Resilience of the Internet to random breakdowns

Abstract

A common property of many large networks, including the Internet, is that the connectivity of the various nodes follows a scale-free power-law distribution, P(k)=ck^-a. We study the stability of such networks with respect to crashes, such as random removal of sites. Our approach, based on percolation theory, leads to a general condition for the critical fraction of nodes, p_c, that need to be removed before the network disintegrates. We show that for a<=3 the transition never takes place, unless the network is finite. In the special case of the Internet (a=2.5), we find that it is impressively robust, where p_c is approximately 0.99.

latex, 3 pages, 1 figure (eps), explanations added, Phys. Rev. Lett., in press

References in corpus (4)

• Emergence of scaling in random networks
• Error and attack tolerance of complex networks
• The diameter of the world wide web
• Mean-field solution of the small-world network model

Cited by in corpus (707)

• Statistical mechanics of complex networks
• The structure and function of complex networks
• Community detection in graphs
• Assortative mixing in networks
• Catastrophic cascade of failures in interdependent networks
• The architecture of complex weighted networks
• Epidemic processes in complex networks
• The spread of epidemic disease on networks
• Identification of influential spreaders in complex networks
• Multilayer Networks
• Evolution of networks
• Mixing patterns in networks
• Benchmark graphs for testing community detection algorithms
• The structure and dynamics of multilayer networks
• Network robustness and fragility: Percolation on random graphs
• Critical phenomena in complex networks
• Evolution of the social network of scientific collaborations
• Structure and tie strengths in mobile communication networks
• Cascade-based attacks on complex networks
• Epidemic dynamics and endemic states in complex networks
• Breakdown of the Internet under intentional attack
• Structural Vulnerability of the North American Power Grid
• Why social networks are different from other types of networks
• Efficient Immunization Strategies for Computer Networks and Populations
• Vital nodes identification in complex networks
• Dynamics of Rumor Spreading in Complex Networks
• Reaction-diffusion processes and metapopulation models in heterogeneous networks
• Statistical physics of vaccination
• What's in a crowd? Analysis of face-to-face behavioral networks
• Scale-Free Networks are Ultrasmall
• Interdependent networks: Reducing the coupling strength leads to a change from a first to second order percolation transition
• Cascade control and defense in complex networks
• New Model of Internet Topology Using k-shell Decomposition
• MEDUSA - New Model of Internet Topology Using k-shell Decomposition
• Generation of uncorrelated random scale-free networks
• Thresholds for epidemic spreading in networks
• Modeling Cascading Failures in the North American Power Grid
• k-core organization of complex networks
• Robustness of a Network of Networks
• Control Principles of Complex Networks
• Modeling the Internet's Large-Scale Topology
• Social physics
• Origins of fractality in the growth of complex networks
• Measuring preferential attachment for evolving networks
• Halting viruses in scale-free networks
• Exact Controllability of Complex Networks
• Emergence of network features from multiplexity
• Clique percolation in random networks
• Absence of epidemic threshold in scale-free networks with connectivity correlations
• Pseudofractal Scale-free Web
• A fast Monte Carlo algorithm for site or bond percolation
• Robustness of interdependent networks under targeted attack
• Coevolution of dynamical states and interactions in dynamic networks
• Random graphs with clustering
• Spectra of "Real-World" Graphs: Beyond the Semi-Circle Law
• Understanding the spreading patterns of mobile phone viruses
• Properties of highly clustered networks
• Random Geometric Graphs
• Weighted Evolving Networks
• Statistical properties of sampled networks
• Are randomly grown graphs really random?
• Cascade of failures in coupled network systems with multiple support-dependent relations
• Controlling edge dynamics in complex networks
• Spectral methods for network community detection and graph partitioning
• Highly clustered scale-free networks
• Voter Models on Heterogeneous Networks
• Ising Model on Networks with an Arbitrary Distribution of Connections
• Classification of scale-free networks
• Unification of theoretical approaches for epidemic spreading on complex networks
• Percolation on complex networks: Theory and application
• Power grids vulnerability: a complex network approach
• Why Nestedness in Mutualistic Networks?
• Analysis of a large-scale weighted network of one-to-one human communication
• Average path length in random networks
• Average path length in uncorrelated random networks with hidden variables
• Invasion threshold in heterogeneous metapopulation networks
• Percolation Critical Exponents in Scale-Free Networks
• Stability and topology of scale-free networks under attack and defense strategies
• The extreme vulnerability of interdependent spatially embedded networks
• Cut-offs and finite size effects in scale-free networks
• Percolation on sparse networks
• The critical effect of dependency groups on the function of networks
• Generic scale of the "scale-free" growing networks
• Giant strongly connected component of directed networks
• Modeling the evolution of weighted networks
• Threshold effects for two pathogens spreading on a network
• Resilience of public transport networks against attacks
• Bipartite Graphs as Models of Complex Networks
• Behaviors of susceptible-infected epidemics on scale-free networks with identical infectivity
• Spectrum of Controlling and Observing Complex Networks
• Robustness of the European power grids under intentional attack
• A k-shell decomposition method for weighted networks
• Maximal planar networks with large clustering coefficient and power-law degree distribution
• Improved routing strategies for Internet traffic delivery
• Characterizing the community structure of complex networks
• Resilience to damage of graphs with degree correlations
• Network robustness of multiplex networks with interlayer degree correlations
• Characterizing the dynamical importance of network nodes and links
• Epidemic spreading in complex networks with degree correlations
• Competing epidemics on complex networks
• Scaling properties of scale-free evolving networks: Continuous approach
• Sandpile on Scale-Free Networks
• Interdependent networks with correlated degrees of mutually dependent nodes
• Incomplete ordering of the voter model on small-world networks
• Dynamics of jamming transitions in complex networks
• Construction and properties of assortative random networks
• Spectra and eigenvectors of scale-free networks
• Instability of scale-free networks under node-breaking avalanches
• Robustness of a Tree-like Network of Interdependent Networks
• Percolation in real interdependent networks
• Dynamical Patterns of Cattle Trade Movements
• Network resilience
• Complex network analysis of water distribution systems
• Network Geometry
• Ferromagnetic Phase Transition in Barabasi-Albert Networks
• Percolation in Directed Scale-Free Networks
• Epidemic Incidence in Correlated Complex Networks
• Subgraphs in random networks
• Bootstrap Percolation on Complex Networks
• Fractal and Transfractal Recursive Scale-Free Nets
• Percolation in Interdependent and Interconnected Networks: Abrupt Change from Second to First Order Transition
• Optimal Allocation of Interconnecting Links in Cyber-Physical Systems: Interdependence, Cascading Failures and Robustness
• Evolution of cooperation on scale-free networks subject to error and attack
• Epidemic threshold for the SIS model on networks
• Random graphs containing arbitrary distributions of subgraphs
• Critical load and congestion instabilities in scale-free networks
• Entropy Optimization of Scale-Free Networks Robustness to Random Failures
• Mean field solution of the Ising model on a Barabasi-Albert network
• Optimization of Robustness of Complex Networks
• Percolation and Epidemic Thresholds in Clustered Networks
• Optimal Paths in Disordered Complex Networks
• Spatially localized attacks on interdependent networks: the existence of a finite critical attack size
• Emergence of multilevel selection in the prisoner's dilemma game on coevolving random networks
• Enhancing network robustness for malicious attacks
• k-core (bootstrap) percolation on complex networks: Critical phenomena and nonlocal effects
• Percolation of localized attack on complex networks
• Analytical Solution of the Voter Model on Disordered Networks
• Optimization of Network Robustness to Waves of Targeted and Random Attack
• Analytical computation of the epidemic threshold on temporal networks
• Explosive percolation in scale-free networks
• Citation Networks in High Energy Physics
• Scale-free Networks on Lattices
• The synchronized dynamics of time-varying networks
• Efficiency and reliability of epidemic data dissemination in complex networks
• Statistical physics approaches to the complex Earth system
• Correlated multiplexity and connectivity of multiplex random networks
• Towards real-world complexity: an introduction to multiplex networks
• Transport in weighted networks: Partition into superhighways and roads
• Percolation of Partially Interdependent Scale-free Networks
• Spectral centrality measures in complex networks
• Risk perception in epidemic modeling
• Non mean-field behavior of the contact process on scale-free networks
• Percolation in networks composed of connectivity and dependency links
• Approximating the largest eigenvalue of network adjacency matrices
• Percolation of a general network of networks
• Containing epidemic outbreaks by message-passing techniques
• Improving immunization strategies
• From Scale-free to Erdos-Renyi Networks
• Successful strategies for competing networks
• Critical phenomena in networks
• Anomalous percolating properties of growing networks
• Multiple routes transmitted epidemics on multiplex networks
• Percolation of partially interdependent networks under targeted attack
• Structure of shells in complex networks
• Geography in a Scale-Free Network Model
• The effects of spatial constraints on the evolution of weighted complex networks
• Resilience of networks with community structure behaves as if under an external field
• How to make a fragile network robust and vice versa
• Explosive percolation: a numerical analysis
• Competing activation mechanisms in epidemics on networks
• Clustering in complex networks. II. Percolation properties
• Two-Peak and Three-Peak Optimal Complex Networks
• How clustering affects the bond percolation threshold in complex networks
• Core percolation on complex networks
• Eradicating Catastrophic Collapse in Interdependent Networks via Reinforced Nodes
• Spatiotemporal patterns and predictability of cyberattacks
• A Universal Model of Global Civil Unrest
• Articulation Points in Complex Networks
• Predicting percolation thresholds in networks
• Higher-order percolation processes on multiplex hypergraphs
• Robustness of onion-like correlated networks against targeted attacks
• Kinetics of Social Contagion
• Rate equation approach for correlations in growing network models
• The effect of network topology on the stability of discrete state models of genetic control
• Diffusion-annihilation processes in complex networks
• Onion structure and network robustness
• Synchronizability determined by coupling strengths and topology on Complex Networks
• Polynomial growth in age-dependent branching processes with diverging reproductive number
• Spectra of random graphs with arbitrary expected degrees
• Self-similar scaling of density in complex real-world networks
• Number of loops of size h in growing scale-free networks
• Growth and structure of Slovenia's scientific collaboration network
• Correlation between centrality metrics and their application to the opinion model
• Multi-node basin stability in complex dynamical networks
• Standard random walks and trapping on the Koch network with scale-free behavior and small-world effect
• Emergence of large cliques in random scale-free network
• A High Robustness and Low Cost Model for Cascading Failures
• Information filtering in complex weighted networks
• Small-world networks and the conformation space of a lattice polymer chain
• Geographical threshold graphs with small-world and scale-free properties
• WiFi Epidemiology: Can Your Neighbors' Router Make Yours Sick?
• Anomalous Transport in Complex Networks
• Error and Attack Tolerance of Layered Complex Networks
• Percolation in Hierarchical Scale-Free Nets
• Immunization of networks with community structure
• Robustness of partially interdependent network formed of clustered networks
• Solving the Dynamic Correlation Problem of the Susceptible-Infected-Susceptible Model on Networks
• Extreme self-organization in networks constructed from gene expression data
• Multi-state epidemic processes on complex networks
• Intermittent social distancing strategy for epidemic control
• Vulnerability of weighted networks
• Recent advances of percolation theory in complex networks
• Sandpile avalanche dynamics on scale-free networks
• Metric structure of random networks
• Subgraphs and network motifs in geometric networks
• Evolution of scale-free random graphs: Potts model formulation
• Rockfall susceptibility and network-ranked susceptibility along the Italian railway
• Immunization of Real Complex Communication Networks
• Bicomponents and the robustness of networks to failure
• Spatio-temporal networks: reachability, centrality and robustness
• Quarantine generated phase transition in epidemic spreading
• Optimal Path and Minimal Spanning Trees in Random Weighted Networks
• Inducing Effect on the Percolation Transition in Complex Networks
• Supercondutor-Insulator Transition on Annealed Complex Networks
• Resource allocation pattern in infrastructure networks
• Complex networks vulnerability to module-based attacks
• The influence of the broadness of the degree distribution on network's robustness: comparing localized attack and random attack
• A tale of two cities. Vulnerabilities of the London and Paris transit networks
• Multiple resource demands and viability in multiplex networks
• A limited resource model of fault-tolerant capability against cascading failure of complex network
• Epidemic Threshold in Continuous-Time Evolving Networks
• Epidemics, disorder, and percolation
• A dynamic model of time-dependent complex networks
• Group percolation in interdependent networks
• Let Your CyberAlter Ego Share Information and Manage Spam
• $k$-core percolation on complex networks: Comparing random, localized and targeted attacks
• Dynamic networks and directed percolation
• Percolation transition in networks with degree-degree correlation
• Resilience of Complex Networks to Random Breakdown
• The structural role of weak and strong links in a financial market network
• Compensating for population sampling in simulations of epidemic spread on temporal contact networks
• Breaking of the site-bond percolation universality in networks
• Randomized reference models for temporal networks
• Synchronization in Weighted Uncorrelated Complex Networks in a Noisy Environment: Optimization and Connections with Transport Efficiency
• Epidemic variability in complex networks
• Phase Transition with the Berezinskii--Kosterlitz--Thouless Singularity in the Ising Model on a Growing Network
• Tricritical point in heterogeneous k-core percolation
• Rare events and discontinuous percolation transitions
• Emergence of synchronization in multiplex networks of mobile Rössler oscillators
• Climate network percolation reveals the expansion and weakening of the tropical component under global warming
• Optimal Paths in Complex Networks with Correlated Weights: The World-wide Airport Network
• Quasiuniversal connectedness percolation of polydisperse rod systems
• Weighted percolation on directed networks
• Explosive Percolation in the Human Protein Homology Network
• Tight lower bound for percolation threshold on a quasi-regular graph
• Beyond the locally tree-like approximation for percolation on real networks
• Trapping in complex networks
• Synchronization transition of heterogeneously coupled oscillators on scale-free networks
• Impact of embedding on predictability of failure-recovery dynamics in networks
• Immunization of Susceptible-Infected Model on Scale-Free networks
• The resilience of interdependent transportation networks under targeted attack
• VANET Connectivity Analysis
• Width of percolation transition in complex networks
• A Condition for Cooperation in a Game on Complex Networks
• Localization transition on complex networks via spectral statistics
• The cost of attack in competing networks
• Non-consensus opinion models on complex networks
• Inverse targeting -- an effective immunization strategy
• Reciprocity and the Emergence of Power Laws in Social Networks
• Stability of shortest paths in complex networks with random edge weights
• Statistics of Cycles: How Loopy is your Network?
• Extreme events and event size fluctuations in biased random walks on networks
• Generation of arbitrarily two-point correlated random networks
• Stochastic Opinion Formation in Scale-Free Networks
• Geographical Embedding of Scale-Free Networks
• Critical noise of majority-vote model on complex networks
• Optimization of robustness of scale-free network to random and targeted attacks
• Critical fluctuations in spatial complex networks
• Enhancing resilience of interdependent networks by healing
• Emergent failures and cascades in power grids: a statistical physics perspective
• Number of cliques in random scale-free network ensembles
• Networking - A Statistical Physics Perspective
• Localization Transition of Biased Random Walks on Random Networks
• Fluctuations in percolation of sparse complex networks
• Fractal Boundaries of Complex Networks
• Nonlinear spread of rumor and inoculation strategies in the nodes with degree dependent tie strength in complex networks
• Modeling of Information Diffusion on Social Networks with Applications to WeChat
• Enhancing the spectral gap of networks by node removal
• The Effect of Disease-induced Mortality on Structural Network Properties
• Itineration of the Internet over Nonequilibrium Stationary States in Tsallis Statistics
• Spin Glass Phase Transition on Scale-Free Networks
• Giant Clusters in Random Ad Hoc Networks
• Quasi-stationary analysis of the contact process on annealed scale-free networks
• Quantum statistics in complex networks
• Biased Percolation on Scale-free Networks
• Percolation in self-similar networks
• Cascading failures in scale-free interdependent networks
• Geographical effects on cascading breakdowns of scale-free networks
• The localization of non-backtracking centrality in networks and its physical consequences
• A message-passing approach to epidemic tracing and mitigation with apps
• The diminishing role of hubs in dynamical processes on complex networks
• Mapping Koch curves into scale-free small-world networks
• Modeling Structure and Resilience of the Dark Network
• Impact of link deletions on public cooperation in scale-free networks
• Sparse random graphs with clustering
• Rare region effects in the contact process on networks
• Optimization of Network Robustness to Random Breakdowns
• Transportation Network Stability: a Case Study of City Traffic
• The spread of gossip in American schools
• Complex Quantum Networks: a Topical Review
• A statistical mechanics approach for scale-free networks and finite-scale networks
• Satellite-based photonic quantum networks are small-world
• Large-deviation properties of the largest biconnected component for random graphs
• Degree Correlation in Scale-Free Graphs
• Optimization of network structure to random failures
• Percolation theory applied to measures of fragmentation in social networks
• Clique percolation in random graphs
• Epidemic spreading on modular networks: The fear to declare a pandemic
• Simple and efficient self-healing strategy for damaged complex networks
• Message passing methods on complex networks
• Robustness of Noisy Quantum Networks
• Percolation in real multiplex networks
• Limited path percolation in complex networks
• Scaling in Small-World Resistor Networks
• The Bak-Sneppen Model on Scale-Free Networks
• Finite-size scaling of synchronized oscillation on complex networks
• Reaction-diffusion processes on correlated and uncorrelated scale-free networks
• Clustering determines the dynamics of complex contagions in multiplex networks
• Disassortativity of percolating clusters in random networks
• Efficient network immunization under limited knowledge
• Estimation of global network statistics from incomplete data
• Corporate competition: A self-organized network
• The synchronizability of highly clustered scale-free networks
• The combined effect of connectivity and dependency links on percolation of networks
• Deriving an underlying mechanism for discontinuous percolation
• Search in Complex Networks : a New Method of Naming
• Effect of Disorder Strength on Optimal Paths in Complex Networks
• Structurally Robust Control of Complex Networks
• Mean first-passage time for maximal-entropy random walks in complex networks
• Asymmetry in interdependence makes a multilayer system more robust against cascading failures
• Zero temperature Glauber dynamics on complex networks
• Complex networks embedded in space: Dimension and scaling relations between mass, topological distance and Euclidean distance
• Cavity analysis on the robustness of random networks against targeted attacks: Influences of degree-degree correlations
• Fluctuation-driven capacity distribution in complex networks
• Underestimated cost of targeted attacks on complex networks
• Epidemic spreading driven by biased random walks
• Homological percolation transitions in growing simplicial complexes
• Bridges in Complex Networks
• Information Theory Perspective on Network Robustness
• Diverse types of percolation transitions
• Strategy of Competition between Two Groups based on a Contrarian Opinion Model
• Temporal percolation of the susceptible network in an epidemic spreading
• Limited path entanglement percolation in quantum complex networks
• Nation-State Routing: Censorship, Wiretapping, and BGP
• Transport in networks with multiple sources and sinks
• The bridges to consensus: Network effects in a bounded confidence opinion dynamics model
• Epidemic spreading in lattice-embedded scale-free networks
• Families and clustering in a natural numbers network
• Robustness and modular structure in networks
• Enhancing the robustness of scale-free networks
• Effect of edge removal on topological and functional robustness of complex networks
• Small-world behavior in a system of mobile elements
• Infection propagator approach to compute epidemic thresholds on temporal networks: impact of immunity and of limited temporal resolution
• Building Damage-Resilient Dominating Sets in Complex Networks against Random and Targeted Attacks
• Resumption of dynamism in damaged networks of coupled oscillators
• Effects of local population structure in a reaction-diffusion model of a contact process on metapopulation networks
• Diffusion in scale-free networks with annealed disorder
• Probabilistic heuristics for disseminating information in networks
• Optimization of scale-free network for random failures
• Why are there six degrees of separation in a social network?
• Analytical results for bond percolation and k-core sizes on clustered networks
• Multifractal properties of growing networks
• Enhancement of Tc in the Superconductor-Insulator Phase Transition on Scale-Free Networks
• Directedness of information flow in mobile phone communication networks
• On the topologic structure of economic complex networks: Empirical evidence from large scale payment network of Estonia
• Spreading paths in partially observed social networks
• Resilience in Hierarchical Fluid Flow Networks
• Phase diagram of the Bose-Hubbard Model on Complex Networks
• Network Overload due to Massive Attacks
• Highly intensive data dissemination in complex networks
• Multicomponent reaction-diffusion processes on complex networks
• Improvement of the robustness on geographical networks by adding shortcuts
• Network Risk and Forecasting Power in Phase-Flipping Dynamical Networks
• Fluctuation-dissipation relations for complex networks
• SIS Epidemic Spreading with Correlated Heterogeneous Infection Rates
• Phase transition of light on complex quantum networks
• Numerical evaluation of the upper critical dimension of percolation in scale-free networks
• Shortest path and Schramm-Loewner Evolution
• Sequential Defense Against Random and Intentional Attacks in Complex Networks
• Power Grid Network Evolutions for Local Energy Trading
• Conservation laws for voter-like models on directed networks
• Epidemic Extinction Paths in Complex Networks
• Interfaces and the edge percolation map of random directed networks
• Equitable random graphs
• Sampling properties of directed networks
• Correlations in Scale-Free Networks: Tomography and Percolation
• Diffusive Capture Process on Complex Networks
• Scaling of percolation transitions on Erdös-Rényi networks under centrality-based attacks
• Network percolation reveals adaptive bridges of the mobility network response to COVID-19
• Effectiveness of dismantling strategies on moderated vs. unmoderated online social platforms
• Cascading collapse of online social networks
• Criterion for explosive percolation transitions on complex networks
• Dynamical robustness of network of oscillators
• Different thresholds of bond percolation in scale-free networks with identical degree sequence
• Renormalization group for evolving networks
• Coevolving complex networks in the model of social interactions
• Neighborhood Information-based Probabilistic Algorithm for Network Disintegration
• Correlations in complex networks under attack
• Protecting infrastructure networks from cost-based attacks
• Self-Organization of Dragon Kings
• Interconnections between networks act like an external field in first-order percolation transitions
• Robustness of skeletons and salient features in networks
• Spin glass behavior of the antiferromagnetic Ising model on a scale-free network
• Omori's law in the Internet traffic
• Multiple outbreaks in epidemic spreading with local vaccination and limited vaccines
• A comprehensive statistical study of metabolic and protein-protein interaction network properties
• Evolution of Vocabulary on Scale-free and Random Networks
• Energy-landscape network approach to the glass transition
• Abrupt transition due to non-local cascade propagation in multiplex systems
• History-dependent percolation on multiplex networks
• Effects of local and global network connectivity on synergistic epidemics
• New universality class in percolation on multifractal scale-free planar stochastic lattice
• On the Tomography of Networks and Multicast Trees
• Quantifying long-range correlations in complex networks beyond nearest neighbors
• Discrete surface growth process as a synchronization mechanism for scale free complex networks
• Epidemic Spreading and Digital Contact Tracing: Effects of Heterogeneous Mixing and Quarantine Failures
• Probabilistic prediction in scale-free networks: Diameter changes
• Distribution of infected mass in disease spreading in scale-free networks
• Designer Nets from Local Strategies
• Resilience of spatial networks with inter-links behaving as an external field
• Percolation Framework of the Earth's Topography
• Renormalization group for link percolation on planar hyperbolic manifolds
• Percolation on interdependent networks with a fraction of antagonistic interactions
• Optimal modularity for nucleation in network-organized Ising model
• Scale-Free Networks Emerging from Weighted Random Graphs
• Restoring site percolation on a damaged square lattice
• Path Percolation in Quantum Communication Networks
• Modeling social resilience: Questions, answers, open problems
• Emergence of double scaling law in complex system
• Non-consensus opinion model on directed networks
• The Blind Watchmaker Network: Scale-freeness and Evolution
• Social distancing strategies against disease spreading
• A topological approach to neural complexity
• Percolation on bipartite scale-free networks
• Statistical analysis of articulation points in configuration model networks
• Hierarchy in directed random networks
• Scale-Free Networks beyond Power-Law Degree Distribution
• A dissemination strategy for immunizing scale-free networks
• Synchronization of networks with variable local properties
• Resonant response of forced complex networks: the role of topological disorder
• Emergence of Blind Areas in Information Spreading
• Deterministic Entanglement Distribution on Series-Parallel Quantum Networks
• A general model of hierarchical fractal scale-free networks
• Classes of critical avalanche dynamics in complex networks
• Effect of degree correlations above the first shell on the percolation transition
• Color-avoiding percolation
• Percolation of Hierarchical Networks and Networks of Networks
• Optimal redundancy against disjoint vulnerabilities in networks
• Hierarchical scale-free network is fragile against random failure
• Universality class of site and bond percolation on multi-multifractal scale-free planar stochastic lattice
• Network Growth with Arbitrary Initial Conditions: Analytical Results for Uniform and Preferential Attachment
• Goal-directed graph construction using reinforcement learning
• Network resilience against intelligent attacks constrained by degree dependent node removal cost
• Shortest-path percolation on random networks
• Enhancing Robustness and Immunization in geographical networks
• Systemic risk in dynamical networks with stochastic failure criterion
• Variability of Contact Process in Complex Networks
• Robustness of Spatial Micronetworks
• Dynamical Scaling Behavior of Percolation Clusters in Scale-free Networks
• A study of the edge-switching Markov-chain method for the generation of random graphs
• Approximating the Largest Eigenvalue of the Modified Adjacency Matrix of Networks with Heterogeneous Node Biases
• Percolation and Loop Statistics in Complex Networks
• Correlations in interacting systems with a network topology
• Analysis of the convergence of the degree distribution of contracting random networks towards a Poisson distribution using the relative entropy
• Explosive percolation in graphs
• Shortest path discovery of complex networks
• Random geometric graph description of connectedness percolation in rod systems
• Diffusive capture processes for information search
• How to design multi-target drugs: Target search options in cellular networks
• Sizing the length of complex networks
• Epidemic Spreading in a Social Network with Facial Masks wearing Individuals
• Spreaders in the Network SIR Model: An Empirical Study
• Resilience of networks to environmental stress: From regular to random networks
• The Complex Topology of Chemical Plants
• Generalized theory for node disruption in finite size complex networks
• Convergence towards an Erd{\H o}s-Rényi graph structure in network contraction processes
• Percolation transition in correlated hypergraphs
• Scale-free networks resistant to intentional attacks
• Thermodynamic forces, flows, and Onsager coefficients in complex networks
• Partition of Networks into Basins of Attraction
• Synchronization in Scale-free Dynamical Networks: Robustness and Fragility
• Bond and site color-avoiding percolation in scale free networks
• Oscillations of complex networks
• Efficiently Controllable Graphs
• Profile and scaling of the fractal exponent of percolations in complex networks
• Controlling percolation with limited resources
• Highly Clustered Complex Networks in the Configuration Model: Random Regular Small-World Network
• Mean-field theory for double-well systems on degree-heterogeneous networks
• Dynamical robustness of complex networks subject to long-range connectivity
• The effect of randomness for dependency map on the robustness of interdependent lattices
• Anomalous electrical and frictionless flow conductance in complex networks
• Percolation on spatial anisotropic networks
• Universality of noise-induced resilience restoration in spatially-extended ecological systems
• Quantum Internet under random breakdowns and intentional attacks
• The structure of networks that evolve under a combination of growth, via node addition and random attachment, and contraction, via random node deletion
• Random field Ising model on networks with inhomogeneous connections
• Uncertainty of Resilience in Complex Networks with Nonlinear Dynamics
• Exotic phase transitions of k-cores in clustered networks
• Spectral estimation of the percolation transition in clustered networks
• Optimal cost for strengthening or destroying a given network
• Disentangling Giant Component and Finite Cluster Contributions in Sparse Matrix Spectra
• Effects of aging and links removal on epidemic dynamics in scale-free networks
• Diluted antiferromagnet in a ferromagnetic enviroment
• Derivation of the percolation threshold for the network model of Barabasi and Albert
• How Does the Vulnerability of an Evolving Power Grid Change?
• Preferential Detachment in Broadcast Signaling Networks: Connectivity and Cost Trade-off
• Percolation transition and distribution of connected components in generalized random network ensembles
• Critical phenomena of a hybrid phase transition in cluster merging dynamics
• Graph Partitioning Induced Phase Transitions
• Given enough choice, simple local rules percolate discontinuously
• Stability of glassy hierarchical networks
• Propinquity drives the emergence of network structure and density
• Impact of noise and damage on collective dynamics of scale-free neuronal networks
• Phase transitions with infinitely many absorbing states in complex networks
• Duality and free energy analyticity bounds for few-body Ising models with extensive homology rank
• Geographical networks stochastically constructed by a self-similar tiling according to population
• Spontaneous symmetry breaking and discontinuous phase transition for spreading dynamics in multiplex networks
• Bypass rewiring and robustness of complex networks
• Link deletion in directed complex networks
• Size of quantum networks
• Robustness analysis of bimodal networks in the whole range of degree correlation
• Nucleation in scale-free networks
• Binary dynamics on star networks under external perturbations
• On extreme points of the diffusion polytope
• Clarifying How Degree Entropies and Degree-Degree Correlations Relate to Network Robustness
• Percolation of networks with directed dependency links
• Double phase transition of the Ising model in core-periphery networks
• Extended $s$-wave pairing symmetry on the triangular lattice heavy fermion system
• Limits and trade-offs of topological network robustness
• Integrating fluctuations into distribution of resources in transportation networks
• Universality class of the percolation in two-dimensional lattices with distortion
• Statistical analysis of edges and bredges in configuration model networks
• Criticality and Continuity of Explosive Site Percolation in Random Networks
• K-Core Minimization: A Game Theoretic Approach
• Network science Ising states of matter
• Kauffman Boolean model in undirected scale free networks
• Two faces of greedy leaf removal procedure on graphs
• Dynamic Behavior of Interacting between Epidemics and Cascades on Heterogeneous Networks
• Mesoscopics and fluctuations in networks
• Robustness of correlated networks against propagating attacks
• Random node reinforcement and $K$-core structure of complex networks
• Scale-Free Random SAT Instances
• Assessing Percolation Threshold Based on High-Order Non-Backtracking Matrices
• A network-specific approach to percolation in networks with bidirectional links
• Towards Decentralized Trading: A Topological Investigation of the Dutch Medium and Low Voltage Grids
• Percolation of randomly distributed growing clusters
• Emergence of Clusters in Growing Networks with Aging
• Impact of random and targeted disruptions on information diffusion during outbreaks
• Exponential rate of epidemic spreading on complex networks
• The Impact of Social Networks on Multi-Agent Recommender Systems
• Inhomogeneous percolation models for spreading phenomena in random graphs
• Characterizing the intrinsic correlations of scale-free networks
• Percolation on Networks with Antagonistic and Dependent Interactions
• SIS Epidemic Spreading with Heterogeneous Infection Rates
• Selective pressure on metabolic network structures as measured from the random blind-watchmaker network
• Activity patterns on random scale-free networks: Global dynamics arising from local majority rules
• Susceptible-Infected-Recovered model on Euclidean network
• The emergence of Logarithmic-periodic oscillations in Contact process with the topological disorder
• Degree-targeted cascades in modular, degree-heterogeneous networks
• Decoding the structure of the WWW: facts versus sampling biases
• Sufficient conditions of endemic threshold on metapopulation networks
• Robustness of networks against propagating attacks under vaccination strategies
• Preservation of network Degree Distributions from non-uniform failures
• Generalized $k$-core pruning process on directed networks
• Performance of attack strategies on modular networks
• The Fate of Articulation Points and Bredges in Percolation
• Catching the head, tail, and everything in between: a streaming algorithm for the degree distribution
• Weighted projected networks: mapping hypergraphs to networks
• Crossover from weak to strong disorder regime in the duration of epidemics
• Epidemics in networks: A master equation approach
• Local heuristics and the emergence of spanning subgraphs in complex networks
• Robustness and Resilience of cities around the world
• Exactly solvable tight-binding models on two scale-free networks with identical degree distribution
• Degree and component size distributions in generalized uniform recursive tree
• Error Propagation Through a Network With Non-Uniform Failure
• Underestimating extreme events in power-law behavior due to machine-dependent cutoffs
• Vitality of Neural Networks under Reoccurring Catastrophic Failures
• Robustness of Regular Graphs Based on Natural Connectivity
• Degree-ordered percolation on hierarchical scale-free network
• Impact of shocks to economies on the efficiency and robustness of the international pesticide trade networks
• Random Walk Sampling in Social Networks Involving Private Nodes
• Robustness Assessment of Complex Networks using the Idle Network
• The behavior of noise-resilient Boolean networks with diverse topologies
• Branching process approach for Boolean bipartite networks of metabolic reactions
• Faster calculation of the percolation correlation length on spatial networks
• Potts Model On Random Trees
• Phase transition in evolving networks that combine preferential attachment and random node deletion
• Spontaneous repulsion in the $A+B\to 0$ reaction on coupled networks
• Organic Design of Massively Distributed Systems: A Complex Networks Perspective
• Percolation on a maximally disassortative network
• Fundamental limitations of network reconstruction
• Robustness of Random Networks with Selective Reinforcement against Attacks
• Geometric detection of hierarchical backbones in real networks
• Economy importance and structural robustness of the international pesticide trade networks
• Giant component in a configuration-model power-law graph with a variable number of links
• Compression-based inference of network motif sets
• Degree-dependent network growth: From preferential attachment to explosive percolation
• Percolation of the Site Random-Cluster Model by Monte Carlo Method
• A measure for characterizing heavy-tailed networks
• Scaling and universality for percolation in random networks: A unified view
• Design of endurable networks in the presence of aging
• How does degree heterogeneity affect nucleation of Ising model on complex networks?
• Dynamical processes on metric networks
• Improving robustness of spatial networks via reinforced nodes
• Critical Percolation on Random Networks with Prescribed Degrees
• Heterogeneity in Outcomes of Repeated Instances of Percolation Experiments
• Context-Independent Centrality Measures Underestimate the Vulnerability of Power Grids
• Degree distributions under general node removal: Power-law or Poisson?
• Quantifying plasticity: a network-based framework linking structure to dynamical regimes
• Cumulative Merging Percolation: A long-range percolation process in networks
• Optimal shattering of complex networks
• Spin glasses and percolation
• Finding the proper node ranking method for complex networks
• Phase transitions in Ising model induced by weight redistribution on weighted regular networks
• The Atlas for the Aspiring Network Scientist
• Network Science approach to Modelling Emergence and Topological Robustness of Supply Networks: A Review and Perspective
• Percolation transitions with nonlocal constraint
• Describing and Simulating Internet Routes
• Degree-Ordered-Percolation on uncorrelated networks
• Inhomogeneous substructures hidden in random networks
• Random Walks in Local Dynamics of Network Losses
• Disorder Induced Limited Path Percolation
• No-exclaves percolation on random networks
• Heterogeneous Recovery Rates against SIS Epidemics in Directed Networks
• A hybrid percolation transition at a finite transition point in scale-free networks
• Quantifying the temporal stability of international fertilizer trade networks
• Quantitative modeling of degree-degree correlation in complex networks
• Structural and temporal heterogeneities on networks
• Induced Percolation on Networked Systems
• Random Apollonian networks with tailored clustering coefficient
• Lattice model for percolation on a plane of partially aligned sticks with length dispersity
• A statistical mechanics investigation of Unfolded Protein Response across organisms
• On the monotonicity of the critical time in the Constrained-degree percolation model
• Null models for multi-optimized large-scale network structures
• New nodes attack strategies for real complex weighted networks
• Emergent networks in fractional percolation
• Usage leading to an abrupt collapse of connectivity
• The network structure of mathematical knowledge according to the Wikipedia, MathWorld, and DLMF online libraries
• Efficiency of message transmission using biased random walks in complex networks in the presence of traps
• Predicting Lifetime of Dynamical Networks Experiencing Persistent Random Attacks
• Large deviations of connected components in the stochastic block model
• Ising model for distribution networks
• Degree and connectivity of the Internet's scale-free topology
• Distributed Algorithm to Locate Critical Nodes to Network Robustness based on Spectral Analysis
• Deep Learning Super-Diffusion in Multiplex Networks
• Range-based attacks on links in random scale-free networks
• Temporal Correlations of Local Network Losses
• From the Grid to the Smart Grid, Topologically
• Critical line in undirected Kauffman boolean networks - the role of percolation
• Revisiting small-world network models: Exploring technical realizations and the equivalence of the Newman-Watts and Harary models
• Optimal counter-current exchange networks
• Protein Interaction Networks are Fragile against Random Attacks and Robust against Malicious Attacks
• External field and critical exponents in controlling dynamics on complex networks
• Reaction-diffusion on random spatial networks with scale-free jumping rates via effective medium theory
• On the distances within cliques in a soft random geometric graph
• Barely supercritical percolation on Poissonian scale-free networks
• Flux of information in scale-free networks
• Epidemics on evolving networks with varying degrees
• Universality in percolation of arbitrary Uncorrelated Nested Subgraphs
• Fault Tolerance of Random Graphs with respect to Connectivity: Mean-field Approximation for Semi-dense Random Graphs
• A machine learning procedure to detect network attacks
• Detrended fluctuation analysis on the correlations of complex networks under attack and repair strategy
• Janus Percolation in Anisotropic Limited-Degree Networks
• Identifying Central Nodes in Multiplex Networks by Embracing Layer-Specific Heterogeneity via DomiRank
• Polynomial Observables in the Graph Partitioning Problem
• On the Degree Distribution of Faulty Peer-to-Peer Overlays
• Mandala Networks: ultra-robust, ultra-small-world and highly sparse graphs
• Carrying capacity in growing networks
• The Truth about Power Laws: Theory and Reality
• Dimensions, Structures and Security of Networks
• General Connectivity Distribution Functions for Growing Networks with Preferential Attachment of Fractional Power
• Effects of missing data in social networks
• Spreading processes in "post-epidemic" environments. II. Safety patterns on scale-free networks
• Continuum Percolation on Disoriented Surfaces: the Problem of Permeable Disks on a Klein Bottle
• Resilience of Social Networks Under Different Attack Strategies
• Network-ensemble comparisons with stochastic rewiring and von Neumann entropy
• Starvation suppression in dense scale-free metabolic networks: Dynamical mean-field analysis of catalytic reaction networks
• Coping with Physical Attacks on Random Network Structures
• Interdependent Security with Strategic Agents and Cascades of Infection
• Impact of Random Failures and Attacks on Poisson and Power-Law Random Networks
• Homophyly and Randomness Resist Cascading Failure in Networks
• The effect of preferential node deletion on the structure of networks that evolve via preferential attachment
• On the shape of Gaussian scale-free polymer networks
• An integrated model of traffic, geography and economy in the Internet
• Complex Design Networks: Structure and Dynamics
• The strategy of survival for a competition between normal and anomalous diffusion
• Assessing the significance of knockout cascades in metabolic networks
• Phase transition in the diffusion and bootstrap percolation models on regular random and Erdős-Rényi networks
• Controlling network dynamics
• Recovery of Interdependent Networks
• Macroscopic and Microscopic Characteristics of Networks with Time-variant Functionality for Evaluating Resilience to External Perturbations
• Network robustness assessed within a dual connectivity perspective
• Predicting effects of structural stress in a genome-reduced model bacterial metabolism
• Percolation Processes and Wireless Network Resilience to Degree-Dependent and Cascading Node Failures
• Structural Changes and Percolation Transition in Networks after Aging Processes
• Graph Metrics for Internet Robustness -- A Survey
• Random Walks on Complex Networks