Erdős number

Erdős number
Name	Erdős number
Field	Mathematics, Bibliometrics

Erdős number is a collaborative distance metric originating in mathematical research that measures coauthorship proximity to the prolific Hungarian mathematician Paul Erdős. It functions as an informal indicator within scholarly networks, reflecting connections among mathematicians, statisticians, physicists, and computer scientists across institutions such as Princeton University, University of Cambridge, University of Oxford, Stanford University, and University of Chicago.

Definition and concept

The concept assigns an integer distance based on joint authorship with Paul Erdős, where direct coauthors receive distance one and subsequent coauthors increase the distance incrementally; notable participants include Paul Erdős himself, collaborators at University of Manchester, faculty at Massachusetts Institute of Technology, researchers at Bell Labs, and members of Institute for Advanced Study. The measure is analogous to network metrics used in analyses of coauthorship graphs involving nodes representing individuals such as Alfréd Rényi, Ronald Graham, Vaughan Jones, László Lovász, and Paul Turán and edges representing publications in journals like Annals of Mathematics, Journal of Combinatorial Theory, Proceedings of the American Mathematical Society, Inventiones Mathematicae, and Communications in Mathematical Physics. As a small-world example, it parallels metrics applied to social networks examined in studies by Stanley Milgram, Duncan Watts, Steven Strogatz, Mark Granovetter, and research groups at IBM Research.

History and origin

The origin traces to informal lists circulated in the mathematics community during the late 20th century among collaborators of Paul Erdős, including figures such as Alfréd Rényi, Erdős Ginzburg Ziv, Paul Turán, Miklós Schweitzer and later curators at institutions like California Institute of Technology, University of California, Berkeley, and Hungarian Academy of Sciences. The metric gained wider attention through publications and popular accounts by journalists and authors such as Philip J. Davis, Reuben Hersh, Arthur Benjamin, Siobhán Roberts, and popularizers referencing conferences at International Congress of Mathematicians and symposia at Mathematical Sciences Research Institute. Lists and databases maintained by librarians and archivists at Royal Society, American Mathematical Society, Mathematical Reviews, and Zentralblatt MATH helped formalize computation and dissemination.

Calculation and methodology

Computation treats coauthorship networks as graphs where vertices correspond to researchers like John Nash, Alonzo Church, András Sárközy, Endre Szemerédi, and Gillian K. Pierce and edges correspond to joint publications in venues such as Journal of Number Theory, Combinatorica, European Journal of Combinatorics, and SIAM Journal on Discrete Mathematics; shortest-path algorithms such as Dijkstra's algorithm, breadth-first search, and variants used in software from Mathematica, MATLAB, SageMath, and libraries developed at Google and Microsoft Research compute distances. Databases from arXiv, Web of Science, Scopus, MathSciNet, and institutional repositories provide bibliographic records allowing disambiguation of names like Paul Erdős collaborators Ronald L. Graham, Alain Connes, Endre Ernő Molnár and resolution of homonyms via authority files from Library of Congress and ORCID.

Notable Erdős numbers and examples

Prominent mathematicians, computer scientists, and physicists with short distances include Ronald L. Graham, László Lovász, Alfréd Rényi, John H. Conway, Richard M. Karp, and Richard P. Stanley, while interdisciplinary figures such as Noam Chomsky and Stephen Hawking have been discussed in relation to collaborative paths through coauthorship networks and edited volumes. Applied examples appear in collaborations involving researchers at Bell Laboratories, Los Alamos National Laboratory, CERN, NASA, and Microsoft Research, where coauthorship with combinatorialists, graph theorists, and probabilists like Paul Erdős’s circle produced short collaborative distances; editorial connections via journals such as Science, Nature, Proceedings of the Royal Society A, and Physical Review Letters also generate notable linkages. Collective works from conferences at International Congress of Mathematicians, workshops at Banff Centre, and seminars at Institut des Hautes Études Scientifiques contributed to networks yielding many low-distance examples.

Variations and related metrics

Related measures include collaborative distances to other central figures such as the Bacon number associated with Kevin Bacon, hybrid metrics combining entertainment and scientific collaboration like the Bacon–Erdős concept discussed in media outlets and studies by scholars at University of Southern California, and centrality measures developed in network science by Linton C. Freeman, Albert-László Barabási, Jon Kleinberg, M. E. J. Newman, and Derek J. de Solla Price. Bibliometric indices such as h-index used by Eugene Garfield, citation counts catalogued by Clarivate, and coauthorship clustering techniques applied in work from Stanford Network Analysis Project provide complementary perspectives. Tools and platforms like Google Scholar, ResearchGate, ORCID, and Semantic Scholar implement or enable computation of analogous collaborative metrics.

Cultural impact and usage

The metric entered popular culture via articles in The New York Times, features in Nature, profiles in Scientific American, and mentions on programs by BBC Radio 4 and NPR, where journalists connected mathematical collaboration networks to popular figures and institutions such as Harvard University, Yale University, Princeton University, and Columbia University. It influenced hiring, outreach, and networking practices at conferences like SIGGRAPH, NeurIPS, IEEE Symposium on Foundations of Computer Science, and Society for Industrial and Applied Mathematics meetings, and has been referenced in biographies, documentaries, and web resources maintained by academic societies such as American Mathematical Society and European Mathematical Society.

Criticism and limitations

Critics from bibliometrics and sociology, including researchers at Max Planck Society, Ludwig Maximilian University of Munich, CNRS, and University of Amsterdam, note that the metric overemphasizes coauthorship frequency, overlooks contribution quality highlighted by awards like the Fields Medal and Abel Prize, and can be distorted by multidisciplinary coauthorship practices at laboratories like Bell Labs and collaborative projects at CERN; concerns also arise regarding name disambiguation challenges faced by ORCID and authority control at Library of Congress and VIAF. Methodological critiques reference network model limitations discussed by Mark Newman, Barabási, and Derek J. de Solla Price, arguing that simple path-length metrics inadequately capture scientific influence compared to citation-based alternatives used by Clarivate and Scopus.

Category:Mathematics