Rich-club and page-club coefficients for directed graphs

Rich-club and page-club coefficients and their null models are introduced for directed graphs. Null models allow for a quantitative discussion of the rich-club and page-club phenomena. These coefficients are computed for four directed real-world networks: Arxiv High Energy Physics paper citation network, Web network (released from Google), Citation network among US Patents, and Email network from a EU research institution. The results show a high correlation between rich-club and page-club ordering. For journal paper citation network, we identify both rich-club and page-club ordering, showing that {}"elite" papers are cited by other {}"elite" papers. Google web network shows partial rich-club and page-club ordering up to some point and then a narrow declining of the corresponding normalized coefficients, indicating the lack of rich-club ordering and the lack of page-club ordering, i.e. high in-degree (PageRank) pages purposely avoid sharing links with other high in-degree (PageRank) pages. For UC patents citation network, we identify page-club and rich-club ordering providing a conclusion that {}"elite" patents are cited by other {}"elite" patents. Finally, for e-mail communication network we show lack of both rich-club and page-club ordering. We construct an example of synthetic network showing page-club ordering and the lack of rich-club ordering.Comment: 18 pages, 6 figure

Characterising Web Site Link Structure

The topological structures of the Internet and the Web have received considerable attention. However, there has been little research on the topological properties of individual web sites. In this paper, we consider whether web sites (as opposed to the entire Web) exhibit structural similarities. To do so, we exhaustively crawled 18 web sites as diverse as governmental departments, commercial companies and university departments in different countries. These web sites consisted of as little as a few thousand pages to millions of pages. Statistical analysis of these 18 sites revealed that the internal link structure of the web sites are significantly different when measured with first and second-order topological properties, i.e. properties based on the connectivity of an individual or a pairs of nodes. However, examination of a third-order topological property that consider the connectivity between three nodes that form a triangle, revealed a strong correspondence across web sites, suggestive of an invariant. Comparison with the Web, the AS Internet, and a citation network, showed that this third-order property is not shared across other types of networks. Nor is the property exhibited in generative network models such as that of Barabasi and Albert.Comment: To appear at IEEE/WSE0

Characterization of complex networks: A survey of measurements

Each complex network (or class of networks) presents specific topological features which characterize its connectivity and highly influence the dynamics of processes executed on the network. The analysis, discrimination, and synthesis of complex networks therefore rely on the use of measurements capable of expressing the most relevant topological features. This article presents a survey of such measurements. It includes general considerations about complex network characterization, a brief review of the principal models, and the presentation of the main existing measurements. Important related issues covered in this work comprise the representation of the evolution of complex networks in terms of trajectories in several measurement spaces, the analysis of the correlations between some of the most traditional measurements, perturbation analysis, as well as the use of multivariate statistics for feature selection and network classification. Depending on the network and the analysis task one has in mind, a specific set of features may be chosen. It is hoped that the present survey will help the proper application and interpretation of measurements.Comment: A working manuscript with 78 pages, 32 figures. Suggestions of measurements for inclusion are welcomed by the author

Structure and dynamics of core-periphery networks

Recent studies uncovered important core/periphery network structures characterizing complex sets of cooperative and competitive interactions between network nodes, be they proteins, cells, species or humans. Better characterization of the structure, dynamics and function of core/periphery networks is a key step of our understanding cellular functions, species adaptation, social and market changes. Here we summarize the current knowledge of the structure and dynamics of "traditional" core/periphery networks, rich-clubs, nested, bow-tie and onion networks. Comparing core/periphery structures with network modules, we discriminate between global and local cores. The core/periphery network organization lies in the middle of several extreme properties, such as random/condensed structures, clique/star configurations, network symmetry/asymmetry, network assortativity/disassortativity, as well as network hierarchy/anti-hierarchy. These properties of high complexity together with the large degeneracy of core pathways ensuring cooperation and providing multiple options of network flow re-channelling greatly contribute to the high robustness of complex systems. Core processes enable a coordinated response to various stimuli, decrease noise, and evolve slowly. The integrative function of network cores is an important step in the development of a large variety of complex organisms and organizations. In addition to these important features and several decades of research interest, studies on core/periphery networks still have a number of unexplored areas.Comment: a comprehensive review of 41 pages, 2 figures, 1 table and 182 reference

Core-Periphery in Networks: An Axiomatic Approach

Recent evidence shows that in many societies worldwide the relative sizes of the economic and social elites are continuously shrinking. Is this a natural social phenomenon? What are the forces that shape this process? We try to address these questions by studying a Core-Periphery social structure composed of a social elite, namely, a relatively small but well-connected and highly influential group of powerful individuals, and the rest of society, the periphery. Herein, we present a novel axiom-based model for the forces governing the mutual influences between the elite and the periphery. Assuming a simple set of axioms, capturing the elite's dominance, robustness, compactness and density, we are able to draw strong conclusions about the elite-periphery structure. In particular, we show that a balance of powers between elite and periphery and an elite size that is sub-linear in the network size are universal properties of elites in social networks that satisfy our axioms. We note that the latter is in controversy to the common belief that the elite size converges to a linear fraction of society (most recently claimed to be 1%). We accompany these findings with a large scale empirical study on about 100 real-world networks, which supports our results

Climate Dynamics: A Network-Based Approach for the Analysis of Global Precipitation

Precipitation is one of the most important meteorological variables for defining the climate dynamics, but the spatial patterns of precipitation have not been fully investigated yet. The complex network theory, which provides a robust tool to investigate the statistical interdependence of many interacting elements, is used here to analyze the spatial dynamics of annual precipitation over seventy years (1941-2010). The precipitation network is built associating a node to a geographical region, which has a temporal distribution of precipitation, and identifying possible links among nodes through the correlation function. The precipitation network reveals significant spatial variability with barely connected regions, as Eastern China and Japan, and highly connected regions, such as the African Sahel, Eastern Australia and, to a lesser extent, Northern Europe. Sahel and Eastern Australia are remarkably dry regions, where low amounts of rainfall are uniformly distributed on continental scales and small-scale extreme events are rare. As a consequence, the precipitation gradient is low, making these regions well connected on a large spatial scale. On the contrary, the Asiatic South-East is often reached by extreme events such as monsoons, tropical cyclones and heat waves, which can all contribute to reduce the correlation to the short-range scale only. Some patterns emerging between mid-latitude and tropical regions suggest a possible impact of the propagation of planetary waves on precipitation at a global scale. Other links can be qualitatively associated to the atmospheric and oceanic circulation. To analyze the sensitivity of the network to the physical closeness of the nodes, short-term connections are broken. The African Sahel, Eastern Australia and Northern Europe regions again appear as the supernodes of the network, confirming furthermore their long-range connection structure. Almost all North-American and Asian nodes vanish, revealing that extreme events can enhance high precipitation gradients, leading to a systematic absence of long-range patterns

Module hierarchy and centralisation in the anatomy and dynamics of human cortex

Systems neuroscience has recently unveiled numerous fundamental features of the macroscopic architecture of the human brain, the connectome, and we are beginning to understand how characteristics of brain dynamics emerge from the underlying anatomical connectivity. The current work utilises complex network analysis on a high-resolution structural connectivity of the human cortex to identify generic organisation principles, such as centralised, modular and hierarchical properties, as well as specific areas that are pivotal in shaping cortical dynamics and function. After confirming its small-world and modular architecture, we characterise the cortex’ multilevel modular hierarchy, which appears to be reasonably centralised towards the brain’s strong global structural core. The potential functional importance of the core and hub regions is assessed by various complex network metrics, such as integration measures, network vulnerability and motif spectrum analysis. Dynamics facilitated by the large-scale cortical topology is explored by simulating coupled oscillators on the anatomical connectivity. The results indicate that cortical connectivity appears to favour high dynamical complexity over high synchronizability. Taking the ability to entrain other brain regions as a proxy for the threat posed by a potential epileptic focus in a given region, we also show that epileptic foci in topologically more central areas should pose a higher epileptic threat than foci in more peripheral areas. To assess the influence of macroscopic brain anatomy in shaping global resting state dynamics on slower time scales, we compare empirically obtained functional connectivity data with data from simulating dynamics on the structural connectivity. Despite considerable micro-scale variability between the two functional connectivities, our simulations are able to approximate the profile of the empirical functional connectivity. Our results outline the combined characteristics a hierarchically modular and reasonably centralised macroscopic architecture of the human cerebral cortex, which, through these topological attributes, appears to facilitate highly complex dynamics and fundamentally shape brain function

Reconfiguration of dominant coupling modes in mild traumatic brain injury mediated by δ-band activity: a resting state MEG study

During the last few years, rich-club (RC) organization has been studied as a possible brain-connectivity organization model for large-scale brain networks. At the same time, empirical and simulated data of neurophysiological models have demonstrated the significant role of intra-frequency and inter-frequency coupling among distinct brain areas. The current study investigates further the importance of these couplings using recordings of resting-state magnetoencephalographic activity obtained from 30 mild traumatic brain injury (mTBI) subjects and 50 healthy controls. Intra-frequency and inter-frequency coupling modes are incorporated in a single graph to detect group differences within individual rich-club subnetworks (type I networks) and networks connecting RC nodes with the rest of the nodes (type II networks). Our results show a higher probability of inter-frequency coupling for (δ–γ1), (δ–γ2), (θ–β), (θ–γ2), (α–γ2), (γ1–γ2) and intra-frequency coupling for (γ1–γ1) and (δ–δ) for both type I and type II networks in the mTBI group. Additionally, mTBI and control subjects can be correctly classified with high accuracy (98.6%), whereas a general linear regression model can effectively predict the subject group using the ratio of type I and type II coupling in the (δ, θ), (δ, β), (δ, γ1), and (δ, γ2) frequency pairs. These findings support the presence of an RC organization simultaneously with dominant frequency interactions within a single functional graph. Our results demonstrate a hyperactivation of intrinsic RC networks in mTBI subjects compared to controls, which can be seen as a plausible compensatory mechanism for alternative frequency-dependent routes of information flow in mTBI subjects