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Network centrality: an introduction
Francisco Aparecido Rodrigues
TL;DR
The chapter addresses the lack of a general definition of network centrality and the need to understand how centrality relates to dynamical processes. It reviews centrality measures and their features and limitations, examines epidemic spreading and synchronization, and discusses applications and extensions. It reports that central nodes are the most influential disease spreaders and that degree-correlated oscillator frequencies produce a first-order synchronization transition.
Problem
No general definition of centrality exists, despite its relevance to network organization and dynamical processes.
Method
The chapter reviews centrality measures, discusses their advantages and limitations, examines epidemic spreading and synchronization, and presents applications to biological and cortical networks.
Results
Central nodes are the most influential disease spreaders, while degree-correlated natural frequencies produce a first-order phase transition instead of the second-order transition seen when frequencies are independent of network structure.
Takeaways & Limitations
Centrality measures provide a framework for studying complex-system organization and dynamical behavior across biological, cortical, epidemic, and synchronization networks.
Takeaways & Limitations
Identifying the most influential spreaders for any kind of network remains an open problem.
Abstract
from arXiv · showhide
Centrality is a key property of complex networks that influences the behavior of dynamical processes, like synchronization and epidemic spreading, and can bring important information about the organization of complex systems, like our brain and society. There are many metrics to quantify the node centrality in networks. Here, we review the main centrality measures and discuss their main features and limitations. The influence of network centrality on epidemic spreading and synchronization is also pointed out in this chapter. Moreover, we present the application of centrality measures to understand the function of complex systems, including biological and cortical networks. Finally, we discuss some perspectives and challenges to generalize centrality measures for multilayer and temporal networks.
1 Introduction
Complex-network structure shapes dynamical processes, while heterogeneous connectivity makes some nodes more central than others. This chapter reviews centrality measures, their limitations, and applications to spreading, synchronization, biology, and neuroscience.
- Motivation: Complex systems are represented as networks whose heterogeneous structure influences processes including epidemic spreading, synchronization, cascade failures, percolation, and voter models.Scale-free heterogeneity can bring the epidemic threshold close to zero, while synchronization depends on a critical coupling strength that also varies with heterogeneity.
- Centrality measures: Centrality quantifies node importance, but no general definition exists, motivating multiple measures tailored to different applications.The chapter reviews the main measures together with their advantages and limitations.
- Epidemic spreading: Central nodes are identified as the most influential disease spreaders, although identifying such nodes remains dependent on network structure.The chapter discusses centrality’s influence on epidemic spreading and addresses spreader identification through centrality measures.
- Synchronization: When Kuramoto oscillators’ natural frequencies correlate with node degree, synchronization undergoes a first-order phase transition rather than the second-order transition observed without that structural dependence.The relevant correlation is expressed as ω_i = k_i.
- Applications and perspectives: The chapter presents centrality applications in systems biology and neuroscience and discusses extensions to multilayer and temporal networks.These extensions address networks with multiple interaction types and time-dependent structure.
2 Centrality measures
Centrality has no single general definition, so different measures capture different structural notions of importance. The chapter reviews degree, k-core, shortest-path, load, random-walk, accessibility, eigenvector, and non-backtracking approaches, emphasizing their distinct interpretations and limitations.
- Centrality is application-dependent because networks contain highly heterogeneous structures and no consensus exists on a general definition.
- Degree centrality: Degree centrality counts each node’s connections but is local: peripheral hubs may have high degree without being central.
- k-core: The k-core assigns higher centrality to nodes with greater coreness after iterative removal of nodes below the degree threshold k.
- Closeness centrality: Closeness centrality uses average shortest-path distance to other nodes, but its narrow variation limits discrimination in small-diameter networks.
- Betweenness centrality: Betweenness or load centrality measures the fraction of shortest paths passing through a node, but excludes information traveling along longer routes.
- Eigenvector centrality: Eigenvector centrality assigns importance through neighboring importance, while localization can concentrate high values in a small subset of nodes.
- Non-backtracking centrality: The non-backtracking matrix addresses a limitation of eigenvector centrality and supports community detection above a detectability threshold.
- Because centrality measures can identify different central nodes, the most suitable analysis depends on the problem and may require multiple metrics.
3 Centrality and dynamical processes in networks
Centrality shapes epidemic spreading and synchronization, but its predictive value depends on network structure and oscillator-frequency relationships. In epidemic networks, central nodes often spread disease most effectively; in synchronization, heterogeneity and frequency–degree correlations alter the transition.
- Identification of influential spreaders: In the E-road network, most central nodes are the most influential spreaders, while generalized random walk accessibility is the best predictor.The E-road network is spatially organized and lacks hubs, which limits the predictive performance of degree, K-core, and betweenness centrality.
- Identification of influential spreaders: In the US airport network, eigenvector centrality, degree, and K-core predict average outbreak size; accessibility works only at high centrality, whereas betweenness does not.The airport analysis uses 50 SIR simulations starting from each node and considers the largest component of the undirected network.
- Identification of influential spreaders: The suitable centrality measure for identifying influential spreaders depends on network organization: accessibility suits spatial networks, whereas K-core and eigenvector centrality suit scale-free networks.The identification of the most influential spreaders for any kind of network remains an open problem.
- Synchronization: Greater network heterogeneity lowers the critical coupling for synchronization, while frequency–degree correlations can change the transition from continuous to discontinuous.For scale-free networks, unimodal symmetric frequencies produce a continuous transition, whereas faster hub oscillations produce a first-order transition with hysteresis.
- Synchronization: When natural frequency correlates with node degree, system behavior changes, showing that centrality affects synchronization in uncorrelated networks.The scale-free example assigns oscillator frequencies according to node degree and observes hysteretic behavior indicating a discontinuous transition.
4 Applications
Centrality measures characterize complex systems across climate, transportation, biological, and cortical networks. These applications connect centrality patterns with network structure, biological function, disease diagnosis, brain organization, and robustness.
- Biological networks: Highly connected proteins are more likely to be essential, while disruption of the central tumor-suppressor P53 gene has severe cellular consequences.These examples connect network position with protein lethality and genetic-network function.
- Biological networks: Centrality measures support biological discovery by identifying genes related to prostate cancer and enabling disease-gene predictions and potential individualized therapies.Eigenvector and degree centrality identified prostate-cancer-related genes with high accuracy.
- Cortical networks: Cortical-network centrality reveals highly connected neocortical hubs with long-distance links, which are associated with robustness and cognition-related large-scale circuits.Regional hubs represent neocortical-thalamic circuits related to human cognition and consciousness.
- Cortical networks: Different centrality measures capture distinct cortical connectivity patterns: degree centrality decreases with age in precuneus and posterior cingulate regions, whereas eigenvector centrality remains constant.The comparison used data from 1003 subjects.
- Cortical networks: Cortical centrality measures distinguish schizophrenia from healthy subjects, achieving 90% sensitivity and 74% specificity while four of 54 measures differed between groups.Three of the four differing measures were related to centrality, alongside higher closeness-centrality variance and accessibility and lower average k-core.
- Climate and transportation networks: Centrality identifies structural backbones in climate networks and distinguishes connected cities from central cities in multi-community air transportation networks.Regions with the largest betweenness centrality relate to global surface ocean currents, while the most connected cities are not necessarily the most central.
5 Perspectives
Perspectives on centrality focus on selecting application-dependent measures and extending them to multilayer and temporal networks. Broader generalizations are needed to connect centrality more effectively with dynamical processes and complex-system evolution.
- Measure selection: The most suitable centrality measure is application dependent, so different concepts such as random walks and connectivity motivate distinct metrics.The paper frames measure selection as dependent on the network application.
- Dynamical processes: Centrality remains underexplored in dynamical processes such as cooperation and opinion formation, leaving opportunities for new measures and analyses.Only a few studies have examined centrality’s importance for improving cooperative processes.
- Multilayer networks: Multilayer centrality must account for edges within each layer and connections between layers when interactions have multiple types.Multilayer networks represent systems such as social networks with more than one interaction type.
- Open applications: New centrality metrics remain a research opportunity for characterizing complex systems in ecology, economy, and social networks and studying central nodes in dynamics.The paper identifies applications of these metrics and their influence on dynamical processes as open research areas.
- Temporal networks: Temporal centrality adapts static-network measures to changing connections, but existing approaches have mostly generalized static measures and need further improvement.Changing contact intervals affect behavior in processes such as epidemic spreading in social networks.
- Dynamical processes: Better links between centrality metrics and dynamics could provide tools to control and forecast processes such as epidemic spreading and synchronization.The stated scope concerns networks whose connections are not static and the evolution of complex systems.