Eigenvector Centrality Mapping for Analyzing Connectivity Patterns in fMRI Data of the Human Brain

Functional magnetic resonance data acquired in a task-absent condition (“resting state”) require new data analysis techniques that do not depend on an activation model. In this work, we introduce an alternative assumption- and parameter-free method based on a particular form of node centrality called eigenvector centrality. Eigenvector centrality attributes a value to each voxel in the brain such that a voxel receives a large value if it is strongly correlated with many other nodes that are themselves central within the network. Google''s PageRank algorithm is a variant of eigenvector centrality. Thus far, other centrality measures - in particular “betweenness centrality” - have been applied to fMRI data using a pre-selected set of nodes consisting of several hundred elements. Eigenvector centrality is computationally much more efficient than betweenness centrality and does not require thresholding of similarity values so that it can be applied to thousands of voxels in a region of interest covering the entire cerebrum which would have been infeasible using betweenness centrality. Eigenvector centrality can be used on a variety of different similarity metrics. Here, we present applications based on linear correlations and on spectral coherences between fMRI times series. This latter approach allows us to draw conclusions of connectivity patterns in different spectral bands. We apply this method to fMRI data in task-absent conditions where subjects were in states of hunger or satiety. We show that eigenvector centrality is modulated by the state that the subjects were in. Our analyses demonstrate that eigenvector centrality is a computationally efficient tool for capturing intrinsic neural architecture on a voxel-wise level.     

Fast Fragmentation of Networks Using Module-Based Attacks

In the multidisciplinary field of Network Science, optimization of procedures for efficiently breaking complex networks is attracting much attention from a practical point of view. In this contribution, we present a module-based method to efficiently fragment complex networks. The procedure firstly identifies topological communities through which the network can be represented using a well established heuristic algorithm of community finding. Then only the nodes that participate of inter-community links are removed in descending order of their betweenness centrality. We illustrate the method by applying it to a variety of examples in the social, infrastructure, and biological fields. It is shown that the module-based approach always outperforms targeted attacks to vertices based on node degree or betweenness centrality rankings, with gains in efficiency strongly related to the modularity of the network. Remarkably, in the US power grid case, by deleting 3% of the nodes, the proposed method breaks the original network in fragments which are twenty times smaller in size than the fragments left by betweenness-based attack.     

Fast computing betweenness centrality with virtual nodes on large sparse networks

Betweenness centrality is an essential index for analysis of complex networks. However, the calculation of betweenness centrality is quite time-consuming and the fastest known algorithm uses O(N(M + N log N)) time and O(N + M) space for weighted networks, where N and M are the number of nodes and edges in the network, respectively. By inserting virtual nodes into the weighted edges and transforming the shortest path problem into a breadth-first search (BFS) problem, we propose an algorithm that can compute the betweenness centrality in O(wDN2) time for integer-weighted networks, where w is the average weight of edges and D is the average degree in the network. Considerable time can be saved with the proposed algorithm when w < log N/D + 1, indicating that it is suitable for lightly weighted large sparse networks. A similar concept of virtual node transformation can be used to calculate other shortest path based indices such as closeness centrality, graph centrality, stress centrality, and so on. Numerical simulations on various randomly generated networks reveal that it is feasible to use the proposed algorithm in large network analysis.     

Nodes occupying central positions in human tissue specific PPI networks are enriched with many splice variants

The functional repertoire of genes in the eukaryotic organisms is enhanced by the phenomenon of alternative splicing. Hence, a node in a tissue specific protein–protein interaction (TS PPIN) network can be thought of as an ensemble of various spliced protein products of the corresponding gene expressed in that tissue. Here we demonstrate that the nodes that occupy topologically central positions characterized by high degree, betweenness, closeness, and eigenvector centrality values in TS PPINs of Homo sapiens are associated with high number of splice variants. We also show that the high “centrality” of these genes/nodes could in part be explained by the presence of a large number of promiscuous domains.     

The organisational structure of protein networks: revisiting the centrality–lethality hypothesis

Protein networks, describing physical interactions as well as functional associations between proteins, have been unravelled for many organisms in the recent past. Databases such as the STRING provide excellent resources for the analysis of such networks. In this contribution, we revisit the organisation of protein networks, particularly the centrality–lethality hypothesis, which hypothesises that nodes with higher centrality in a network are more likely to produce lethal phenotypes on removal, compared to nodes with lower centrality. We consider the protein networks of a diverse set of 20 organisms, with essentiality information available in the Database of Essential Genes and assess the relationship between centrality measures and lethality. For each of these organisms, we obtained networks of high-confidence interactions from the STRING database, and computed network parameters such as degree, betweenness centrality, closeness centrality and pairwise disconnectivity indices. We observe that the networks considered here are predominantly disassortative. Further, we observe that essential nodes in a network have a significantly higher average degree and betweenness centrality, compared to the network average. Most previous studies have evaluated the centrality–lethality hypothesis for Saccharomyces cerevisiae and Escherichia coli; we here observe that the centrality–lethality hypothesis hold goods for a large number of organisms, with certain limitations. Betweenness centrality may also be a useful measure to identify essential nodes, but measures like closeness centrality and pairwise disconnectivity are not significantly higher for essential nodes.     

Social Structure of a Semi-Free Ranging Group of Mandrills (Mandrillus sphinx): A Social Network Analysis

The difficulty involved in following mandrills in the wild means that very little is known about social structure in this species. Most studies initially considered mandrill groups to be an aggregation of one-male/multifemale units, with males occupying central positions in a structure similar to those observed in the majority of baboon species. However, a recent study hypothesized that mandrills form stable groups with only two or three permanent males, and that females occupy more central positions than males within these groups. We used social network analysis methods to examine how a semi-free ranging group of 19 mandrills is structured. We recorded all dyads of individuals that were in contact as a measure of association. The betweenness and the eigenvector centrality for each individual were calculated and correlated to kinship, age and dominance. Finally, we performed a resilience analysis by simulating the removal of individuals displaying the highest betweenness and eigenvector centrality values. We found that related dyads were more frequently associated than unrelated dyads. Moreover, our results showed that the cumulative distribution of individual betweenness and eigenvector centrality followed a power function, which is characteristic of scale-free networks. This property showed that some group members, mostly females, occupied a highly central position. Finally, the resilience analysis showed that the removal of the two most central females split the network into small subgroups and increased the network diameter. Critically, this study confirms that females appear to occupy more central positions than males in mandrill groups. Consequently, these females appear to be crucial for group cohesion and probably play a pivotal role in this species.     

Being in the thick of things: context-dependent network centrality in a captive flock of American flamingos

We test the hypothesis that the relationship between networks resulting from unresolved agonistic interactions (URI) and social dominance was context-dependent in a captive group of American Flamingos (Phoenicopterus ruber ruber). URI formed networks that differed substantially from the dominance network and depended on the context in which the interactions occurred. Betweenness centrality in the URI network on the region (“the island”) where nesting took place was strongly correlated with dominance score. This correlation was particularly strong in the case of males, but, in both sexes, the dominant individual in a dyad was likely to have a higher betweenness centrality in the island URI. On the other hand, betweenness centrality in URI networks at the feeder did not show any relationship with dominance, but was associated with observed visits to the feeder. Thus, centrality in both the island URI network and the feeder URI network was associated with access to a resource over which individuals competed. Along with other results suggesting that non-dominance interactions may form social networks distinct from, but related to, dominance networks, our results support the hypothesis that relationships within animal social groups can be modeled as a series of distinct but inter-related networks.     

The importance of bottlenecks in protein networks: correlation with gene essentiality and expression dynamics

It has been a long-standing goal in systems biology to find relations between the topological properties and functional features of protein networks. However, most of the focus in network studies has been on highly connected proteins (“hubs”). As a complementary notion, it is possible to define bottlenecks as proteins with a high betweenness centrality (i.e., network nodes that have many “shortest paths” going through them, analogous to major bridges and tunnels on a highway map). Bottlenecks are, in fact, key connector proteins with surprising functional and dynamic properties. In particular, they are more likely to be essential proteins. In fact, in regulatory and other directed networks, betweenness (i.e., “bottleneck-ness”) is a much more significant indicator of essentiality than degree (i.e., “hub-ness”). Furthermore, bottlenecks correspond to the dynamic components of the interaction network—they are significantly less well coexpressed with their neighbors than nonbottlenecks, implying that expression dynamics is wired into the network topology.     

Mapping the structural core of human cerebral cortex

Structurally segregated and functionally specialized regions of the human cerebral cortex are interconnected by a dense network of cortico-cortical axonal pathways. By using diffusion spectrum imaging, we noninvasively mapped these pathways within and across cortical hemispheres in individual human participants. An analysis of the resulting large-scale structural brain networks reveals a structural core within posterior medial and parietal cerebral cortex, as well as several distinct temporal and frontal modules. Brain regions within the structural core share high degree, strength, and betweenness centrality, and they constitute connector hubs that link all major structural modules. The structural core contains brain regions that form the posterior components of the human default network. Looking both within and outside of core regions, we observed a substantial correspondence between structural connectivity and resting-state functional connectivity measured in the same participants. The spatial and topological centrality of the core within cortex suggests an important role in functional integration.     

Fitness consequences of centrality in mutualistic individual-based networks

The relationships among the members of a population can be visualized using individual networks, where each individual is a node connected to each other by means of links describing the interactions. The centrality of a given node captures its importance within the network. We hypothesize that in mutualistic networks, the centrality of a node should benefit its fitness. We test this idea studying eight individual-based networks originated from the interaction between Erysimum mediohispanicum and its flower visitors. In these networks, each plant was considered a node and was connected to conspecifics sharing flower visitors. Centrality indicates how well connected is a given E. mediohispanicum individual with the rest of the co-occurring conspecifics because of sharing flower visitors. The centrality was estimated by three network metrics: betweenness, closeness and degree. The complex relationship between centrality, phenotype and fitness was explored by structural equation modelling. We found that the centrality of a plant was related to its fitness, with plants occupying central positions having higher fitness than those occupying peripheral positions. The structural equation models (SEMs) indicated that the centrality effect on fitness was not merely an effect of the abundance of visits and the species richness of visitors. Centrality has an effect even when simultaneously accounting for these predictors. The SEMs also indicated that the centrality effect on fitness was because of the specific phenotype of each plant, with attractive plants occupying central positions in networks, in relation to the distribution of conspecific phenotypes. This finding suggests that centrality, owing to its dependence on social interactions, may be an appropriate surrogate for the interacting phenotype of individuals.     

Structural vulnerability of the French swine industry trade network to the spread of infectious diseases

The networks generated by live animal movements are the principal vector for the propagation of infectious agents between farms, and their topology strongly affects how fast a disease may spread. The structural characteristics of networks may thus provide indicators of network vulnerability to the spread of infectious disease. This study applied social network analysis methods to describe the French swine trade network. Initial analysis involved calculating several parameters to characterize networks and then identifying high-risk subgroups of holdings for different time scales. Holding-specific centrality measurements (‘degree’, ‘betweenness’ and ‘ingoing infection chain’), which summarize the place and the role of holdings in the network, were compared according to the production type. In addition, network components and communities, areas where connectedness is particularly high and could influence the speed and the extent of a disease, were identified and analysed. Dealer holdings stood out because of their high centrality values suggesting that these holdings may control the flow of animals in part of the network. Herds with growing units had higher values for degree and betweenness centrality, representing central positions for both spreading and receiving disease, whereas herds with finishing units had higher values for in-degree and ingoing infection chain centrality values and appeared more vulnerable with many contacts through live animal movements and thus at potentially higher risk for introduction of contagious diseases. This reflects the dynamics of the swine trade with downward movements along the production chain. But, the significant heterogeneity of farms with several production units did not reveal any particular type of production for targeting disease surveillance or control. Besides, no giant strong connected component was observed, the network being rather organized according to communities of small or medium size (<20% of network size). Because of this fragmentation, the swine trade network appeared less structurally vulnerable than ruminant trade networks. This fragmentation is explained by the hierarchical structure, which thus limits the structural vulnerability of the global trade network. However, inside communities, the hierarchical structure of the swine production system would favour the spread of an infectious agent (especially if introduced in breeding herds).     

Topographic Hub Maps of the Human Structural Neocortical Network

Hubs within the neocortical structural network determined by graph theoretical analysis play a crucial role in brain function. We mapped neocortical hubs topographically, using a sample population of 63 young adults. Subjects were imaged with high resolution structural and diffusion weighted magnetic resonance imaging techniques. Multiple network configurations were then constructed per subject, using random parcellations to define the nodes and using fibre tractography to determine the connectivity between the nodes. The networks were analysed with graph theoretical measures. Our results give reference maps of hub distribution measured with betweenness centrality and node degree. The loci of the hubs correspond with key areas from known overlapping cognitive networks. Several hubs were asymmetrically organized across hemispheres. Furthermore, females have hubs with higher betweenness centrality and males have hubs with higher node degree. Female networks have higher small-world indices.     

Grooming network cohesion and the role of individuals in a captive chimpanzee group

Social network analysis offers new tools to study the social structure of primate groups. We used social network analysis to investigate the cohesiveness of a grooming network in a captive chimpanzee group (N = 17) and the role that individuals may play in it. Using data from a year-long observation, we constructed an unweighted social network of preferred grooming interactions by retaining only those dyads that groomed above the group mean. This choice of criterion was validated by the finding that the properties of the unweighted network correlated with the properties of a weighted network (i.e. a network representing the frequency of grooming interactions) constructed from the same data. To investigate group cohesion, we tested the resilience of the unweighted grooming network to the removal of central individuals (i.e. individuals with high betweenness centrality). The network fragmented more after the removal of individuals with high betweenness centrality than after the removal of random individuals. Central individuals played a pivotal role in maintaining the network's cohesiveness, and we suggest that this may be a typical property of affiliative networks like grooming networks. We found that the grooming network correlated with kinship and age, and that individuals with higher social status occupied more central positions in the network. Overall, the grooming network showed a heterogeneous structure, yet did not exhibit scale-free properties similar to many other primate networks. We discuss our results in light of recent findings on animal social networks and chimpanzee grooming.     

Characterization of topological keystone species: Local,global and “meso-scale” centralities in food webs

An important question in the network representation of ecological systems is to determine how direct and indirect interactions between species determine the positional importance of species in the ecosystem. Here we present a quantitative analysis of the similarities and differences of six different topological centrality measures as indicators of keystone species in 17 food webs. These indicators account for local, global and “meso-scale” – intermediate between local and global – topological information about species in the food webs. Using factor analysis we shown that most of these centrality indices share a great deal of topological information, which range from 75% to 96%. A generalized keystone indicator is then proposed by considering the factor loadings of the six-centrality measures, which contains most of the information encoded by these indices. However, the individual ordering of species according to these criteria display significant differences in most food webs. We simulate the effects of species extinction by removing species ranked according to a local and a “meso-scale” centrality indicator. The differences observed on three network characteristics – size, average distance and clustering coefficient of the largest component – after the removal of the most central nodes indicate that the consideration of these indices have different impacts for the ranking of species with conservational biology purposes. The “meso-scale” indicator appears to play an important role in determining the relative importance of species in epidemic spread and parasitism rates.     

Failure tolerance of motif structure in biological networks

Complex networks serve as generic models for many biological systems that have been shown to share a number of common structural properties such as power-law degree distribution and small-worldness. Real-world networks are composed of building blocks called motifs that are indeed specific subgraphs of (usually) small number of nodes. Network motifs are important in the functionality of complex networks, and the role of some motifs such as feed-forward loop in many biological networks has been heavily studied. On the other hand, many biological networks have shown some degrees of robustness in terms of their efficiency and connectedness against failures in their components. In this paper we investigated how random and systematic failures in the edges of biological networks influenced their motif structure. We considered two biological networks, namely, protein structure network and human brain functional network. Furthermore, we considered random failures as well as systematic failures based on different strategies for choosing candidate edges for removal. Failure in the edges tipping to high degree nodes had the most destructive role in the motif structure of the networks by decreasing their significance level, while removing edges that were connected to nodes with high values of betweenness centrality had the least effect on the significance profiles. In some cases, the latter caused increase in the significance levels of the motifs.     

Looking for Robust Properties in the Growth of an Academic Network: The Case of the Uruguayan Biological Research Community

Graph-theoretical methods have recently been used to analyze certain properties of natural and social networks. In this work, we have investigated the early stages in the growth of a Uruguayan academic network, the Biology Area of the Programme for the Development of Basic Science (PEDECIBA). This transparent social network is a territory for the exploration of the reliability of clustering methods that can potentially be used when we are confronted with opaque natural systems that provide us with a limited spectrum of observables (happens in research on the relations between brain, thought and language). From our social net, we constructed two different graph representations based on the relationships among researchers revealed by their co-participation in Master’s thesis committees. We studied these networks at different times and found that they achieve connectedness early in their evolution and exhibit the small-world property (i.e. high clustering with short path lengths). The data seem compatible with power law distributions of connectivity, clustering coefficients and betweenness centrality. Evidence of preferential attachment of new nodes and of new links between old nodes was also found in both representations. These results suggest that there are topological properties observed throughout the growth of the network that do not depend on the representations we have chosen but reflect intrinsic properties of the academic collective under study. Researchers in PEDECIBA are classified according to their specialties. We analysed the community structure detected by a standard algorithm in both representations. We found that much of the pre-specified structure is recovered and part of the mismatches can be attributed to convergent interests between scientists from different sub-disciplines. This result shows the potentiality of some clustering methods for the analysis of partially known natural systems.     

The genetic network of greater sage‐grouse: Range‐wide identification of keystone hubs of connectivity

Genetic networks can characterize complex genetic relationships among groups of individuals, which can be used to rank nodes most important to the overall connectivity of the system. Ranking allows scarce resources to be guided toward nodes integral to connectivity. The greater sage‐grouse (Centrocercus urophasianus) is a species of conservation concern that breeds on spatially discrete leks that must remain connected by genetic exchange for population persistence. We genotyped 5,950 individuals from 1,200 greater sage‐grouse leks distributed across the entire species’ geographic range. We found a small‐world network composed of 458 nodes connected by 14,481 edges. This network was composed of hubs—that is, nodes facilitating gene flow across the network—and spokes—that is, nodes where connectivity is served by hubs. It is within these hubs that the greatest genetic diversity was housed. Using indices of network centrality, we identified hub nodes of greatest conservation importance. We also identified keystone nodes with elevated centrality despite low local population size. Hub and keystone nodes were found across the entire species’ contiguous range, although nodes with elevated importance to network‐wide connectivity were found more central: especially in northeastern, central, and southwestern Wyoming and eastern Idaho. Nodes among which genes are most readily exchanged were mostly located in Montana and northern Wyoming, as well as Utah and eastern Nevada. The loss of hub or keystone nodes could lead to the disintegration of the network into smaller, isolated subnetworks. Protecting both hub nodes and keystone nodes will conserve genetic diversity and should maintain network connections to ensure a resilient and viable population over time. Our analysis shows that network models can be used to model gene flow, offering insights into its pattern and process, with application to prioritizing landscapes for conservation.     

Making Large-Scale Networks from fMRI Data

Pairwise correlations are currently a popular way to estimate a large-scale network (> 1000 nodes) from functional magnetic resonance imaging data. However, this approach generally results in a poor representation of the true underlying network. The reason is that pairwise correlations cannot distinguish between direct and indirect connectivity. As a result, pairwise correlation networks can lead to fallacious conclusions; for example, one may conclude that a network is a small-world when it is not. In a simulation study and an application to resting-state fMRI data, we compare the performance of pairwise correlations in large-scale networks (2000 nodes) against three other methods that are designed to filter out indirect connections. Recovery methods are evaluated in four simulated network topologies (small world or not, scale-free or not) in scenarios where the number of observations is very small compared to the number of nodes. Simulations clearly show that pairwise correlation networks are fragmented into separate unconnected components with excessive connectedness within components. This often leads to erroneous estimates of network metrics, like small-world structures or low betweenness centrality, and produces too many low-degree nodes. We conclude that using partial correlations, informed by a sparseness penalty, results in more accurate networks and corresponding metrics than pairwise correlation networks. However, even with these methods, the presence of hubs in the generating network can be problematic if the number of observations is too small. Additionally, we show for resting-state fMRI that partial correlations are more robust than correlations to different parcellation sets and to different lengths of time-series.     

Social networks in the Little Scrub Island ground lizard (<Emphasis Type="Italic">Ameiva corax</Emphasis>)

Ameiva corax is a diurnal, widely foraging lizard endemic to a small (<2 ha) Caribbean island and is known for social foraging, whereby individuals aggregate at large food items (e.g., bird eggs and cactus fruits). We characterized the social network for A. corax through focal observations and surveys, which delineated associations for 82 known individuals. Lizards varied greatly in the extent to which they were linked to the social network. Approximately 31 % of individuals were not observed in any associations while one individual associated with 22 % (n = 18) of the animals in the study area. Larger lizards tended to be more central to the social network; body size was positively correlated with number of associations (degree) and centrality (betweenness), but negatively correlated with average distance between an individual and its associates (mean path length). Larger individuals were also associated with lower clustering coefficients, indicating that their associates were less closely interconnected. Sex was not related to number of associations (degree), but did help explain some patterns. The extent to which a lizard’s associates were of the same sex (homophily) was related to both sex and body size. Females had lower homophily scores than males; within each sex larger lizards tended to have lower homophily.