A network model of social contacts with small-world and scale-free features, tunable connectivity, and geographic restrictions.

Small-world networks and scale-free networks are well-known theoretical models within the realm of complex graphs. These models exhibit "low" average shortest-path length; however, key distinctions are observed in their degree distributions and average clustering coefficients: in small-world networks, the degree distribution is bell-shaped and the clustering is "high"; in scale-free networks, the degree distribution follows a power law and the clustering is "low". Here, a model for generating scale-free graphs with "high" clustering is numerically explored, since these features are concurrently identified in networks representing social interactions. In this model, the values of average degree and exponent of the power-law degree distribution are both adjustable, and spatial limitations in the creation of links are taken into account. Several topological metrics are calculated and compared for computer-generated graphs. Unexpectedly, the numerical experiments show that, by varying the model parameters, a transition from a power-law to a bell-shaped degree distribution can occur. Also, in these graphs, the degree distribution is most accurately characterized by a pure power-law for values of the exponent typically found in real-world networks.

The structure and function of antagonistic ties in village social networks.

Negative or antagonistic relationships are common in human social networks, but they are less often studied than positive or friendly relationships. The existence of a capacity to have and to track antagonistic ties raises the possibility that they may serve a useful function in human groups. Here, we analyze empirical data gathered from 24,770 and 22,513 individuals in 176 rural villages in Honduras in two survey waves 2.5 y apart in order to evaluate the possible relevance of antagonistic relationships for broader network phenomena. We find that the small-world effect is more significant in a positive world with negative ties compared to an otherwise similar hypothetical positive world without them. Additionally, we observe that nodes with more negative ties tend to be located near network bridges, with lower clustering coefficients, higher betweenness centralities, and shorter average distances to other nodes in the network. Positive connections tend to have a more localized distribution, while negative connections are more globally dispersed within the networks. Analysis of the possible impact of such negative ties on dynamic processes reveals that, remarkably, negative connections can facilitate the dissemination of information (including novel information experimentally introduced into these villages) to the same degree as positive connections, and that they can also play a role in mitigating idea polarization within village networks. Antagonistic ties hold considerable importance in shaping the structure and function of social networks.

Small World derived index to distinguish Alzheimer's type dementia and healthy subjects.

BACKGROUND: This article introduces a novel index aimed at uncovering specific brain connectivity patterns associated with Alzheimer's disease (AD), defined according to neuropsychological patterns. METHODS: Electroencephalographic (EEG) recordings of 370 people, including 170 healthy subjects and 200 mild-AD patients, were acquired in different clinical centres using different acquisition equipment by harmonising acquisition settings. The study employed a new derived Small World (SW) index, SWcomb, that serves as a comprehensive metric designed to integrate the seven SW parameters, computed across the typical EEG frequency bands. The objective is to create a unified index that effectively distinguishes individuals with a neuropsychological pattern compatible with AD from healthy ones. RESULTS: Results showed that the healthy group exhibited the lowest SWcomb values, while the AD group displayed the highest SWcomb ones. CONCLUSIONS: These findings suggest that SWcomb index represents an easy-to-perform, low-cost, widely available and non-invasive biomarker for distinguishing between healthy individuals and AD patients.

Quantifying synergy and redundancy between networks.

Understanding how different networks relate to each other is key for understanding complex systems. We introduce an intuitive yet powerful framework to disentangle different ways in which networks can be (dis)similar and complementary to each other. We decompose the shortest paths between nodes as uniquely contributed by one source network, or redundantly by either, or synergistically by both together. Our approach considers the networks' full topology, providing insights at multiple levels of resolution: from global statistics to individual paths. Our framework is widely applicable across scientific domains, from public transport to brain networks. In humans and 124 other species, we demonstrate the prevalence of unique contributions by long-range white-matter fibers in structural brain networks. Across species, efficient communication also relies on significantly greater synergy between long-range and short-range fibers than expected by chance. Our framework could find applications for designing network systems or evaluating existing ones.

Modeling COVID-19 spread using multi-agent simulation with small-world network approach.

BACKGROUND: The rapid global spread of COVID-19 has seriously impacted people's daily lives and the social economy while also posing a threat to their lives. The analysis of infectious disease transmission is of significant importance for the rational allocation of epidemic prevention and control resources, the management of public health emergencies, and the improvement of future public health systems. METHODS: We propose a spatiotemporal COVID-19 transmission model with a neighborhood as an agent unit and an urban spatial network with long and short edge connections. The spreading model includes a network of defined agent attributes, transformation rules, and social relations and a small world network representing agents' social relations. Parameters for each stage are fitted by the Runge-Kutta method combined with the SEIR model. Using the NetLogo development platform, accurate dynamic simulations of the spatial and temporal evolution of the early epidemic were achieved. RESULTS: Experimental results demonstrate that the fitted curves from the four stages agree with actual data, with only a 12.27% difference between the average number of infected agents and the actual number of infected agents after simulating 1 hundred times. Additionally, the model simulates and compares different "city closure" scenarios. The results showed that implementing a 'lockdown' 10 days earlier would lead to the peak number of infections occurring 7 days earlier than in the normal scenario, with a reduction of 40.35% in the total number of infections. DISCUSSION: Our methodology emphasizes the crucial role of timely epidemic interventions in curbing the spread of infectious diseases, notably in the predictive assessment and evaluation of lockdown strategies. Furthermore, this approach adeptly forecasts the influence of varying intervention timings on peak infection rates and total case numbers, accurately reflecting real-world virus transmission patterns. This highlights the importance of proactive measures in diminishing epidemic impacts. It furnishes a robust framework, empowering policymakers to refine epidemic response strategies based on a synthesis of predictive modeling and empirical data.

Can social network analysis contribute to supply chain management? A systematic literature review and bibliometric analysis.

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.

Modeling Misinformation Spread in a Bounded Confidence Model: A Simulation Study.

Misinformation has posed significant threats to all aspects of people's lives. One of the most active areas of research in misinformation examines how individuals are misinformed. In this paper, we study how and to what extent agents are misinformed in an extended bounded confidence model, which consists of three parts: (i) online selective neighbors whose opinions differ from their own but not by more than a certain confidence level; (ii) offline neighbors, in a Watts-Strogatz small-world network, whom an agent has to communicate with even though their opinions are far different from their own; and (iii) a Bayesian analysis. Furthermore, we introduce two types of epistemically irresponsible agents: agents who hide their honest opinions and focus on disseminating misinformation and agents who ignore the messages received and follow the crowd mindlessly. Simulations show that, in an environment with only online selective neighbors, the misinforming is more successful with broader confidence intervals. Having offline neighbors contributes to being cautious of misinformation, while employing a Bayesian analysis helps in discovering the truth. Moreover, the agents who are only willing to listen to the majority, regardless of the truth, unwittingly help to bring about the success of misinformation attempts, and they themselves are, of course, misled to a greater extent.

Fractured columnar small-world functional network organization in volumes of L2/3 of mouse auditory cortex.

The sensory cortices of the brain exhibit large-scale functional topographic organization, such as the tonotopic organization of the primary auditory cortex (A1) according to sound frequency. However, at the level of individual neurons, layer 2/3 (L2/3) A1 appears functionally heterogeneous. To identify if there exists a higher-order functional organization of meso-scale neuronal networks within L2/3 that bridges order and disorder, we used in vivo two-photon calcium imaging of pyramidal neurons to identify networks in three-dimensional volumes of L2/3 A1 in awake mice. Using tonal stimuli, we found diverse receptive fields with measurable colocalization of similarly tuned neurons across depth but less so across L2/3 sublayers. These results indicate a fractured microcolumnar organization with a column radius of ∼50 µm, with a more random organization of the receptive field over larger radii. We further characterized the functional networks formed within L2/3 by analyzing the spatial distribution of signal correlations (SCs). Networks show evidence of Rentian scaling in physical space, suggesting effective spatial embedding of subnetworks. Indeed, functional networks have characteristics of small-world topology, implying that there are clusters of functionally similar neurons with sparse connections between differently tuned neurons. These results indicate that underlying the regularity of the tonotopic map on large scales in L2/3 is significant tuning diversity arranged in a hybrid organization with microcolumnar structures and efficient network topologies.

An application of small-world network on predicting the behavior of infectious disease on campus.

Networks haven been widely used to understand the spread of infectious disease. This study examines the properties of small-world networks in modeling infectious disease on campus. Two different small-world models are developed and the behaviors of infectious disease in the models are observed through numerical simulations. The results show that the behavior pattern of infectious disease in a small-world network is different from those in a regular network or a random network. The spread of the infectious disease increases as the proportion of long-distance connections p increasing, which indicates that reducing the contact among people is an effective measure to control the spread of infectious disease. The probability of node position exchange in a network (p2) had no significant effect on the spreading speed, which suggests that reducing human mobility in closed environments does not help control infectious disease. However, the spreading speed is proportional to the number of shared nodes (s), which means reducing connections between different groups and dividing students into separate sections will help to control infectious disease. In the end, the simulating speed of the small-world network is tested and the quadratic relationship between simulation time and the number of nodes may limit the application of the SW network in areas with large populations.

Acute Changes in the Resting Brain Networks in Concussion Patients: Small-World Topology Perspective.

BACKGROUND: The acute changes that occur in the small-world topology of the brain in concussion patients remain unclear. Here, we investigated acute changes in the small-world organization of brain networks in concussion patients and their influence on persistent post-concussion symptoms. METHODS: Eighteen concussion patients and eighteen age-matched controls were enrolled in this study. All participants underwent computed tomography, magnetic resonance imaging (MRI), susceptibility weighted imaging, and blood oxygen level-dependent functional MRI. A complex network analysis method based on graph theory was used to calculate the parameters of small-world networks under different degrees of network sparsity. All subjects were evaluated using the Glasgow Coma Scale and Rivermead Postconcussion Symptom Questionnaire. RESULTS: Compared with the controls, the normalized cluster coefficient (γ) of whole brain networks in patients and the "small-world" index (σ) was slightly enhanced, whereas the standardized minimum path (λ) was slightly shorter. Whole brain effect (Eglobal) and local effect (Elocal) changes were not pronounced. Under the condition of minimum network sparsity (Dmin = 0.13), the numbers of nodes in the "right intraorbital superior frontal gyrus" (Anatomical Automatic Labeling, AAL26), right globus pallidus (AAL76), and bilateral temporal transverse gyrus (AAL79,80) in brain concussion patients were significantly lower. The numbers of nodes in the left subcapital lobe (AAL61) and left occipital gyrus (AAL51) were significantly higher, and the normalized cluster coefficients of the right intraorbital supraphalus (AAL26) and left posterior cingulate gyrus (AAL35) were significantly increased. The normalized clustering coefficients of the right triangular subfrontal gyrus (AAL55) (based on the normalized clustering coefficients of nodes in AAL14) and left sub-parietal lobes (AAL61) were significantly reduced. The mean local effects of nodes in the right intraorbital upper frontal gyrus (AAL26), left posterior cingulate gyrus (AAL35), and bilateral auxiliary motor cortex (AAL19, 20) were enhanced, whereas the mean local effects of the bilateral triangular inferior frontal gyrus (AAL13,14) and left insular cap (AAL11) were reduced (p < 0.05). CONCLUSIONS: The overall trend of network topology abnormalities in patients was random, and generalized and local functional abnormalities were seen. Changes in the function and affective circuitry of the resting default network were particularly pronounced in these patients, which we speculate may be one of the main drivers of the cognitive dysfunction and mood changes seen in concussion patients.

Impacts of the Middle Route of China's South-to-North Water Diversion Project on the water network structure in the receiving basin.

The Middle Route of South-to-North Water Diversion Project (MRSNWD) is the main skeleton of China's National Water Network, its construction has changed the structure of the original water network, and analyzing the topological change of the water network in context with MRSNWD is significant for water network planning and management. In this study, the overall network characteristics of the water network in 2010 and 2020 were analyzed based on the small-world and scale-free characteristics of complex network theory. The topological changes of the water network from a node perspective were examined using three network centrality indexes: degree centrality (DC), closeness centrality (CC), and betweenness centrality (BC), while assessing the important nodes of the water network and recognizing functional areas of cold-hot spots. The results show that the water network's centrality in the study area improved after the project construction, with the average degree of the water network increasing from 2.39 to 2.42 and the average path length decreasing from 111.81 to 97.08. The propagation efficiency and network stability also increased, with a rise in important node proportion from 9.8 to 14.4%. The nodes in the DC hotspot zone along the project route have increased by 1.5%, implying an increase in the connectivity of the water network, while MRSNWD optimizes its north-south hub propagation path. "Small-worldness" indicates that most nodes of a network can be accessed and connected over shorter paths. The water network has a significant "small-worldness" and has been enhanced by the MRSNWD's construction. Approximating the water network as a scale-free network can impact its security by identifying critical nodes. The results of this research can provide the necessary technical support and reference significance for China's National Water Network.

The free energy landscape of small-world networks of cells.

The properties of organs, tissues, organoids, and other systems of cells, are influenced by the spatial localization and distribution of their elements. Here, we used networks to describe distributions of cells on a surface where the small-world coefficient (SW) of the networks was varied between SW~1 (random uniform distributions) and SW~10 (clustered distributions). The small-world coefficient is a topological measure of graphs: networks with SW>1 are topologically biased to transmit information. For each system configuration, we then determined the total energy U as the sum of the energies that describe cell-cell interactions - approximated by a harmonic potential. The graph of energy (U) across the configuration space of the networks (SW) is the energy landscape: it indicates which configuration a system of cells will likely assume over time. We found that, depending on the model parameters, the energy landscapes of 2D distributions of cells may be of different types: from type I to type IV. Type I and type II systems have high probability to evolve into random distributions. Type III and type IV systems have a higher probability to form clustered architectures. A great many of simulations indicated that cultures of cells with high initial density and limited sensing range could evolve into clustered configurations with enhanced topological characteristics. Moreover, the strongest the binding between cells, the greater the likelihood that they will assume configurations characterized by finite values of SW. Results of the work are relevant for those working the field of tissue engineering, regenerative medicine, the formation of in-vitro-models, the analysis of neuro-degenerative diseases.

Topological properties and connectivity patterns in brain networks of patients with refractory epilepsy combined with intracranial electrical stimulation.

Objective: Although intracranial electrical stimulation has emerged as a treatment option for various diseases, its impact on the properties of brain networks remains challenging due to its invasive nature. The combination of intracranial electrical stimulation and whole-brain functional magnetic resonance imaging (fMRI) in patients with refractory epilepsy (RE) makes it possible to study the network properties associated with electrical stimulation. Thus, our study aimed to investigate the brain network characteristics of RE patients with concurrent electrical stimulation and obtain possible clinical biomarkers. Methods: Our study used the GRETNA toolbox, a graph theoretical network analysis toolbox for imaging connectomics, to calculate and analyze the network topological attributes including global measures (small-world parameters and network efficiency) and nodal characteristics. The resting-state fMRI (rs-fMRI) and the fMRI concurrent electrical stimulation (es-fMRI) of RE patients were utilized to make group comparisons with healthy controls to identify the differences in network topology properties. Network properties comparisons before and after electrode implantation in the same patient were used to further analyze stimulus-related changes in network properties. Modular analysis was used to examine connectivity and distribution characteristics in the brain networks of all participants in study. Results: Compared to healthy controls, the rs-fMRI and the es-fMRI of RE patients exhibited impaired small-world property and reduced network efficiency. Nodal properties, such as nodal clustering coefficient (NCp), betweenness centrality (Bc), and degree centrality (Dc), exhibited differences between RE patients (including rs-fMRI and es-fMRI) and healthy controls. The network connectivity of RE patients (including rs-fMRI and es-fMRI) showed reduced intra-modular connections in subcortical areas and the occipital lobe, as well as decreased inter-modular connections between frontal and subcortical regions, and parieto-occipital regions compared to healthy controls. The brain networks of es-fMRI showed a relatively weaker small-world structure compared to rs-fMRI. Conclusion: The brain networks of RE patients exhibited a reduced small-world property, with a tendency toward random networks. The network connectivity patterns in RE patients exhibited reduced connections between cortical and subcortical regions and enhanced connections among parieto-occipital regions. Electrical stimulation can modulate brain network activity, leading to changes in network connectivity patterns and properties.

A novel conceptual model of heart rate autonomic modulation based on a small-world modular structure of the sinoatrial node.

The present view on heartbeat initiation is that a primary pacemaker cell or a group of cells in the sinoatrial node (SAN) center paces the rest of the SAN and the atria. However, recent high-resolution imaging studies show a more complex paradigm of SAN function that emerges from heterogeneous signaling, mimicking brain cytoarchitecture and function. Here, we developed and tested a new conceptual numerical model of SAN organized similarly to brain networks featuring a modular structure with small-world topology. In our model, a lower rate module leads action potential (AP) firing in the basal state and during parasympathetic stimulation, whereas a higher rate module leads during ß-adrenergic stimulation. Such a system reproduces the respective shift of the leading pacemaker site observed experimentally and a wide range of rate modulation and robust function while conserving energy. Since experimental studies found functional modules at different scales, from a few cells up to the highest scale of the superior and inferior SAN, the SAN appears to feature hierarchical modularity, i.e., within each module, there is a set of sub-modules, like in the brain, exhibiting greater robustness, adaptivity, and evolvability of network function. In this perspective, our model offers a new mainframe for interpreting new data on heterogeneous signaling in the SAN at different scales, providing new insights into cardiac pacemaker function and SAN-related cardiac arrhythmias in aging and disease.

Upregulation of a Small-World Brain Network Improves Inhibitory Control: An fNIRS Neurofeedback Training Study.

The aim of this study was to investigate the inner link between the small-world brain network and inhibitory control. Functional near-infrared spectroscopy (fNIRS) was used to construct a neurofeedback (NF) training system and regulate the frontal small-world brain network. The small-world network downregulation group (DOWN, n = 17) and the small-world network upregulation group (UP, n = 17) received five days of fNIRS-NF training and performed the color-word Stroop task before and after training. The behavioral and functional brain network topology results of both groups were analyzed by a repeated-measures analysis of variance (ANOVA), which showed that the upregulation training helped to improve inhibitory control. The upregulated small-world brain network exhibits an increase in the brain network regularization, links widely dispersed brain resources, and reduces the lateralization of brain functional networks between hemispheres. This suggests an inherent correlation between small-world functional brain networks and inhibitory control; moreover, dynamic optimization under cost efficiency trade-offs provides a neural basis for inhibitory control. Inhibitory control is not a simple function of a single brain region or connectivity but rather an emergent property of a broader network.

Abnormal Topological Organization of White Matter Structural Networks in Normal Tension Glaucoma Revealed via Diffusion Tensor Tractography.

BACKGROUND: Normal tension glaucoma (NTG) is considered a neurodegenerative disease with glaucomatous damage extending to diffuse brain areas. Therefore, this study aims to explore the abnormalities in the NTG structural network to help in the early diagnosis and course evaluation of NTG. METHODS: The structural networks of 46 NTG patients and 19 age- and sex-matched healthy controls were constructed using diffusion tensor imaging, followed by graph theory analysis and correlation analysis of small-world properties with glaucoma clinical indicators. In addition, the network-based statistical analysis (NBS) method was used to compare structural network connectivity differences between NTG patients and healthy controls. RESULTS: Structural brain networks in both NTG and NC groups exhibited small-world properties. However, the small-world index in the severe NTG group was reduced and correlated with a mean deviation of the visual field (MDVF) and retinal nerve fiber layer (RNFL) thickness. When compared to healthy controls, degree centrality and nodal efficiency in visual brain areas were significantly decreased, and betweenness centrality and nodal local efficiency in both visual and nonvisual brain areas were also significantly altered in NTG patients (all p < 0.05, FDR corrected). Furthermore, NTG patients exhibited increased structural connectivity in the occipitotemporal area, with the left fusiform gyrus (FFG.L) as the hub (p < 0.05). CONCLUSIONS: NTG exhibited altered global properties and local properties of visual and cognitive-emotional brain areas, with enhanced structural connections within the occipitotemporal area. Moreover, the disrupted small-world properties of white matter might be imaging biomarkers for assessing NTG progression.

The effective enhancement of information in 3D small-world networks of biological neuronal cells.

The cardiovascular system, the kidney, or the brain, are examples of complex systems - where the properties of the systems arise because of the layout of cells in those systems. One way to characterize systems is using networks, where elements and interactions of the systems are represented as nodes and links of a graph. Network's topology can be, in turn, measured by the small-world coefficient. Small world networks feature increased clustering and shorter paths compared to random or periodic networks of the same size. This suggests that systems with small world attributes can also efficiently transport signals, nutrients, or information within their body. While several reports in literature have illustrated that real biological systems are small-world, yet little is known about how information varies as a function of the small-world-ness (sw) of three dimensional graphs. Here, we used a model of the brain to estimate quantitatively the information processed in 3D networks. In the model, nodes of the graph are neuronal units capable to receive, integrate and transmit signals to other neurons of the system in parallel. The information encoded in the signals was then extracted using the techniques of information theory. In simulations where the topology of networks of400nodes was varied over large intervals, we found that in the0-9swrange information scales linearly with the small world coefficient, with a five-fold increase. Results of the paper and review of the existing literature on model organisms suggest that a small-world architecture may offer an evolutionary advantage to biological systems.

[Changes of brain network in patients with insomnia following radiotherapy for cervical cancer and their correlation with IT-TMS treatment efficacy: a graph-theory analysis].

OBJECTIVE: To analyze the changes of brain small-world and node function network properties in patients with insomnia following radiotherapy for cervical cancer based on graph theory analysis and explore the correlation between functional networks and the clinical efficacy of individual-target transcranial magnetic stimulation (IT-TMS) for treatment of insomnia. METHODS: The resting state functional magnetic resonance imaging (rs-fMRI) data were collected from 30 patients with insomnia following radiotherapy for cervical cancer and 30 matched healthy individuals. All the patients received accelerated intelligent neuromodulation TMS therapy. Using graph theory analysis and GRETNA software, the functional connectivity matrices were constructed and the attribute features were extracted. The scores on the Pittsburgh Sleep Quality Index (PSQI), Insomnia Severity Index (ISI), Self-Rating Anxiety Scale (SAS), and Self-Rating Depression Scale (SDS) of the participants were collected before and after IT-TMS, and the correlation between improvement in insomnia and the functional network was investigated. RESULTS: The two groups matched for age, gender, and education level (P>0.05) showed significant differences in PSQI, ISI, SAS and SDS scores (P<0.05). Both groups showed attributes of the small-world network. Compared with the healthy individuals, the patients showed significantly decreased σ, EI, Cp and Lp (P<0.05) and increased Eg (P<0.05) at baseline, which, along with insomnia symptoms, were all improved after IT-TMS treatment. The patients showed reduced functional connections of the node network at follow-up compared with the baseline, and the low functional connectivity between the right insula and left superior frontal gyrus was correlated with the improvement of ISI scores. CONCLUSION: The patients with insomnia following radiotherapy for cervical cancer have impaired information integration ability of the brain network, IT-TMS can significantly improve insomnia symptoms by reducing the hyperconnectivity between the default mode network and the salience network.

Hierarchical network structure as the source of hierarchical dynamics (power-law frequency spectra) in living and non-living systems: How state-trait continua (body plans, personalities) emerge from first principles in biophysics.

Living systems are hierarchical control systems that display a small world network structure. In such structures, many smaller clusters are nested within fewer larger ones, producing a fractal-like structure with a 'power-law' cluster size distribution (a mereology). Just like their structure, the dynamics of living systems shows fractal-like qualities: the timeseries of inner message passing and overt behavior contain high frequencies or 'states' (treble) that are nested within lower frequencies or 'traits' (bass), producing a power-law frequency spectrum that is known as a 'state-trait continuum' in the behavioral sciences. Here, we argue that the power-law dynamics of living systems results from their power-law network structure: organisms 'vertically encode' the deep spatiotemporal structure of their (anticipated) environments, to the effect that many small clusters near the base of the hierarchy produce high frequency signal changes and fewer larger clusters at its top produce ultra-low frequencies. Such ultra-low frequencies exert a tonic regulatory pressure that produces morphological as well as behavioral traits (i.e., body plans and personalities). Nested-modular structure causes higher frequencies to be embedded within lower frequencies, producing a power-law state-trait continuum. At the heart of such dynamics lies the need for efficient energy dissipation through networks of coupled oscillators, which also governs the dynamics of non-living systems (e.q., earthquakes, stock market fluctuations). Since hierarchical structure produces hierarchical dynamics, the development and collapse of hierarchical structure (e.g., during maturation and disease) should leave specific traces in system dynamics (shifts in lower frequencies, i.e. morphological and behavioral traits) that may serve as early warning signs to system failure. The applications of this idea range from (bio)physics and phylogenesis to ontogenesis and clinical medicine.

Brain functional connectivity and network characteristics changes after vagus nerve stimulation in patients with refractory epilepsy.

Objective: This study aims to investigate the impact of vagus nerve stimulation (VNS) on the connectivity and small-world metrics of brain functional networks during seizure periods. Methods: Ten refractory epilepsy patients underwent video encephalographic monitoring before and after VNS treatment. The 2-min electroencephalogram segment containing the ictal was selected for each participant, resulting in a total of 20 min of seizure data. The weighted phase lag index (wPLI) and small-world metrics were calculated for the whole frequency band and different frequency bands (delta, theta, alpha, beta, and gamma). Finally, the relevant metrics were statistically analyzed, and the false discovery rate was used to correct for differences after multiple comparisons. Results: In the whole band, the wPLI was notably enhanced, and the network metrics, including degree (D), clustering coefficient (CC), and global efficiency (GE), increased, while characteristic path length (CPL) decreased (P < 0.01). In different frequency bands, the wPLI between the parieto-occipital and frontal regions was significantly strengthened in the delta and beta bands, while the wPLI within the frontal region and between the frontal and parieto-occipital regions were significantly reduced in the beta and gamma bands (P < 0.01). In the low-frequency band (<13 Hz), the small-world metrics demonstrated significantly increased CC, D, and GE, with a significantly decreased CPL, indicating a more efficient network organization. In contrast, in the gamma band, the GE decreased, and the CPL increased, suggesting a shift toward less efficient network organization. Conclusion: VNS treatment can significantly change the wPLI and small-world metrics. These findings contribute to a deeper understanding of the impact of VNS therapy on brain networks and provide objective indicators for evaluating the efficacy of VNS.