Principles of statistical mechanics of random networks

Abstract

We develop a statistical mechanics approach for random networks with uncorrelated vertices. We construct equilibrium statistical ensembles of such networks and obtain their partition functions and main characteristics. We find simple dynamical construction procedures that produce equilibrium uncorrelated random graphs with an arbitrary degree distribution. In particular, we show that in equilibrium uncorrelated networks, fat-tailed degree distributions may exist only starting from some critical average number of connections of a vertex, in a phase with a condensate of edges.

14 pages, an extended version

References in corpus (4)

• Evolution of networks
• Statistical ensemble of scale-free random graphs
• Correlated random networks
• Wealth Condensation in Pareto Macro-Economies

Cited by in corpus (53)

• The structure and function of complex networks
• Critical phenomena in complex networks
• Hyperbolic Geometry of Complex Networks
• The statistical mechanics of networks
• Nonequilibrium Statistical Mechanics of the Zero-Range Process and Related Models
• Spectra of complex networks
• The entropy of network ensembles
• The entropy of randomized network ensembles
• The origin of degree correlations in the Internet and other networks
• Canonical Analysis of Condensation in Factorised Steady State
• Curvature and temperature of complex networks
• Current fluctuations in the zero-range process with open boundaries
• Zero-range process with open boundaries
• Recent advances of percolation theory in complex networks
• Evolution of scale-free random graphs: Potts model formulation
• Exact Solution for the Time Evolution of Network Rewiring Models
• Homogeneous complex networks
• Scale-free networks with exponent one
• Condensation transition in the late-time position of a Run-and-Tumble particle
• A statistical mechanics approach for scale-free networks and finite-scale networks
• The Grand Canonical ensemble of weighted networks
• Critical phase in non-conserving zero-range processes and equilibrium networks
• Fluctuation-dissipation relations for complex networks
• Organization of complex networks without multiple connections
• Competition for Popularity in Bipartite Networks
• Condensation transitions in a model for a directed network with weighted links
• Coevolving complex networks in the model of social interactions
• Finite size effects in Barabasi-Albert growing networks
• The two-star model: exact solution in the sparse regime and condensation transition
• Fluctuations analysis in complex networks modeled by hidden variable models. Necessity of a large cut-off in hidden-variable models
• Statistical mechanics of scale-free networks at a critical point: Complexity without irreversibility?
• Phase transitions on Markovian bipartite graphs - an application of the zero-range process
• Power laws in zero-range processes on random networks
• Network community structure and resilience to localized damage: application to brain microcirculation
• Evolving network - simulation study. From regular lattice to scale free network
• Coarsening dynamics in a two-species zero-range process
• Condensation of degrees emerging through a first-order phase transition in classical random graphs
• Hamiltonian Dynamics of Preferential Attachment
• Ising spin glass models versus Ising models: an effective mapping at high temperature II. Applications to graphs and networks
• Analysis of network by generalized mutual entropies
• Mechanism for linear preferential attachment in growing networks
• Preferential attachment without vertex growth: emergence of the giant component
• Structural phase transition in evolving networks
• A Statistical Mechanical Load Balancer for the Web
• Ising spin glass models versus Ising models: an effective mapping at high temperature III. Rigorous formulation and detailed proof for general graphs
• Approaching the thermodynamic limit in equilibrated scale-free networks
• Organic Design of Massively Distributed Systems: A Complex Networks Perspective
• Fluctuations of motifs and non self-averaging in complex networks. A self- vs non-self-averaging phase transition scenario
• Condensation phase transition in nonlinear fitness networks
• Finite-size scaling of percolation on scale-free networks
• Reverse engineering of linking preferences from network restructuring
• A program generating homogeneous random graphs with given weights
• Factorised steady states for multi-species mass transfer models