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Fermat–Wiener index based on topological Wiener index is the total sum of Fermat distance over all the triplets for vertices. In this paper, we construct a class of hierarchical graphs based on hierarchical product generalized from Cartesian product. We study some critical properties of the hierarchical networks by investigating its topological indices. Applying the finite pattern method, we analytically deduce the dominant term of average Fermat distance and obtain its asymptotic formula, which implies small-world property. We finally exhibit a close connection between Fermat–Wiener index and related graph invariants like average geodesic distance, Wiener index and eigenvalues of Laplacian matrix. [ABSTRACT FROM AUTHOR]

Background: Network theory is largely applied in real-world systems to assess landscape connectivity using empirical or theoretical networks. Empirical networks are usually built from discontinuous individual movement trajectories without knowing the effect of relocation frequency on the assessment of landscape connectivity while theoretical networks generally rely on simple movement rules. We investigated the combined effects of relocation sampling frequency and landscape fragmentation on the assessment of landscape connectivity using simulated trajectories and empirical high-resolution (1 Hz) trajectories of Alpine ibex (Capra ibex). We also quantified the capacity of commonly used theoretical networks to accurately predict landscape connectivity from multiple movement processes. Methods: We simulated forager trajectories from continuous correlated biased random walks in simulated landscapes with three levels of landscape fragmentation. High-resolution ibex trajectories were reconstructed using GPS-enabled multi-sensor biologging data and the dead-reckoning technique. For both simulated and empirical trajectories, we generated spatial networks from regularly resampled trajectories and assessed changes in their topology and information loss depending on the resampling frequency and landscape fragmentation. We finally built commonly used theoretical networks in the same landscapes and compared their predictions to actual connectivity. Results: We demonstrated that an accurate assessment of landscape connectivity can be severely hampered (e.g., up to 66% of undetected visited patches and 29% of spurious links) when the relocation frequency is too coarse compared to the temporal dynamics of animal movement. However, the level of landscape fragmentation and underlying movement processes can both mitigate the effect of relocation sampling frequency. We also showed that network topologies emerging from different movement behaviours and a wide range of landscape fragmentation were complex, and that commonly used theoretical networks accurately predicted only 30–50% of landscape connectivity in such environments. Conclusions: Very high-resolution trajectories were generally necessary to accurately identify complex network topologies and avoid the generation of spurious information on landscape connectivity. New technologies providing such high-resolution datasets over long periods should thus grow in the movement ecology sphere. In addition, commonly used theoretical models should be applied with caution to the study of landscape connectivity in real-world systems as they did not perform well as predictive tools. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global clustering coefficient (a.k.a transitivity) and clique number of graphs generated by a preferential attachment random graph model with an additional feature of allowing edge connections between existing vertices. Specifically, at each time step t, either a new vertex is added with probability f(t), or an edge is added between two existing vertices with probability 1 - f (t) . We establish concentration inequalities for the global clustering and clique number of the resulting graphs under the assumption that f(t) is a regularly varying function at infinity with index of regular variation - γ , where γ ∈ [ 0 , 1) . We also demonstrate an inverse relation between these two statistics: the clique number is essentially the reciprocal of the global clustering coefficient. [ABSTRACT FROM AUTHOR]

Many complex networks have scale-free and small-world effects. In this paper, a family of evolving networks is constructed modeled by a non-symmetric self-similar planar fractal, using the encoding method in fractal geometry. Based on the self-similar structure, we study the degree distribution, clustering coefficient and average path length of our evolving network to verify their scale-free and small-world characteristics. [ABSTRACT FROM AUTHOR]

Social Network Analysis (SNA) is a modeling technique and analytical approach well-suited for identifying and examining the structural features of supply networks and the patterns of connections between members within the network. This paper aims to present a systematic review and bibliometric analysis by investigating network structural properties and metrics in supply chain management (SCM) research. The approach involved combining a systematic literature review with a bibliometric analysis, forming a two-part methodology to examine 113 articles published between 2008 and 2023 in 62 journals. Our systematic thematic analysis reveals how SCM researchers have applied SNA techniques in terms of the reported node-level and network-level structural metrics, including network configuration description metrics, centrality measures, supply network subgroups, and models of supply network structure and formation. We identify the gaps in the existing body of literature and propose potential directions for future research. By quantitatively analyzing, classifying, and visualizing bibliographic data of previous studies, this paper provides further insights into the application of network structural properties in SCM research. Furthermore, our findings contribute to a deeper understanding of the significance of the supply network's relational structure and configuration. Considering the disruptions to global supply chains caused by the COVID-19 pandemic and the Russo-Ukrainian War, our findings can contribute to a better understanding of strategic supply network design.

Functional connectivity (FC) estimated from functional magnetic resonance imaging (fMRI) signals is important in understanding neural representation and information processing in cortical networks. However, due to a lack of "ground truth" FC pattern, the reliability and robustness of FC estimates are usually examined in downstream FC analysis tasks, such as performing participant's identification (also known as "fingerprinting"). In this paper, we propose to learn FC via a smooth graph learning framework. In particular, we treat each time frame of the fMRI time series as a graph signal on an underlying functional brain graph, and estimate the smooth graph functional connectivity (SGFC) by learning the weighted graph adjacency matrix based on graph signal smoothness assumption. We demonstrate that our approach gives rise to a natural and sparse graph representation of FC from which reliable graph measures can be extracted. Reliability of SGFC is evaluated in the context of fingerprinting and compared to correlation FC (CFC). SGFC achieves higher fingerprinting accuracy across several different experiment settings; the improvement is even more significant when a shorter fMRI scanning length is used for FC estimation. In addition to being reliable, we also validate the cognitive relevance of SGFC by using it to predict fluid intelligence. Finally, in evaluating topological measures of the sparse graph, SGFC reveals a more small-world and modular structure compared to CFC. Together, our results suggest that the smooth graph learning framework produces a naturally sparse, reliable, and cognitive-relevant representation of functional connectivity.

The development of the cerebral cortex during the fetal period is a complex yet well-coordinated process. MRI-based morphological brain network provides a powerful tool for describing this process at a network level. Due to the challenges of in-utero MRI acquisition and image processing, the fetal morphological brain network has not been established. In this study, utilizing high-resolution in-utero MRI data, we constructed an individual morphometric similarity network for each fetus based on multiple cortical features. The spatiotemporal development of morphological connections was described at the level of edge, node, and lobe, respectively. Based on graph theoretical method, the topology structure of fetal morphological network was characterized. Edge analysis demonstrated an increase of morphological dissimilarity between hemispheres with gestational age, especially for the parietal cortex. The limbic and parieto-occipital regions exhibited the most drastic changes of morphological connections at both the edge and node levels. Between- and within-lobe analysis illustrated that the limbic lobe became more similar to other lobes, while the parietal and occipital lobes became more dissimilar to other lobes. Graph theoretical analysis indicated that the small-world structure of the fetal morphological network appeared as early as 22 weeks and that the network topology exhibited an enhanced integration and reduced segregation during prenatal development. The findings obtained from the preterm-born neonates agreed well with those of the fetuses. In summary, this study fills a gap in prenatal morphological brain network research and provides a piece of important evidence for understanding the normal development of fetal brain connectome during the second-third trimester.

For patients in early coma after cardiopulmonary resuscitation (CPR), quantitative electroencephalogram (EEG) and brain network analysis was performed to identify relevant indicators of awakening. A prospective cohort study was conducted on comatose patients after CPR in the neuro-critical care unit. The included patients received clinical evaluation. The bedside high-density (64-lead) EEG monitoring was performed for visual grading and calculation of power spectrum and brain network parameters. A 3-month prognostic assessment was performed and the patients were dichotomized into the awakening group and the unawakening group. A total of 25 patients were included. The awakening group had higher GCS score, more slow wave pattern and reactive EEG than the unawakening group (P = 0.003, P < 0.001, P < 0.001, respectively). Compared with the unawakening group, (1) the awakening group had significantly higher absolute and relative θ power and slow/fast band ratio of the whole brain (P < 0.05), (2) the awakening group had stronger connection based on coherence, phase synchronization, phase lag index and cross-correlation (P < 0.05), (3) the awakening group had higher small-worldness, clustering coefficient and average path length based on graph theory (P < 0.05). The power spectrum and brain network characteristics in patients in early coma after CPR have predictive value for recovery. [ABSTRACT FROM AUTHOR]

The Fermat problem is a crucial topological issue corresponding to fractal networks. In this paper, we discuss the average Fermat distance (AFD) of the Vicsek polygon network and analyze structural properties. We construct the Vicsek polygon network based on Vicsek fractal in an iterative way. Given the structure of network, we present an elaborate analysis of the Fermat point under various situations. The special network structure allows a way to calculate the AFD based on average geodesic distance (AGD). Moreover, we introduce the Vicsek polygon fractal and calculate its AGD and AFD. Its relationship with the network enables us to deduce the above two indices of the network directly. The results show that both in network and fractal, the ratio of AFD and AGD tends to 3/2, which demonstrates that both of them can serve as indicators of small-world property of complex networks. In fact, in Vicsek polygon network, the AFD grows linearly with network order, implying that our evolving network does not possess the small-world property. [ABSTRACT FROM AUTHOR]

El Niño-Southern Oscillation (ENSO) is a complex climate phenomenon that results from ocean–atmosphere interactions and exhibits nonlinearity in both spatial and temporal evolution. The directed horizontal visibility graph (DHVG) and Kullback–Leibler divergence (KLD) efficiently characterize the nonlinearity of complex systems without requiring additional symbolization processes. Our study utilizes this method to quantify sea surface temperature (SST) irreversibility across multiple El Niño events and identified similar fluctuation patterns. Subsequently, we map the irreversibility series into visibility graphs and analyze multiple topological properties to investigate the fluctuation structure of SST irreversibility. The irreversibility fluctuation structure effectively explains the periodic changes of El Niño and its nonlinear evolution over time. Results show that KLD exhibits a sharp increase during the strengthening and ending phases of El Niño events and remains in a lower value after the peak month of Niño events; the visibility graph of SST irreversibility exhibits scale-free and small-world properties, indicating that the KLD series has scale invariance and self-similarity, and its fluctuations are not random but correlated; Hurst exponent analysis revealed long-range anti-persistence and mean reversion characteristics in the KLD series. This study utilizes complex network and information theory to investigate the fluctuation pattern and structure of SST irreversibility during El Niño events, providing novel insights into the evolution of ENSO over time scales. [ABSTRACT FROM AUTHOR]

In this paper, we construct a class of growing networks by the encoding method of the iterated function system based on a planar self-similar fractal, and demonstrate that the networks have small-world and scale-free effects. [ABSTRACT FROM AUTHOR]

The Boolean satisfiability problem (SAT) is a fundamental NP-complete decision problem in automated reasoning and mathematical logic. As evidenced by the results of SAT competitions, the performance of SAT solvers varies substantially between different SAT categories (random, crafted, and industrial). A suggested explanation is that SAT solvers may exploit the underlying structure inherent to SAT instances. There have been attempts to define the structure of SAT in terms of structural measures such as phase transition, backbones, backdoors, small-world, scale-free, treewidth, centrality, community, self-similarity, and entropy. Still, the empirical evidence of structural measures for SAT has been provided for only some SAT categories. Furthermore, the evidence has not been theoretically proven. Also, the impact of structural measures on the behavior of SAT solvers has not been extensively examined. This work provides a comprehensive study on structural measures for SAT that have been presented in the literature. We provide an overview of the works on structural measures for SAT and their relatedness to the performance of SAT solvers. Accordingly, a taxonomy of structural measures for SAT is presented. We also review in detail important applications of structural measures for SAT, focusing mainly on enhancing SAT solvers, generating SAT instances, and classifying SAT instances. [ABSTRACT FROM AUTHOR]

Fermat point of a triangle is the point with the minimal total distance from the three vertices in a triangle. In this paper, we discuss the average Fermat distance for a class of hierarchical networks. First, the unweighted hierarchical scale-free network is established in an iterative way. Applying the recursive method, we deduce the analytical expression of average Fermat distance and average geodesic distance. Then we reveal the linear relation of the leading terms for average Fermat distance and average geodesic distance. Finally, we obtain the small-world property of the hierarchical scale-free network, which indicates that average Fermat distance can be a valuable index of small-word property. [ABSTRACT FROM AUTHOR]

Infectious disease outbreaks bring new challenges to campus operations at institutions of higher education (IHEs). At the beginning of the COVID-19 pandemic, as a precaution against COVID-19 transmission on their campuses, many IHEs shifted to entirely remote courses in the Spring 2020 semester. As these institutions prepared for the Fall 2020 semester, their administrations evaluated the extent to which remote courses should continue. While remote courses may reduce the number of students who interact on campus and thus mitigate the risk of disease spread, there may be academic and financial downsides. Using de-identified enrollment, course schedule, and student profile data from the Fall 2019 semester at Georgia Institute of Technology (Georgia Tech), we construct a co-enrollment network , which is a representation of student connections through courses. In this co-enrollment network, we observe the small-world property in which the average path length among reachable student pairs is short and nearly all students are connected to each other in the main component of the network through multiple independent paths. We also observe high connectivity between different majors. Then, we apply various hybrid instructional mode strategies to this co-enrollment network and use network connectivity and in-person instruction metrics to compare these strategies from health and academic perspectives, by their impacts on various groups of students, and based on the trade-off between reducing network connectivity and delivering in-person instruction. Although these strategies result in reduced network connectivity, they do not remove the small-world property. When comparing the strategies in terms of reducing network connectivity, we find three different strategy types to be the most favorable. Even under these strategies, the percentage of students with in-person courses that could be exposed rises quickly with initial infection rates. When comparing the strategies in terms of delivering in-person instruction, the most favorable strategies select a fixed percentage of courses for each subject to be delivered remotely. We then consider the impacts of these strategies on various groups of students to evaluate if there may be disparate impacts across these groups. In general, we find that strategies with more remote courses result in less disparate impacts on network connectivity and in-person instruction across these groups. In addition, we generally see less in-person instruction for lower-level students (i.e., first-year students and sophomores) compared to upper-level students. Regarding the trade-off between reducing network connectivity and delivering in-person instruction, the only strategy that is dominated under all scenarios in this study is the 1000s and 4000s only strategy, which keeps only the 1000- and 4000-level (i.e., first-year and senior level, respectively) courses in-person. The reason for this is that even though the university offers less in-person instruction due to this strategy's large quantity of remote courses, this strategy results in more connected networks because the 1000- and 4000-level courses tend to be large in size, the 1000-level courses are highly central, and the first-year students and seniors have high connectivity. By quantitatively comparing network connectivity and in-person instruction for these various strategies, campus administrators can weigh the physical health, academic performance, and mental health of students, along with the financial well-being of their academic institutions when responding to infectious disease outbreaks. • Co-enrollment networks exhibit the small-world property, even after various hybrid instructional mode strategies are applied. • We evaluate these strategies from health and academic perspectives with network connectivity and in-person instruction metrics, respectively. • Different strategies are favorable in terms of reducing network connectivity, delivering in-person instruction, impacts across various groups of students, and the trade-off between reducing network connectivity and delivering in-person instruction. • strategy that keeps only the first-year and senior level courses in-person does not achieve an efficient trade-off between reducing network connectivity and delivering in-person instruction. [ABSTRACT FROM AUTHOR]

Community detection has attracted many attentions recently. Considering the effect of current network structure on the result of the recent community detection methods is useful to yield a probable performance trade-off for future algorithm selection. In this paper, we first offer a new ranking method with 3 levels for small-world and scale-free networks to measure such properties more accurately, in determining their influences on the methods performance. Thereafter, we examine 12 popular community detection methods and 43 related datasets. The results show that 24 datasets have small-world properties, 5 datasets have scale-free properties, and 9 datasets have both. However, 5 of them have no features of small-world or scale-free networks. It is also observable that 4 methods work better for networks with small-world features and 8 for both small-world and scale free. Finally, we propose a flexible community detection method based on the detected network type.

Knowledge synchronization in robot swarm systems is a challenging task under communication constraints. Swarm robots cannot maintain the complete communication structure with classic fixed communication network, which leads to the decline of motion decision effects. It is expected to improve the performance of swarm robot motion decisions under communication constraints with more researches on dynamic complex networks. This study established a feedback search control model for robot swarm systems for target search tasks in a 3D environment. Inspired by the WS model, which showed good synchronization performance in complex systems owing to its small average shortest path length, a novel dynamic small-world network model called decay small-world was proposed for negative impact of communication constraints. Decay small-world model reconnects robot communication links by controlling the rewired probability based on the half-life formula. It realizes dynamic network topology by decentralized computing. This new model maintains a small-world pattern over time without degenerating it into a random network. Simulations show that good knowledge synchronization of swarm robots can be realized using the decay small-world model. And the results also show that target search performances are promoted by this method.

Covering neuroimaging evidence has demonstrated that epileptic symptoms are associated with the disrupted topological architecture of the brain network. Infantile spasms (IS) as an age-specific epileptic encephalopathy also showed abnormal structural or functional connectivity in specific brain regions or specific networks. However, little is known about the topological alterations of whole-brain functional networks in patients with IS. To fill this gap, we used the graph theoretical analysis to investigate the topological properties (whole-brain small-world property and modular interaction) in 17 patients with IS and 34 age- and gender-matched healthy controls. The functional networks in both groups showed efficient small-world architecture over the sparsity range from 0.05 to 0.4. While patients with IS showed abnormal global properties characterized by significantly decreased normalized clustering coefficient, normalized path length, small-worldness, local efficiency, and significantly increased global efficiency, implying a shift toward a randomized network. Modular analysis revealed decreased intramodular connectivity within the default mode network (DMN) and frontoparietal network but increased inter-modular connectivity between the cingulo-opercular network and occipital network. Moreover, the decreased intra-modular connectivity in DMN was significantly negatively correlated with seizure frequency. The inter-modular connectivity between the cinguloopercular and occipital network also showed a significant correlation with epilepsy frequency. Together, the current study revealed the disrupted topological organization of the whole-brain functional network, which greatly advances our understanding of neuronal architecture in IS and may contribute to predict the prognosis of IS as disease biomarkers. [ABSTRACT FROM AUTHOR]

Neuroimaging studies have identified alterations in functional connectivity between specific brain regions in patients with unilateral hearing loss (UHL) and different influence of the side of UHL on neural plasticity. However, little is known about changes of whole-brain functional networks in patients with UHL and whether differences exist in topological organization between right-sided UHL (RUHL) and left-sided UHL (LUHL). To address this issue, we employed resting-state fMRI (rs-fMRI) and graph-theoretical approaches to investigate the topological alterations of brain functional connectomes in patients with RUHL and LUHL. Data from 44 patients with UHL (including 22 RUHL patients and 22 LUHL patients) and 37 healthy control subjects (HCs) were collected. Functional brain networks were constructed for each participant, following by graph-theoretical network analyses at connectional and global (e.g., small-worldness) levels. The correlations between brain network topologies and clinical variables were further studied. Using network-based analysis, we found a subnetwork in the visual cortex which had significantly lower connectivity strength in patients with RUHL as compared to HCs. At global level, all participants showed small-world architecture in functional brain networks, however, significantly lower normalized clustering coefficient and small-worldness were observed in patients with RUHL than in HCs. Moreover, these abnormal network metrics were demonstrated to be correlated with the clinical variables and cognitive performance of patients with RUHL. Notably, no significant alterations in the functional brain networks were found in patients with LUHL. Our findings demonstrate that RUHL (rather than LUHL) is accompanied with aberrant topological organization of the functional brain connectome, indicating different pathophysiological mechanisms between RUHL and LUHL from a viewpoint of network topology. [ABSTRACT FROM AUTHOR]

We investigate the dynamic behavior of lattices with disorder introduced through non-local network connections. Inspired by the Wattsâ€"Strogatz small-world model, we employ a single parameter to determine the probability of local connections being re-wired, and to induce transitions between regular and disordered lattices. These connections are added as non-local springs to underlying periodic one-dimensional (1D) and two-dimensional (2D) square, triangular and hexagonal lattices. Eigenmode computations illustrate the emergence of spectral gaps in various representative lattices for increasing degrees of disorder. These gaps manifest themselves as frequency ranges where the modal density goes to zero, or that are populated only by localized modes. In both cases, we observe low transmission levels of vibrations across the lattice. Overall, we find that these gaps are more pronounced for lattice topologies with lower connectivity, such as the 1D lattice or the 2D hexagonal lattice. We then illustrate that the disordered lattices undergo transitions from ballistic to super-diffusive or diffusive transport for increasing levels of disorder. These properties, illustrated through numerical simulations, unveil the potential for disorder in the form of non-local connections to enable additional functionalities for metamaterials. These include the occurrence of disorder-induced spectral gaps, which is relevant to frequency filtering devices, as well as the possibility to induce diffusive-type transport which does not occur in regular periodic materials, and that may find applications in dynamic stress mitigation. [ABSTRACT FROM AUTHOR]

Frontotemporal dementia (FTD) is a spectrum of clinical syndromes that affects personality, behavior, language, and cognition. The current diagnostic criteria recognize three main clinical subtypes: the behavioral variant of FTD (bvFTD), the semantic variant of primary progressive aphasia (svPPA), and the non-fluent/agrammatic variant of PPA (nfvPPA). Patients with FTD display heterogeneous clinical and neuropsychological features that highly overlap with those presented by psychiatric syndromes and other types of dementia. Moreover, up to now there are no reliable disease biomarkers, which makes the diagnosis of FTD particularly challenging. To overcome this issue, different studies have adopted metrics derived from magnetic resonance imaging (MRI) to characterize structural and functional brain abnormalities. Within this field, a growing body of scientific literature has shown that graph theory analysis applied to MRI data displays unique potentialities in unveiling brain network abnormalities of FTD subtypes. Here, we provide a critical overview of studies that adopted graph theory to examine the topological changes of large-scale brain networks in FTD. Moreover, we also discuss the possible role of information arising from brain network organization in the diagnostic algorithm of FTD-spectrum disorders and in investigating the neural correlates of clinical symptoms and cognitive deficits experienced by patients. [ABSTRACT FROM AUTHOR]

The well-known Sierpiński square is a fractal generated by iterated function system (IFS). In this paper, we focus on a class of fractal networks created by IFS. We show a universal approach to solve the average geodesic distance of these fractal networks. [ABSTRACT FROM AUTHOR]

Every blood vessel is lined by a single layer of highly specialized, yet adaptable and multifunctional endothelial cells. These cells, the endothelium, control vascular contractility, hemostasis, and inflammation and regulate the exchange of oxygen, nutrients, and waste products between circulating blood and tissue. To control each function, the endothelium processes endlessly arriving requests from multiple sources using separate clusters of cells specialized to detect specific stimuli. A well-developed but poorly understood communication system operates between cells to integrate multiple lines of information and coordinate endothelial responses. Here, the nature of the communication network has been addressed using single-cell Ca2+ imaging across thousands of endothelial cells in intact blood vessels. Cell activities were cross-correlated and compared to a stochastic model to determine network connections. Highly correlated Ca2+ activities occurred in scattered cell clusters, and network communication links between them exhibited unexpectedly short path lengths. The number of connections between cells (degree distribution) followed a power-law relationship revealing a scale-free network topology. The path length and degree distribution revealed an endothelial network with a “small-world” configuration. The small-world configuration confers particularly dynamic endothelial properties including high signal-propagation speed, stability, and a high degree of synchronizability. Local activation of small clusters of cells revealed that the short path lengths and rapid signal transmission were achieved by shortcuts via connecting extensions to nonlocal cells. These findings reveal that the endothelial network design is effective for local and global efficiency in the interaction of the cells and rapid and robust communication between endothelial cells in order to efficiently control cardiovascular activity. [ABSTRACT FROM AUTHOR]

This work aims to establish whether astroturfing was used during the Covid-19 pandemic to manipulate Spanish public opinion through Twitter. This study analyzes tweets published in Spanish and geolocated in the Philippines, and its first objective is to determine the existence of an organized network that directs its messages mainly towards Spain. To determine the non-existence of a random network, a preliminary collection of 1,496,596 tweets was carried out. After determining its 14 main clusters, 280 users with a medium-low profile of participation and micro- and nano-influencer traits were randomly selected and followed for 103 days, for a total of 309,947 tweets. Network science, text mining, sentiment and emotion, and bot probability analyses were performed using Gephi and R. Their network structure suggests an ultra-small-world phenomenon, which would determine the existence of a possible organized network that tries not to be easily identifiable. The data analyzed confirm a digital communication scenario in which astroturfing is used as a strategy aimed at manipulating public opinion through non-influencers (cybertroops). These users create and disseminate content with proximity and closeness to different groups of public opinion, mixing topics of general interest with disinformation or polarized content. [ABSTRACT FROM AUTHOR]

In this paper, we use a kind of Sierpiński-like carpet to construct a special evolving network, whose nodes are all the solid squares in the construction of the carpet up to stage t. In our network, two nodes are neighbors if and only if the intersection of their corresponding squares is a line segment. Using the encoding method in fractal geometry, we show that our evolving network is scale-free and has the small-world effect, but is not fractal scaling. [ABSTRACT FROM AUTHOR]

Frontotemporal dementia (FTD) is a spectrum of clinical syndromes that affects personality, behavior, language, and cognition. The current diagnostic criteria recognize three main clinical subtypes: the behavioral variant of FTD (bvFTD), the semantic variant of primary progressive aphasia (svPPA), and the non-fluent/agrammatic variant of PPA (nfvPPA). Patients with FTD display heterogeneous clinical and neuropsychological features that highly overlap with those presented by psychiatric syndromes and other types of dementia. Moreover, up to now there are no reliable disease biomarkers, which makes the diagnosis of FTD particularly challenging. To overcome this issue, different studies have adopted metrics derived from magnetic resonance imaging (MRI) to characterize structural and functional brain abnormalities. Within this field, a growing body of scientific literature has shown that graph theory analysis applied to MRI data displays unique potentialities in unveiling brain network abnormalities of FTD subtypes. Here, we provide a critical overview of studies that adopted graph theory to examine the topological changes of large-scale brain networks in FTD. Moreover, we also discuss the possible role of information arising from brain network organization in the diagnostic algorithm of FTD-spectrum disorders and in investigating the neural correlates of clinical symptoms and cognitive deficits experienced by patients.

The clinical treatment and prognosis of major depressive disorder (MDD) are limited by the high degree of disease heterogeneity. It is unclear whether there is a potential network mechanism for age-related heterogeneity. We aimed to uncover the heterogeneity of the white matter (WM) network at different ages of onset and its correlation with different symptom characteristics. 85 first-episode MDD patients and 84 corresponding healthy controls (HCs) were recruited and underwent diffusion tensor imaging scans. Structural network characteristics were analyzed using graph theory methods. We observed an accelerated age-related decline of the WM network in MDD patients compared with HCs. Distinct symptom-related networks were identified in three MDD groups with different onset-age. For early-onset MDD (18-29 years; EOD), higher guilt and loss of interest were correlated with the insula, and inferior parietal lobe which in default mode network and salience network. For mid-term-onset MDD (30-44 years; MOD), higher somatic symptoms were correlated with thalamus which in cortico-striatal-thalamic-cortical circuit. For later-onset MDD (45-60 years; LOD), poor sleep symptoms were correlated with the caudate in the basal ganglia, which suggests the cingulate operculum network in the control of sleep. These results supported a circuit-based heterogeneity associated with the age of onset in MDD. Understanding this circuit-based heterogeneity might help to develop a new target for clinical treatment strategies. [ABSTRACT FROM AUTHOR]

Background: The efficacy and prognosis of major depressive disorder (MDD) are limited by its heterogeneity. MDD with melancholic features is an important subtype of MDD. The present study aimed to reveal the white matter (WM) network changes in melancholic depression. Materials and Methods: Twenty-three first-onset, untreated melancholic MDD, 59 non-melancholic MDD patients and 63 health controls underwent diffusion tensor imaging (DTI) scans. WM network analysis based on graph theory and support vector machine (SVM) were used for image data analysis. Results: Compared with HC, small-worldness was reduced and abnormal node attributes were in the right orbital inferior frontal gyrus, left orbital superior frontal gyrus, right caudate nucleus, right orbital superior frontal gyrus, right orbital middle frontal gyrus, left rectus gyrus, and left median cingulate and paracingulate gyrus of MDD patients. Compared with non-melancholic MDD, small-worldness was reduced and abnormal node attributes were in right orbital inferior frontal gyrus, left orbital superior frontal gyrus and right caudate nucleus of melancholic MDD. For correlation analysis, the 7th item score of the HRSD-17 (work and interest) was positively associated with increased node betweenness centrality (aBC) values in right orbital inferior frontal gyrus, while negatively associated with the decreased aBC in left orbital superior frontal gyrus. SVM analysis results showed that abnormal aBC in right orbital inferior frontal gyrus and left orbital superior frontal gyrus showed the highest accuracy of 81.0% (69/83), the sensitivity of 66.3%, and specificity of 85.2% for discriminating MDD patients with or without melancholic features. Conclusion: There is a significant difference in WM network changes between MDD patients with and without melancholic features. [ABSTRACT FROM AUTHOR]

Community detection has attracted many attentions recently. Considering the effect of current network structure on the result of the recent community detection methods is useful to yield a probable performance trade-off for future algorithm selection. In this paper, we first offer a new ranking method with 3 levels for small-world and scale-free networks to measure such properties more accurately, in determining their influences on the methods performance. Thereafter, we examine 12 popular community detection methods and 43 related datasets. The results show that 24 datasets have small-world properties, 5 datasets have scale-free properties, and 9 datasets have both. However, 5 of them have no features of small-world or scale-free networks. It is also observable that 4 methods work better for networks with small-world features and 8 for both small-world and scale free. Finally, we propose a flexible community detection method based on the detected network type. [ABSTRACT FROM AUTHOR]

Innovation in higher education teaching is essential to respond to global challenges and actively improve teaching (e.g. through new technologies), necessitating the implementation of educational reform programmes that fund educational innovations. Although currently deployed strategies frequently promote networks between innovators to diffuse educational innovations, little is known about the efficiency of these networks or whether they promote innovation diffusion. This study investigates a network comprising 88 higher education teachers who received funding for their educational innovations in a German university. We collected longitudinal data by asking higher education teachers from whom they adopted innovative teaching ideas and requesting self-reports on innovative teaching climate. Our findings show that the teachers' social network had a smaller path length and more clustering than might be expected by chance. This observation might indicate that the examined educational innovation network exhibits a small-world property and allows efficient exchange of ideas among the teachers. In line with our hypotheses, the network's initial tendency toward hierarchy and homophily decreased over time in response to strategies and network interventions. In summary, this study provides initial empirical support that educational reform programmes can create efficient educational innovation networks, facilitating innovation diffusion and promoting change in higher education teaching. [ABSTRACT FROM AUTHOR]

BackgroundThe efficacy and prognosis of major depressive disorder (MDD) are limited by its heterogeneity. MDD with melancholic features is an important subtype of MDD. The present study aimed to reveal the white matter (WM) network changes in melancholic depression.Materials and MethodsTwenty-three first-onset, untreated melancholic MDD, 59 non-melancholic MDD patients and 63 health controls underwent diffusion tensor imaging (DTI) scans. WM network analysis based on graph theory and support vector machine (SVM) were used for image data analysis.ResultsCompared with HC, small-worldness was reduced and abnormal node attributes were in the right orbital inferior frontal gyrus, left orbital superior frontal gyrus, right caudate nucleus, right orbital superior frontal gyrus, right orbital middle frontal gyrus, left rectus gyrus, and left median cingulate and paracingulate gyrus of MDD patients. Compared with non-melancholic MDD, small-worldness was reduced and abnormal node attributes were in right orbital inferior frontal gyrus, left orbital superior frontal gyrus and right caudate nucleus of melancholic MDD. For correlation analysis, the 7th item score of the HRSD-17 (work and interest) was positively associated with increased node betweenness centrality (aBC) values in right orbital inferior frontal gyrus, while negatively associated with the decreased aBC in left orbital superior frontal gyrus. SVM analysis results showed that abnormal aBC in right orbital inferior frontal gyrus and left orbital superior frontal gyrus showed the highest accuracy of 81.0% (69/83), the sensitivity of 66.3%, and specificity of 85.2% for discriminating MDD patients with or without melancholic features.ConclusionThere is a significant difference in WM network changes between MDD patients with and without melancholic features.

Node centrality measures are among the most commonly used analytical techniques for networks. They have long helped analysts to identify “important” nodes that hold power in a social context, where damages could have dire consequences for transportation applications, or who should be a focus for prevention in epidemiology. Given the ubiquity of network data, new measures have been proposed, occasionally motivated by emerging applications or by the ability to interpolate existing measures. Before analysts use these measures and interpret results, the fundamental question is: are these measures likely to complete within the time window allotted to the analysis? In this paper, we comprehensively examine how the time necessary to run 18 new measures (introduced from 2005 to 2020) scales as a function of the number of nodes in the network. Our focus is on giving analysts a simple and practical estimate for sparse networks. As the time consumption depends on the properties in the network, we nuance our analysis by considering whether the network is scale-free, small-world, or random. Our results identify that several metrics run in the order of O(nlogn) and could scale to large networks, whereas others can require O(n2) or O(n3) and may become prime targets in future works for approximation algorithms or distributed implementations.

The human brain contains billions of neurons that flexibly interconnect to support local and global computational spans. As neuronal activity propagates through the neural medium, it approaches a critical state hedged between ordered and disordered system regimes. Recent work demonstrates that this criticality coincides with the small-world topology, a network arrangement that accommodates both local (subcritical) and global (supercritical) system properties. On one hand, operating near criticality is thought to offer several neurocomputational advantages, e.g., high-dynamic range, efficient information capacity, and information transfer fidelity. On the other hand, aberrations from the critical state have been linked to diverse pathologies of the brain, such as post-traumatic epileptiform seizures and disorders of consciousness. Modulation of brain activity, through neuromodulation, presents an attractive mode of treatment to alleviate such neurological disorders, but a tractable neural framework is needed to facilitate clinical progress. Using a variation on the generative small-world model of Watts and Strogatz and Kuramoto's model of coupled oscillators, we show that the topological and dynamical properties of the small-world network are divided into two functional domains based on the range of connectivity, and that these domains play distinct roles in shaping the behavior of the critical state. We demonstrate that short-range network connections shape the dynamics of the system, e.g., its volatility and metastability, whereas long-range connections drive the system state, e.g., a seizure. Together, these findings lend support to combinatorial neuromodulation approaches that synergistically normalize the system dynamic while mobilizing the system state.

The human brain contains billions of neurons that flexibly interconnect to support local and global computational spans. As neuronal activity propagates through the neural medium, it approaches a critical state hedged between ordered and disordered system regimes. Recent work demonstrates that this criticality coincides with the small-world topology, a network arrangement that accommodates both local (subcritical) and global (supercritical) system properties. On one hand, operating near criticality is thought to offer several neurocomputational advantages, e.g., high-dynamic range, efficient information capacity, and information transfer fidelity. On the other hand, aberrations from the critical state have been linked to diverse pathologies of the brain, such as post-traumatic epileptiform seizures and disorders of consciousness. Modulation of brain activity, through neuromodulation, presents an attractive mode of treatment to alleviate such neurological disorders, but a tractable neural framework is needed to facilitate clinical progress. Using a variation on the generative small-world model of Watts and Strogatz and Kuramoto's model of coupled oscillators, we show that the topological and dynamical properties of the small-world network are divided into two functional domains based on the range of connectivity, and that these domains play distinct roles in shaping the behavior of the critical state. We demonstrate that short-range network connections shape the dynamics of the system, e.g., its volatility and metastability, whereas long-range connections drive the system state, e.g., a seizure. Together, these findings lend support to combinatorial neuromodulation approaches that synergistically normalize the system dynamic while mobilizing the system state. [ABSTRACT FROM AUTHOR]

We study Kleinberg's navigation in small-world networks with fractal geometry (Sierpinski Carpet) using the dynamic rejection sampling algorithm adapted for this purpose. It is a methodology that dispenses with the application of periodic boundary conditions and consequently does not deform the toroid network. Instead, the technique relies on acceptance masks overlaid on the square network to regulate the distribution of long-range links. As a result, the complexity of the algorithm is drastically reduced. In this research, we propose an adaptation of the acceptance masks, originally conceived for two-dimensional (2D) Euclidean networks, in order to make them capable of recognizing fractal geometry. This is done by implementing an auxiliary routine (fractal_search) capable of identifying the fractal topology. After an extensive validation process, we concluded that the adaptation attempt was successful and that it performed considerably better than traditional methods (torus) in all parameters analyzed. [ABSTRACT FROM AUTHOR]

Purpose: Previous studies have indicated that the global topology of the brain functional network in patients with major depressive disorder (MDD) differs from that of those with normal controls (NCs). However, the relationship between an altered global topology and the response to antidepressants remains unclear. Here, we investigated whether differences in global topology affect the efficacy of antidepressants in MDD patients.Methods: 108 MDD patients and 61 NCs were recruited. A magnetic resonance imaging (MRI) scan was performed at the baseline, and the Hamilton Depression Scale-24 (HAMD-24) was assessed at baseline and after 2 and 8 weeks of antidepressant treatment. Seven global topological parameters of the brain functional network were measured and compared between groups. A correlation analysis was performed to identify the relationships between global topological parameters and antidepressant efficacy.Results: The brain networks of MDD patients and NCs were both small-world networks. The clustering coefficient (Cp) and local efficiency (Eloc) were significantly smaller in MDD patients compared with those in NCs. The characteristic path length (Lp) were negatively correlated with the 8-week reductive rate of HAMD-24 in the MDD group.Conclusion: The present research found that the brain functional network of MDD patients still had a small-world organization but with a lower Cp and Eloc than the NCs. In addition, the brain network global topology might have an impact on the antidepressant response and thus had the potential to become a treatment predictor of MDD. [ABSTRACT FROM AUTHOR]

Activity observed in biological neural networks is determined by anatomical connectivity between cortical areas. The monkey frontoparietal network facilitates cognitive functions, but the organization of its connectivity is unknown. Here, a new connectivity matrix is proposed which shows that the network utilizes a small-world architecture and the 3-node M9 motif. Its areas exhibit relatively homogeneous connectivity with no suggestion of the hubs seen in scale-free networks. Crucially, its M9 dynamical relay motif is optimally arranged for near-zero and non-zero phase synchrony to arise in support of cognition, serving as a candidate topological mechanism for previously reported findings. These results can serve as a benchmark to be used in the treatment of neurological disorders where the types of cognition the frontoparietal network supports are impaired.

Most real-world networks are endowed with the small-world property, by means of which the maximal distance between any two of their nodes scales logarithmically rather than linearly with their size. The evidence sparkled a wealth of studies trying to reveal possible mechanisms through which the pairwise interactions amongst the units of a network are structured in a way to determine such observed regularity. Here we show that smallworldness occurs also when interactions are of higher order. Namely, by considering Q -uniform hypergraphs and a process through which connections can be randomly rewired with given probability p , we find that such systems may exhibit prominent clustering properties in connection with small average path lengths for a wide range of p values, in analogy to the case of dyadic interactions. The nature of small-world transition remains the same at different orders Q ( ${=}2, 3, 4, 5,$ and 6) of the interactions, however, the increase in the hyperedge order reduces the range of rewiring probability for which smallworldness emerge.

Functional connectivity (FC) estimated from functional magnetic resonance imaging (fMRI) signals is important in understanding neural representation and information processing in cortical networks. However, due to a lack of ”ground truth” FC pattern, the reliability and robustness of FC estimates are usually examined in downstream FC analysis tasks, such as performing participant’s identification (also known as ”fingerprinting”). In this paper, we propose to learn FC via a smooth graph learning framework. In particular, we treat each time frame of the fMRI time series as a graph signal on an underlying functional brain graph, and estimate the smooth graph functional connectivity (SGFC) by learning the weighted graph adjacency matrix based on graph signal smoothness assumption. We demonstrate that our approach gives rise to a natural and sparse graph representation of FC from which reliable graph measures can be extracted. Reliability of SGFC is evaluated in the context of fingerprinting and compared to correlation FC (CFC). SGFC achieves higher fingerprinting accuracy across several different experiment settings; the improvement is even more significant when a shorter fMRI scanning length is used for FC estimation. In addition to being reliable, we also validate the cognitive relevance of SGFC by using it to predict fluid intelligence. Finally, in evaluating topological measures of the sparse graph, SGFC reveals a more small-world and modular structure compared to CFC. Together, our results suggest that the smooth graph learning framework produces a naturally sparse, reliable, and cognitive-relevant representation of functional connectivity.

Aneurysmal subarachnoid hemorrhage (aSAH) is a common disease causing vascular dementia. Survivors often suffer from cognitive impairment especially working memory deficit. Currently, lack of theoretical support limits the improvement of cognitive intervention or rehabilitation. It is unclear how the large-scale network differs and to what extent is the brain network affected? Our study aims to provide novel information about the topological characteristics of brain organization, especially “small-world” property. A total of 62 aSAH patients are enrolled in this study. They are divided into two groups according to the syndrome of working memory deficit. Their working memory function is evaluated by TMT-B and AVLT (Chinese version). Functional MRI scan is also performed for detecting resting-state cortical plasticity. We utilized ICA to extract functional sub-networks including working memory network from imaging data. And then we establish binarized network and calculate the small-worldness property as well as local and global efficiency of networks. aSAH group with working memory deficit shows no significant difference of clustering coefficient with control group. Our study discovered significant decrease of characteristic path length indicating an increase of overall routing efficiency. We reason that patients with working memory deficit have to recruit more neuronal resources and thus develops higher overall routing efficiency of local network. This study provides novel information about the neural alterations of aSAH patients with working memory deficit. It might contribute to the understanding of neural mechanism and the improvement of current intervention for vascular dementia.

Microvascular complications can accelerate cognitive impairment in patients with type 2 diabetes mellitus (T2DM) and have a high impact on their quality of life; however, the underlying mechanism is still unclear. The complex network in the human brain is the physiological basis for information processing and cognitive expression. Therefore, this study explored the relationship between the functional network topological properties and cognitive function in T2DM patients with and without microvascular complications (T2DM-C and T2DM-NC, respectively). Sixty-seven T2DM patients and 41 healthy controls (HCs) underwent resting-state functional MRI and neuropsychological assessment. Then, graph theoretical network analysis was performed to explore the global and nodal topological alterations in the functional whole brain networks of T2DM patients. Correlation analyses were performed to investigate the relationship between the altered topological parameters and cognitive/clinical variables. The T2DM-C group exhibited significantly higher local efficiency (Eloc), normalized cluster coefficient (γ), and small-world characteristics (σ) than the HCs. Patients with T2DM at different clinical stages (T2DM-C and T2DM-NC) showed varying degrees of abnormalities in node properties. In addition, compared with T2DM-NC patients, T2DM-C patients showed nodal properties disorders in the occipital visual network, cerebellum and middle temporal gyrus. The Eloc metrics were positively correlated with HbA1c level (P = 0.001, r = 0.515) and the NE values in the right paracentral lobule were negatively related with serum creatinine values (P = 0.001, r = −0.517) in T2DM-C patients. This study found that T2DM-C patients displayed more extensive changes at different network topology scales. The visual network and cerebellar may be the central vulnerable regions of T2DM-C patients.

Mild behavioral impairment (MBI), characterized by the late-life onset of sustained and meaningful neuropsychiatric symptoms, is increasingly recognized as a prodromal stage of dementia. However, the underlying neural mechanisms of MBI remain unclear. Here, we examined alterations in the topological organization of the structural covariance networks of patients with MBI (N = 32) compared with normal controls (N = 38). We found that the gray matter structural covariance networks of both the patients with MBI and controls exhibited a small-world topology evidenced by sigma value larger than one. The patients with MBI had significantly decreased clustering coefficients at several network densities and local efficiency at densities ranging from 0.05 to 0.26, indicating decreased local segregation. No significant differences in the characteristic path length, gamma value, sigma value, or global efficiency were detected. Locally, the patients with MBI showed significantly decreased nodal betweenness centrality in the left middle frontal gyrus, right inferior frontal gyrus (opercular part), and left Heschl gyrus and increased betweenness centrality in the left gyrus rectus, right insula, bilateral precuneus, and left thalamus. Moreover, the difference in the bilateral precuneus survived after correcting for multiple comparisons. In addition, a different number and distribution of hubs was identified in patients with MBI, showing more paralimbic hubs than observed in the normal controls. In conclusion, we revealed abnormal topological patterns of the structural covariance networks in patients with MBI and offer new insights into the network dysfunctional mechanisms of MBI.