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| Random Graphs of Erdős–Rényi | |
|---|---|
| Name | Random Graphs of Erdős–Rényi |
| Introduced | 1959 |
| Creators | Paul Erdős, Alfréd Rényi |
| Field | Paul Erdős; Alfréd Rényi; Probabilistic method (combinatorics); Graph theory |
| Key models | G(n,p); G(n,M) |
| Notable results | Phase transition; Giant component; Threshold functions |
Random Graphs of Erdős–Rényi are foundational models in Paul Erdős–Alfréd Rényi probabilistic graph theory that formalize random networks through the ensembles G(n,p) and G(n,M). Originating in the collaboration between Paul Erdős and Alfréd Rényi in the mid‑20th century, these models underpin modern work by researchers such as Béla Bollobás, Szemerédi, Joel Spencer, and Jeff Kahn. They connect to results in André Weil‑era probability, influence studies by Kolmogorov, and provide bridges to applications cited by Erdős collaborators like Ronald Graham, Endre Szemerédi, László Babai, and Ádám Marcus.
The classical constructions G(n,p) and G(n,M) were formalized by Paul Erdős and Alfréd Rényi and later elaborated by Béla Bollobás, Joel Spencer, Noga Alon, Van H. Vu, and Boris Pittel. In G(n,p) each unordered pair of n labeled vertices is included independently with probability p; in G(n,M) a graph is chosen uniformly from all labeled graphs with M edges. These models link to combinatorial foundations developed by Pál Erdős‑era collaborators and are compared with deterministic families studied by Kőnig, Pólya, Egon Balas, and Donald Knuth. Variants include inhomogeneous models influenced by work of Erdős coauthors such as Miklós Simonovits and later generalizations like the Chung–Lu model developed by Fan Chung and Linyuan Lu as well as models studied by Remco van der Hofstad and Christina Goldschmidt.
Fundamental probabilistic techniques applied to these models were pioneered by Paul Erdős, Alfréd Rényi, Béla Bollobás, and Joel Spencer, and use tools from Andrey Kolmogorov‑style probability and concentration inequalities due to Sergey Bernstein, Sergey Nagaev, Vladimir Vovk, Azuma Hashimoto, and the Chernoff bounds associated historically with Herman Chernoff. Key results include expected degree distributions, concentration of edge counts addressed by Alfréd Rényi and Béla Bollobás, and coupling arguments advanced by Paul Erdős and Béla Bollobás. The probabilistic method developed by Paul Erdős and Joel Spencer yields combinatorial existence proofs subsequently extended by Noga Alon and Joel Spencer in the book by Noga Alon and Joel H. Spencer.
The emergence of a giant component at p ~ 1/n is a hallmark discovered in papers by Paul Erdős and Alfréd Rényi and refined by Béla Bollobás, Svante Janson, Oliver Riordan, Joel Spencer, and Remco van der Hofstad. Near criticality the component size distribution follows branching process approximations attributed to Thingvellir‑era work analogous to branching process theory by Andrey Kolmogorov and Aleksandr Lyapunov; modern rigorous treatments are due to Svante Janson and Oliver Riordan. Supercritical and subcritical regimes were analyzed by Béla Bollobás, Eugène Wigner‑style spectral intuitions by Friedrich Hund‑era researchers and by later contributors such as Fan Chung and Van H. Vu for spectral gap behavior.
Degree distribution in G(n,p) follows binomial laws studied by Alfréd Rényi, Paul Erdős, and Béla Bollobás and asymptotically approaches Poisson in sparse regimes, a fact used by Svante Janson and Oliver Riordan in component analysis. Connectivity thresholds and k‑connectivity are classical results due to Paul Erdős, Alfréd Rényi, and refinements by Béla Bollobás, Klaus Wagner, László Lovász, and Paul Seymour. Diameter estimates have been developed by Béla Bollobás, Fan Chung, Linyuan Lu, and Van H. Vu, with regimes showing logarithmic diameters linked to random regular graphs studied by W. T. Tutte and Brett B. Sudakov. Clustering coefficients in Erdős–Rényi graphs are small compared to models by Duncan Watts and Steven Strogatz or Albert‑László Barabási, an observation that motivated network scientists such as Mark Newman and M. E. J. Newman to propose alternatives.
The random graph process, where edges are added sequentially, was introduced by Paul Erdős and Alfréd Rényi and studied through the lens of hitting times by Béla Bollobás, Joel Spencer, Noga Alon, and Michael Krivelevich. Hitting time results connect to monotone graph properties and threshold phenomena explored by Béla Bollobás and Jeff Kahn; related differential equation methods were developed by Nicola Wormald and applied by Svante Janson and Tom Bohman. The Achlioptas processes studied by David Achlioptas and analyzed by Oliver Riordan and Brendan McKay generalize edge selection rules; their study relates to percolation theory by H. Eugene Stanley and stochastic processes literature from William Feller and Frank Spitzer.
Threshold functions for monotone properties were systematically studied by Paul Erdős, Alfréd Rényi, Béla Bollobás, Joel Spencer, and later by Ehud Friedgut, Jeff Kahn, and Gil Kalai. The sharp threshold phenomenon, proven in general by Ehud Friedgut and Jeff Kahn, connects to influences and hypercontractivity tools developed by János Komlós‑era analysts and later used by Ryan O'Donnell and Elchanan Mossel. Specific thresholds such as Hamiltonicity, connectivity, and appearance of given subgraphs were established by Béla Bollobás, Klaus Wagner, Paul Seymour, and Endre Szemerédi.
Erdős–Rényi models inform work in network science by Mark Newman, Duncan Watts, Steven Strogatz, and Albert‑László Barabási and influence algorithms studied by Leslie Valiant, Noga Alon, Vladimir Levenshtein, and Éva Tardos. Extensions include inhomogeneous random graphs by Béphane Bordenave, configuration models by M. Molloy and B. Reed, and preferential attachment models by Albert‑László Barabási, Adrián R. West. Applications span combinatorial optimization studied by Michel Goemans and David Johnson, epidemiological models linked with Roy M. Anderson and Robert May, and statistical physics connections explored by Giorgio Parisi and Remco van der Hofstad.