Small-world Properties Research Articles

Impact of moderate aerobic exercise on small-world topology and characteristics of brain networks after sport-related concussion: an exploratory study

Sport-related concussion (SRC) is a complex brain injury. By applying graph-theoretical analysis to networks derived from neuroimaging techniques, studies have shown that despite an overall retention of small-world topology, changes in small-world properties occur after brain injury. Less is known about how exercise during athletes’ return to sport (RTS) influences these brain network properties. Therefore, in the present study dense electroencephalography (EEG) datasets were collected pre- and post-moderate aerobic exercise. Small-world properties of whole brain (WB) and the default mode network (DMN) were extracted from the EEG datasets of 21 concussed athletes and 21 healthy matched controls. More specifically, path length (LP), clustering coefficient (CP), and small-world index (SWI) in binary and weighted graphs were calculated in the alpha frequency band (7–13 Hz). Pre-exercise, SRC athletes had higher DMN-CP values compared to controls, while post-exercise SRC athletes had higher WB-LP compared to controls. Weighted WB analysis revealed a significant association between SRC and the absence of small-world topology (SWI ≤ 1) post-exercise. This explorative study provides preliminary evidence that moderate aerobic exercise during athletes’ RTS induces an altered network response. Furthermore, this altered response may be related to the clinical characteristics of the SRC athlete.

Error and attack vulnerability of Apollonian networks

Abstract This article examines the resilience of different Apollonian network (AN) types—deterministic, random, and evolutionary—to systematic attacks. ANs, members of the family of maximal planar graphs, possess unique properties such as high clustering coefficients, small-world properties, scale-free behavior, Euclidean and space-filling properties, and modularity. These peculiarities require a thorough investigation of their robustness. This work presents a novel approach to studying ANs by implementing evolutionary Apollonian networks (EANs). These EANs include various probabilities distribution functions, including exponential, degenerate, logistic, Pareto, and stable (Cauchy, Lévy, Normal) distributions. To improve the robustness of these networks, we propose a novel edge rewiring mechanism using a genetic algorithm (GA). The GA aims to optimize a combined metric that includes the Flow Robustness of Degree (SFRD), Betweenness (SFRB), and Dangalchev's closeness (SFRC) centralities while preserving the original degree distribution and structural properties of the network. To evaluate the effectiveness of this approach, we use various robustness measures to assess the resilience of different AN types. The results show that SFRB, SFRD, and SFRC effectively rank ANs based on their robustness.

Exploring Spatiotemporal Interaction Patterns of Dockless Shared Bicycle Networks: A Case Study of Shenzhen, China

Abstract. The explosive development of shared bicycles has changed the way people travel. In parallel with the urbanization process, shared bicycles have emerged as a significant mode of transportation, especially for the first and last mile transit. With the aim of obtaining a comprehensive understanding of the spatiotemporal patterns underlying shared bicycle usage, this study takes Shenzhen, China as a case study. Drawing on the principles and techniques of complex network analysis, the research adopts a spatial grid framework, treating each grid as a node, the flow of shared bicycles as an edge, and the flow frequency as the corresponding weight. Subsequently, the spatiotemporal interaction networks of different periods of working days and weekends are constructed respectively, based on which the topological characteristics are analysed. The results show that the spatiotemporal interaction networks exhibit high regional connectivity and node interdependence, suggesting the presence of a small-world phenomenon.

Small world properties of schizophrenia and OCD patients derived from fNIRS based functional brain network connectivity metrics

Individuals suffering from obsessive compulsive disorder (OCD) and schizophrenia (SCZ) frequently exhibit symptoms of cognitive disassociations, which are linked to poor functional integration among brain regions. The loss of functional integration can be assessed using graph metrics computed from functional connectivity matrices (FCMs) derived from neuroimaging data. A healthy brain at rest is known to exhibit small-world features with high clustering coefficients and shorter path lengths in contrast to random networks. The aim of this study was to compare the small-world properties of prefrontal cortical functional networks of healthy subjects with OCD and SCZ patient groups by use of hemodynamic data obtained with functional near infrared spectroscopy (fNIRS). 13 healthy subjects and 47 patients who were clinically diagnosed with either OCD (N = 21) or SCZ (N = 26) completed a Stroop test while their prefrontal cortex (PFC) hemodynamics were monitored with fNIRS. The Stroop test had a block design consisting of neutral, congruent and incongruent stimuli. For each subject and stimuli type, FCMs were derived separately which were then used to compute small world features that included (i) global efficiency (GE), (ii) clustering coefficient (CC), (iii) modularity (Q), and (iv) small-world parameter (σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document}). Small-world features of patients exhibited random networks which were indicated by higher GE and lower CC values when compared to healthy controls, implying a higher neuronal operational cost.

Patients with temporomandibular disorders and chronic pain of myofascial origin display reduced alpha power density and altered small-world properties of brain networks

BACKGROUND: Chronic pain is one of the most common symptoms of temporomandibular disorders (TMD). Although its pathophysiology is still a challenge, TMD has been associated with changes in central nervous system activity related to pain modulatory capacity. OBJECTIVE: To assess the cortical activity of patients with temporomandibular disorders and chronic pain of myofascial origin using quantitative electroencephalography (qEEG) in different mental states. METHOD: This study consists of a cross-sectional study. Individuals with TMD and chronic pain and healthy controls were evaluated using qEEG in four consecutive conditions, all with closed eyes: 1) initial resting condition; 2) non-painful motor imagery task of hand movement; 3) painful motor imagery task of clenching the teeth; 4) final resting condition. RESULTS: Participants with TMD and chronic pain overall presented decreased alpha power density during baseline at rest, non-painful and painful motor imagery tasks when compared to healthy controls. Furthermore, functional brain connectivity was distinct between groups, with TMD and chronic pain showing lower small-world values for the delta (all conditions), theta (painful and non-painful motor imagery task), and alpha bands (painful motor imagery task), and an increase in the beta band (all conditions). CONCLUSION: These results suggest that TMD and chronic pain could be associated with maladaptive plasticity in the brain, which may correspond to a reduced ability to modify brain activity during different mental tasks, including painful and non-painful motor imagery.

Assessment of Ventilation Potential and Construction of Wind Corridors in Chengdu City Based on Multi-Source Data and Multi-Model Analysis

The establishment of urban ventilation corridors (UVCs) aims to mitigate the urban heat island effect. While most studies focus on the construction and assessment of the environmental benefit of UVCs, they often overlook the analysis of UVCs’ topological features. This research integrates multi-source data including 3D urban buildings, historical meteorological observations, high-resolution remote sensing, and land use planning, combined with multiple models, including geographic information system spatial analysis, circuit theory, and complex networks. Based on an assessment of urban ventilation potential, the circuit model was applied to extract UVCs aligned with the prevailing wind direction for both summer and winter seasons. Complex network modeling was employed to analyze the topological features of the ventilation network. From the analytical results, a multi-level wind corridor system for Chengdu was quantitatively developed. The results indicate that the city’s overall ventilation resistance is high, with notable spatial clustering, and the southeastern region faces substantial ventilation obstructions. A total of 143 critical ventilation nodes were identified, with the number of air inlets and outlets in summer being significantly fewer than in winter. However, the cooling effect of ventilation corridors in the prevailing summer wind direction is superior to that in winter. The ventilation network comprises 16 communities with distinct ventilation characteristics, exhibiting moderate connectivity, lacking small-world properties, and showing congestion and instability.

Using network analysis to provide evidence for brain health as a unified construct relevant to aging with HIV.

Brain health is a dynamic state involving cognitive, emotional, and motor domains. Measuring brain health is a challenge owing to the uncertainty as to whether it is one or many constructs. This study aimed to contribute evidence for brain health as a unified construct by estimating the strength of relationships between and among patient-reported items related to the brain health construct in a population with brain vulnerability owing to HIV. Data for this cross-sectional analysis came from a Canadian cohort of people aging with HIV. The sample included 710 men recruited between 2014 and 2016 from five Canadian cities. A network analysis was conducted with 30 items selected from the brain-related domains of fatigue, cognition, depression, sleep, anxiety, and motivation. Node centrality measures were used to determine the most critical items in the network. The network showed small-world properties, that is, most nodes can be reached from other nodes with few hops," indicating strong connectivity. The most central symptoms were "How much do you enjoy life?" and "How often do you have negative feelings?". The small-world properties of the network structure indicate that brain health items are interconnected and may be influenced by shared underlying factors. The centrality indices suggest that items related to enjoyment of life and negative feelings may be particularly important for understanding brain health in this population. Future research should aim to replicate these findings in larger and more diverse samples to confirm their robustness and generalizability.

Aberrant individual large-scale functional network connectivity and topology in chronic insomnia disorder with and without depression

Insomnia is increasingly prevalent with significant associations with depression. Delineating specific neural circuits for chronic insomnia disorder (CID) with and without depressive symptoms is fundamental to develop precision diagnosis and treatment. In this study, we examine static, dynamic and network topology changes of individual large-scale functional network for CID with (CID-D) and without depression to reveal their specific neural underpinnings. Seventeen individual-specific functional brain networks are obtained using a regularized nonnegative matrix factorization technique. Disorders-shared and -specific differences in static and dynamic large-scale functional network connectivities within or between the cognitive control network, dorsal attention network, visual network, limbic network, and default mode network are found for CID and CID-D. Additionally, CID and CID-D groups showed compromised network topological architecture including reduced small-world properties, clustering coefficients and modularity indicating decreased network efficiency and impaired functional segregation. Moreover, the altered neuroimaging indices show significant associations with clinical manifestations and could serve as effective neuromarkers to distinguish among healthy controls, CID and CID-D. Taken together, these findings provide novel insights into the neural basis of CID and CID-D, which may facilitate developing new diagnostic and therapeutic approaches.

Developmental dynamics of brain network modularity and temporal co-occurrence diversity in childhood

ObjectiveBrain development during childhood involves significant structural, functional, and connectivity changes, reflecting the interplay between modularity, information interaction, and functional segregation. This study aims to understand the dynamic properties of brain connectivity and their impact on cognitive development, focusing on temporal co-occurrence diversity patterns. MethodsWe recruited 481 children aged 6 to 12 years from the Healthy Brain Network database. Functional MRI data were used to construct dynamic functional connectivity matrices with a sliding window approach. Modular structures were identified using multilayer network community detection, and the Dagum Gini coefficient decomposition technique, which uniquely allows for multi-faceted exploration of modular temporal co-occurrence diversities, quantified these diversities. Mediation analysis assessed the impact on small-world properties. ResultsTemporal co-occurrence diversity in brain networks increased with age, especially in the default mode, frontoparietal, and salience networks. These changes were driven by disparities within and between communities. The small-world coefficient increased with age, indicating improved information processing efficiency. To validate the impact of changes in spatiotemporal interaction disparities during childhood on information transmission within brain networks, we used mediation analysis to verify its effect on alterations in small-world properties. ConclusionThis study highlights the critical developmental changes in brain modularity and spatiotemporal interaction patterns during childhood, emphasizing their role in cognitive maturation. These insights into neural mechanisms can inform the diagnosis and intervention of developmental disorders.

Advancing the complex adaptive systems approach to enterprise risk management with quantified risk networks (QRNs)

Business enterprises are complex adaptive systems (CAS) subject to fragility caused by the non-linear effects of uncertain risk events. The regional bank collapses in the United States are a case in point. Recent papers have highlighted a shortcoming of the traditional risk management process, which focuses on compliance but considers neither the interdependence between complex risks nor the mechanism of their non-linear impact on the organization. A new perspective on Enterprise Risk Management (ERM) has instead called for a shift of mindset from mitigating risk to building resilience by honing a CAS view to contextualize, assess, and manage complex risks. But the building blocks needed for a CAS representation of ERM have not yet been systematically developed. Specifically, we are missing a typological inventory of risks and a mapping of risks onto the general structure of an enterprise. In this paper, we build an industry-agnostic inventory of plausible risk factors using information extraction on large scale text data. Additionally, we develop an understanding of risk-function and function-function interdependencies through a survey of top business managers. The result is a novel complex network view of enterprise risk called a Quantified Risk Network (QRN) that displays small-world properties and highlights internal company functions central to non-linear risk propagation mechanisms within the enterprise. The QRN draws attention to vulnerabilities in the enterprise structure such as risk-function connections measured by Edge Betweenness centrality. The generic QRN developed herein is a proof of concept and we advocate that enterprises build their own company-specific QRNs to identify highly connected and central functions in their company structure that could lead to cascading failure when specific risks arise. QRNs can contribute to the objective of building enterprise resilience.

Neural network architecture of a mammalian brain

Connectomics research is making rapid advances, although models revealing general principles of connectional architecture are far from complete. Our analysis of 106 published connection reports indicates that the adult rat brain interregional connectome has about 76,940 of a possible 623,310 axonal connections between its 790 gray matter regions mapped in a reference atlas, equating to a network density of 12.3%. We examined the sexually dimorphic network using multiresolution consensus clustering that generated a nested hierarchy of interconnected modules/subsystems with three first-order modules and 157 terminal modules in females. Top-down hierarchy analysis suggests a mirror-image primary module pair in the central nervous system's rostral sector (forebrain-midbrain) associated with behavior control, and a single primary module in the intermediate sector (rhombicbrain) associated with behavior execution; the implications of these results are considered in relation to brain development and evolution. Bottom-up hierarchy analysis reveals known and unfamiliar modules suggesting strong experimentally testable hypotheses. Global network analyses indicate that all hubs are in the rostral module pair, a rich club extends through all three primary modules, and the network exhibits small-world attributes. Simulated lesions of all regions individually enabled ranking their impact on global network organization, and the visual path from the retina was used as a specific example, including the effects of cyclic connection weight changes from the endogenous circadian rhythm generator, suprachiasmatic nucleus. This study elucidates principles of interregional neuronal network architecture for a mammalian brain and suggests a strategy for modeling dynamic structural connectivity.

Type-II Apollonian network: More robust and more efficient Apollonian network

The family of planar graphs is a particularly important family and models many networks including the layout of printed circuits. The widely-known Apollonian packing process has been used as guideline to create the typical Apollonian network with planarity. In this paper, we propose a new principled framework based on the Apollonian packing process to generate model as complex network, and obtain a family of new networks called Type-II Apollonian network At. While our network and the typical Apollonian network are maximal planar, the former turns out to be Hamiltonian and Eulerian, however, the latter is not. Then, we in-depth study some fundamental structural properties on network At, and verify that network At is sparse, has scale-free feature and small-world property, and exhibits disassortative mixing structure. Next, we derive the asymptotic solution of the spanning tree entropy of network At by designing an effective algorithm, which suggests that Type-II Apollonian network is more robust to a random removal of edges than the typical Apollonian network. Additionally, we study trapping problem on network At, and use average trapping time as metric to show that Type-II Apollonian network At has more efficient underlying structure for fast information diffusion than the typical Apollonian network.

Cascading failure model and resilience-based sequential recovery strategy for complex networks

Complex networks, which exhibit high connectivity, self-organization, small-world properties, and heterogeneity, are susceptible to the rapid spread of local failures, often resulting in cascading effects throughout the entire system. The paper introduces a cascading failure model based on biased random walks that incorporate betweenness centrality and the power-law distribution of node degrees. This model is used to investigate cascade failures triggered by extreme fluctuations in load that follow a Poisson distribution. Furthermore, we propose a resilience-based sequential recovery strategy that accounts for varying node recovery time and resource limitations, setting an upper limit on the number of nodes that can be in recovery simultaneously. The network’s robustness improves, and the variation in the power-law exponent during cascading failures and recovery decreases when the betweenness bias parameter is set to 1 instead of -1. The capacity parameter has the most significant and direct effect on improving the network’s robustness. Reducing node recovery time can improve the network’s initial invulnerability; however, its impact on final residual resilience remains limited. The power-law exponent of the initial network significantly affects residual resilience during the recovery process, with higher exponents leading to improved network performance. An appropriate increase in the number of nodes that can be in recovery simultaneously can enhance the overall recovery performance of the network. Extensive comparative simulations reveal substantial advantages of our proposed recovery strategy in enhancing network recovery.

The small-world phenomenon: a model, explanations, characterizations, and examples

The small-world phenomenon: a model, explanations, characterizations, and examples

Laplacian renormalization group: an introduction to heterogeneous coarse-graining

The renormalization group (RG) constitutes a fundamental framework in modern theoretical physics. It allows the study of many systems showing states with large-scale correlations and their classification into a relatively small set of universality classes. The RG is the most powerful tool for investigating organizational scales within dynamic systems. However, the application of RG techniques to complex networks has presented significant challenges, primarily due to the intricate interplay of correlations on multiple scales. Existing approaches have relied on hypotheses involving hidden geometries and based on embedding complex networks into hidden metric spaces. Here, we present a practical overview of the recently introduced Laplacian RG (LRG) for heterogeneous networks. First, we present a brief overview that justifies the use of the Laplacian as a natural extension of well-known field theories to analyze spatial disorder. We then draw an analogy to traditional real-space RG procedures, explaining how the LRG generalizes the concept of ‘Kadanoff supernodes’ as block nodes that span multiple scales. These supernodes help mitigate the effects of cross-scale correlations due to small-world properties. Additionally, we rigorously define the LRG procedure in momentum space in the spirit of the Wilson RG. Finally, we show different analyses for the evolution of network properties along the LRG flow following structural changes when the network is properly reduced.

Abnormal Topological Organization of Human Brain Connectome in Primary Dysmenorrhea Patients Using Graph Theoretical Analysis.

Accumulating studies have revealed altered brain function and structure in regions linked to sensory, pain and emotion in individuals with primary dysmenorrhea (PD). However, the changes in the topological properties of the brain's functional connectome in patients with PD experiencing chronic pain remain poorly understood. Our study aimed to explore the mechanism of functional brain network impairment in individuals withPD through a graph-theoretic analysis. This study was a randomized controlled trial that included individuals with PD and healthy controls (HC) from June 2021 to June 2022. The experiment took place in the magnetic resonance imaging facility at Jiangxi Provincial People's Hospital. Static MRI scans were conducted on 23 female patients with PD and 23 healthy female controls. A two-sample t-test was conducted to compare the global and nodal indices between the two groups, while the Network-Based Statistics (NBS) method was utilized to explore the functional connectivity alterations between the groups. In the global index, The PD group exhibited decreased Sigma (p = 0.0432) and Gamma (p = 0.0470) compared to the HC group among the small-world network properties.(p<0.05) In the nodal index, the PD group displayed reduced betweenness centrality and increased degree centrality in the default mode network (DMN), along with decreased nodal efficiency and increased degree centrality in the visual network (VN). (P < 0.05, Bonferroni-corrected) Furthermore, in the connection analysis, PD patients showed altered functional connectivity in the basal ganglia network (BGN), VN, and DMN.(NBS corrected). Our results indicate that individuals with PD showed abnormal brain network efficiency and abnormal connection within DMN, VN and BGN related to pain matrix. These findings have important references for understanding the neural mechanism of pain in PD.

Weaving social networks from cultural similarities on the neolithisation process in the Western Mediterranean: Evolutionary trajectories using projectile tools.

In this paper, we concentrate on the neolithisation process in Mediterranean Iberia through a diachronic view (from 8600-6800 cal. BP), focusing on social interaction as a factor in articulating new cultural ties. To do this, we apply techniques centred on similarities in material culture by applying Social Network Analysis (SNA). For the first time, we point to the geometric projectiles, taking into account their recurrence in both Mesolithic and Neolithic groups as part of their characteristic hunting equipment. We hypothesise that patterns of cultural variability would express the changing flow of information between communities according to their mobility strategies (last hunter-gatherer groups), including economic and social behaviour, and that these relationships will be restructured with the arrival of the newcomer farmers and herders and their new spatial and social arrangement. The results obtained allow us to describe a connected and homogeneous Late Mesolithic network dramatically structured by the Neolithic arrival. Since then, a heterogenous pattern emerged, involving connected periods, network ruptures, and small-world phenomena. The emergence of this characteristic could support the flow of information when the network presents a clustered structure, the last probably due to regionalisation events. These diachronic dynamics fit well with demographic and socioecological trends observed from regional literature.

Analysis of Topological Properties and Robustness of Urban Public Transport Networks

With the acceleration of urbanization, public transport networks are an important part of urban transport systems, and their robustness is critical for city operation. The objective of this study is to analyze the topological properties and robustness of an urban public transport network (UPTN) with a view to enhancing the sustainability of urbanization. In order to present the topological structure of the UPTN, the L-Space complex network modeling method is used to construct a model. Topological characteristics of the network are calculated. Based on single evaluation indices of station significance, a comprehensive evaluation index is proposed as the basis for selecting critical stations. The UPTN cascading failure model is established. Using the proportion of the maximum connected subgraph as the evaluation index, the robustness of the UPTN is analyzed using different station significance indices and deliberate attack strategies. The public transport network of Xuzhou city is selected for instance analysis. The results show that the UPTN in Xuzhou city has small-world effects and scale-free characteristics. Although the network has poor connectivity, it is a convenient means to travel for residents with many independent communities. The network’s dynamic robustness is demonstrably inferior to its static robustness due to the prevalence of cascading failure phenomena. Specifically, the failure of important stations has a wider impact on the network performance. Improving their load capacity and distributing the routes via them will help bolster the network resistance against contingencies. This study provides a scientific basis and strategic recommendations for urban planners and public transport managers to achieve a more sustainable public transport system.

Minimal specialization: Coevolution of network structure and dynamics

The changing topology of a network is driven by the need to maintain or optimize network function. As this function is often related to moving quantities such as traffic, information, etc., efficiently through the network, the structure of the network and the dynamics on the network directly depend on the other. To model this interplay of network structure and dynamics we use the dynamics on the network, or the dynamical processes the network models, to influence the dynamics of the network structure, i.e., to determine where and when to modify the network structure. We model the dynamics on the network using Jackson network dynamics and the dynamics of the network structure using minimal specialization, a variant of the more general network growth model known as specialization. The resulting model, which we refer to as the integrated specialization model, coevolves both the structure and the dynamics of the network. We show that by reducing dynamic bottlenecks, i.e. reducing the network’s maximal asymptotic load, this model simultaneously produces networks with real-world like properties. This includes right-skewed degree distributions, sparsity, the small-world property, and non-trivial equitable partitions. Additionally, when compared to other growth models, the integrated specialization model creates networks with small diameter, minimizing distances across the network. The result are networks that have both dynamic and structural features that allow quantities to more efficiently move through the network.

Brain Responses Difference between Sexes for Strong Desire to Void: A Functional Magnetic Resonance Imaging Study in Adults Based on Graph Theory.

Background: The alternations of brain responses to a strong desire to void were unclear, and the gender differences under the strong desire to void remain controversial. The present study aims to identify the functional brain network's topologic property changes evoked by a strong desire to void in healthy male and female adults with synchronous urodynamics using a graph theory analysis. Methods: The bladders of eleven healthy males and eleven females were filled via a catheter using a specific infusion and withdrawal pattern. A resting-state functional magnetic resonance imaging (fMRI) was performed on the enrolled subjects, scanning under both the empty bladder and strong desire to void states. An automated anatomical labeling (AAL) atlas was used to identify the ninety cortical and subcortical regions. Pearson's correlation calculations were performed to establish a brain connection matrix. A paired t-test (p < 0.05) and Bonferroni correction were applied to identify the significant statistical differences in topological properties between the two states, including small-world network property parameters [gamma (γ) and lambda (λ)], characteristic path length (Lp), clustering coefficient (Cp), global efficiency (Eglob), local efficiency (Eloc), and regional nodal efficiency (Enodal). Results: The final data suggested that females and males had different brain response patterns to a strong desire to void, compared with an empty bladder state. Conclusions: More brain regions involving emotion, cognition, and social work were active in females, and males might obtain a better urinary continence via a compensatory mechanism.