Short Characteristic Path Length Research Articles

Associations of resilience, white matter topological organization, and cognitive functions in first-episode, drug-naïve schizophrenia patients: A moderated mediation analysis

BackgroundCognitive deficits are core symptoms of schizophrenia (SZ) and are associated with impaired resilience to stress. Different cognitive functions appeared to be interrelated, and the mechanism may involve neural alterations. The disrupted topological organization indicated abnormalities in the segregation and integration of brain networks that support various cognitive processes in SZ patients. Therefore, this study aimed to assess the direct and indirect effects of resilience on cognitive functions. We hypothesized that topological properties would moderate these associations. MethodsForty-nine SZ patients and fifty-two healthy controls (HCs) were recruited in this study. The Connor-Davidson Resilience Scale and the MATRICS Consensus Cognitive Battery were used to examine resilience and cognitive functions, respectively, and a graph theory approach was used to assess white matter topological organization. ResultsCompared to HCs, SZ patients showed lower levels of resilience and cognitive functions in multiple domains as well as abnormal global properties and nodal metrics. In addition, shorter characteristic path length was associated with a stronger indirect effect of resilience on working memory through processing speed in SZ patients. ConclusionCharacteristic path length might moderate the mediating effects of processing speed in the relationship between resilience and working memory in schizophrenia patients.

Inherent seizure susceptibility in patients with antihistamine-induced acute symptomatic seizure: a resting-state EEG analysis

We compared neural activities and network properties between the antihistamine-induced seizures (AIS) and seizure-free groups, with the hypothesis that patients with AIS might have inherently increased neural activities and network properties that are easily synchronized. Resting-state electroencephalography (EEG) data were collected from 27 AIS patients and 30 healthy adults who had never had a seizure. Power spectral density analysis was used to compare neural activities in each localized region. Functional connectivity (FC) was measured using coherence, and graph theoretical analyses were performed to compare network properties between the groups. Machine learning algorithms were applied using measurements found to be different between the groups in the EEG analyses as input features. Compared with the seizure-free group, the AIS group showed a higher spectral power in the entire regions of the delta, theta, and beta bands, as well as in the frontal areas of the alpha band. The AIS group had a higher overall FC strength, as well as a shorter characteristic path length in the theta band and higher global efficiency, local efficiency, and clustering coefficient in the beta band than the seizure-free group. The Support Vector Machine, k-Nearest Neighbor, and Random Forest models distinguished the AIS group from the seizure-free group with a high accuracy of more than 99%. The AIS group had seizure susceptibility considering both regional neural activities and functional network properties. Our findings provide insights into the underlying pathophysiological mechanisms of AIS and may be useful for the differential diagnosis of new-onset seizures in the clinical setting.

A thresholding method based on society modularity and role division for functional connectivity analysis

Objective. Inferring the optimized and sparse network structure from the fully connected matrix is a key step in functional connectivity (FC) analysis. However, it is still an urgent problem to be solved, how to exclude the weak and spurious connections contained in functional networks objectively. Most existing binarization methods assume that the network has some certain constraint structures, which lead to changes in the original topology of the network. Approach. To solve this problem, we develop a Trade-off Model between Cost and Topology under Role Division (MCT), which consists of three crucial strategies, including modularity detection, definition of node role, and E-cost optimization algorithm. This algorithm weighs the physical cost and adaptive value of the network while preserving the network structure. Reliability and validity of MCT were evaluated by comparing different binarization methods (efficiency cost optimization, cluster-span threshold, threshold method, and MCT) on synthetic and real data sets. Main results. Experiment results demonstrated that the recovery rate of MCT for networks under noise interference is superior to other methods. In addition, brain networks filtered with MCT had higher network efficiency and shorter characteristic path length, which is more in line with the small world characteristics. Finally, applying MCT to resting-state electroencephalography data from patients with major depression reveals abnormal topology of the patients’ connectivity networks, manifested as lower clustering coefficient (CC) and higher global efficiency (GE). Significance. This study provides an objective method for complex network analysis, which may contribute to the future of FC research.

The processing network of high-frequency acoustoelectric signal in the living rat brain

Objective. Acoustoelectric brain imaging (ABI) is a potential noninvasive electrophysiological neuroimaging method with high spatiotemporal resolution. At the focal spot of the focused ultrasound, with the couple of acoustic and electric fields, high-frequency acoustoelectric (HF AE) signal is generated. Because the brain is a volume conductor, HF AE signal can be detected in other brain cortex. The processing of HF AE signal is critical for improving decoding precision, further improving the spatial resolution performance of ABI. This study investigates the processing network of HF AE signal in the living rat brain. Approach. When HF AE generated on the left primary visual cortex (V1-L), low-frequency (LF) electroencephalography and HF AE signals on different cortex were recorded at the same time. Firstly, AE signal on different sides of the brain cortex were compared, including prefrontal cortex (FrA) and primary somatosensory cortex (S1FL). Then, we constructed and analyzed functional networks of two signals. Main results. In the same cortex, HF AE signal on the right side had stronger intensity. And compared with LF networks, HF AE network had larger global efficiency and shorter characteristic path length, denoting the stronger processing and transmission of AE signal. Additionally, in HF AE network, the node had significantly increased local properties and the connection were concentrated in the occipital lobe, reflecting the occipital lobe plays an important role in the processing. Significance. Experiment results demonstrate that, compared with LF network, HF AE network is more efficient and had stronger transmission capabilities. And the connection of HF AE network is concentrated in the occipital lobe. This work preliminarily reveals the HF AE signal processing, which is significant for improving the ABI quality and provides a new insight for understanding the brain HF signal.

Left Frontotemporal Region Plays a Key Role in Letter Fluency Task-Evoked Activation and Functional Connectivity in Normal Subjects: A Functional Near-Infrared Spectroscopy Study.

Letter fluency task (LFT) is a tool that measures memory, executive function, and language function but lacks a definite cutoff value to define abnormalities. We used the optical signals of functional near-infrared spectroscopy (fNIRS) to study the differences in power and connectivity between the high-functioning and low-functioning participants while performing three successive LFTs, as well as the relationships between the brain network/power and LFT performance. We found that the most differentiating factor between these two groups was network topology rather than activation power. The high-functional group (7 men and 10 women) displayed higher left intra-hemispheric global efficiency, nodal strength, and shorter characteristic path length in the first section. They then demonstrated a higher power over the left Broca's area than the right corresponding area in the latter two sections. The low-LFT group (9 men and 11 women) displayed less left-lateralized connectivity and activation power. LFT performance was only related to the network topology rather than the power values, which was only presented in the low-functioning group in the second section. The direct correlation between power and connectivity primarily existed in the inter-hemispheric network, with the timing relationship also seeming to be present. In conclusion, the high-functioning group presented more prominent left-lateralized intra-hemispheric network connectivity and power activation, particularly in the Broca's area. The low-functioning group seemed to prefer using other networks, like the inter-hemispheric, rather than having a single focus on left intra-hemispheric connectivity. The network topology seemed to better reflect the LFT performance than did the power values.

Pregnancy leads to changes in the brain functional network: a connectome analysis.

Pregnancy leads to long-lasting changes in human brain structure; however, little is known regarding alterations in the topological organization of functional networks. In this study, we investigated the effect of pregnancy on human brain function networks. Resting-state fMRI data was collected from eighteen primiparous mothers and twenty-four nulliparous control women of similar age, education level and body mass index (BMI). The functional brain network and topological properties were calculated by using GRETNA toolbox. The demographic data differences between two groups were computed by the independent two sample t-test. We tested group differences in network metrics' area under curve (AUC) using non-parametric permutation test of 1,000 permutations and corrected for false discovery rate (FDR). Differences in regional networks between groups were evaluated using non-parametric permutation tests by network-based statistical analysis (NBS). Compared with the nulliparous control women, a hub node changed from left inferior temporal gyrus to right precentral gyrus in primiparous mothers, while primiparous mothers showed enhanced network global efficiency (p = 0.247), enhanced local efficiency (p = 0.410), larger clustering coefficient (p = 0.410), but shorter characteristic path length (p = 0.247), smaller normalized clustering coefficient (p = 0.111), and shorter normalized characteristic path length (p = 0.705). Although both groups of functional networks have small-world property (σ > 1), the σ values of primiparous mothers were decreased significantly. NBS evaluation showed the majority of altered connected sub-network in the primiparous mothers occurred in the bilateral frontal lobe gyrus (p < 0.05). Altered functional network metrics and an abnormal sub-network were found in primiparous mothers, suggested that pregnancy may lead to changes in the brain functional network.

Hyper-Activated Brain Resting-State Network and Mismatch Negativity Deficit in Schizophrenia With Auditory Verbal Hallucination Revealed by an Event-Related Potential Evidence

Schizophrenia is a holergasia with unclear mechanism and high heterogeneity. Auditory verbal hallucination (AVH) study might help in understanding schizophrenia from the perspective of individual symptoms. This study aimed to investigate the activities of the resting-state networks (RSN) in the electroencephalogram (EEG) and mismatch negativity (MMN) in task-related state of schizophrenia patients with AVH. We recruited 30 schizophrenia patients without any medication for more than 4 weeks (15 AVH patients and 15 Non-AVH patients) and 15 healthy controls. We recorded the EEG data of the participants in the resting-state for 7 min and the event-related potential (ERP) data under an auditory oddball paradigm. In the resting-state EEG network, AVH patients exhibited a higher clustering coefficient than Non-AVH patients and healthy controls on delta and beta bands and a shorter characteristic path length than Non-AVH patients and healthy controls on all frequency bands. For ERP data, AVH patients showed a lower MMN amplitude than healthy controls (p = 0.017) and Non-AVH patients (p = 0.033). What’s more, MMN amplitude was positively correlated with clustering coefficient, and negatively correlated with characteristic path length on delta, theta, beta and gamma band in AVH patients. Our results indicate that AVH patients showed a hyper-activity in resting-state and may have impaired higher-order auditory expectations in the task-related state than healthy controls and Non-AVH patients. And it seems reasonable to conclude that the formation of AVH may occupy certain brain resources and compete for brain resources with external auditory stimuli.

The Maximum Eigenvalue of the Brain Functional Network Adjacency Matrix: Meaning and Application in Mental Fatigue Evaluation.

The maximum eigenvalue of the adjacency matrix (AM) has been supposed to contain rich information about the corresponding network. An experimental study focused on revealing the meaning and application of the maximum eigenvalue is missing. To this end, AM was constructed using mutual information (MI) to determine the functional connectivity with electroencephalogram (EEG) data recorded with a mental fatigue model, and then was converted into both binary and weighted brain functional network (BFN) and corresponding random networks (RNs). Both maximum eigenvalue and corresponding network characters in BFNs and RNs were considered to explore the changes during the formation of mental fatigue. The results indicated that large maximum eigenvalue means more edges in the corresponding network, along with a high degree and a short characteristic path length both in weighted and binary BFNs. Interestingly, the maximum eigenvalue of AM was always a little larger than that of the corresponding random matrix (RM), and had an obvious linearity with the sum of the AM elements, indicating that the maximum eigenvalue can be able to distinguish the network structures which have the same mean degree. What is more, the maximum eigenvalue, which increased with the deepening of mental fatigue, can become a good indicator for mental fatigue estimation.

Predicting individual decision-making responses based on the functional connectivity of resting-state EEG

Objective. Despite increasing evidence revealing the relationship between task-related brain activity and decision-making, the association between resting-state functional connectivity and decision-making remains unknown. Approach. In this study, we investigated the potential relationship between the network revealed in the resting-state electroencephalogram (EEG) and decision responses and further predicted individuals’ acceptance rates during the ultimatum game (UG) based on the functional connectivity revealed in the resting-state EEG. Main results. The results of this study demonstrated a significant relationship between the resting-state frontal-occipital connectivity and the UG acceptance rate in the alpha band. Increased acceptance rates were accompanied by a larger clustering coefficient and global and local efficiency as well as a shorter characteristic path length. Compared to the low-acceptance group, the high-acceptance group exhibited stronger frontal-occipital linkages. Finally, a multiple linear regression model based on the resting-state EEG network properties was adopted to predict the acceptance rates when subjects made their decision in the UG task. Significance. Together, the findings of this study may deepen our knowledge of decision-making and provide a potential physiological biomarker to predict the decision-making responses of subjects.

Co-contagion diffusion on multilayer networks

This study examines the interface of three elements during co-contagion diffusion: the synergy between contagions, the dormancy rate of each individual contagion, and the multiplex network topology. Dormancy is defined as a weaker form of “immunity,” where dormant nodes no longer actively participate in diffusion, but are still susceptible to infection. The proposed model extends the literature on threshold models, and demonstrates intricate interdependencies between different graph structures. Our simulations show that first, the faster contagion induces branching on the slower contagion; second, shorter characteristic path lengths diminish the impact of dormancy in lowering diffusion. Third, when two long-range graphs are paired, the faster contagion depends on both dormancy rates, whereas the slower contagion depends only on its own; fourth, synergistic contagions are less sensitive to dormancy, and have a wider window to diffuse. Furthermore, when long-range and spatially constrained graphs are paired, ring vaccination occurs on the spatial graph and produces partial diffusion, due to dormant, surrounding nodes. The spatial contagion depends on both dormancy rates whereas the long-range contagion depends on only its own.

Increased electroencephalography connectivity precedes epileptic spasm onset in infants with tuberous sclerosis complex.

To identify whether abnormal electroencephalography (EEG) connectivity is present before the onset of epileptic spasms (ES) in infants with tuberous sclerosis complex (TSC). Scalp EEG recordings were collected prospectively in infants diagnosed with TSC in the first year of life. This study compared the earliest recorded EEG from infants prior to ES onset (n=16) and from infants who did not develop ES (n=28). Five minutes of stage II or quiet sleep was clipped and filtered into canonical EEG frequency bands. Mutual information values between each pair of EEG channels were compared directly and used as a weighted graph to calculate graph measures of global efficiency, characteristic path length, average clustering coefficient, and modularity. At the group level, infants who later developed ES had increased EEG connectivity in sleep. They had higher mutual information values between most EEG channels in all frequency bands adjusted for age. Infants who later developed ES had higher global efficiency and average clustering coefficients, shorter characteristic path lengths, and lower modularity across most frequency bands adjusted for age. This suggests that infants who went on to develop ES had increased local and long-range EEG connectivity with less segregation of graph regions into distinct modules. This study suggests that increased neural connectivity precedes clinical ES onset in a cohort of infants with TSC. Overconnectivity may reflect progressive pathologic network synchronization culminating in generalized ES. Further research is needed before scalp EEG connectivity measures can be used as a potential biomarker of ES risk and treatment response in pre-symptomatic infants with TSC.

Long-term Chinese calligraphic handwriting training has a positive effect on brain network efficiency.

As a visual art form, Chinese calligraphic handwriting (CCH) has been found to correlate with certain brain activity and to induce functional connectivity reorganization of the brain. This study investigated the effect of long-term CCH training on brain functional plasticity as assessed with network measures. With the resting-state fMRI data from 31 participants with at least five years of CCH training and 40 controls, we constructed brain functional networks, examined group differences at both the whole brain and modular levels, and correlated the topological characteristics with calligraphy skills. We found that, compared to the control group, the CCH group showed shorter characteristic path lengths and higher local efficiency in certain brain areas in the frontal and parietal cortices, limbic system, basal ganglia, and thalamus. Moreover, these network measures in the cingulate cortex, caudate nucleus, and thalamus were associated with CCH performance (i.e., copying and creating skills). These results suggest that long-term CCH training has a positive effect on the topological characteristics of brain networks.

Neuroticism and conscientiousness respectively positively and negatively correlated with the network characteristic path length in dorsal lateral prefrontal cortex: A resting-state fNIRS study.

BackgroundAccumulating evidence shows that the dorsal lateral prefrontal cortex (dlPFC) is implicated in personality traits. In this study, resting‐state functional near infrared spectroscopy (fNIRS) combined with small‐world analysis was utilized to examine the relationship between the network properties of dlPFC and personality traits.MethodsThirty college students (aged between 20 and 29) were recruited from the University of Macau campus, whose personality scores were accessed with the NEO‐FFT questionnaire. Graph theory combined with resting‐state fNIRS data was used to quantify the network properties of dlPFC, whereas Pearson correlation analysis was performed to generate the relationship between the small‐world indicators and personality scores.ResultsCompared to matched random networks, the resting‐state brain networks exhibited a larger clustering coefficient (C p, 0.1–0.66), shorter characteristic path length (L p, 0.1–0.66), and higher global (E g, 0.1–0.66) and local efficiency (E loc, 0.1–0.65). In particular, conscientiousness (r = −0.63) and neuroticism (r = 0.40) respectively showed negative and positive correlation with the L p.ConclusionsThe resting‐state functional brain networks in dlPFC exhibited the small‐world properties. In addition, participants with higher conscientiousness scores showed a shorter L p.

An Efficient Method of Generating Deterministic Small-World and Scale-Free Graphs for Simulating Real-World Networks

We propose a novel mechanism to generate a family of deterministic small-world and scale-free networks by inserting new nodes into old nodes. These models can characterize the distinguishing properties of many real-world networks because this novel class of networks incorporates some key properties characterizing a majority of real-world networked systems, namely a high clustering coefficient and a short characteristic path length. We also obtain some accurate results for their properties, including degree distribution, clustering coefficient and network diameter, exactly according to the proposed generation algorithm of the networks considered, and discuss them. The network representation approach provided here can be used to investigate the complexity of many real-world systems from the perspective of complex networks.

Increased efficiency of brain connectivity networks in veterans with suicide attempts.

BackgroundSuicide is a public health concern for United States veterans and civilians. Prior research has shown neurobiological factors in suicide. However, studies of neuroimaging correlates of suicide risk have been limited. This study applied complex weighted network analyses to characterize the neural connectivity in white matter in veterans with suicide behavior.MethodsTwenty-eight veterans without suicide behavior (NS), 29 with a history of suicidal ideation only (SI), and 23 with prior suicide attempt (SA) completed diffusion tensor brain imaging, the Columbia Suicide Severity Rating Scale and Barratt Impulsiveness Scale (BIS). Structural connectivity networks among 82 parcellated brain regions were produced using whole-brain tractography. Global and nodal metrics of network topology have been calculated.ResultsSA had shorter characteristic path length and greater global efficiency and mean weighted degree of global network metrics (p < 0.024). SA had more hub nodes than NS and SI. The left posterior cingulate cortex (PCC) showed significantly greater weighted degree in SA relative to others (p < 0.0003). Nonplanning subscale of BIS correlated with the weighted degrees of the left PCC within SA. In rich club connectivity, SA had higher local connections than others (p = 0.001).ConclusionVeterans with prior suicide attempt had altered connectivity networks characteristics in the white matter. These findings may be distinctive neurobiological markers for individuals with suicide attempt. Strong connectivity in the left PCC may be implicated in impulsivity in veterans with suicide attempt.

Temporal slice registration and robust diffusion-tensor reconstruction for improved fetal brain structural connectivity analysis

Diffusion weighted magnetic resonance imaging, or DWI, is one of the most promising tools for the analysis of neural microstructure and the structural connectome of the human brain. The application of DWI to map early development of the human connectome in-utero, however, is challenged by intermittent fetal and maternal motion that disrupts the spatial correspondence of data acquired in the relatively long DWI acquisitions. Fetuses move continuously during DWI scans. Reliable and accurate analysis of the fetal brain structural connectome requires careful compensation of motion effects and robust reconstruction to avoid introducing bias based on the degree of fetal motion. In this paper we introduce a novel robust algorithm to reconstruct in-vivo diffusion-tensor MRI (DTI) of the moving fetal brain and show its effect on structural connectivity analysis. The proposed algorithm involves multiple steps of image registration incorporating a dynamic registration-based motion tracking algorithm to restore the spatial correspondence of DWI data at the slice level and reconstruct DTI of the fetal brain in the standard (atlas) coordinate space. A weighted linear least squares approach is adapted to remove the effect of intra-slice motion and reconstruct DTI from motion-corrected data. The proposed algorithm was tested on data obtained from 21 healthy fetuses scanned in-utero at 22–38 weeks gestation. Significantly higher fractional anisotropy values in fiber-rich regions, and the analysis of whole-brain tractography and group structural connectivity, showed the efficacy of the proposed method compared to the analyses based on original data and previously proposed methods. The results of this study show that slice-level motion correction and robust reconstruction is necessary for reliable in-vivo structural connectivity analysis of the fetal brain. Connectivity analysis based on graph theoretic measures show high degree of modularity and clustering, and short average characteristic path lengths indicative of small-worldness property of the fetal brain network. These findings comply with previous findings in newborns and a recent study on fetuses. The proposed algorithm can provide valuable information from DWI of the fetal brain not available in the assessment of the original 2D slices and may be used to more reliably study the developing fetal brain connectome.

Structural properties of the Chinese air transportation multilayer network

Recently multilayer networks are attracting great attention because the properties of many real-world systems cannot be well understood without considering their different layers. In this paper, we investigate the structural properties of the Chinese air transportation multilayer network (ATMN) by progressively merging layers together, where each commercial airline (company) defines a layer. The results show that the high clustering coefficient, short characteristic path length and large collection of reachable destinations of the Chinese ATMN can only emerge when several layers are merged together. Moreover, we compare two main types of layers corresponding to major and low-cost airlines. It is found that the small-world property and the rich-club effect of the Chinese ATMN are mainly caused by those layers corresponding to major airlines. Our work will highlight a better understanding of the Chinese air transportation network.

Divergent Network Patterns of Amyloid-β Deposition in Logopenic and Amnestic Alzheimer’s Disease Presentations

BackgroundDespite divergent clinical features, language and amnestic presentations of Alzheimer’s disease (AD) appear to show comparable regional amyloid-β (Aβ) burden. By using a statistical network approach, we aimed to identify complex network patterns of Aβ deposition and explore the effect of apolipoprotein E (APOE) ε4 allele on cortical Aβ burden across AD phenotypes. MethodsSixteen amnestic AD participants and 18 cases with logopenic-variant of primary progressive aphasia (lv-PPA) with a high cortical Aβ burden were selected. A comprehensive clinical assessment, Aβ imaging, and APOE genotyping were performed in all cases. Statistical network analysis was undertaken based on the estimation of sparse partial correlations of Aβ burden between cortical regions. Global and regional network statistical parameters as well as the effect of APOEε4 genotype on cortical Aβ were explored. ResultsThe two groups showed equivalent distribution of cortical amyloid burden and frequency of APOEε4 genotype. Statistical network analysis, however, demonstrated divergent connectivity properties. The lv-PPA group demonstrated higher mean network degree and shorter characteristic path length than the amnestic AD group. Amnestic AD cases showed connectivity hubs confined to the mesial temporal and prefrontal lobes bilaterally, whereas lv-PPA cases showed hubs scattered across the whole cortical mantle. An interaction effect on total Aβ burden between APOE genotype and AD presentations was also detected. ConclusionsThe network analysis reveals interregional network differences not evident using a simple comparison of Aβ burden. This suggests that regional neurotoxic effects may explain the phenotypical differences in AD presentation and that these can be modulated by APOE genotype.

Genetic network properties of the human cortex based on regional thickness and surface area measures.

We examined network properties of genetic covariance between average cortical thickness (CT) and surface area (SA) within genetically-identified cortical parcellations that we previously derived from human cortical genetic maps using vertex-wise fuzzy clustering analysis with high spatial resolution. There were 24 hierarchical parcellations based on vertex-wise CT and 24 based on vertex-wise SA expansion/contraction; in both cases the 12 parcellations per hemisphere were largely symmetrical. We utilized three techniques—biometrical genetic modeling, cluster analysis, and graph theory—to examine genetic relationships and network properties within and between the 48 parcellation measures. Biometrical modeling indicated significant shared genetic covariance between size of several of the genetic parcellations. Cluster analysis suggested small distinct groupings of genetic covariance; networks highlighted several significant negative and positive genetic correlations between bilateral parcellations. Graph theoretical analysis suggested that small world, but not rich club, network properties may characterize the genetic relationships between these regional size measures. These findings suggest that cortical genetic parcellations exhibit short characteristic path lengths across a broad network of connections. This property may be protective against network failure. In contrast, previous research with structural data has observed strong rich club properties with tightly interconnected hub networks. Future studies of these genetic networks might provide powerful phenotypes for genetic studies of normal and pathological brain development, aging, and function.

Spatially embedded growing small-world networks.

Networks in nature are often formed within a spatial domain in a dynamical manner, gaining links and nodes as they develop over time. Motivated by the growth and development of neuronal networks, we propose a class of spatially-based growing network models and investigate the resulting statistical network properties as a function of the dimension and topology of the space in which the networks are embedded. In particular, we consider two models in which nodes are placed one by one in random locations in space, with each such placement followed by configuration relaxation toward uniform node density, and connection of the new node with spatially nearby nodes. We find that such growth processes naturally result in networks with small-world features, including a short characteristic path length and nonzero clustering. We find no qualitative differences in these properties for two different topologies, and we suggest that results for these properties may not depend strongly on the topology of the embedding space. The results do depend strongly on dimension, and higher-dimensional spaces result in shorter path lengths but less clustering.