The dichotomy in degree correlation of biological networks

Most complex networks from different areas such as biology, sociology or technology, show a correlation on node degree where the possibility of a link between two nodes depends on their connectivity. It is widely believed that complex networks are either disassortative (links between hubs are systematically suppressed) or assortative (links between hubs are enhanced). In this paper, we analyze a variety of biological networks and find that they generally show a dichotomous degree correlation. We find that many properties of biological networks can be explained by this dichotomy in degree correlation, including the neighborhood connectivity, the sickle-shaped clustering coefficient distribution and the modularity structure. This dichotomy distinguishes biological networks from real disassortative networks or assortative networks such as the Internet and social networks. We suggest that the modular structure of networks accounts for the dichotomy in degree correlation and vice versa, shedding light on the source of modularity in biological networks. We further show that a robust and well connected network necessitates the dichotomy of degree correlation, suggestive of an evolutionary motivation for its existence. Finally, we suggest that a dichotomous degree correlation favors a centrally connected modular network, by which the integrity of network and specificity of modules might be reconciled.     

Dynamics and integrity of wetland network in estuary

The increasing fragmentation and degradation tendency of estuarine wetlands due to rapid urbanization in coastal areas exert the urgent need to quantify the structural characteristic of wetland network, which involves the fragmentized wetland patches for wetland restoration and conservation. Although wetland networks have already been used for conservation planning on a regional scale, the variation of connectivity and structural characteristic has not been well studied. Therefore, this study gathered data from Landsat TM/ETM + images to construct the wetland network and employed the graph theoretical analysis to characterize the network evolution, structural connectivity changes, and network integrity alteration in the Pearl River Estuary, South China. The wetland network integrity was defined in the study, including structural connectivity and several functional attributes, the variation of six generated networks during 1979 to 2009 was analyzed as well. The wetland network exhibited a weakening topology over time, which is indicated by a significant decrease in the degree distribution, connectivity, and clustering coefficient. The diminishing trends of wetland network integrity reveal that although the latest wetland network can still protect the regional environment, the weakening phenomenon in its topology and integrity would reduce the ability of estuarine wetland network to resist external interference.     

Impact of the topology of global macroeconomic network on the spreading of economic crises

Throughout economic history, the global economy has experienced recurring crises. The persistent recurrence of such economic crises calls for an understanding of their generic features rather than treating them as singular events. The global economic system is a highly complex system and can best be viewed in terms of a network of interacting macroeconomic agents. In this regard, from the perspective of collective network dynamics, here we explore how the topology of the global macroeconomic network affects the patterns of spreading of economic crises. Using a simple toy model of crisis spreading, we demonstrate that an individual country's role in crisis spreading is not only dependent on its gross macroeconomic capacities, but also on its local and global connectivity profile in the context of the world economic network. We find that on one hand clustering of weak links at the regional scale can significantly aggravate the spread of crises, but on the other hand the current network structure at the global scale harbors higher tolerance of extreme crises compared to more "globalized" random networks. These results suggest that there can be a potential hidden cost in the ongoing globalization movement towards establishing less-constrained, trans-regional economic links between countries, by increasing vulnerability of the global economic system to extreme crises.     

Resolving Structural Variability in Network Models and the Brain

Large-scale white matter pathways crisscrossing the cortex create a complex pattern of connectivity that underlies human cognitive function. Generative mechanisms for this architecture have been difficult to identify in part because little is known in general about mechanistic drivers of structured networks. Here we contrast network properties derived from diffusion spectrum imaging data of the human brain with 13 synthetic network models chosen to probe the roles of physical network embedding and temporal network growth. We characterize both the empirical and synthetic networks using familiar graph metrics, but presented here in a more complete statistical form, as scatter plots and distributions, to reveal the full range of variability of each measure across scales in the network. We focus specifically on the degree distribution, degree assortativity, hierarchy, topological Rentian scaling, and topological fractal scaling—in addition to several summary statistics, including the mean clustering coefficient, the shortest path-length, and the network diameter. The models are investigated in a progressive, branching sequence, aimed at capturing different elements thought to be important in the brain, and range from simple random and regular networks, to models that incorporate specific growth rules and constraints. We find that synthetic models that constrain the network nodes to be physically embedded in anatomical brain regions tend to produce distributions that are most similar to the corresponding measurements for the brain. We also find that network models hardcoded to display one network property (e.g., assortativity) do not in general simultaneously display a second (e.g., hierarchy). This relative independence of network properties suggests that multiple neurobiological mechanisms might be at play in the development of human brain network architecture. Together, the network models that we develop and employ provide a potentially useful starting point for the statistical inference of brain network structure from neuroimaging data.     

A comparison of network and clustering methods to detect biogeographical regions

Bioregions are an important concept in biogeography, and are key to our understanding of biodiversity patterns across the world. The use of networks in biogeography to produce bioregions is a relatively novel approach that has been proposed to improve upon current methods. However, it remains unclear if they may be used in place of current methods and/or offer additional biogeographic insights. We compared two network methods to detect bioregions (modularity and map equation) with the conventional distance‐based clustering method. We also explored the relationship between network and biodiversity metrics. For the analysis we used two datasets of iconic Australian plant groups at a continental scale, Acacia and eucalypts, as example groups. The modularity method detected fewer large bioregions produced the most succinct bioregionalisation for both plant groups corresponding to Australian biomes, while map equation detected many small bioregions including interzones at a natural scale of one. The clustering method was less sensitive than network methods in detecting bioregions. The network metric called participation coefficient from both network partition methods identified interzones or transition zones between bioregions. Furthermore, another network metric (betweenness) was highly correlated to richness and endemism. We conclude that the application of networks to biogeography offers a number of advantages and provides novel insights. More specifically, our study showed that these network partition methods were more efficient than the clustering method for bioregionalisation of continental‐scale data in: 1) the identification of bioregions and 2) the quantification of biogeographic transition zones using the participation coefficient metric. The use of network methods and especially the participation coefficient metric adds to bioregionalisation by identifying transition zones which could be useful for conservation purposes and identifying biodiversity hotspots.     

Revisiting the variation of clustering coefficient of biological networks suggests new modular structure

ABSTRACT: A central idea in biology is the hierarchical organization of cellular processes. A commonly used method to identify the hierarchical modular organization of network relies on detecting a global signature known as variation of clustering coefficient (so-called modularity scaling). Although several studies have suggested other possible origins of this signature, it is still widely used nowadays to identify hierarchical modularity, especially in the analysis of biological networks. Therefore, a further and systematical investigation of this signature for different types of biological networks is necessary. RESULTS: We analyzed a variety of biological networks and found that the commonly used signature of hierarchical modularity is actually the reflection of spoke-like topology, suggesting a different view of network architecture. We proved that the existence of super-hubs is the origin that the clustering coefficient of a node follows a particular scaling law with degree k in metabolic networks. To study the modularity of biological networks, we systematically investigated the relationship between repulsion of hubs and variation of clustering coefficient. We provided direct evidences for repulsion between hubs being the underlying origin of the variation of clustering coefficient, and found that for biological networks having no anti-correlation between hubs, such as gene co-expression network, the clustering coefficient doesn't show dependence of degree. CONCLUSIONS: Here we have shown that the variation of clustering coefficient is neither sufficient nor exclusive for a network to be hierarchical. Our results suggest the existence of spoke-like modules as opposed to "deterministic model" of hierarchical modularity, and suggest the need to reconsider the organizational principle of biological hierarchy.     

From Caenorhabditis elegans to the human connectome: a specific modular organization increases metabolic,functional and developmental efficiency

The connectome, or the entire connectivity of a neural system represented by a network, ranges across various scales from synaptic connections between individual neurons to fibre tract connections between brain regions. Although the modularity they commonly show has been extensively studied, it is unclear whether the connection specificity of such networks can already be fully explained by the modularity alone. To answer this question, we study two networks, the neuronal network of Caenorhabditis elegans and the fibre tract network of human brains obtained through diffusion spectrum imaging. We compare them to their respective benchmark networks with varying modularities, which are generated by link swapping to have desired modularity values. We find several network properties that are specific to the neural networks and cannot be fully explained by the modularity alone. First, the clustering coefficient and the characteristic path length of both C. elegans and human connectomes are higher than those of the benchmark networks with similar modularity. High clustering coefficient indicates efficient local information distribution, and high characteristic path length suggests reduced global integration. Second, the total wiring length is smaller than for the alternative configurations with similar modularity. This is due to lower dispersion of connections, which means each neuron in the C. elegans connectome or each region of interest in the human connectome reaches fewer ganglia or cortical areas, respectively. Third, both neural networks show lower algorithmic entropy compared with the alternative arrangements. This implies that fewer genes are needed to encode for the organization of neural systems. While the first two findings show that the neural topologies are efficient in information processing, this suggests that they are also efficient from a developmental point of view. Together, these results show that neural systems are organized in such a way as to yield efficient features beyond those given by their modularity alone.     

Altered small-world,functional brain networks in patients with lower back pain

In this study, we aimed to investigate the functional network changes that occur in patients with lower back pain(LBP). We also investigated the link between LBP and the small-world properties of functional networks within the brain. Functional MRI(fMRI) was performed on 20 individuals with LBP and 17 age and gender-matched normal controls during the resting state. The severity of the pain in the individuals with LBP ranged from 5 to 8 on a 0–10 scale, with 0 indicating no pain. Network-based statistics were performed to investigate the differences between the brain networks of individuals with LBP and those of normal controls. Several small-world parameters of brain networks were calculated, including the clustering coefficient, characteristic path length, local efficiency, and global efficiency. These criteria reflect the overall network efficiency. The brain networks in the individuals with LBP due to herniation of a lumbar disc demonstrated a significantly longer characteristic path length as well as a lower clustering coefficient, global efficiency, and local efficiency compared to those in control subjects. We found that LBP patients tended to have unstable and inefficient brain networks when compared with healthy controls. In addition, LBP individuals showed significantly decreased functional connectivity in the anterior cingulate cortex, middle cingulate cortex, post cingulate cortex, inferior frontal gyrus, middle temporal gyrus, occipital gyrus, postcentral gyrus, precentral gyrus, supplementary motor area, thalamus, fusiform, caudate, and cerebellum. We believe that these regions may be involved in the pathophysiology of lower back pain.     

GraphCrunch 2: Software tool for network modeling,alignment and clustering

Background  

Recent advancements in experimental biotechnology have produced large amounts of protein-protein interaction (PPI) data. The topology of PPI networks is believed to have a strong link to their function. Hence, the abundance of PPI data for many organisms stimulates the development of computational techniques for the modeling, comparison, alignment, and clustering of networks. In addition, finding representative models for PPI networks will improve our understanding of the cell just as a model of gravity has helped us understand planetary motion. To decide if a model is representative, we need quantitative comparisons of model networks to real ones. However, exact network comparison is computationally intractable and therefore several heuristics have been used instead. Some of these heuristics are easily computable "network properties," such as the degree distribution, or the clustering coefficient. An important special case of network comparison is the network alignment problem. Analogous to sequence alignment, this problem asks to find the "best" mapping between regions in two networks. It is expected that network alignment might have as strong an impact on our understanding of biology as sequence alignment has had. Topology-based clustering of nodes in PPI networks is another example of an important network analysis problem that can uncover relationships between interaction patterns and phenotype.     

Copying nodes versus editing links: the source of the difference between genetic regulatory networks and the WWW

We study two kinds of networks: genetic regulatory networks and the World Wide Web. We systematically test microscopic mechanisms to find the set of such mechanisms that optimally explain each networks' specific properties. In the first case we formulate a model including mainly random unbiased gene duplications and mutations. In the second case, the basic moves are website generation and rapid surf-induced link creation (/destruction). The different types of mechanisms reproduce the appropriate observed network properties. We use those to show that different kinds of networks have strongly system-dependent macroscopic experimental features. The diverging properties result from dissimilar node and link basic dynamics. The main non-uniform properties include the clustering coefficient, small-scale motifs frequency, time correlations, centrality and the connectivity of outgoing links. Some other features are generic such as the large-scale connectivity distribution of incoming links (scale-free) and the network diameter (small-worlds). The common properties are just the general hallmark of autocatalysis (self-enhancing processes), while the specific properties hinge on the specific elementary mechanisms. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics Online.     

Topological evolution of coexpression networks by new gene integration maintains the hierarchical and modular structures in human ancestors

We analyze the global structure and evolution of human gene coexpression networks driven by new gene integration. When the Pearson correlation coefficient is greater than or equal to 0.5, we find that the coexpression network consists of 334 small components and one "giant" connected subnet comprising of 6317 interacting genes. This network shows the properties of power-law degree distribution and small-world. The average clustering coefficient of younger genes is larger than that of the elderly genes(0.6685 vs. 0.5762). Particularly, we find that the younger genes with a larger degree also show a property of hierarchical architecture. The younger genes play an important role in the overall pivotability of the network and this network contains few redundant duplicate genes. Moreover, we find that gene duplication and orphan genes are two dominant evolutionary forces in shaping this network. Both the duplicate genes and orphan genes develop new links through a "rich-gets-richer"mechanism. With the gradual integration of new genes into the ancestral network, most of the topological structure features of the network would gradually increase. However, the exponent of degree distribution and modularity coefficient of the whole network do not change significantly, which implies that the evolution of coexpression networks maintains the hierarchical and modular structures in human ancestors.     

Analyzing the International Exergy Flow Network of Ferrous Metal Ores

This paper employs an un-weighted and weighted exergy network to study the properties of ferrous metal ores in countries worldwide and their evolution from 2002 to 2012. We find that there are few countries controlling most of the ferrous metal ore exports in terms of exergy and that the entire exergy flow network is becoming more heterogeneous though the addition of new nodes. The increasing of the average clustering coefficient indicates that the formation of an international exergy flow system and regional integration is improving. When we contrast the average out strength of exergy and the average out strength of currency, we find both similarities and differences. Prices are affected largely by human factors; thus, the growth rate of the average out strength of currency has fluctuated acutely in the eleven years from 2002 to 2012. Exergy is defined as the maximum work that can be extracted from a system and can reflect the true cost in the world, and this parameter fluctuates much less. Performing an analysis based on the two aspects of exergy and currency, we find that the network is becoming uneven.     

Brain Network Organization in Focal Epilepsy: A Systematic Review and Meta-Analysis

Normal brain functioning is presumed to depend upon interacting regions within large-scale neuronal networks. Increasing evidence exists that interictal network alterations in focal epilepsy are associated with cognitive and behavioral deficits. Nevertheless, the reported network alterations are inconclusive and prone to low statistical power due to small sample sizes as well as modest effect sizes. We therefore systematically reviewed the existing literature and conducted a meta-analysis to characterize the changes in whole-brain interictal focal epilepsy networks at sufficient power levels. We focused on the two most commonly used metrics in whole-brain networks: average path length and average clustering coefficient. Twelve studies were included that reported whole-brain network average path length and average clustering coefficient characteristics in patients and controls. The overall group difference, quantified as the standardized mean average path length difference between epilepsy and control groups, corresponded to a significantly increased average path length of 0.29 (95% confidence interval (CI): 0.12 to 0.45, p = 0.0007) in the epilepsy group. This suggests a less integrated interictal whole-brain network. Similarly, a significantly increased standardized mean average clustering coefficient of 0.35 (CI: 0.05 to 0.65, p = 0.02) was found in the epilepsy group in comparison with controls, pointing towards a more segregated interictal network. Sub-analyses revealed similar results for functional and structural networks in terms of effect size and directionality for both metrics. In addition, we found individual network studies to be prone to low power due to the relatively small group differences in average path length and average clustering coefficient in combination with small sample sizes. The pooled network characteristics support the hypothesis that focal epilepsy has widespread detrimental effects, that is, reduced integration and increased segregation, on whole brain interictal network organization, which may relate to the co-morbid cognitive and behavioral impairments often reported in patients with focal epilepsy.     

Axelrod's metanorm games on networks

Metanorms is a mechanism proposed to promote cooperation in social dilemmas. Recent experimental results show that network structures that underlie social interactions influence the emergence of norms that promote cooperation. We generalize Axelrod's analysis of metanorms dynamics to interactions unfolding on networks through simulation and mathematical modeling. Network topology strongly influences the effectiveness of the metanorms mechanism in establishing cooperation. In particular, we find that average degree, clustering coefficient and the average number of triplets per node play key roles in sustaining or collapsing cooperation.     

Biological Motion Coding in the Brain: Analysis of Visually Driven EEG Functional Networks

Herein, we address the time evolution of brain functional networks computed from electroencephalographic activity driven by visual stimuli. We describe how these functional network signatures change in fast scale when confronted with point-light display stimuli depicting biological motion (BM) as opposed to scrambled motion (SM). Whereas global network measures (average path length, average clustering coefficient, and average betweenness) computed as a function of time did not discriminate between BM and SM, local node properties did. Comparing the network local measures of the BM condition with those of the SM condition, we found higher degree and betweenness values in the left frontal (F7) electrode, as well as a higher clustering coefficient in the right occipital (O2) electrode, for the SM condition. Conversely, for the BM condition, we found higher degree values in central parietal (Pz) electrode and a higher clustering coefficient in the left parietal (P3) electrode. These results are discussed in the context of the brain networks involved in encoding BM versus SM.     

Structure-Dynamics Relationships in Bursting Neuronal Networks Revealed Using a Prediction Framework

The question of how the structure of a neuronal network affects its functionality has gained a lot of attention in neuroscience. However, the vast majority of the studies on structure-dynamics relationships consider few types of network structures and assess limited numbers of structural measures. In this in silico study, we employ a wide diversity of network topologies and search among many possibilities the aspects of structure that have the greatest effect on the network excitability. The network activity is simulated using two point-neuron models, where the neurons are activated by noisy fluctuation of the membrane potential and their connections are described by chemical synapse models, and statistics on the number and quality of the emergent network bursts are collected for each network type. We apply a prediction framework to the obtained data in order to find out the most relevant aspects of network structure. In this framework, predictors that use different sets of graph-theoretic measures are trained to estimate the activity properties, such as burst count or burst length, of the networks. The performances of these predictors are compared with each other. We show that the best performance in prediction of activity properties for networks with sharp in-degree distribution is obtained when the prediction is based on clustering coefficient. By contrast, for networks with broad in-degree distribution, the maximum eigenvalue of the connectivity graph gives the most accurate prediction. The results shown for small () networks hold with few exceptions when different neuron models, different choices of neuron population and different average degrees are applied. We confirm our conclusions using larger () networks as well. Our findings reveal the relevance of different aspects of network structure from the viewpoint of network excitability, and our integrative method could serve as a general framework for structure-dynamics studies in biosciences.     

EEG-Based Functional Brain Networks: Does the Network Size Matter?

Functional connectivity in human brain can be represented as a network using electroencephalography (EEG) signals. These networks--whose nodes can vary from tens to hundreds--are characterized by neurobiologically meaningful graph theory metrics. This study investigates the degree to which various graph metrics depend upon the network size. To this end, EEGs from 32 normal subjects were recorded and functional networks of three different sizes were extracted. A state-space based method was used to calculate cross-correlation matrices between different brain regions. These correlation matrices were used to construct binary adjacency connectomes, which were assessed with regards to a number of graph metrics such as clustering coefficient, modularity, efficiency, economic efficiency, and assortativity. We showed that the estimates of these metrics significantly differ depending on the network size. Larger networks had higher efficiency, higher assortativity and lower modularity compared to those with smaller size and the same density. These findings indicate that the network size should be considered in any comparison of networks across studies.     

Evolution Characteristics of the Network Core in the Facebook

Statistical properties of the static networks have been extensively studied. However, online social networks are evolving dynamically, understanding the evolving characteristics of the core is one of major concerns in online social networks. In this paper, we empirically investigate the evolving characteristics of the Facebook core. Firstly, we separate the Facebook-link(FL) and Facebook-wall(FW) datasets into 28 snapshots in terms of timestamps. By employing the k-core decomposition method to identify the core of each snapshot, we find that the core sizes of the FL and FW networks approximately contain about 672 and 373 nodes regardless of the exponential growth of the network sizes. Secondly, we analyze evolving topological properties of the core, including the k-core value, assortative coefficient, clustering coefficient and the average shortest path length. Empirical results show that nodes in the core are getting more interconnected in the evolving process. Thirdly, we investigate the life span of nodes belonging to the core. More than 50% nodes stay in the core for more than one year, and 19% nodes always stay in the core from the first snapshot. Finally, we analyze the connections between the core and the whole network, and find that nodes belonging to the core prefer to connect nodes with high k-core values, rather than the high degrees ones. This work could provide new insights into the online social network analysis.     

Identification and classification of hubs in brain networks

Brain regions in the mammalian cerebral cortex are linked by a complex network of fiber bundles. These inter-regional networks have previously been analyzed in terms of their node degree, structural motif, path length and clustering coefficient distributions. In this paper we focus on the identification and classification of hub regions, which are thought to play pivotal roles in the coordination of information flow. We identify hubs and characterize their network contributions by examining motif fingerprints and centrality indices for all regions within the cerebral cortices of both the cat and the macaque. Motif fingerprints capture the statistics of local connection patterns, while measures of centrality identify regions that lie on many of the shortest paths between parts of the network. Within both cat and macaque networks, we find that a combination of degree, motif participation, betweenness centrality and closeness centrality allows for reliable identification of hub regions, many of which have previously been functionally classified as polysensory or multimodal. We then classify hubs as either provincial (intra-cluster) hubs or connector (inter-cluster) hubs, and proceed to show that lesioning hubs of each type from the network produces opposite effects on the small-world index. Our study presents an approach to the identification and classification of putative hub regions in brain networks on the basis of multiple network attributes and charts potential links between the structural embedding of such regions and their functional roles.