Critical phenomena in complex networks

Abstract

The combination of the compactness of networks, featuring small diameters, and their complex architectures results in a variety of critical effects dramatically different from those in cooperative systems on lattices. In the last few years, researchers have made important steps toward understanding the qualitatively new critical phenomena in complex networks. We review the results, concepts, and methods of this rapidly developing field. Here we mostly consider two closely related classes of these critical phenomena, namely structural phase transitions in the network architectures and transitions in cooperative models on networks as substrates. We also discuss systems where a network and interacting agents on it influence each other. We overview a wide range of critical phenomena in equilibrium and growing networks including the birth of the giant connected component, percolation, k-core percolation, phenomena near epidemic thresholds, condensation transitions, critical phenomena in spin models placed on networks, synchronization, and self-organized criticality effects in interacting systems on networks. We also discuss strong finite size effects in these systems and highlight open problems and perspectives.

Review article, 79 pages, 43 figures, 1 table, 508 references, extended

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• Networking - A Statistical Physics Perspective
• Mean-field analysis of the q-voter model on networks
• Generalization of core percolation on complex networks
• Bosonic reaction-diffusion processes on scale-free networks
• Network Evolution Based on Centrality
• Quasi-stationary analysis of the contact process on annealed scale-free networks
• Maximizing Influence Propagation in Networks with Community Structure
• Emergent Criticality Through Adaptive Information Processing in Boolean Networks
• Controlling the uncertain response of real multiplex networks to random damage
• Biased Percolation on Scale-free Networks
• Percolation in self-similar networks
• The localization of non-backtracking centrality in networks and its physical consequences
• Percolation on hyperbolic lattices
• Scientific success from the perspective of the strength of weak ties
• Stochastic pair approximation treatment of the noisy voter model
• Evolutionary dynamics of time-resolved social interactions
• A message-passing approach to epidemic tracing and mitigation with apps
• The diminishing role of hubs in dynamical processes on complex networks
• Metastable Localization of Diseases in Complex Networks
• Rhythmogenic neuronal networks, pacemakers, and k-cores
• Recovery rate affects the effective epidemic threshold with synchronous updating
• Effects of heterogeneous social interactions on flocking dynamics
• Trapping in scale-free networks with hierarchical organization of modularity
• Rare region effects in the contact process on networks
• Estimating the value of containment strategies in delaying the arrival time of an influenza pandemic: A case study of travel restriction and patient isolation
• Ramp Reversal Memory and Phase-Boundary Scarring in Transition Metal Oxides
• Complex Quantum Networks: a Topical Review
• Does the brain behave like a (complex) network? I. Dynamics
• Critical behavior of the Ising model in annealed scale-free networks
• Message-passing theory for cooperative epidemics
• Transportation Network Stability: a Case Study of City Traffic
• Explosive synchronization transitions in complex neural network
• Effective epidemic containment strategy in hypergraphs
• Network cloning unfolds the effect of clustering on dynamical processes
• Controllability and maximum matchings of complex networks
• Close or connected? Distance and connectivity effects on transport in networks
• Percolation in real multiplex networks
• Machine learning dynamical phase transitions in complex networks
• Generalized epidemic process on modular networks
• Dynamical organization towards consensus in the Axelrod model on complex networks
• Epidemic centrality - is there an underestimated epidemic impact of network peripheral nodes?
• Transition to synchrony in degree-frequency correlated Sakaguchi-Kuramoto model
• Random walks in modular scale-free networks with multiple traps
• Local Empathy provides Global Minimization of Congestion in Communication Networks
• Neural Ordinary Differential Equation Control of Dynamics on Graphs
• Effects of degree-frequency correlations on network synchronization: universality and full phase-locking
• Synchronization of phase oscillators on complex hypergraphs
• Epidemic Spreading and Aging in Temporal Networks with Memory
• Robustness and fragility of the susceptible-infected-susceptible epidemic models on complex networks
• How Many Nodes are Effectively Accessed in Complex Networks?
• From incoherence to synchronicity in the network Kuramoto model
• Multilayer Network Science: from Cells to Societies
• Networks in Motion
• Griffiths effects of the susceptible-infected-susceptible epidemic model on random power-law networks
• Large deviation theory of percolation on multiplex networks
• Ising model on the Apollonian network with node dependent interactions
• Exotic Critical Behavior of Weak Multiplex Percolation
• Spontaneous ordering against an external field in nonequilibrium systems
• Percolation-based precursors of transitions in extended systems
• Mean first-passage time for maximal-entropy random walks in complex networks
• An efficient strategy to suppress epidemic explosion in heterogeneous metapopulation networks
• Social contagions with communication channels alternation on multiplex networks
• Complex networks embedded in space: Dimension and scaling relations between mass, topological distance and Euclidean distance
• Fragmentation transitions in multi-state voter models
• Solution of the explosive percolation quest. II. Infinite-order transition produced by the initial distributions of clusters
• Mutually cooperative epidemics on power-law networks
• Epidemic spreading driven by biased random walks
• Cavity analysis on the robustness of random networks against targeted attacks: Influences of degree-degree correlations
• Steady-State Dynamics of the Forest Fire Model on Complex Networks
• Six Susceptible-Infected-Susceptible Models on Scale-free Networks
• Decongestion of urban areas with hotspot-pricing
• Optimal and suboptimal networks for efficient navigation measured by mean-first passage time of random walks
• Homological percolation transitions in growing simplicial complexes
• First-order phase transitions in outbreaks of co-infectious diseases and the extended general epidemic process
• Criticality and the Onset of Ordering in the Standard Vicsek Model
• Cooperation percolation in spatial prisoner's dilemma game
• Monte-Carlo simulation study of the two-stage percolation transition in enhanced binary trees
• Complex Quantum Network Manifolds in Dimension $d>2$ are Scale-Free
• Bridges in Complex Networks
• Role of fractal dimension in random walks on scale-free networks
• Structural stability of interaction networks against negative external fields
• Vortices and the entrainment transition in the 2D Kuramoto model
• Cascading failures in interdependent networks with finite functional components
• Critical behavior of the contact process in annealed scale-free networks
• Temporal percolation of the susceptible network in an epidemic spreading
• Mean first-passage time for random walks in general graphs with a deep trap
• Limited path entanglement percolation in quantum complex networks
• Phase synchronization between collective rhythms of globally coupled oscillator groups: noisy identical case
• Critical behavior of the contact process on small-world networks
• Slave nodes and the controllability of metabolic networks
• Traffic-driven SIR epidemic model on networks
• Robustness and modular structure in networks
• Ranking influential nodes in networks from partial information
• Dynamic structural and topological phase transitions on the Warsaw Stock Exchange: A phenomenological approach
• Sensitivity of directed networks to the addition and pruning of edges and vertices
• Growth of graph states in quantum networks
• General and exact approach to percolation on random graphs
• Revealing the Micro-Structure of the Giant Component in Random Graph Ensembles
• Spectral density of random graphs with topological constraints
• Violation of Lee-Yang circle theorem for Ising phase transitions on complex networks
• Discriminating Topology in Galaxy Distributions using Network Analysis
• Resumption of dynamism in damaged networks of coupled oscillators
• Anomalous behavior of trapping on a fractal scale-free network
• Explicit construction of the eigenvectors and eigenvalues of the graph Laplacian on the Cayley tree
• Effects of local population structure in a reaction-diffusion model of a contact process on metapopulation networks
• Contagion dynamics on hypergraphs with nested hyperedges
• Generating hierarchial scale free graphs from fractals
• Geometric Frustration of Colloidal Dimers on a Honeycomb Magnetic Lattice
• Spectral analysis and slow spreading dynamics on complex networks
• Bond percolation on a class of correlated and clustered random graphs
• Percolation on branching simplicial and cell complexes and its relation to interdependent percolation
• Analytical results for bond percolation and k-core sizes on clustered networks
• Driving interconnected networks to supercriticality
• Enhancement of Tc in the Superconductor-Insulator Phase Transition on Scale-Free Networks
• Fixed points and stability in the two-network frustrated Kuramoto model
• On $P$-adic Ising-Vannimenus model on an arbitrary order Cayley tree
• Universal transient behavior in large dynamical systems on networks
• Phase diagram of the Bose-Hubbard Model on Complex Networks
• q-state Potts model on the Apollonian network
• Extreme events in a complex network: interplay between degree distribution and repulsive interaction
• Network Risk and Forecasting Power in Phase-Flipping Dynamical Networks
• Complex networks with tuneable dimensions as a universality playground
• Levy random walks on multiplex networks
• Critical phenomena and noise-induced phase transitions in neuronal networks
• The nature and nurture of network evolution
• Impact of degree heterogeneity on the behavior of trapping in Koch networks
• Meta-validation of bipartite network projections
• An overview of epidemic models with phase transitions to absorbing states running on top of complex networks
• Phase diagrams of interacting spreading dynamics in complex networks
• Percolation on random networks with arbitrary k-core structure
• Entropy production at criticality in a nonequilibrium Potts model
• Concurrency measures in the era of temporal network epidemiology: A review
• Universality at Breakdown of Quantum Transport on Complex Networks
• Epidemic spreading on preferred degree adaptive networks
• Interplay of network dynamics and ties heterogeneity on spreading dynamics
• Critical behaviour of the XY -rotors model on regular and small world networks
• Phase transition of light on complex quantum networks
• Multifractal Characterization of Protein Contact Networks
• Pair approximation for the $q$-voter model with independence on multiplex networks
• Percolation-induced PT symmetry breaking
• Network-complement transitions, symmetries, and cluster synchronization
• Non massive immunization to contain spreading on complex networks
• Fixed Point Properties of the Ising Ferromagnet on the Hanoi Networks
• Physical states in the canonical tensor model from the perspective of random tensor networks
• Superconductor-insulator transition in a network of 2d percolation clusters
• Continuous Percolation Phase Transitions of Two-dimensional Lattice Networks under a Generalized Achlioptas Process
• Slow dynamics and rare-region effects in the contact process on weighted tree networks
• Nature of synchronization transitions in random networks of coupled oscillators
• Local Dirac Synchronization on Networks
• Ising model in clustered scale-free networks
• Constraining Cosmology with Big Data Statistics of Cosmological Graphs
• Relevance of backtracking paths in epidemic spreading on networks
• Recursive solutions for Laplacian spectra and eigenvectors of a class of growing treelike networks
• Effectiveness of dismantling strategies on moderated vs. unmoderated online social platforms
• The nature of hypergraph $k$-core percolation problems
• Outbreaks in susceptible-infected-removed epidemics with multiple seeds
• Generating-function approach for bond percolations in hierarchical networks
• Epidemic Extinction Paths in Complex Networks
• Dynamics and Processing in Finite Self-Similar Networks
• Phase diagram of an Ising model with competitive interactions on a Husimi tree and its disordered counterpart
• Hybrid dynamics in delay-coupled swarms with "mothership" networks
• Dynamical robustness of network of oscillators
• Criterion for explosive percolation transitions on complex networks
• Fractional random walk lattice dynamics
• Synchronisation Properties of Trees in the Kuramoto Model
• Synchronization centrality and explosive synchronization in complex networks
• Different thresholds of bond percolation in scale-free networks with identical degree sequence
• Collective fluctuations in networks of noisy components
• High prevalence regimes in the pair quenched mean-field theory for the susceptible-infected-susceptible model on networks
• Modeling self-sustained activity cascades in socio-technical networks
• Ising critical behavior of inhomogeneous Curie-Weiss models and annealed random graphs
• Dynamics to equilibrium in Network Games: individual behavior and global response
• Cumulative Merging Percolation and the epidemic transition of the Susceptible-Infected-Susceptible model in networks
• Scaling of mean first-passage time as efficiency measure of nodes sending information on scale-free Koch networks
• Characterization of multiple topological scales in multiplex networks through supra-Laplacian eigengaps
• Multistability, local pattern formation, and global collective firing in a small-world network of non-leaky integrate-and-fire neurons
• Influential spreaders for recurrent epidemics on networks
• The Impact of Competing Time Delays in Coupled Stochastic Systems
• Contagion spreading on complex networks with local deterministic dynamics
• Phase of Ising spins on modular networks analogous to social polarization
• Layer-layer competition in multiplex complex networks
• A generalised mean-field approximation for the Deffuant opinion dynamics model on networks
• Complex Network Structure of Flocks in the Standard Vicsek Model
• Energy-landscape network approach to the glass transition
• Topological structure and the H-index in complex networks
• Homophily-based social group formation in a spin-glass self-assembly framework
• Controlling self-organized criticality in complex networks
• Using synchronous Boolean networks to model several phenomena of collective behavior
• Scaling and universality in glass transition
• Slow, bursty dynamics as the consequence of quenched network topologies
• Epidemic Spreading and Digital Contact Tracing: Effects of Heterogeneous Mixing and Quarantine Failures
• Continuous-time quantum walk spatial search on the Bollobás scale-free network
• Generalized Voter-like model on activity driven networks with attractiveness
• Heterogenous mean-field analysis of a generalized voter-like model on networks
• Glass transition and random walks on complex energy landscapes
• Self-Organised Criticality and Emergent Hyperbolic Networks -- Blueprint for Complexity in Social Dynamics
• Clustering in random line graphs
• Growing community networks with local events
• FastSIR Algorithm: A Fast Algorithm for simulation of epidemic spread in large networks by using SIR compartment model
• Extreme variability in convergence to structural balance in frustrated dynamical systems
• Fractal fluctuations at mixed-order transitions in interdependent networks
• Evolution Features and Behavior Characters of Friendship Networks on Campus Life
• Quantum walks on complex networks with connection instabilities and community structure
• Critical exponents of the explosive percolation transition
• Gaussian noise and the two-network frustrated Kuramoto model
• On recurrence of random walks with long-range steps generated by fractional Laplacian matrices on regular networks and simple cubic lattices
• Renormalization group for link percolation on planar hyperbolic manifolds
• Transition-type change between an inverted Berezinskii-Kosterlitz-Thouless transition and an abrupt transition in the bond percolation on a random hierarchical small-world network
• Site-resolved imaging of beryllium ion crystals in a high-optical-access Penning trap with inbore optomechanics
• Rare regions of the Susceptible Infected Susceptible model on Barabási-Albert networks
• Quantum-Inspired Hierarchy for Rank-Constrained Optimization
• Quantifying Networks Complexity from Information Geometry Viewpoint
• Transition from small to large world in growing networks
• Optimal modularity for nucleation in network-organized Ising model
• Percolation on interdependent networks with a fraction of antagonistic interactions
• Phase Transitions in the Quadratic Contact Process on Complex Networks
• Critical phenomena on scale-free networks: logarithmic corrections and scaling functions
• Spectra of Random Stochastic Matrices and Relaxation in Complex Systems
• Amplification of explosive width in complex networks
• Statistical pairwise interaction model of stock market
• Lower bound of assortativity coefficient in scale-free networks
• Universal scaling in sports ranking
• Dynamical replica analysis of processes on finitely connected random graphs II: Dynamics in the Griffiths phase of the diluted Ising ferromagnet
• Heterogeneous Mean Field for neural networks with short term plasticity
• Influence, originality and similarity in directed acyclic graphs
• Structure and stability of online chat networks built on emotion-carrying links
• Tracing the Evolution of Physics on the Backbone of Citation Networks
• Scaling of Clusters near Discontinuous Percolation Transitions in Hyperbolic Networks
• Order parameter analysis of synchronization transitions on star networks
• A unified approach to percolation processes on multiplex networks
• Critical stretching of mean-field regimes in spatial networks
• Reconstructing dynamical networks via feature ranking
• On the numerical study of percolation and epidemic critical properties in networks
• Emergence of scale-free close-knit friendship structure in online social networks
• Percolation on bipartite scale-free networks
• Driven synchronization in random networks of oscillators
• Network Analysis of Cosmic Structures : Network Centrality and Topological Environment
• Phase transitions in Ising models on directed networks
• Geometry, Topology and Simplicial Synchronization
• Excess reciprocity distorts reputation in online social networks
• Synchronized and mixed outbreaks of coupled recurrent epidemics
• Statistically consistent coarse-grained simulations for critical phenomena in complex networks
• Quenched central limit theorems for the Ising model on random graphs
• Extended Anderson Criticality in Heavy-Tailed Neural Networks
• Emergence of Blind Areas in Information Spreading
• Mean-Field Solution for Critical Behavior of Signed Networks in Competitive Balance Theory
• Binary-state dynamics on complex networks: Stochastic pair approximation and beyond
• Classes of critical avalanche dynamics in complex networks
• Scale-Free Networks beyond Power-Law Degree Distribution
• Enhancing network synchronization by phase modulation
• Phase synchronization between collective rhythms of fully locked oscillator groups
• Learning epidemic threshold in complex networks by Convolutional Neural Network
• Effect of degree correlations above the first shell on the percolation transition
• Optimal community structure for social contagions
• Random walks in small-world exponential treelike networks
• Multitarget search on complex networks: A logarithmic growth of global mean random cover time
• Fractality and degree correlations in scale-free networks
• A general model of hierarchical fractal scale-free networks
• Classification of critical phenomena in hierarchical small-world networks
• Core organization of directed complex networks
• Link-disorder fluctuation effects on synchronization in random networks
• A framework for cascade size calculations on random networks
• Continuous- and discrete-time Glauber dynamics. First- and second-order phase transitions in mean-field Potts models
• Double transition of information spreading in a two-layered network
• Non-Markovian Majority-Vote model
• Observability transitions in correlated networks
• Using complex networks towards information retrieval and diagnostics in multidimensional imaging
• Suppressed epidemics in multi-relational networks
• Hierarchical scale-free network is fragile against random failure
• Hidden geometry and dynamics of complex networks: Spin reversal in nanoassemblies with pairwise and triangle-based interactions
• Communication and correlation among communities
• Exploring complex networks by means of adaptive walkers
• Generalized scaling theory for critical phenomena including essential singularity and infinite dimensionality
• Uncertainty Reduction for Stochastic Processes on Complex Networks
• Symmetric configurations highlighted by collective quantum coherence
• The dynamic nature of conflict in Wikipedia
• Enhancement of large fluctuations to extinction in adaptive networks
• Critical Phase of Bond Percolations on Growing Networks
• Dynamics of diffusion on monoplex and multiplex networks: A message-passing approach
• The low-temperature phase in the two-dimensional long-range diluted XY model
• Voter model on a directed network: Role of bidirectional opinion exchanges
• Directed Percolation in Temporal Networks
• Scale-free networks emerging from multifractal time series
• Analysis of heat kernel highlights the strongly modular and heat-preserving structure of proteins
• Generalized Ising Model on a Scale-Free Network: An Interplay of Power Laws
• Fundamental interactions in self-organized critical dynamics on higher-order networks
• Criticality governed by the stable renormalization fixed point of the Ising model in the hierarchical small-world network
• Synchronization and long-time memory in neural networks with inhibitory hubs and synaptic plasticity
• Dynamics of condensation in growing complex networks
• Explosive percolation in graphs
• Statistical Mechanics of the Chinese Restaurant Process: lack of self-averaging, anomalous finite-size effects and condensation
• Equation of motion of canonical tensor model and Hamilton-Jacobi equation of general relativity
• Not all interventions are equal for the height of the second peak
• Cooperation in changing environments: Irreversibility in the transition to cooperation in networks
• Superradiant phase transition in complex networks
• Majority-vote model on spatially embedded networks: crossover from mean-field to Ising universality classes
• Effective field theory for models defined over small-world networks. First and second order phase transitions
• Percolation transition in correlated hypergraphs
• Exact eigenvalue spectrum of a class of fractal scale-free networks
• Critical behavior of the SIS epidemic model with time-dependent infection rate
• Towards a Universal Measure of Complexity
• Synergistic signatures of group mechanisms in higher-order systems
• Exploring Quantum Annealing Architectures: A Spin Glass Perspective
• Pair approximation for the q-voter models with quenched disorder on networks
• Structure of percolating clusters in random clustered networks
• Phase transitions in a multistate majority-vote model on complex networks
• Rapid decay in the relative efficiency of quarantine to halt epidemics in networks
• Critical behavior of a two-step contagion model with multiple seeds
• Explosive Phase Transition in a Majority-Vote Model with Inertia
• Synchronization in the random field Kuramoto model on complex networks
• Scaling functions for systems with finite range of interaction
• Percolation properties of growing networks under an Achlioptas process
• Spectral renormalization group theory on networks
• Tailored graph ensembles as proxies or null models for real networks II: results on directed graphs
• Irreversible Aggregation and Network Renormalization
• Scaling of load in communications networks
• Absorbing Random Walks Interpolating Between Centrality Measures on Complex Networks
• Highly Clustered Complex Networks in the Configuration Model: Random Regular Small-World Network
• Bond and site color-avoiding percolation in scale free networks
• Adversarial decision strategies in multiple network phased oscillators: the Blue-Green-Red Kuramoto-Sakaguchi model
• Dynamical robustness of complex networks subject to long-range connectivity
• Critical behavior and correlations on scale-free small-world networks. Application to network design
• Spontaneous Formation of Dynamical Groups in an Adaptive Networked System
• Condensation and topological phase transitions in a dynamical network model with rewiring of the links
• Profile and scaling of the fractal exponent of percolations in complex networks
• Triadic percolation induces dynamical topological patterns in higher-order networks
• Global disorder transition in the community structure of large-q Potts systems
• Potts Model with Invisible States: A Review
• Incorporating Heterogeneous Interactions for Ecological Biodiversity
• Phase transitions in the Potts model on complex networks
• The stability to instability transition in the structure of large scale networks
• Synergistic information in a dynamical model implemented on the human structural connectome reveals spatially distinct associations with age
• Large order fluctuations, switching, and control in complex networks
• Antiferromagnetic Ising model in scale-free networks
• Ising model with variable spin/agent strengths
• Multifractal cross-correlation effects in two-variable time series of complex network vertex observables
• Nested Subgraphs of Complex Networks
• Statistical mechanics of random geometric graphs: Geometry-induced first order phase transition
• Structural and spectral properties of a family of deterministic recursive trees: Rigorous solutions
• A Random Laser as a Dynamical Network
• Scaling behavior of the contact process in networks with long-range connections
• The symbiotic contact process: phase transitions, hysteresis cycles, and bistability
• Ergodic-localized junctions in periodically-driven systems
• Noise-enhanced nonlinear response and the role of modular structure for signal detection in neuronal networks
• Lumping of Degree-Based Mean Field and Pair Approximation Equations for Multi-State Contact Processes
• Large deviation induced phase switch in an inertial majority-vote model
• Quenched mean-field theory for the majority-vote model on complex networks
• Asymptotic absorption-time distributions in extinction-prone Markov processes
• Complex networks and glassy dynamics: walks in the energy landscape
• Synchronization Invariance Under Network Structural Transformations
• A geometric entropy detecting the Erdös-Rényi phase transition
• Infinite randomness critical behavior of the contact process on networks with long-range connections
• Quantum correlated cluster mean-field theory applied to the transverse Ising model
• Interdependent transport via percolation backbones in spatial networks
• Enhancing Transport Efficiency by Hybrid Routing Strategy
• Impact of Committed Minorities: Unveiling Critical Mass of Cooperation in the Iterated Prisoner's Dilemma Game
• Sudden spreading of infections in an epidemic model with a finite seed fraction
• Riemannian-geometric entropy for measuring network complexity
• Uncovering hidden dependency in weighted networks via information entropy
• A neuronal network model of interictal and recurrent ictal activity
• Scale-free network clustering in hyperbolic and other random graphs
• Effect of overlap on spreading dynamics on multiplex networks
• Inverting the Achlioptas rule for explosive percolation
• An efficient approach to suppress the negative role of contrarian oscillators in synchronization
• Impact of noise and damage on collective dynamics of scale-free neuronal networks
• Unified framework for hybrid percolation transitions based on microscopic dynamics
• Solution of the random field $XY$ magnet on a fully connected graph
• To synchronize or not to synchronize, that is the question: finite-size scaling and fluctuation effects in the Kuramoto model
• Reconstruction scheme for excitatory and inhibitory dynamics with quenched disorder: application to zebrafish imaging
• Percolation transition and distribution of connected components in generalized random network ensembles
• Contact graphs of disk packings as a model of spatial planar networks
• Understanding the temporal pattern of spreading in heterogeneous networks: Theory of the mean infection time
• Ising model on a 2D additive Small-World Network
• From explosive to infinite-order transitions on a hyperbolic network
• Phase transitions with infinitely many absorbing states in complex networks
• Optimal forwarding ratio on dynamical networks with heterogeneous mobility
• Convexity in complex networks
• Models, Entropy and Information of Temporal Social Networks
• Finite Size Effects for the Ising Model on Random Graphs with Varying Dilution
• Bypass rewiring and robustness of complex networks
• Non-equilibrium mean-field theories on scale-free networks
• Statistics of Two-point Correlation and Network Topology for Lyman Alpha Emitters at $z\approx 2.67$
• Promoting cooperation by preventing exploitation: The role of network structure
• Zero-temperature Glauber dynamics on small-world networks
• Spontaneous symmetry breaking and discontinuous phase transition for spreading dynamics in multiplex networks
• Dynamical Complex Network Models of the Turbulent Cascade
• Double phase transition of the Ising model in core-periphery networks
• Emergent entanglement structures and self-similarity in quantum spin chains
• Nucleation in scale-free networks
• A 3-states magnetic model of binary decisions in sociophysics
• Modeling interacting dynamic networks: II. Systematic study of the statistical properties of cross-links between two networks with preferred degrees
• Entropic equation of state and scaling functions near the critical point in scale-free networks
• Real-Space Renormalization Group for Spectral Properties of Hierarchical Networks
• Spectrum of the tight-binding model on Cayley Trees and comparison with Bethe Lattices
• Condensation of degrees emerging through a first-order phase transition in classical random graphs
• Self-avoiding walks and connective constants in clustered scale-free networks
• Evolutionary network games: equilibria from imitation and best-response dynamics
• Multistage onset of epidemics in heterogeneous networks
• The lower bound of the network connectivity guaranteeing in-phase synchronization
• Rank the spreading influence of nodes using dynamic Markov process
• Non-Markovian random walks characterize network robustness to nonlocal cascades
• Non-equilibrium Ising Model on a 2D Additive Small-World Network
• Fundamental Structural Constraint of Random Scale-Free Networks
• Connectivity and Dynamics of Neuronal Networks as Defined by the Shape of Individual Neurons
• The Diffusive Epidemic Process on Barabasi-Albert Networks
• Renormalization procedure for random tensor networks and the canonical tensor model
• Conductivity of Coniglio-Klein clusters
• Criticality and Continuity of Explosive Site Percolation in Random Networks
• Multi-scale Laplacian community detection in heterogeneous networks
• Self-Organization and Complex Networks
• Agreement dynamics on directed random graphs
• Bifurcations in synergistic epidemics on random regular graphs
• Network science Ising states of matter
• Robustness of correlated networks against propagating attacks
• Ferromagnetic and spin-glass like transition in the $q$-neighbor Ising model on random graphs
• Exact solution of bond percolation on small arbitrary graphs
• Two faces of greedy leaf removal procedure on graphs
• Graph's Topology and Free Energy of a Spin Model on the Graph
• Upstreamness and downstreamness in input-output analysis from local and aggregate information
• High-dimensional percolation criticality and hints of mean-field-like caging of the random Lorentz gas
• Link overlap influences opinion dynamics on multiplex networks: spin model approach
• Absence of the non-percolating phase for percolation on the non-planar Hanoi network
• Higher-order triadic percolation on random hypergraphs
• Creating and controlling overlap in two-layer networks. Application to a mean-field SIS epidemic model with awareness dissemination
• Analogue neural networks on correlated random graphs
• Control of excitable systems is optimal near criticality
• Dynamics on Spatial Networks and the Effect of Distance Coarse Graining
• Characterizing the intrinsic correlations of scale-free networks
• Fractional dynamics on circulant multiplex networks: optimal coupling and long-range navigation for continuous-time random walks
• Statistical mechanics of high-density bond percolation
• Quantum Chaos in Random Ising Networks
• Ising model with invisible states on scale-free networks
• Analytic solution to space-fractional Fokker-Planck equations for tempered-stable Lévy distributions with spatially linear, time-dependent drift
• Generalized Markov stability of network communities
• Human mobility patterns at the smallest scales
• Unevenness of Loop Location in Complex Networks
• The relationship between structure and function in locally observed complex networks
• General theory for extended-range percolation on simple and multiplex networks
• Directed percolation criticality due to Stochastic switching between Inhibitory and Excitatory Coupling in Coupled Circle maps
• Sample-to-sample fluctuations in real-network ensembles
• Susceptible-Infected-Recovered model on Euclidean network
• Emergence of Clusters in Growing Networks with Aging
• Lattice splitting under intermittent flows
• Dynamical fluctuations of random walks in higher-order networks
• Effectiveness of Contact Tracing on Networks with Cliques
• State Concentration Exponent as a Measure of Quickness in Kauffman-type Networks
• Optimizing synchronization in multiplex networks of phase oscillators
• Phase diagram of epidemic spreading - unimodal vs. bimodal probability distributions
• Quantum phase transitions in the triangular coupled-top model
• Network-based confidence scoring system for genome-scale metabolic reconstructions
• Scalar model of flocking dynamics on complex social networks
• Generalized $k$-core pruning process on directed networks
• Inferring network topology via the propagation process
• Reentrant phase transition in a predator-prey model
• Spectral properties, localization transition and multifractal eigenvectors of the Laplacian on heterogeneous networks
• Reactive explorers to unravel network topology
• Structural phase transition in evolving networks
• Criterions for locally dense subgraphs
• $Q$-voter model with independence on signed random graphs: homogeneous approximations
• Fragility and anomalous susceptibility of weakly interacting networks
• Epidemics in networks: A master equation approach
• Robustness of networks against propagating attacks under vaccination strategies
• Impact of temporal correlations on high risk outbreaks of independent and cooperative SIR dynamics
• Emergent classical geometries on boundaries of randomly connected tensor networks
• Emergent information dynamics in many-body interconnected systems
• Polymer models with competing collapse interactions on Husimi and Bethe lattices
• Social Network Heterogeneity Promotes Depolarization of Multidimensional Correlated Opinions
• Quantum Monte Carlo study of the transverse-field quantum Ising model on infinite-dimensional structures
• Laplacian Renormalization Group: An introduction to heterogeneous coarse-graining
• Sufficient conditions of endemic threshold on metapopulation networks
• Diffusion of complex contagions is shaped by a trade-off between reach and reinforcement
• Surface and bulk criticality in midpoint percolation
• Weak percolation on multiplex networks with overlapping edges
• Homophily Within and Across Groups
• Ising model on a $restricted$ scale-free network
• Ferromagnetic transition in a simple variant of the Ising model on multiplex networks
• Generating random networks that consist of a single connected component with a given degree distribution
• Node Diversification in Complex Networks by Decentralized Coloring
• Local cluster-size statistics in the critical phase of bond percolation on the Cayley tree
• Critical exponents in coupled phase-oscillator models on small-world networks
• Amplitude death and restoration in networks of oscillators with random-walk diffusion
• Degree dependent transmission rates in epidemic processes
• Universal scaling in real dimension
• Nonequilibrium dynamics of the Ising model on heterogeneous networks with an arbitrary distribution of threshold noise
• Spectral Renormalization Group for the Gaussian model and ${\psi }^4$ theory on non-spatial networks
• Asymmetry of social interactions and its role in link predictability: the case of coauthorship networks
• Strongly clustered random graphs via triadic closure: An exactly solvable model
• Ferromagnetic Ising spin systems on the growing random tree
• Lightcone Bounds for Quantum Circuit Mapping via Uncomplexity
• Strange Attractors in Complex Networks
• Efficient network exploration by means of resetting self-avoiding random walkers
• Quantal Response Equilibria in Binary Choice Games on Graphs
• Inverse transitions in a spin-glass model on a scale-free network
• Underestimating extreme events in power-law behavior due to machine-dependent cutoffs
• Moment-Closure Approximations for Discrete Adaptive Networks
• The cause of universality in growth fluctuations
• Large deviations for the annealed Ising model on inhomogeneous random graphs: spins and degrees
• Degree product rule tempers explosive percolation in the absence of global information
• Heterogeneous information network model for equipment-standard system
• Degree-dependent network growth: From preferential attachment to explosive percolation
• Phase diagram and metastability of the Ising model on two coupled networks
• Rank-dependent deactivation in network evolution
• Observability transitions in clustered networks
• Measuring the dimension of partially embedded networks
• Exact evaluation of the causal spectrum and localization properties of electronic states on a scale-free network
• Inheritances, social classes, and wealth distribution
• Giant component in a configuration-model power-law graph with a variable number of links
• How does degree heterogeneity affect nucleation of Ising model on complex networks?
• The effect of network structure on phase transitions in queuing networks
• Percolation in Networks of Liquid Diodes
• Boolean decision problems with competing interactions on scale-free networks: Critical thermodynamics
• Quantum phase transition of the transverse-field quantum Ising model on scale-free networks
• Fundamental limitations of network reconstruction
• Approaching the thermodynamic limit in equilibrated scale-free networks
• Distinct dynamical behavior in Erdős-Rényi networks, regular random networks, ring lattices, and all-to-all neuronal networks
• Design of endurable networks in the presence of aging
• Phase transition without global ordering in a hierarchical scale-free network
• Elements of Consciousness and Cognition. Biology, Mathematic, Physics and Panpsychism: an Information Topology Perspective
• Detecting and modelling real percolation and phase transitions of information on social media
• The precursor of the critical transitions in majority vote model with the noise feedback from the vote layer
• Realizing interdependent couplings as thermal or higher-order interactions
• Response to targeted perturbations for random walks on networks
• Fluctuations of motifs and non self-averaging in complex networks. A self- vs non-self-averaging phase transition scenario
• Two network Kuramoto-Sakaguchi model under tempered stable Lévy noise
• Ageing of complex networks
• Spontaneous repulsion in the $A+B\to 0$ reaction on coupled networks
• Scaling and universality for percolation in random networks: A unified view
• Around the gap between sociophysics and sociology
• Cumulative Merging Percolation: A long-range percolation process in networks
• Condensation phase transition in nonlinear fitness networks
• Symmetry-based coarse-graining of evolved dynamical networks
• Coarse-grained Monte Carlo simulations of the phase transition of Potts model on weighted networks
• Random walk in degree space and the time-dependent Watts-Strogatz model
• Fluctuation-induced traffic congestion in heterogeneous networks
• Extracting phase coupling functions between collectively oscillating networks directly from time-series data
• Sparse Random Block Matrices
• Renormalization-group theory of the abnormal singularities at the critical-order transition in bond percolation on pointed hierarchical graphs
• Generating functions for message-passing on weighted networks: directed bond percolation and SIR epidemics
• Scaling of spin avalanches in growing networks
• Correlation networks from random walk time series
• Triadic percolation on multilayer networks
• Ensemble inequivalence and absence of quasi-stationary states in long-range random networks
• Detecting Structure of Complex Network by Quantum Bosonic Dynamics
• Finite-size scaling of percolation on scale-free networks
• Slow dynamics of the contact process on complex networks
• Collective decision-making with heterogeneous biases: Role of network topology and susceptibility
• Critical Network Cascades with Re-excitable nodes: Why tree-like approximations usually work, when they breakdown, and how to correct them
• Validity of annealed approximation in a high-dimensional system
• Potts model with q=3 and 4 states on directed Small-World network
• Scaling up real networks by geometric branching growth
• Structural Properties of Networks Grown via an Achlioptas Process
• The $q$-neighbor Ising model on multiplex networks with partial overlap of nodes
• Potts model with invisible states on a scale-free network
• Complex ordering in spin networks: Critical role of adaptation rate for dynamically evolving interactions
• An introduction to spectral distances in networks (extended version)
• Improving robustness of spatial networks via reinforced nodes
• Degree-Ordered-Percolation on uncorrelated networks
• Pseudo-Darwinian evolution of physical flows in complex networks
• Local topological moves determine global diffusion properties of hyperbolic higher-order networks
• Clusters' size-degree distribution for bond percolation
• Degree distributions under general node removal: Power-law or Poisson?
• Numerical assessment of the percolation threshold using complement networks
• Topologically protected synchronization in networks
• Effect of disorder on condensation in the lattice gas model on a random graph
• Representation of degree correlation using eigenvalue decomposition and its application to epidemic models
• The Fate of Ernst Ising and the Fate of his Model
• Critical avalanches of Susceptible-Infected-Susceptible dynamics in finite networks
• Induced Percolation on Networked Systems
• Phase transitions in atypical systems induced by a condensation transition on graphs
• Community Detection in the Hyperbolic Space
• Continuous spin models on annealed generalized random graphs
• Ising model for distribution networks
• Tsallis mapping in growing complex networks with fitness
• Viable flux distribution in metabolic networks
• Quantum magnetism on small-world networks
• Functional brain networks: great expectations, hard times, and the big leap forward
• Spin-glass behaviour on random lattices
• Complex Contagions and hybrid phase transitions in unclustered and clustered random networks
• Self-organized model of cascade spreading
• Stopping explosion by penalising transmission to hubs in scale-free spatial random graphs
• Finding Non-Zero Stable Fixed Points of the Weighted Kuramoto model is NP-hard
• Winding number excitation detects phase transition in one-dimensional XY model with variable interaction range
• Evolution of cooperation in networked heterogeneous fluctuating environments
• Absence of small-world effects at the quantum level and stability of the quantum critical point
• Large deviation and anomalous fluctuations scaling in degree assortativity on configuration networks
• Theory of Interface: Category Theory, Directed Networks and Evolution of Biological Networks
• Probing Black Hole Microstate Evolution with Networks and Random Walks
• Can Spatiality Promote Diversity?
• Critical phenomena on heterogeneous small-world networks
• Communities in networks - a continuous approach
• Degree and connectivity of the Internet's scale-free topology
• Asymmetric simple exclusion process in one-dimensional chains with long-range links
• Structural instability of large-scale functional networks
• Universality of critical dynamics on a complex network
• Une étude de physique sur les élections
• Fault Tolerance of Random Graphs with respect to Connectivity: Mean-field Approximation for Semi-dense Random Graphs
• Organization of cooperation in fractal structures
• Global decomposition of networks into multiple cores formed by local hubs
• Single Polygon Counting for $m$ Fixed Nodes in Cayley Tree: Two Extremal Cases
• Dynamical leaps due to microscopic changes in multiplex networks
• Small-world of communities: communication and correlation of the meta-network
• Numerical Integration of the KPZ and Related Equations on Networks: The Case of the Cayley Tree
• Magnetism of frustrated regular networks
• External field and critical exponents in controlling dynamics on complex networks
• Abstract Phase-space Networks Describing Reactive Dynamics
• Cooperation in Evolutionary Public Goods Game on Complex Networks with Topology Change
• Growing Networks with Super-Joiners
• Critical Percolation Phase and Thermal BKT Transition in a Scale-Free Network with Short-Range and Long-Range Random Bonds
• Scaling exponents and phase separation in a nonlinear network model inspired by the gravitational accretion
• Relevance of the minimum degree to dynamic fluctuation in strongly heterogeneous networks
• Barely supercritical percolation on Poissonian scale-free networks
• Locating influential nodes via dynamics-sensitive centrality
• Abrupt transition of the efficient vaccination strategy in a population with heterogeneous fatality rates
• Janus Percolation in Anisotropic Limited-Degree Networks
• Effects of long-range links on metastable states in a dynamic interaction network
• Mean-field approach for frequency synchronization in complex networks of two oscillator types
• The free and freer XY models
• Scaling functions and amplitude ratios for the Potts model on an uncorrelated scale-free network
• Magnetization of Antiferromagnetic Cactus Graph Model with Exact Dimer Ground State
• Long-time tails in sheared fluids
• Reaction-diffusion on random spatial networks with scale-free jumping rates via effective medium theory
• Exact results for the Ising model on a small-world network
• Prisoner's dilemma on directed networks
• Non-equilibrium random walks on multiplex networks
• Mother canonical tensor model
• Epidemics on evolving networks with varying degrees
• Network-Growth Rule Dependence of Fractal Dimension of Percolation Cluster on Square Lattice
• Inferring, comparing and exploring ecological networks from time-series data through R packages constructnet, disgraph and dynet
• Empirical symptoms of catastrophic bifurcation transitions on financial markets: A phenomenological approach
• Emergence of Multivariate Extremes in Multilayer Inhomogeneous Random Graphs
• Shaping Magnetic Order by Local Frustration for Itinerant Fermions on a Graph
• Tropical approximation to finish time of activity networks
• Heat-Bath and Metropolis Dynamics in Ising-like Models on Directed Regular Random Graphs
• Flocking dynamics mediated by weighted social networks
• Ising model with varying spin strength on a scale-free network: scaling functions and critical amplitude ratios
• Hierarchical Coarse-grained Approach to the Duration-dependent Spreading Dynamics in Complex Networks
• Reconstructing propagation networks with temporal similarity metrics
• Exact statistical mechanics of the Ising model on networks
• Latent geometry emerging from network-driven processes
• How universal is the mean-field universality class for percolation in complex networks?
• Unveiling hidden features of the Kitaev model through a complex-network analysis
• Network topology and collapse of collective stable chaos
• Ensemble order parameter equations in star network
• Mandala Networks: ultra-robust, ultra-small-world and highly sparse graphs
• Multiscale genesis of a tiny giant for percolation on scale-free random graphs
• Renormalization Group Solution of the Chutes&Ladder Model
• Filtering Statistics on Networks
• Rewiring driven evolution of quenched frustrated signed network
• Improving the accuracy of the k-shell method by removing redundant links-from a perspective of spreading dynamics
• Stochastic hybrid systems in equilibrium: moment closure, finite-time blowup, and exact solutions
• On the discontinuity of the specific heat of the Ising model on a scale-free network
• The prevalence of small world networks explained by modeling the competing dynamics of local signaling events in geometric networks
• Frustration and collectivity in spatial networks
• Average synaptic activity and neural networks topology: a global inverse problem
• The spreading ability of nodes towards localized targets in complex networks
• Recovery of Interdependent Networks
• On the shape of Gaussian scale-free polymer networks
• Entropy of random graph ensembles constrained with generalised degrees
• Manageable to unmanageable transition in a fractal model of project networks
• Modeling for the Dynamics of Human Innovative Behaviors
• Heterogeneous Network Topology Induces the Widom Line
• Cut-off and Escape Behaviors for Birth and Death Chains on Trees
• Irregular Metamaterial Networks
• Starvation suppression in dense scale-free metabolic networks: Dynamical mean-field analysis of catalytic reaction networks
• Partitioning networks into clusters of synchronized nodes via the message-passing algorithm: an unbiased scalable approach
• Stability via symmetry breaking in interacting driven systems
• Hierarchy of percolation patterns in a kinetic replication model
• Ising model on a Galton-Watson tree with a sparse random external field
• Unveiling the Relationship Between Structure and Dynamics in Complex Networks
• Two Categories of Indoor Interactive Dynamics of a Large-scale Human Population in a WiFi covered university campus
• Mean-field theory of vector spin models on networks with arbitrary degree distributions
• Metanetworks of artificially evolved regulatory networks
• Transition to congestion in communication/computation networks: from ideal to realistic resource allocation via Montecarlo simulations
• The effect of delayed awareness and fatigue on the efficacy of self-isolation in epidemic control
• Promotion of Cooperation in Coevolutionary Public Goods Game on Complex Networks with and without Topology Change
• Correlation-Weighted Communicability Curvature as a Structural Driver of Dengue Spread: A Bayesian Spatial Analysis of Recife (2015-2024)
• Shortest-path percolation on scale-free networks
• Spin-glass transition in the Ising model on multiplex networks
• Activity Dynamics in Collaboration Networks
• Modelling the Self-similarity in Complex Networks Based on Coulomb's Law
• Approximating nonbacktracking centrality and localization phenomena in large networks