Detecting rich-club ordering in complex networks

Abstract

Uncovering the hidden regularities and organizational principles of networks arising in physical systems ranging from the molecular level to the scale of large communication infrastructures is the key issue for the understanding of their fabric and dynamical properties [1-5]. The ``rich-club'' phenomenon refers to the tendency of nodes with high centrality, the dominant elements of the system, to form tightly interconnected communities and it is one of the crucial properties accounting for the formation of dominant communities in both computer and social sciences [4-8]. Here we provide the analytical expression and the correct null models which allow for a quantitative discussion of the rich-club phenomenon. The presented analysis enables the measurement of the rich-club ordering and its relation with the function and dynamics of networks in examples drawn from the biological, social and technological domains.

1 table, 3 figures

References in corpus (1)

• The Rich-Club Phenomenon In The Internet Topology

Cited by in corpus (184)

• Link Prediction in Complex Networks: A Survey
• Critical phenomena in complex networks
• Predicting Missing Links via Local Information
• Vital nodes identification in complex networks
• Structure and dynamics of molecular networks: A novel paradigm of drug discovery. A comprehensive review
• Scale-free networks are rare
• The network of global corporate control
• Analyzing and Modeling Real-World Phenomena with Complex Networks: A Survey of Applications
• Communicability in complex networks
• Random walks and diffusion on networks
• Emergence of network features from multiplexity
• Inferring Social Status and Rich Club Effects in Enterprise Communication Networks
• Understanding brain networks and brain organization
• Classes of complex networks defined by role-to-role connectivity profiles
• Module identification in bipartite and directed networks
• Prominence and control: The weighted rich-club effect
• The Statistical Physics of Real-World Networks
• Structure and dynamics of core-periphery networks
• Community landscapes: an integrative approach to determine overlapping network module hierarchy, identify key nodes and predict network dynamics
• The multilayer connectome of Caenorhabditis elegans
• Empirical analysis of the worldwide maritime transportation network
• Measuring and modelling correlations in multiplex networks
• Statistically validated networks in bipartite complex systems
• Weighted network modules
• Optimally controlling the human connectome: the role of network topology
• Network resilience
• Topological strata of weighted complex networks
• Analytical maximum-likelihood method to detect patterns in real networks
• A fitness model for the Italian Interbank Money Market
• Identification of core-periphery structure in networks
• Link prediction in complex networks: a local na\"ıve Bayes model
• Quantifying randomness in real networks
• Enhancing network robustness for malicious attacks
• The Diverse Club: The Integrative Core of Complex Networks
• The International Trade Network: weighted network analysis and modelling
• Control of Dynamics in Brain Networks
• The rich-club phenomenon across complex network hierarchies
• Spectral entropies as information-theoretic tools for complex network comparison
• Accuracy of Mean-Field Theory for Dynamics on Real-World Networks
• Complex Networks from Classical to Quantum
• The unreasonable effectiveness of tree-based theory for networks with clustering
• Community structure and ethnic preferences in school friendship networks
• Triple Point in Correlated Interdependent Networks
• Correlations in weighted networks
• Multicores-periphery structure in networks
• Modular gateway-ness connectivity and structural core organization in maritime network science
• Social goods dilemmas in heterogeneous societies
• Geometric renormalization unravels self-similarity of the multiscale human connectome
• From modular to centralized organization of synchronization in functional areas of the cat cerebral cortex
• Detection of Core-Periphery Structure in Networks Using Spectral Methods and Geodesic Paths
• Socioeconomic correlations and stratification in social-communication networks
• On the rich-club effect in dense and weighted networks
• A Unifying Framework for Measuring Weighted Rich Clubs
• The Weakness of Weak Ties in the Classroom
• Rich-club connectivity dominates assortativity and transitivity of complex networks
• Chaperones as integrators of cellular networks: Changes of cellular integrity in stress and diseases
• Rich-cores in networks
• Scaling of degree correlations and the influence on diffusion in scale-free networks
• Emergence of assortative mixing between clusters of cultured neurons
• Models of communication and control for brain networks: distinctions, convergence, and future outlook
• From the origin of life to pandemics: Emergent phenomena in complex systems
• Structural constraints in complex networks
• Community Detection in Large Hypergraphs
• Core-periphery structure requires something else in the network
• How structure sculpts function: unveiling the contribution of anatomical connectivity to the brain's spontaneous correlation structure
• Complex stock trading network among investors
• Topological implications of negative curvature for biological and social networks
• Multiplex core-periphery organization of the human connectome
• Community characterization of heterogeneous complex systems
• A new dominance concept and its application to diversity-stability analysis
• Rich-club vs rich-multipolarization phenomena in weighted networks
• Inferring network properties from time series using transfer entropy and mutual information: validation of multivariate versus bivariate approaches
• Behavior patterns of online users and the effect on information filtering
• Fast and scalable likelihood maximization for Exponential Random Graph Models with local constraints
• Complex Network Analysis of the Bitcoin Transaction Network
• Hyper-cores promote localization and efficient seeding in higher-order processes
• Enhanced Gravity Model of trade: reconciling macroeconomic and network models
• Modeling Structure and Resilience of the Dark Network
• Estimating the impact of structural directionality: How reliable are undirected connectomes?
• Dynamics on Modular Networks with Heterogeneous Correlations
• Does the brain behave like a (complex) network? I. Dynamics
• Topological Structure of Urban Street Networks from the Perspective of Degree Correlations
• Effects of network topology on wealth distributions
• Multiscale core-periphery structure in a global liner shipping network
• A network perspective on intermedia agenda-setting
• The Temporal Rich Club Phenomenon
• Tie Strength Distribution in Scientific Collaboration Networks
• Influence of reciprocal arcs on the degree distribution and degree correlations
• Evolution of the Internet k-dense structure
• Scale-free networks as preasymptotic regimes of superlinear preferential attachment
• Statistical significance of rich-club phenomena in complex networks
• Revealing and exploiting hierarchical material structure through complex atomic networks
• Spectral analysis of deformed random networks
• Meta-validation of bipartite network projections
• Introduction to correlation networks: Interdisciplinary approaches beyond thresholding
• Information Diffusion in Computer Science Citation Networks
• Spontaneous Self-Assembly of Transcription Factor Based Gene Regulation Networks
• Rich-Club Ordering and the Dyadic Effect: Two Interrelated Phenomena
• Generalized Rich-Club Ordering in Networks
• The World-Trade Web: Topological Properties, Dynamics, and Evolution
• Longitudinal structural connectomic and rich-club analysis in adolescent mTBI reveals persistent, distributed brain alterations acutely through to one year post-injury
• Randomizing growing networks with a time-respecting null model
• Rich-club and page-club coefficients for directed graphs
• Geometric explanation of the rich-club phenomenon in complex networks
• Topological Centrality and Its Applications
• Interplay between $k$-core and community structure in complex networks
• Uncovering functional brain signature via random matrix theory
• Measuring topological descriptors of complex networks under uncertainty
• Networks with arbitrary edge multiplicities
• Structural Bounds on the Dyadic Effect
• Revealing dynamics, communities and criticality from data
• Observability transitions in correlated networks
• Path-dependent connectivity, not modularity, consistently predicts controllability of structural brain networks
• Assortative and disassortative mixing investigated using the spectra of graphs
• Local rewiring rules for evolving complex networks
• Weighted hypersoft configuration model
• Reactive immunization on complex networks
• The Drivers of Global News Spreading Patterns
• Impulse-Induced Optimum Signal Amplification in Scale-Free Networks
• Random Networks with given Rich-club Coefficient
• Higher-order rich-club phenomenon in collaborative research grants
• Using Complex Network Theory for Temporal Locality in Network Traffic Flows
• Detection of the elite structure in a virtual multiplex social system by means of a generalized $K$-core
• Optimizing and controlling functions of complex networks by manipulating rich-club connections
• Impact of noise and damage on collective dynamics of scale-free neuronal networks
• The topology of a discussion: the #occupy case
• Prediction of next career moves from scientific profiles
• Growing homophilic networks are natural navigable small worlds
• Drastic disorded-induced reduction of signal amplification in scale-free networks
• Explaining the emergence of complex networks through log-normal fitness in a Euclidean node similarity space
• Connectedness matters: Construction and exact random sampling of connected graphs
• Structure constrained by metadata in networks of chess players
• Statistical clustering of temporal networks through a dynamic stochastic block model
• Sampling networks by nodal attributes
• From Caesar to Twitter: An Axiomatic Approach to Elites of Social Networks
• Reconstructing topological properties of complex networks using the fitness model
• Sufficient conditions of endemic threshold on metapopulation networks
• Characterising Web Site Link Structure
• Social Network Heterogeneity Promotes Depolarization of Multidimensional Correlated Opinions
• Characterizing cycle structure in complex networks
• Weighted cycle-based identification of influential node groups in complex networks
• Rich-clubness test: how to determine whether a complex network has or doesn't have a rich-club?
• Geometric randomization of real networks with prescribed degree sequence
• The political economy of big data leaks: Uncovering the skeleton of tax evasion
• Stochastic resetting in a networked multiparticle system with correlated transitions
• Diverging network architecture of the $\text{C. elegans}$ connectome and signaling network
• Internet data packet transport: from global topology to local queueing dynamics
• Incorporating uncertainty into the study of animal social networks
• Ensembles related to the rich-club coefficient for non-evolving networks
• Influence measures in subnetworks using vertex centrality
• Recovering Zipf's law in intercontinental scientific collaboration
• Finding Influential Cores via Normalized Ricci Flows in Directed and Undirected Hypergraphs with Applications
• Determinable and interpretable network representation for link prediction
• Influential scientists shape knowledge flows between science and IGO policy
• Information content based model for the topological properties of the gene regulatory network of Escherichia coli
• The behavior of rich-club coefficient in scale-free networks
• The Atlas for the Aspiring Network Scientist
• Role of inter-hemispheric connections in functional brain networks
• On the location and the strength of controllers to desynchronize coupled Kuramoto oscillators
• The NonHuman Primate Neuroimaging & Neuroanatomy Project
• Network Null Model based on Maximal Entropy and the Rich-Club
• Multiscale modeling of brain network organization
• Communities and Patterns of Scientific collaboration
• Functional brain networks: great expectations, hard times, and the big leap forward
• Robustness of Transcriptional Regulation in Yeast-like Model Boolean Networks
• Temporal rich club phenomenon and its formation mechanisms
• Ensembles based on the Rich-Club and how to use them to build soft-communities
• The growing topology of the C. elegans connectome
• Hierarchical route to the emergence of leader nodes in real-world networks
• Bipartite Communities
• Spreader events and the limitations of projected networks for capturing dynamics on multipartite networks
• Diversity of Skills and Collective Intelligence in GitHub
• Computational Human Dynamics
• Unifying Modular and Core-Periphery Structure in Functional Brain Networks over Development
• Macroscopic cortical dynamics: Spatially uncorrelated but temporally coherent rich-club organisations in source-space resting-state EEG
• Global Networks of Trade and Bits
• Scale-Invariant Strength Assortativity of Streaming Butterflies
• The Formation and Imprinting of Network Effects Among the Business Elite
• Predicting Critical Nodes in Temporal Networks by Dynamic Graph Convolutional Networks
• Friendship-paradox paradox: Do most people's friends really have more friends than they do?
• The Structure and Dynamics of Gene Regulation Networks
• The Function of Communities in Protein Interaction Networks at Multiple Scales
• Funding shapes the anatomy of scientific research
• Graph Metrics for Internet Robustness -- A Survey