster, is Gaussian noise. rather than entirely belonging to one cluster as k-means. This Frontiers in Neuroscience | 5 June 2019 | Volume 13 | Article 585 Farahani et al. Graph Theory and Brain Networks algorithm performs optimization by updating memberships and relationships in BOLD signals rather than neuronal responses cluster centers until convergence (Lee et al., 2012; Lahijanian (Smith et al., 2011, 2012). To tackle this issue, GC analysis et al., 2016). It’s worth noting that, given the non-Euclidean is typically performed by fitting a linear vector autoregressive nature of MRI data, the use of Euclidean distance in FCM- (VAR) to the time series (Seth, 2010; Friston et al., 2013; Seth based algorithms may lead to an invalid result (Farahani et al., et al., 2015). However, linear methods are not suitable for testing 2015, 2018). van den Heuvel and Hulshoff Pol (2010) compared GC in higher moments (e.g., the variance). Non-linear and non- the results of clustering algorithms to those of decomposition- parametric models are used to solve this problem (Dhamala et al., based methods and reported a high level of overlap. Future 2008; Roebroeck et al., 2011). Wen et al. (2013) pointed out that studies may, therefore, pay more attention to these algorithms several factors may hamper the neural interpretability of GC, and, by eliminating the above issues, achieve more acceptable such as low sampling rates (Lin et al., 2014), latency mismatches performance in human neuroscience. in HRF across distinct brain regions, and the presence of noise. Their findings reflect that GC is a viable method for analyzing Mutual information (MI) fMRI signals when associated confounds are controlled. MI is an information theoretic concept that quantifies the shared information (undirected) between two random variables Dynamic causal modeling (DCM) (Grassberger et al., 1991; Kraskov et al., 2004). Equivalently, DCM is based on a general bilinear state equation that quantifies the MI is a model-free technique that does not require any how variations in neural activity in one node are affected by a priori assumptions about the connectivity patterns among the activation in another node under predefined stimuli (Friston variables, thus, it can be applied to detect both linear and non- et al., 2003; Stephan et al., 2010). This equation involves a variety linear correlations (Wilmer et al., 2012). Tsai et al. (1999) were of information including the coupling between brain regions, among the first to present a theoretical framework for using MI changes in the coupling strength as a result of experimental to calculate the fMRI activation map. To further explore the conditions, and the direct effects on a region (Friston, 2009). strengths and pitfalls of this method in comparison to other DCM provides a powerful statistical platform that estimates functional connectivity measures, refer to Wang et al. (2014a) the experimental modulation of both intrinsic and extrinsic and Bastos and Schoffelen (2016). connections in the brain, and the Bayesian model comparison is executed to choose the best-fitted model (Goldenberg and Effective Connectivity Galván, 2015). Perhaps the biggest disadvantage of DCM is that The primary goal of effective connectivity analysis is to assess it is not exploratory and requires prior knowledge about the causal interactions between neuronal units of the brain network hypotheses and model specification to be implemented. However, (Friston, 1994). Studies in this area help researchers better a recent trend has emerged for comparing numerous models in understand the mechanisms underlying neuronal dynamics (Wu a more exploratory manner using a post hoc analysis, wherein et al., 2014; Farahani and Karwowski, 2018). In the following, we only the largest model is inverted while all of the reduced models review the existing effective connectivity methods with their pros would be searched quickly (Friston et al., 2011). Friston et al. and cons in greater detail. pointed out that GC and DCM play complementary roles in analyzing the causal interactions (Friston et al., 2013). In fact, GC Model-Based Methods can be used generically to any specified time series to identify the Granger causality (Granger, 1969) is the most traditional model- coupling between neuronal units, making helpful insights into based method for directional interactions that can be easily the dynamic behavior of the human brain in different situations. implemented. However, Granger causality appears to encounter One might then continue effective connectivity analyses in a difficulties when applied to fMRI data due to the underlying hypothesis-driven manner to obtain a further interpretation of assumptions in its modeling (Wen et al., 2013; Dang et al., the neuronal interactions using DCM (Daunizeau et al., 2011). 2017). Two other model-based methods for analyzing effective Notably, although both build upon model selection, they have connectivity are dynamic causal modeling (Friston et al., 2003) a fundamental difference. Model selection in DCM is based on and structural equations modeling (McIntosh and Gonzalez- a direct comparison between all models (Penny, 2012), whereas Lima, 1994). Despite the coherent interpretations provided by in GC this involves testing for the presence of GC followed these methods, they are highly dependent on prior knowledge, by selecting the VAR model order using Akaike or Bayesian so their application in analysis of rs-fMRI data is limited information criteria (Bressler and Seth, 2011). (Fox and Raichle, 2007). Granger casualty (GC) Model-Free Methods The core idea behind GC is that X “Granger-causes” Y if Y can Past efforts to detect effective connectivity mostly relied on be better predicted using the histories of both X and Y than the model-based methods such as GC or DCM. Model-free methods past of Y alone (Granger, 1969). Accordingly, past data from including probabilistic Bayesian networks, Markov models, and one brain region can help estimate the current state in another transfer entropy have been developed to determine non-linear region. Due to the time mismatch between sampling interval and forms of directed interactions. These methods do not require neural events, the causality method cannot be applied directly a priori assumptions on connectivity patterns due to their to the fMRI signals because it leads to the prediction of causal exploratory nature (Ramsey et al., 2010), but lagged interactions Frontiers in Neuroscience | 6 June 2019 | Volume 13 | Article 585 Farahani et al. Graph Theory and Brain Networks between fMRI time-courses may be a common shortcoming for most of them (Dang et al., 2017). Bayesian network (BN) BN is a probabilistic model well suited for representing the conditional dependencies over a set of random variables through a directed acyclic graph (DAG) (Friedman et al., 1997). Each edge indicates a dependency between two variables (nodes), where the lack of connection between any pair of nodes reflects conditional independence. Each node has a probability distribution: In root nodes, this is prior probability, while in child nodes this is the conditional probability (Das, 2004; Daly et al., 2011). Gaussian BN (Li et al., 2009b) and discrete dynamic BN (DBN) (Rajapakse and Zhou, 2007; Zeng and Ji, 2010) are the most commonly used techniques in this area. Due to the static nature of Gaussian BNs, they are unable to explicitly model the temporal interactions between multiple processes in different parts of the brain (Rajapakse and Zhou, 2007). Compared with FIGURE 2 | A network can be designed as binary (A) or weighted (B) graphs, Gaussian BN, discrete DBN is not limited by linear assumptions, and can represent the direction of causal effects (C,D) among different regions. and it can model temporal processes via a first-order Markov chain (Rajapakse and Zhou, 2007). However, the presence of multinomial distribution in the nodes of discrete DBN causes discretization of the data, leading to a huge loss of information. biological neural networks (Watts and Strogatz, 1998; Boccaletti To overcome the primary limitations of both methods, Wu et al. et al., 2006; Schweitzer et al., 2009). (2014) proposed a method called Gaussian DBN based on a The turning point of the complex brain network studies using first-order linear dynamic system. graph theory goes back to the introduction of the “Human Connectome” (Sporns et al., 2005). In graph theory, an N×N Transfer entropy (TE) adjacency matrix (also called a connection matrix) with the TE is a non-parametric approach measuring the transfer of elements of zero or non-zero indicates the absence or presence information between joint processes based on information theory of a relationship between the vertices of a network with N nodes. (Schreiber, 2000). Because of its non-linear nature, this method By extracting different metrics from this matrix, one can obtain a is able to properly detect directional connectivity even if there is topological analysis of the desired graph (e.g., the human brain a wide distribution of interaction delays between the two fMRI network). A brain graph may be classified as either directed signals (Vicente et al., 2011; Sharaev et al., 2016). Although TE or undirected (Figure 2) based on whether the links between and GC are relatively equivalent for Gaussian variables (Barnett vertices carry directional information (e.g., causal interaction). and Seth, 2009), TE needs much less computational time than Up to now, most human brain investigations have been devoted GC for high model orders and greater numbers of nodes. In to the undirected networks because of the technical constraints addition, TE does not assume any particular model as underlying surrounding the inference of directional networks (Liao et al., the interactions, therefore, its sensitivity to all order correlations 2017). A brain graph can also be categorized as either weighted or becomes a privilege for exploratory analyzes over GC or other binary (Figure 2) based on whether the links between vertices can model-based methods (Vicente et al., 2011; Montalto et al., take different values. For instance, in a white matter anatomical 2014). However, contrary to the model-based methods, it is network taken by diffusion MRI, we can obtain a weighted more difficult to interpret this measure in functional connectivity network using various information, such as fiber number, analysis due to its generality (Bastos and Schoffelen, 2016). fiber length, and fractional anisotropy (Fornito et al., 2013; Zhong et al., 2015). Graph Theory: Analysis of the Brain as a In 1998, Watts and Strogatz showed that many social, Large, Complex Network biological, and geoscience-based networks have a very striking The first application of graph theory and network analysis can be organization, called “small-world” architecture, that makes them traced back to 1736 when Leonhard Euler solved the Königsberg act as regular networks, while they occasionally experience Bridge Problem (Euler, 1736). In this regard, a graph consists of a random activity (Watts and Strogatz, 1998; Figure 4C). Small- finite set of vertices (or nodes) that are connected by links called world networks represent the shortest path between each pair edges (or arcs). Following the emergence of promising results in of nodes in the network using the minimum number of edges. electrical circuits and chemical structures in its early applications, In small-world networks, the clustering coefficient (also referred graph theory has now become influential in addressing a large to as transitivity) is high, and the average path length is short. number of practical problems in other disciplines, such as These two characteristics are the result of a natural process transportation systems, social networks, big data environments, to satisfy the balance between minimizing the resource cost the internet of things, electrical power infrastructures, and and maximizing the flow of information among the network Frontiers in Neuroscience | 7 June 2019 | Volume 13 | Article 585 Farahani et al. Graph Theory and Brain Networks components (Bassett and Bullmore, 2006; Meunier et al., 2010; Finally, key topological properties that characterize the local Bullmore and Sporns, 2012; Chen et al., 2013; Samu et al., 2014). and global architecture of the brain network connectivity can Liao et al. (2017) explained in detail why the human brain be obtained using the Brain Connectivity Toolbox (http://www. network is expected to have a small-world architecture. The; Rubinov and Sporns, 2010). metabolic and wiring costs in connections among anatomically These characteristics are explained in the following. adjacent brain areas are lower than those among distant brain Computation of Graph Measures regions (Bullmore and Sporns, 2012). Theoretical examinations In this subsection, the most commonly used graph metrics for have pointed out that the brain regions are more likely to interact characterizing the functional brain network are described in two with their neighboring areas to reduce the whole metabolic costs, main groups: global and local properties. Most of these criteria while at the same time they need to have a small number of are applicable to any type of binary, weighted, and directed long-distance connections among themselves to accelerate data networks. In addition to visualizing these properties in Figure 4 transmission (Sik et al., 1995; Karbowski, 2001; Bullmore and (global metrics) and Figure 5 (local metrics), respectively, their Sporns, 2012; Vertes et al., 2012; Chen et al., 2013). In agreement corresponding formulas can be accessed on with theoretical studies, empirical investigations have also proved com/site/bctnet/measures. the dispersion of a few long connections among a plethora of short connections in the human brain network (Salvador et al., Global properties 2005; Hagmann et al., 2007; He et al., 2007). The main capability of graph theory in neuroscience studies Global measures are primarily aimed at revealing: (a) functional segregation and (b) functional integration of information flows is usually unveiled after the construction of a functional brain network. Several measures can be used to assess the topological within the brain network; (c) small-worldness; (d) network resilience against failure (Rubinov and Sporns, 2010; Sporns, patterns of different networks such as clustering coefficient, modularity, average path, small-worldness, assortativity, and 2013a). Segregation refers to the degree to which network elements form specialized communities, and integration provides node centrality, which have been described in detail (Sporns et al., 2004; van den Heuvel et al., 2008b). Typically, one cannot claim insight into the efficiency of global information communication or the ability to combine distributed information (Watts and which measures are more suitable for studying the brain network (Bullmore and Sporns, 2009), but given the complex structure of Strogatz, 1998). Clustering coefficients and modularity are the most common metrics that quantify the properties of the human brain, measures that can represent the small-world topological segregation in brain networks (Newman, 2004; properties of the brain network are of great importance (He and Boccaletti et al., 2006; Rubinov and Sporns, 2010; Figure 4A). Evans, 2010; Liao et al., 2017). This critical property arises with In brain networks, anatomically adjacent or functionally the help of hubs (i.e., highly connected nodes in a network), causing the creation of local clusters (Bullmore and Sporns, 2009; connected areas are generally considered as modules. Various studies have demonstrated that networks based on modular Jain, 2011). In the following, we discuss how to build a brain connectivity network using fMRI data and then explain the main structure generally reflect the properties of small-world networks (Bullmore and Sporns, 2009; Fortunato, 2010; He and Evans, measures that can be extracted from the brain network with the help of graph theory. 2010; Meunier et al., 2010; Sporns and Betzel, 2016). On the other side, functional integration is typically measured by the Construction of Functional Brain Network Using fMRI characteristic path length that quantifies the ability for global information integration (Boccaletti et al., 2006; Rubinov and In Figure 3, we illustrate the main steps used to extract a Sporns, 2010; Figure 4B). The small-world property displays an complex network from fMRI in graph theoretical analysis. optimal balance between network segregation and integration, Initially, a number of pre-processing steps including slice timing and is dedicated to graphs in which most nodes are not neighbors correction, realignment, image co-registration, normalization based on segmentation, and spatial smoothing, are performed but can be reached by any other node with the minimum possible path length (Achard, 2006; Humphries et al., 2006; Humphries on the acquired fMRI data. Note that, the choice and ordering of the preprocessing steps may affect the extent of final graph and Gurney, 2008; Figure 4C). Eventually, assortativity quantifies network resilience against random or deliberate damages in the measures (Gargouri et al., 2018). Then, to explore the large- scale brain network, an appropriate parcellation scheme such main components, which is one of the most significant issues in network science (Noldus and Van Mieghem, 2014; Figure 4D). as anatomical automatic labeling atlas is applied to divide the entire brain into several cortical and subcortical anatomical units (Tzourio-Mazoyer et al., 2002). This is followed by extracting Local properties the time series of each parcel by averaging the time courses In network science, hubs refer to nodes with a high nodal of all voxels within that certain region. Next, one of the centrality and thus profoundly affect the network topology. connectivity methods reviewed in the previous parts, such Hub nodes of a network are divided into two categories, the as cross-correlation, is conducted to determine the pairwise connector or provincial, based on the high or low participation associations between the time series of brain parcels, representing coefficient defined for them, respectively. Connector hubs tend the functional connectivity network (i.e., correlation matrix). to interconnect nodes between different modules, while the A binary connectivity matrix (i.e., adjacency matrix) is then provincial hubs are responsible for linking nodes in the same obtained by thresholding the values of the correlation matrix. module (He et al., 2009; Power et al., 2013; Figure 5A). Frontiers in Neuroscience | 8 June 2019 | Volume 13 | Article 585 Farahani et al. Graph Theory and Brain Networks FIGURE 3 | Schematic representation of brain network construction and graph theoretical analysis using fMRI data. After processing (B) the raw fMRI data (A) and division of the brain into different parcels (C), several time courses are extracted from each region (D) so that they can create the correlation matrix (E). To reduce the complexity and enhance the visual understanding, the binary correlation matrix (F), and the corresponding functional brain network (G) are constructed, respectively. Eventually, by quantifying a set of topological measures, graph analysis is performed on the brain’s connectivity network (H). The easiest way to detect hubs in a network is to calculate be peer reviewed; (c) identify, describe, or use empirical and/or the nodal degree, i.e., counting the edges connected to each modeled graph-based methods to quantify and/or compare node. Also, plotting the degree distribution P(k) of a certain connectivity patterns in the human brain network; (d) be network provides valuable information about the presence applied to fMRI data. Other exclusion criteria were: (a) book of hubs in it, e.g., the existence of several high degree chapters; (b) papers which upon review were not related to the nodes in scale-free networks is accompanied by power-law research questions; (c) opinions, viewpoints, anecdotes, letters, distribution (Barabási and Albert, 1999). Furthermore, other and editorials. Application of inclusion and exclusion criteria at commonly used indexes for measuring the nodal centrality this step yielded 202 eligible articles (roughly 35% of the original include betweenness, closeness, and eigenvector, participation papers). At the fourth step, the full text of these 202 articles coefficient, and PageRank (Boccaletti et al., 2006; Rubinov and were studied in detail to confirm that they met same criteria as Sporns, 2010; Zuo et al., 2012; Figure 5B). the third step. After the fourth step, 163 publications remained for review. RESULTS Study Characteristics Literature Search Sample size across studies ranged from 5 to 763 participants. The Following the PRISMA guidelines (Moher et al., 2010), a mean, mode, median, and standard deviation for the participants summary of the identification, screening, and selection of in all the study samples were 116.73, 40, 60, and 158.87, studies for inclusion in this review is displayed in Figure 6. At respectively. The included studies were published from 1998 the first step, 1,193 papers were identified. Next, 579 papers to 2018 and organized into three taxonomies (Figure 7). The remained after removing duplicates. Papers published before first group deals with the topological concepts of graph theory 2005 accounted for only 5% of all papers, reflecting the novelty of for the discovery of the brain as a large and complex network, the terminology and the research area. In the third step, relevant which account for 34% of the selected articles. Then, papers that scientific articles were selected from the remaining 579 papers have applied graph theory in terms of human cognition and using a formal abstract screening process that incorporated pre- behavior for quantifying or comparing connectivity patterns in determined inclusion and exclusion criteria. Inclusion criteria at the brain network have been considered, accounting for 26% this step required the research to: (a) be written in English; (b) of the selected articles. Finally, applications of graph theory Frontiers in Neuroscience | 9 June 2019 | Volume 13 | Article 585 Farahani et al. Graph Theory and Brain Networks FIGURE 4 | Summary of global graph measures. (A) Segregation measures include clustering coefficient, which quantify how much neighbors of a given node are interconnected and measures the local cliquishness (i.e., the extent to which the neighbors of a node can build a complete graph); modularity, which is related to clusters of nodes, called modules, that have dense interconnectivity within clusters but sparse connections between nodes in different clusters. On the one hand, dense communications within a certain module increase the local clustering and, consequently, enhance the efficiency of information transmission in the given module. On the other hand, a few connections between different modules integrate the global information flow, which is associated with a reduction in the average path length in the graph (B) Integration measure include characteristic path length, which measures the potential for information transmission, determined as the average shortest path length across all pairs of nodes. (C) A regular network (left) displays a high clustering coefficient and a long average path length, while a random network (right) displays a low clustering coefficient and a short average path length. A small-world network (middle) illustrates an intermediate balance between regular and random networks (i.e., they consist of many short-range links alongside a few long-range links), reflecting a high clustering coefficient and a short path length. (D) The assortativity index measures the extent to which a network can resist failures in its main components (i.e., its vertices and edges). Notably, communication between hubs in assortative networks leads to covering each other’s activities when a particular hub crashes, but the performance in disassortative networks will drop sharply due to the presence of vulnerable hubs. in mental disorders were reported, which account for 40% of of authors. Such findings provide a novel perspective on the the selected papers. In particular, the detailed frequency and evolution of computational methods for modeling the brain percentage of the referenced papers in the last two categories are connectivity patterns and the importance of graph theory among shown, separately. them, addressing research questions 1, 2, and 3. To observe the evolution of the theme, Figure 9 displays Quality Assessment the number of reviewed publications, year by year. This The Cochrane collaboration’s tool (Higgins et al., 2011) was figure illustrates the researchers’ special attention to human used to assess the risk of bias in each trial (Figure 8). The connectome studies, especially the emerging role of graph articles were categorized as: (a) low risk of bias, (b) high risk analysis in topological explorations of the complex brain of bias, or (c) unclear risk of bias for each domain. Using connections since 2009. Most articles are concentrated between Cochrane collaboration we judged most domains to be unclear 2009 and 2018 (92% of the selected publications), which is expected to increase dramatically in the next years. Interestingly, or not reported. Eventually, the overall quality of the studies was categorized into weak, fair, or good, if <3, 3, or ≥4 domains were the Human Connectome Project (HCP) was launched in 2009 rated as low risk, respectively. Among 163 studies included in the with the National Institutes of Health sponsorship, which is in systematic review, 52 were categorized as good quality, 39 were line with these findings (, 2009). fair quality, and 72 were low quality. Pareto analysis of the top keywords is shown in Figure 10. Obviously, the words “graph theory,” “fMRI,” “resting-state,” General Overview “functional connectivity,” and “small-world” were among the In this part, a general overview of the selected papers is presented most used keywords in the reviewed papers (50% of the listed in terms of publication trend, keyword analysis, and frequency keywords). By this finding, it can be interpreted that those Frontiers in Neuroscience | 10 June 2019 | Volume 13 | Article 585 Farahani et al. Graph Theory and Brain Networks FIGURE 5 | Basic concept of network centralities. (A) Hubs (connector or provincial) refer to nodes with a high nodal centrality, which can be identified using different measures. (B) The degree centrality is defined as the number of node’s neighbors. The betweenness centrality measures the node’s role in acting as a bridge between separate clusters by computing the ratio of all shortest paths in the network that contain a given node. The closeness centrality quantifies how fast a given node in a connected graph can access all other nodes, hence the more central a node is, the closer it is to all other nodes. The eigenvector centrality is a self-referential measure of centrality that considers the quality of a link, so that being connected to a central node increases one’s centrality in turn; the red colored node is more central than the gray colored node, although their degrees are equal. The participation coefficient of a node represents the distribution of its connections among separate modules. PageRank is a variant of eigenvector centrality, used by Google Search to determine a page’s importance; the PageRank of an undirected graph is statistically similar to the degree centrality, but they are generally distinct. Note that the size of the nodes in all cases is proportional to the node degree, and the red nodes (except in the eigenvalue centrality) are the most central with respect to the corresponding definition of centrality, even though their degree are low. FIGURE 6 | Flow diagram of the methodology and selection processes used in this review. It follows the guidelines of PRISMA (Moher et al., 2010). Frontiers in Neuroscience | 11 June 2019 | Volume 13 | Article 585 Farahani et al. Graph Theory and Brain Networks FIGURE 7 | Categorization of included studies. FIGURE 8 | Assessing the risk of bias using the Cochrane collaboration’s tool. fMRI studies that have benefited from graph theory have: (a) systems,” that complex analysis of human brain connectivity been mostly carried out during resting-state than experimental became widespread in the world. task, which is in line with the HPC claim (Smith et al., 2013); (b) concentrated more on functional connectivity than effective connectivity; (c) considered a pivotal role for the small-world DISCUSSION phenomenon in constructing the human brain architecture. Figure 11 displays a reference analysis through the sample. Deeper discussions about the leading applications of graph The most cited authors by the articles in our sample were theory in cognitive and behavioral topics, as well as different Olaf Sporns, Karl Friston, Yong He, and Edward T Bullmore, neurological and psychiatric illnesses are provided in two with 17, 15, 14, and 13 references, respectively. Unsurprisingly, separate subsections. Considering the weaknesses and strengths Sporns and Bullmore stand out as two of the pioneers of the of these implications provides an insight into how to utilize network neuroscience and connectomics. It was through the graph measures to make neurobiological inferences regarding study of Bullmore and Sporns (2009), entitled “Complex brain the mechanisms underlying neuronal dynamics, in line with networks: graph theoretical analysis of structural and functional questions 4 and 5 of the research. Frontiers in Neuroscience | 12 June 2019 | Volume 13 | Article 585 Farahani et al. Graph Theory and Brain Networks FIGURE 9 | Selected papers per year (publishing trend). FIGURE 10 | Pareto analysis of top keywords. fMRI, Functional magnetic resonance imaging; DMN, default mode network; ADHD, Attention-deficit/hyperactivity disorder; MCI, Mild cognitive impairment; SVM, Support vector machine; ICA independent component analysis. Cognitive and Behavioral Applications of attention, comprehension, memory, decision making, reasoning, judgment, and executive functions (Mesulam, 1998). In the Graph Theory following, some of the applications of graph theory are presented Recent advances in neuroimaging modalities combined with in revealing human behavioral and cognitive performance, as graph theoretical approaches have opened new avenues toward well as the role of different large-scale brain networks in studying the neural mechanisms underlying human cognition various conditions. and behavior from the view of interregional brain interactions (Park and Friston, 2013; Pessoa, 2014; Sporns, 2014; Medaglia et al., 2015; Petersen and Sporns, 2015; Kriegeskorte and Human Intelligence and Brain Topology Douglas, 2018). Cognition involves a range of neuronal actions Human intelligence refers to the marvelous and subtle function for knowledge assimilation and integration through thinking, of human cognition, which is generally characterized by experience, and the senses. Cognition contains manifestations of complex reasoning, conceptual thinking, and learning swiftly Frontiers in Neuroscience | 13 June 2019 | Volume 13 | Article 585 Farahani et al. Graph Theory and Brain Networks FIGURE 11 | Frequency of the authors in the references. from experiences (Guilford, 1967). An early review of brain detected striking age-related alterations in highly connected hub imaging studies has linked human intelligence to the structure areas mainly within the default mode, attentional, sensorimotor, and function of spatially distributed regions (Jung and Haier, and visual regions via rs-fMRI (Meunier et al., 2009; Fransson 2007), indicating the possible importance of interactions between et al., 2011; Hwang et al., 2013; Wu et al., 2013; Betzel et al., several regions, particularly in the frontal and parietal areas. 2014; Cao et al., 2014b; Grayson and Fair, 2017; Finotelli et al., Recently, many studies have focused on the relationship between 2018; Gozdas et al., 2018). Most of them also reported that general intellectual ability and small-world characteristics local efficiency and the rich club coefficient (a metric that in intrinsic functional networks for describing individual measures the extent to which well-connected nodes also connect differences in general intelligence (van den Heuvel et al., 2009; to each other) were incremental until adulthood in healthy Langer et al., 2012; Hilger et al., 2017a). According to these subjects and then dropped with aging, while global efficiency studies, better intellectual performance was associated with remained almost unchanged over the lifetime regardless of the shorter characteristic path length, the nodal centrality of hub early years after birth (Gao et al., 2011). Cao et al. (2014b) further regions in the salience network, as well as the efficiency of identified changes in the number and strength of connections functional integration between the frontal and parietal areas that were created to achieve an optimal balance between the (Jung and Haier, 2007). Through an analysis of rs-fMRI data, wiring costs and communication efficiency over the lifespan Wu et al. (2013) illustrated that intelligence quotient is positively (Bullmore and Sporns, 2012). correlated with nodal properties in the attention-related network Moreover, inverse trajectories of change between long and and is negatively correlated with nodal properties in the default short connections suggest a continuous reorganization in the mode, emotion, and language systems. However, although these functional brain network with aging, leading to significant findings suggest that general intelligence is profoundly affected behavioral and cognitive differences throughout an individual’s by the functional integration of spatially distributed regions, life. Regarding modularity, there are somewhat mixed findings. they could not provide sufficient information as to whether Some have argued for little change in modularity during brain and how human intellectual performance is associated with development (Fair et al., 2009) and aging (Meunier et al., 2009), the brain’s modular architecture. To address this issue, Hilger while Cao et al. reported a linear downward trend (Cao et al., et al. (2017b) proposed that intelligence involves the nodal 2014b). In this regard, combining other functional neuroimaging characteristics of functional connectivity within and between techniques, as well as performing structure-function studies, different brain modules (especially in the parietal and frontal will help elucidate the neural substrates underlying cognitive areas), not global modularity properties or whole-brain ratios of and behavioral differences during developmental stages distinct node types. (Shah et al., 2018). Working Memory Performance and Network Topological Changes Across the Lifespan The human brain goes through remarkable functional changes Efficiency during the lifespan, from birth to adulthood. Modeling the Working memory is a psychological construct for the temporary lifetime trajectory of the functional connectome, multiple studies storage and manipulation of the information required to Frontiers in Neuroscience | 14 June 2019 | Volume 13 | Article 585 Farahani et al. Graph Theory and Brain Networks perform intricate cognitive tasks such as reasoning and decision- with rest in healthy participants. They observed that differences making (Diamond, 2013). Stanley et al. (2015) compared the were generally associated with the language-related and DMN functionality of working memory between young and older regions. More importantly, they found greater intra-modular adults in an n-back experiment by quantifying the local and communication in these regions during decision making (i.e., a global measures in their brain networks. They demonstrated that decrease in distributed connectivity), whereas the inter-modular lower local efficiency corresponds to the better performance of communication was stronger at rest. working memory in both groups. In contrast, increasing global Moreover, Lin et al. (2016) analyzed whether cognitive efficiency has been correlated with high functionality in young behavior correlates with the functional connectivity of the DMN adults but with a slight deficiency in older adults. Seeking to in healthy subjects, both while at rest and during an attentional prove the right intraparietal sulcus as an area responsive to task. Quantifying the static and dynamic nodal properties within manipulations of working memory load, Markett et al. (2018) the DMN, they revealed the importance of the default network, used rs-fMRI to show that centrality measures in this region especially the posterior cingulate areas, on human cognitive correlate inversely with working memory capacity. In another performance. Finally, Sadaghiani et al. (2015) investigated the fMRI study, Gong et al. (2016) analyzed how active learning from relationship between ongoing alterations in baseline connectivity action video games affected the neuroplasticity of the brain by patterns and behavioral performance through a continuous testing the integration of working memory- (central executive) auditory detection task. Interestingly, their results indicated a and attention-related (salience) neural networks. By assessing reduction in modularity (i.e., increasing integration efficiency) the graph theoretical properties between advanced and amateur before misses compared with hits and task-free rest, mostly in players, they revealed that long-term playing would enhance the the DMN areas and visual networks. These findings augment functional integration within and between working memory and our understanding about the key role of the DMN in behavioral attention systems. performance at rest and during a task; however, its association with other brain regions in more complex cognitive tasks, such Effect of Cognitive Loads on the Brain Modularity as reasoning and executive functions, requires further studies. In the last decade, studies on dynamic reconfiguration of human brain topology during different cognitive tasks have attracted Behavioral Performance in Natural Environments and widespread attention. Researchers believe that such functional Everyday Settings brain networks adapt flexibly to their cognitive demands while One of the fascinating areas of cognitive neuroscience in recent preserving the modular structure (Bassett et al., 2011; Fornitoa years is neoroergonomics; that is to say, the behavioral analysis et al., 2012; Braun et al., 2015; Liang et al., 2016). In the course of the human brain performance with regard to environments, of dynamic reorganization, the parietal and frontal brain regions work, technology, and everyday settings (Parasuraman and that hold several connector (inter-modular) hubs are discerned Rizzo, 2008). Qian et al. (2013) studied the topological changes to play crucial roles by regulating their brain-wide connections of the brain connectome during passive hyperthermia using rs- (Cole et al., 2013; Braun et al., 2015). For instance, intensifying fMRI data. Despite maintaining economic small-worldness in cognitive loads during a working memory task is associated with both normal and hyperthermia conditions, the brain networks of increased integration between different modules of the brain heat-exposed subjects exhibited decreased clustering coefficients, network (Kitzbichler et al., 2011; Braun et al., 2015; Liang et al., as well as decreased local efficiency and small-worldness indices, 2016). Furthermore, flexibility and the inter-modular integration suggesting a tendency toward a random network. They also of frontal areas are associated with high performance on working conducted an attention network test (ANT). Their findings memory tasks (Braun et al., 2015). were highly relevant to global measure alterations and pre- Regarding mental state analysis, notable studies have frontal local efficiency, indicating behavioral disorders during shown that modularity corresponds negatively to the level of environmental heat exposure in executive attention but not in consciousness by comparing the functional brain network in alerting or orienting. individuals who experienced non-rapid eye movement sleep and Furthermore, functional imaging analyses on mental fatigue those in wakefulness (Boly et al., 2012; Tagliazucchi et al., 2013). have indicated that declines in performance from fatigue are The common point of all these findings is that an increased associated with brain topological alterations such as a decrease in cognitive load or consciousness level brings about greater global small-world properties and global efficiency, as well as functional integration of the neural networks (i.e., reducing the modularity changes in the fronto-parietal network and connected areas coefficient). However, further studies are needed to make this in the thalamus and the striatum (Petruo et al., 2018). In claim more robust. particular, graph-based investigations using fMRI data express that long-range connectivity is changed when the effects of Role of the Default Mode Network in Behavioral fatigue appear (Sun et al., 2014, 2017). For instance, Sun et al. Performance (2017) studied the effects of a mid-task break on enhancing local Comparing the brain topological alterations during a cognitive efficiency and reported no significant impact of rest breaks on task and resting-state using fMRI data helps identify areas that task performance. In general, such studies help to understand affect human behavioral performance. Desalvo et al. (2014) the neural mechanisms of fatigue; thus, by adopting a suitable used a graph-based approach to explore variations in functional recovery approach, one can try to improve human performance brain organization during semantic decision making compared during cognitive tasks. Frontiers in Neuroscience | 15 June 2019 | Volume 13 | Article 585 Farahani et al. Graph Theory and Brain Networks Disorganization of Brain Networks in it is specialized or serial. The correlation between average path length and intellectual capability has been indicated by Neurological and Psychiatric Disorders other experiments as well (van den Heuvel et al., 2009). To Disconnection in a brain made up of localized but linked conclude, these results support the hypothesis that localization- specialized regions results in functional impairment, associating related epilepsy leads to cognitive impairments by inducing with atypical integration of distributed brain areas. Catani and global changes in the brain network instead of a localized Ffytche (2005) elaborated the rises and fall of disconnection disruption only. syndromes and pointed out that many neurological disorders Apart from TLE, other types of epilepsy such as childhood can be explained via these syndromes, in line with the studies of absence epilepsy (CAE) and sleep-related hypermotor epilepsy pioneers in neurology and psychiatry such as Meynert, Wernicke, (SHE) have recently been investigated by researchers (Wang and Dejerine. Studies in the field of complex brain networks et al., 2017; Evangelisti et al., 2018). CAE is a common generalized have demonstrated that analyzing the network properties and epilepsy syndrome with a presumed genetic cause, characterized metrics derived from brain topology using rs-fMRI can help by episodes of sudden, profound impairment of consciousness neurologists distinguish patient groups from control subjects without loss of body tone, appearing in otherwise healthy school- in mental disorders (Bassett and Bullmore, 2009; Wang et al., aged children. Wang et al. (2017) compared centrality measures 2010; Stam, 2014; Zhou et al., 2017). In the following, several between CAE patients and healthy controls and hypothesized studies that have used graph theory to investigate common that hub nodes inside the DMN and thalamus in CAE patients neurological disorders, comprising epilepsy, Alzheimer’s disease were clearly damaged. In other work, Evangelisti et al. (2018) (AD), multiple sclerosis (MS), autism spectrum disorder reported topological alterations mainly in basal ganglia and (ASD), and attention-deficit/hyperactivity disorder (ADHD), limbic system in SHE patients. are discussed. However, other mental disorders were also found in recent graph-based literature, including schizophrenia, Alzheimer’s Disease Parkinson’s disease, insomnia, major depression, obsessive The AD is a chronic and progressive neurodegenerative disorder compulsive disorder (OCD), borderline personality disorder that leads to deficits in memory and cognitive brain functions (BPD), and bipolar disorder (Armstrong et al., 2016; Kambeitz (Albert et al., 2011). The AD can be described as a disconnection et al., 2016; Manelis et al., 2016; Xu et al., 2016; Algunaid et al., syndrome because of the altered structural and functional 2018; Díez-cirarda et al., 2018; Li et al., 2018; Zhi et al., 2018), but connectivity architecture of the brain in those suffering from their contribution is negligible and more attention is required in this disease (Pievani et al., 2011). Aging is naturally associated future research. with some cognitive decline, but if this inefficiency is exacerbated Epilepsy in an individual’s brain, one could experience mild cognitive Epilepsy is a chronic neurological disorder that is accompanied impairment (MCI), which is an intermediate phase between by aberrations in brain activity, resulting in recurring seizures age-related cognitive decline and dementia (Petersen, 2002). and occasionally loss of consciousness (Hauser and Hesdorffer, Statistical surveys report that ∼15% of adults over 65 years old 1990). Temporal lobe epilepsy (TLE) is the most prevalent form experience MCI (amnestic MCI or non-amnestic MCI) and that of epilepsy with partial seizures (Bernhardt et al., 2015). In two more than half of these cases convert to dementia in 5 years interesting rs-fMRI studies using network analysis, Výtvarová (Farlow, 2009). Early detection of the AD in subjects with MCI et al. (2017) and Dong et al. (2016) described the contribution can prevent the progression of these impairments via disease- of basal ganglia thalamocortical circuitry to the whole-brain modifying treatments (Allison et al., 2014). Fortunately, the functional connectivity in TLE. Although the detection and combination of graph theory and rs-fMRI has been able to act removal of epileptogenic lesions are necessary for the abolition as a disease biomarker and reveal large-scale disconnection that of seizures, many studies have shown that seizures in TLE is present before onset of AD symptoms (Wang et al., 2013; Brier originate from abnormalities in the epileptogenic network rather et al., 2014; Dai and He, 2014; Botha and Jones, 2018). than from lesions (Rosenow and Lüders, 2001; Cooray et al., By examining the brain network characteristics on functional 2015); thus, seizure recurrence is observed following ∼40% of connectivity, researchers concluded that individuals with AD epilepsy surgeries within 5 years (Spencer, 2002). Therefore, exhibited degeneration of specific brain hubs, reduced clustering the application of graph theory, along with clinico-radiological coefficients and path lengths very close to the values of random findings, helps to better understand the network mechanisms networks (Supekar et al., 2008; Sanz-Arigita et al., 2010; Dai behind a cognitive decline in focal epilepsies, particularly TLE, et al., 2015; delEtoile and Adeli, 2017), similar to the results of and offers promising diagnostic biomarkers (Chiang and Haneef, researchers who worked on other imaging modalities (de Haan 2014; Onias et al., 2014; Wang et al., 2014b; Pedersen et al., 2015; et al., 2009, 2012; Stam et al., 2009; Kim et al., 2015; Jalili, 2017). Ridley et al., 2015; Iyer et al., 2018). Also, other studies revealed that cognitive impairment in the AD Vlooswijk et al. examined small-world properties in patients was associated with a weakness in modular interconnectivity and with TLE using rs-fMRI (Vlooswijk et al., 2011). In contrast to hubs destruction (Brier et al., 2014) and significant alterations healthy subjects, they found a disruption of both local segregation within the default network (Toussaint et al., 2014; Zhong et al., [opposed to Wang et al. (2014b)] and global integration in 2014). These findings were in parallel with a global decrease in patients with epilepsy. They confirmed the association between long-distance functional connections especially between frontal the IQ score and information processing performance, whether and caudal brain regions (Sanz-Arigita et al., 2010). On the whole, Frontiers in Neuroscience | 16 June 2019 | Volume 13 | Article 585 Farahani et al. Graph Theory and Brain Networks the degeneration and randomization of the brain functional impaired patients with increased relative importance (centrality) architecture in patients with AD indicates a great loss of global of the DMN. information integration. These results are highly associated with Taken together, major changes in topological parameters of the anterior-posterior disconnection phenomenon and its role in the brain network have been observed in the sensorimotor, the AD. cingulate, and frontotemporal cortex, as well as in the thalamus Moreover, authors combined graph theoretical approaches (Schoonheim et al., 2014, 2015; Tewarie et al., 2015; Faivre et al., with advanced machine learning methods (here, support vector 2016; Rocca et al., 2016; Eijlers et al., 2017). The thalamus is often machines) to explore functional brain network alterations and known as a relay organ between several cortical and subcortical classify individuals with AD using rs-fMRI (Khazaee et al., 2015, regions, taking part in a large variety of neurological functions 2016; Hojjati et al., 2017). Further, by conducting statistical such as motor, sensory, integrative, and higher cortical functions analysis on the brain networks of individuals with MCI who (Minagar et al., 2013). Thus, thalamic degeneration may lead to converted to AD (MCI converter) and those with stable MCI cognitive dysfunction and physical disability in patients with MS, (MCI non-converter), they identified areas underlying this even in the early stages of the disease (Benedict et al., 2013). conversion (Hojjati et al., 2017). To sum up, these papers highlighted the efficiency of combining graph theory and Autism Spectrum Disorder machine learning for early detection of AD based on rs-fMRI ASD is a complex neurodevelopmental disability characterized connectivity analysis. by difficulties in communication and behavior (Roux et al., 2012). The increasing prevalence of ASD over the last decade has underlined the need for medical assessment to identify Multiple Sclerosis the symptoms and signs of this disorder (Johnson and Myers, MS is a chronic, degenerative, and heterogeneous autoimmune 2007). However, there are possible challenges in autism screening disease of the central nervous system, leading to physical, mental, because of the uncertainty associated with the symptoms and or psychiatric problems (Marrie, 2017). Functional recovery in neurobiological properties (Ecker et al., 2013; Mastrovito et al., MS is achieved by repair of damage through remyelination 2018). These properties lead to great heterogeneity in the subjects and functional reorganization, which are the striking hallmarks and are the reason for the spectrum of the disease (Lenroot and of this disease (Filippi and Agosta, 2009). Most studies of Yeung, 2013; Jeste and Geschwind, 2014). functional connectivity based on graph theory in MS include The contribution of rs-fMRI studies based on graph theory analysis of rs-fMRI data (Gamboa et al., 2014). In one such for autism exploration is considerable (Redcay et al., 2013; study, Schoonheim et al. (2014) sorted the brain regions of Rudie et al., 2013; Di Martino et al., 2014; Keown et al., 2017; interest based on their connectivity patterns using eigenvector van den Heuvel et al., 2017; Kazeminejad and Sotero, 2018). centrality mapping (ECM) and reported MS-related differences Authors in Rudie et al. (2013) and Keown et al. (2017) compared for centrality in specific regions. As a result, decreased ECM the brain topology in patients with ASD and healthy controls. values in sensorimotor and ventral stream areas were associated They concluded that modularity, clustering coefficient, and local with clinical disability. In contrast, the thalamus and posterior efficiency are relatively reduced in ASD (i.e., inefficiency of cingulate demonstrated increased centrality as well as higher information transmission in a particular module) while global connectivity to regions with low centrality. To this end, the communication efficiency is increased (shorter average path authors suggested a rerouting of thalamic communications to lengths). As another example, Redcay et al. (2013) observed overcome the continuous inflammatory activity. an increase in betweenness centrality and local connections by In two other studies, Shu et al. (2016) and Liu et al. (2017) analyzing the prefrontal brain areas in adolescents with ASD. compared the topological changes of functional connectome in Moreover, the structure of the hub nodes was significantly individuals with clinically isolated syndrome (i.e., the earliest changed in ASD (Itahashi et al., 2014; Balardin et al., 2015). stage of MS) and MS patients. Their graph-based results Altogether, abnormalities in the functional architecture of the indicated that disrupted network organization emerged in the autistic brain were reported in both local and global metrics. earliest stage of MS, with a lesser degree relative to MS. Considering the huge discrepancies between subjects regarding Also, the extent of network alterations was correlated with local parameters (Finn et al., 2015), it was unclear whether cognitive impairment and physical disability only in MS patients. such local parameters can be applied alone as a biomarker for Importantly, Eijlers et al. (2017) attempted to demonstrate how ASD screening. To answer this question, Sadeghi et al. (2017) abnormalities in functional network hierarchy are related to examined both local and global parameters extracted from rs- cognitive impairment in MS patients. Patients were classified fMRI data and observed that distinctive features were only into three categories: cognitively impaired, mildly cognitively among the local parameters. impaired, and cognitively preserved. The centrality indices indicated that the occipital, sensorimotor, and hippocampal areas Attention-Deficit/Hyperactivity Disorder for all three patient groups became less central than healthy ADHD affects about 3–5% of children globally (Nair et al., 2006). controls, while cognitively impaired patients displayed extensive Wang et al. (2009) were the first to explore the spontaneous centrality growth in areas making up the DMN compared to connectivity patterns of whole-brain functional network in other groups. Their results can be interpreted as reflecting patients with ADHD and healthy controls using graph analysis the hallmark alterations in functional networks of cognitively of rs-fMRI. They reported that the functional networks in both Frontiers in Neuroscience | 17 June 2019 | Volume 13 | Article 585 Farahani et al. Graph Theory and Brain Networks groups represented an economic small-world behavior. However, tendency toward regular configurations (Wang et al., 2009), while the brain networks of ADHD children exhibited more-regular ADHD adults had no significant difference in terms of global configurations with higher local efficiency and a trend toward architecture with healthy individuals (Cocchi et al., 2012). Also, decreased global efficiency relative to healthy subjects, indicating disturbed nodal properties were identified in both children and a developmental delay of whole-brain functional networks in adults, particularly in the attention, default-mode, sensorimotor, this pathology (Wang et al., 2009; Cao et al., 2013, 2014a, 2016; striatum, and cerebellum networks (Wang et al., 2009; Fair et al., van den Heuvel et al., 2017). In addition, by testing nodal 2010, 2013; Cocchi et al., 2012; Tomasi and Volkow, 2012; properties, Wang et al. (2009) claimed that areas such as medial Di Martino et al., 2013). prefrontal, temporal, and occipital cortices experienced regional loss of efficiency, while increased nodal efficiency was found in the inferior frontal gyrus. CHALLENGES AND FUTURE DIRECTIONS Delayed maturation has further been reported in structural MRI studies (Hoogman et al., 2017), as well as in default network In general, the consistency of results across similar experiments connectivity in youth with ADHD (Fair et al., 2010). Maturation that employed a graph theoretical approach indicates that rate differences between brain hemispheres may also characterize this perspective is promising for establishing a comprehensive the ADHD brain, given significantly different interhemispheric and sustainable model in future fMRI studies. However, it is asymmetry patterns recently observed in ADHD youths (Douglas sometimes difficult to integrate all of the reported findings an of et al., 2018). Analyzing rs-fMRI, Fair et al. (2010) scrutinized pathological brain networks because the results do not coincide interregional connectivity patterns within DMN and noticed with each other when the factors affecting the experiments are decreased anterior-posterior connectivity in children with different. For instance, patient demographic factors (such as age, ADHD compared to healthy controls. In another study, Fair et al. gender, educational level, etc.), disease-specific characteristics (2013) conducted a regional connectivity analysis using degree (such as duration, course, severity, disability level, etc.), sample index on the functional networks in children with two different size, and network construction greatly vary across the studies. ADHD presentations, i.e., inattentive and combined. While As an example of network construction, ignoring the negative both subtypes exhibited some overlapping (particularly in the entries in the connectivity matrix is very likely to result in the sensorimotor network), the combined ADHD exhibited atypical loss of valuable information (Shu et al., 2016). To overcome patterns in midline DMN components and the inattentive these heterogeneities and increase the reliability of the findings, ADHD showed atypical connectivity within the dorsolateral more consistent comparisons can be made across the studies. In prefrontal cortex and cerebellum. Contrary to the findings addition, there are several image repositories for pairwise studies of children with ADHD, Cocchi et al. (2012) did not find in the area of brain network connectivity that can be explored by any significant changes in global characteristics of the whole- various packages based on graph theory (Rubinov and Sporns, brain functional networks in adults with ADHD compared to 2010; Hosseini et al., 2012; Kruschwitz et al., 2015; Wang et al., healthy controls. 2015a; Mijalkov et al., 2017; Waller et al., 2018). Apart from the region-wise studies, Tomasi and Volkow Although the importance of computational approaches in (2012) computed the voxel-wise Pearson’s correlations across all fMRI analysis has been evident over the last decade, it pairs of brain voxels in ADHD children and healthy controls has not always matched the richness of fMRI data (Cohen from the ADHD-200 database (Milham et al., 2012). Then, they et al., 2017). Early methods mostly neglected the ability of classified the coefficients into long-range and short-range based predictive models to better understand the distributed and on the anatomical distance, which was followed by constructing dynamic nature of neural representations. Recently, several the corresponding functional connectivity density. As a result, theory-driven techniques have commenced to highlight the they revealed that ADHD children had weaker interconnectivity salient role of machine learning, algorithmic optimization, and (both long- and short-range) in the DMN, dorsal attention parallel computing in fMRI analysis (Cohen et al., 2017). network, and cerebellum, and stronger short-range connectivity Hence, adoption of modern techniques, such as multivoxel within reward network (ventral striatum and orbitofrontal pattern analysis (MVPA), convolutional neural network (CNN), cortex). Alterations in DMN have also been reported in studies generative models, and real-time analysis, then aligning them applying non-negative matrix factorization (Anderson et al., with graph theoretical concepts might open a new generation of 2014). In another study, Di Martino et al. (2013) observed similar experiments that could transform our understanding of complex centrality abnormalities within the precuneus in both ADHD properties in the human brain networks. and ASD groups, whereas ADHD patients exhibited particularly Another challenge in graph theory research is developing higher-degree centrality in the right striatum/pallidum. Finally, a consensus about which of the brain parcellation schemes is Colby et al. (2012) presented a machine learning approach optimal for defining network nodes and constructing the brain using the combination of functional and structural graph-based network (Hayasaka and Laurienti, 2010). Different parcellation features, as well as demographic information, to predict status methods may lead to different topological properties in the of patients with ADHD from healthy children in the ADHD-200 human brain networks, and the results depend on the network database (Milham et al., 2012). resolution. However, for better insight, one can appraise the By interpreting the above findings, it can be concluded reproducibility of the primary findings by applying multiple that the functional connectomes of ADHD children had a parcellation schemes at different spatial scales, particularly Frontiers in Neuroscience | 18 June 2019 | Volume 13 | Article 585 Farahani et al. Graph Theory and Brain Networks those with high resolution (Liu et al., 2017). Moreover, node Therefore, the progression of neurodegenerative disorders may specification in developmental research is extremely important not be well-understood, and subsequently, treatment strategies as it is possible for nodes to be dissimilar across a sample, exhibit poor performance. Madhyastha et al. (2017) reported that which may distort the brain network. Therefore, a fundamental longitudinal fMRI studies with graph theory provide a suitable condition for ensuring the reliability of graph analysis in brain means for understanding the development of pathological connectivity studies is the precise definition of network nodes conditions, as well as tracking temporal correlations between (Stanley et al., 2013), which itself requires the adoption of an topological alterations in the brain network. They also noted appropriate parcellation strategy (Power et al., 2010, 2011). that some development-related issues are still not answered by Although structural pathways are thought to underlie existing software, which should be further explored (Madhyastha functional connectivity patterns (Honey et al., 2009), one et al., 2017). Additionally, longitudinal studies could be employed cannot claim that there is a one-to-one correspondence between in the future for monitoring brain network topological changes topological properties in functional and structural organizations using different therapeutic strategies across longer time durations (Park and Friston, 2013; Wang et al., 2015b; Mash et al., 2018). (Mears and Pollard, 2016). In some neurological diseases such as schizophrenia, small-world network abnormalities may even display opposite directions CONCLUSION over functional and structural organizations. Concerning this matter, van den Heuvel et al. recognized evidence of reduced In this paper, we first reported an in-depth overview of local efficiency and segregation (i.e., clustering and modularity) the computational methods that were proposed to discover together with increased global efficiency in several functional functional and effective connectivity in the human brain network studies of schizophrenia. However, their review of structural using fMRI. In discussing each method, we highlighted their studies resulted in contradictory findings, such as increased strengths and potential drawbacks. Then, as the main focus segregation along with reduced integration and global efficiency of the current paper, comprehensive information on graph (Van Den Heuvel and Fornito, 2014). Moreover, Shu et al. theoretical analysis of connectivity patterns in the complex (2016) examined the structural and functional disruptions in brain network along with its applications in neuroscience the earliest stage of MS and MS patients by combined use of was presented. The brain network topology is expected to DTI and rs-fMRI. Their study exhibited structural changes in be responsive to cognitive performance, behavioral variability, the earliest stage of MS, while functional patterns remained experimental task, and neurological disorders such as epilepsy, stable at that stage. Hence, structure-function relationship Alzheimer’s disease, multiple sclerosis, autism, and attention- studies are needed to help elucidate such existing deviations for deficit/hyperactivity disorder. Graph theoretical metrics such future work. as node degree, clustering coefficient, average path length, The primary features of the small-world organization, i.e., hubs, centrality, modularity, robustness, and assortativity can high local clustering yet short characteristic path length, be utilized to detect the topological patterns of brain networks contribute to the efficient flow of information within and reflect cognitive and behavioral performances (Sporns interconnected complex systems, a pivotal role that can et al., 2004; van den Heuvel et al., 2008b). However, graph reveal discrepancies between groups or across conditions. analysis in human neuroscience faces a number of issues However, most techniques that evaluate small-world properties that remain unaddressed, restricting its interpretation and in real-world systems face significant constraints, such as application (De Vico Fallani et al., 2014). Some examples misdiagnosis of some regular lattices as a small-world structure, are heterogeneity of the results, sensitivity to parcellation lack of attention to weighted graphs, as well as neglecting strategy and node specification, statistical variability of brain the variations in network density and connection strengths. graphs due to noise, lack of attention to the structure-function Fortunately, researchers have made notable efforts in the past relationship, neglecting the variations in network density and decade to resolve these limitations in complex networks by connection strength, and dynamics of the brain network. proposing novel small-world metrics (Rubinov and Sporns, Addressing any of these limitations in future studies will help 2010; Telesford et al., 2011; Bolaños et al., 2013; Muldoon advance our understanding of functional neural networks in the et al., 2016). Applying these newly introduced measures human brain. into future brain connectivity investigations can bring about widespread improvement in knowledge regarding small-world AUTHOR CONTRIBUTIONS brain architecture. The dynamics of brain function seem to result in numerous FF conducted the literature search and prepared the cognitive, emotional, and behavioral changes that occur during initial draft of the paper. WK supervised all aspects of brain development. However, the majority of studies cannot manuscript preparations, revisions, editing, and final interpret brain network dynamics because their design is typically intellectual content. FF and WK were involved in study cross-sectional and the calculated measures of the brain graph conception and contributed to intellectual content. NL are only capable of displaying a snapshot of the disease over contributed to intellectual content and edited the final draft of time (Fleischer et al., 2017; Avena-Koenigsberger et al., 2018). the paper. Frontiers in Neuroscience | 19 June 2019 | Volume 13 | Article 585 Farahani et al. Graph Theory and Brain Networks REFERENCES fMRI. Neuroimage 51, 1126–1139. doi: 10.1016/j.neuroimage.2010. 02.082 Abós, A., Baggio, H. C., Segura, B., García-Díaz, A. I., Compta, Y., Martí, Benedict, R. H., Hulst, H. E., Bergsland, N., Schoonheim, M. M., Dwyer, M. G., M. J., et al. (2017). Discriminating cognitive status in Parkinson’s disease Weinstock-Guttman, B., et al. (2013). Clinical significance of atrophy and white through functional connectomics and machine learning. Sci. Rep. 7:45347. matter mean diffusivity within the thalamus of multiple sclerosis patients. Mult. doi: 10.1038/srep45347 Scler. J. 19, 1478–1484. doi: 10.1177/1352458513478675 Achard, S. (2006). 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Network Analysis of functional brain connectivity in borderline terms. Frontiers in Neuroscience | 27 June 2019 | Volume 13 | Article 585 Frontiers in Neuroscience Pubmed Central

Application of Graph Theory for Identifying Connectivity Patterns in Human Brain Networks: A Systematic Review

Frontiers in Neuroscience , Volume 13 – Jun 6, 2019

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