爱多士数

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保罗·埃尔德什Paul Erdős摄于1992年

The Erdős number (模板:IPA-hu) describes the "collaborative distance" between mathematician 模板:Nobr and another person, as measured by authorship of mathematical papers. The same principle has been applied in other fields where a particular individual has collaborated with a large and broad number of peers.

The Erdős number () describes the "collaborative distance" between mathematician and another person, as measured by authorship of mathematical papers. The same principle has been applied in other fields where a particular individual has collaborated with a large and broad number of peers.

埃尔德什数Erdős number（匈牙利语：[ˈɛrdøːʃ]）描述了数学家保罗·埃尔德什Paul Erdős与另一个人之间的“协作距离”，这是根据数学论文的著作权来衡量的。该原则应用于很多其他领域，意指特定某个人与众多同行之间的合作。

Overview 概况

Paul Erdős (1913–1996) was an influential Hungarian mathematician who in the latter part of his life spent a great deal of time writing papers with a large number of colleagues, working on solutions to outstanding mathematical problems. He published more papers during his lifetime (at least 1,525) than any other mathematician in history. (Leonhard Euler published more total pages of mathematics but fewer separate papers: about 800.) Erdős spent a large portion of his later life living out of a suitcase, visiting his over 500 collaborators around the world.

Paul Erdős (1913–1996) was an influential Hungarian mathematician who in the latter part of his life spent a great deal of time writing papers with a large number of colleagues, working on solutions to outstanding mathematical problems. He published more papers during his lifetime (at least 1,525) than any other mathematician in history. Erdős spent a large portion of his later life living out of a suitcase, visiting his over 500 collaborators around the world.

保罗·埃尔德什Paul Erdős(1913年至1996年）是一位具有很大影响力的匈牙利数学家，他花费了一生中大量的时间与很多同事撰写论文，致力于解决困扰已久的疑难数学问题。他一生中发表的论文（至少1,525件）比历史上其他任何数学家都多。莱昂哈德·欧拉Leonhard Euler发表过更多的数学论文，但单独的论文却较少：大约800篇。而埃尔德什的大部分时间都生活在手提箱里，他拜访过全球500多个合作者。

The idea of the Erdős number was originally created by the mathematician's friends as a tribute to his enormous output. Later it gained prominence as a tool to study how mathematicians cooperate to find answers to unsolved problems. Several projects are devoted to studying connectivity among researchers, using the Erdős number as a proxy. For example, Erdős collaboration graphs can tell us how authors cluster, how the number of co-authors per paper evolves over time, or how new theories propagate.

埃尔德什数的概念最初是由数学家的朋友们提出来的，以赞扬保罗·埃尔德什的巨大成就。后来，它演变为研究数学家如何通过合作来解决问题的的工具而受到重视。有几个项目专门通过使用埃尔德什数作为代理来研究人员之间的连通性。例如，埃尔德什合作图可以告诉我们作者是如何聚集在一起的，每篇论文的共同作者数量随时间变化或新理论的产生又是如何传播的。

Several studies have shown that leading mathematicians tend to have particularly low Erdős numbers. The median Erdős number of Fields Medalists is 3. Only 7,097 (about 5% of mathematicians with a collaboration path) have an Erdős number of 2 or lower. As time passes, the smallest Erdős number that can still be achieved will necessarily increase, as mathematicians with low Erdős numbers die and become unavailable for collaboration. Still, historical figures can have low Erdős numbers. For example, renowned Indian mathematician Srinivasa Ramanujan has an Erdős number of only 3 (through G. H. Hardy, Erdős number 2), even though Paul Erdős was only 7 years old when Ramanujan died.

多项研究表明，领先的数学家往往具有极低的埃尔德什数。费尔兹奖Fields Medalists的埃尔德什中位数是3。只有7,097名（拥有合作经历的数学家中约5％）的埃尔德什数为2或更低。随着时间的流逝，低埃尔德什数的数学家因死亡而无法进行协作，最小埃尔德什数（仍然存在）必然会增加。即使历史人物仍可能一直具有较低的埃尔德什数。例如，印度著名数学家Srinivasa Ramanujan的埃尔德什数仅为3（通过与G. H. Hardy合作，其埃尔德什数为2），尽管Ramanujan去世时保罗·埃尔德什只有7岁。

Definition and application in mathematics 数学的定义与应用

如果爱丽丝在一张纸上与保罗·埃尔德什合作，在另一张纸上与鲍勃合作，但是鲍勃从未与埃尔德什本人合作，那么爱丽丝的埃尔德什数为1，而鲍勃的埃尔德什数为2，因为他离埃尔德什有两步。

To be assigned an Erdős number, someone must be a coauthor of a research paper with another person who has a finite Erdős number. Paul Erdős has an Erdős number of zero. Anybody else's Erdős number is $k + 1$ where $k$ is the lowest Erdős number of any coauthor. The American Mathematical Society provides a free online tool to determine the Erdős number of every mathematical author listed in the Mathematical Reviews catalogue.

To be assigned an Erdős number, someone must be a coauthor of a research paper with another person who has a finite Erdős number. Paul Erdős has an Erdős number of zero. Anybody else's Erdős number is where is the lowest Erdős number of any coauthor. The American Mathematical Society provides a free online tool to determine the Erdős number of every mathematical author listed in the Mathematical Reviews catalogue.

要分配一个埃尔德什数，某人必须与另一个具有有限埃尔德什数的人共同撰写研究论文。保罗·埃尔德什的埃尔德什数为零。其他人的埃尔德什数为k+1，其中k是任何合著者中最低的埃尔德什数。美国数学学会提供免费的在线工具，可确定《数学评论》目录中列出的每个数学作者的埃尔德什数。

Erdős wrote around 1,500 mathematical articles in his lifetime, mostly co-written. He had 511 direct collaborators; these are the people with Erdős number 1. The people who have collaborated with them (but not with Erdős himself) have an Erdős number of 2 (11,009 people as of 2015), those who have collaborated with people who have an Erdős number of 2 (but not with Erdős or anyone with an Erdős number of 1) have an Erdős number of 3, and so forth. A person with no such coauthorship chain connecting to Erdős has an Erdős number of infinity (or an undefined one). Since the death of Paul Erdős, the lowest Erdős number that a new researcher can obtain is 2.

Erdős wrote around 1,500 mathematical articles in his lifetime, mostly co-written. He had 511 direct collaborators;), those who have collaborated with people who have an Erdős number of 2 (but not with Erdős or anyone with an Erdős number of 1) have an Erdős number of 3, and so forth. A person with no such coauthorship chain connecting to Erdős has an Erdős number of infinity (or an undefined one). Since the death of Paul Erdős, the lowest Erdős number that a new researcher can obtain is 2.

埃尔德什一生撰写了约1500篇数学文章，其中大部分是合作的。他有511个直接合作者；这些是埃尔德什数为1的人。与这些人合作（但未与埃尔德什本人合作）的人所拥有的埃尔德什数为2（截至2020年8月7日为12,600人），而与埃尔德什数为2的人合作的人（但与埃尔德什或埃尔德什数为1的任何人无合作关系），其埃尔德什数为3，依此类推。没有此类共同作者链接能指向埃尔德什的人，其埃尔德什数为无穷大（或未定义）。自保罗·埃尔德什逝世以来，新研究员可获得的最低埃尔德什数为2。

There is room for ambiguity over what constitutes a link between two authors. The American Mathematical Society collaboration distance calculator uses data from Mathematical Reviews, which includes most mathematics journals but covers other subjects only in a limited way, and which also includes some non-research publications^{[citation needed]}. The Erdős Number Project web site says:

There is room for ambiguity over what constitutes a link between two authors. The American Mathematical Society collaboration distance calculator uses data from Mathematical Reviews, which includes most mathematics journals but covers other subjects only in a limited way, and which also includes some non-research publications. The Erdős Number Project web site says:

关于具体由什么构成两位作者之间的联系，众说纷纭。美国数学学会的“协作距离计算器”使用的是来自《数学评论》的数据，包括大多数数学期刊，但仅以有限的方式涵盖了其他主题，同时还包括一些非研究出版物。埃尔德什数项目官方网站Erdős Number Project表示：

... Our criterion for inclusion of an edge between vertices u and v is some research collaboration between them resulting in a published work. Any number of additional co-authors is permitted,...

...我们在顶点u和v之间共有的包含边标准是，它们之间的某些研究合作导致了发表的作品。允许任何数量的其他共同作者，...

but they do not include non-research publications such as elementary textbooks, joint editorships, obituaries, and the like. The "Erdős number of the second kind" restricts assignment of Erdős numbers to papers with only two collaborators.

但它们不包括非研究性出版物，例如教科书，联合编辑，讣告等。“第二种埃尔德什数”将其分配给只有两个合作者的论文。

The Erdős number was most likely first defined in print by Casper Goffman, an analyst whose own Erdős number is 2. Goffman published his observations about Erdős' prolific collaboration in a 1969 article entitled "And what is your Erdős number?" See also some comments in an obituary by Michael Golomb.

埃尔德什数很可能最早由卡斯珀·高夫曼Casper Goffman定义，他自己的埃尔德什数为2。高夫曼在1969年发表的一篇文章中表示了他对埃尔德什多产合作的看法，“您的埃尔德什数是多少？”另请参阅迈克尔·哥伦布Michael Golomb在讣告中的一些评论。

The median Erdős number among Fields medalists is as low as 3. Fields medalists with Erdős number 2 include Atle Selberg, Kunihiko Kodaira, Klaus Roth, Alan Baker, Enrico Bombieri, David Mumford, Charles Fefferman, William Thurston, Shing-Tung Yau, Jean Bourgain, Richard Borcherds, Manjul Bhargava, Jean-Pierre Serre and Terence Tao. There are no Fields medalists with Erdős number 1; however, Endre Szemerédi is an Abel Prize Laureate with Erdős number 1.

The median Erdős number among Fields medalists is as low as 3.Fields medalists with Erdős number 2 include Atle Selberg, Kunihiko Kodaira, Klaus Roth, Alan Baker, Enrico Bombieri, David Mumford, Charles Fefferman, William Thurston, Shing-Tung Yau, Jean Bourgain, Richard Borcherds, Manjul Bhargava, Jean-Pierre Serre and Terence Tao. There are no Fields medalists with Erdős number 1; however, Endre Szemerédi is an Abel Prize Laureate with Erdős number 1.

费尔兹Fields奖获得者的埃尔德什中位数低至3。埃尔德什排名第二的奖牌获得者包括Atle Selberg，Kunihiko Kodaira，Klaus Roth，Alan Baker，Enrico Bombieri，David Mumford，Charles Fefferman，William Thurston，Shing-Tung Tung，Jean Bourgain，Richard Borcherds，Manjul Bhargava，Jean-Pierre Serre和陶哲轩。Fields的获得者中没有人的Erdős为1。但是，恩德雷·塞梅雷迪（Endre Szemerédi）是阿贝尔奖获得者，其埃尔德什数为1。

Most frequent Erdős collaborators 最频繁的埃尔德什合作者

While Erdős collaborated with hundreds of co-authors, there were some individuals with whom he co-authored dozens of papers. This is a list of the ten persons who most frequently co-authored with Erdős and their number of papers co-authored with Erdős (i.e. their number of collaborations).

虽然埃尔德什与数百位合著者合作，但其中一些人与他合作过数十篇论文。以下是最经常与埃尔德什合作的十人列表，以及与埃尔德什合作的论文数量（即合作数量）。

Co-author	Number of collaborations
András Sárközy	62
András Hajnal	56
Ralph Faudree	50
Richard Schelp	42
Cecil C. Rousseau	35
Vera T. Sós	35
Alfréd Rényi	32
Pál Turán	30
Endre Szemerédi	29
Ronald Graham	28

Related fields 相关领域

模板:As of, all Fields Medalists have a finite Erdős number, with values that range between 2 and 6, and a median of 3. In contrast, the median Erdős number across all mathematicians (with a finite Erdős number) is 5, with an extreme value of 13. The table below summarizes the Erdős number statistics for Nobel prize laureates in Physics, Chemistry, Medicine and Economics. The first column counts the number of laureates. The second column counts the number of winners with a finite Erdős number. The third column is the percentage of winners with a finite Erdős number. The remaining columns report the minimum, maximum, average and median Erdős numbers among those laureates.

, all Fields Medalists have a finite Erdős number, with values that range between 2 and 6, and a median of 3. In contrast, the median Erdős number across all mathematicians (with a finite Erdős number) is 5, with an extreme value of 13. The table below summarizes the Erdős number statistics for Nobel prize laureates in Physics, Chemistry, Medicine and Economics. The first column counts the number of laureates. The second column counts the number of winners with a finite Erdős number. The third column is the percentage of winners with a finite Erdős number. The remaining columns report the minimum, maximum, average and median Erdős numbers among those laureates.

截至2016年，所有费尔兹奖章获得者都有一个有限的埃尔德什数，其值在2到6之间，中位数为3。相反，所有数学家的埃尔德什数的中位数（有限的埃尔德什数）为5，极限值为13。下表总结了物理，化学，医学和经济学方面的诺贝尔奖获得者的埃尔德什数统计。第一列计算获奖者人数。第二列计算的是具有有限埃尔德什数的获胜者数量。第三列是具有有限埃尔德什数的获胜者的百分比。其余各列表示了这些获奖者中埃尔德什数的最小，最大，平均和中位数。

Statistics on Mathematical Collaboration, 1903-2016
	#Laureates	#Erdős	%Erdős	Min	Max	Average	Median
Fields Medal	56	56	100.0%	2	6	3.36	3
Nobel Economics	76	47	61.84%	2	8	4.11	4
Nobel Chemistry	172	42	24.42%	3	10	5.48	5
Nobel Medicine	210	58	27.62%	3	12	5.50	5
Nobel Physics	200	159	79.50%	2	12	5.63	5

Physics 物理领域

Among the Nobel Prize laureates in Physics, Albert Einstein and Sheldon Lee Glashow have an Erdős number of 2. Nobel Laureates with an Erdős number of 3 include Enrico Fermi, Otto Stern, Wolfgang Pauli, Max Born, Willis E. Lamb, Eugene Wigner, Richard P. Feynman, Hans A. Bethe, Murray Gell-Mann, Abdus Salam, Steven Weinberg, Norman F. Ramsey, Frank Wilczek, and David Wineland. Fields Medal-winning physicist Ed Witten has an Erdős number of 3.

在诺贝尔物理学奖获得者中，爱因斯坦Albert Einstein和谢尔登·李·格拉肖Sheldon Lee Glashow的埃尔德什数为2。诺贝尔奖获得者中埃尔德什数为3的有: Enrico Fermi，Otto Stern，Wolfgang Pauli，Max Born，Willis E.Lamb，Eugene Wigner，Richard P.Feynman，Hans A.Bethe，Murray Gell-Mann，Abdus Salam，Steven Weinberg，Norman F.Ramsey，Frank Wilczek, and David Wineland。获得菲尔兹奖的物理学家Ed Witten的埃尔德什数为3。

Biology 生物学领域

Computational biologist Lior Pachter has an Erdős number of 2. Evolutionary biologist Richard Lenski has an Erdős number of 3, having co-authored a publication with Lior Pachter and with mathematician Bernd Sturmfels, each of whom has an Erdős number of 2.

计算生物学家Lior Pachter的埃尔德什数为2。进化生物学家Richard Lenski的埃尔德什数为3，与Lior Pachter和数学家Bernd Sturmfels共同撰写了出版物的每位作者埃尔德什数为2。

Finance and economics 财经领域

There are at least two winners of the Nobel Prize in Economics with an Erdős number of 2: Harry M. Markowitz (1990) and Leonid Kantorovich (1975). Other financial mathematicians with Erdős number of 2 include David Donoho, Marc Yor, Henry McKean, Daniel Stroock, and Joseph Keller.

至少有两名诺贝尔经济学奖获得者的埃尔德什数为2：哈里·马可维兹Harry M. Markowitz，（1990）和列昂尼德·坎托罗维奇Leonid Kantorovich（1975）。埃尔德什数为2的其他金融数学家包括David Donoho，Marc Yor，Henry McKean，Daniel Stroock和Joseph Keller。

Nobel Prize laureates in Economics with an Erdős number of 3 include Kenneth J. Arrow (1972), Milton Friedman (1976), Herbert A. Simon (1978), Gerard Debreu (1983), John Forbes Nash, Jr. (1994), James Mirrlees (1996), Daniel McFadden (1996), Daniel Kahneman (2002), Robert J. Aumann (2005), Leonid Hurwicz (2007), Roger Myerson (2007), Alvin E. Roth (2012), and Lloyd S. Shapley (2012) and Jean Tirole (2014).

诺贝尔经济学奖获得者的埃尔德什数为3，其中包括Kenneth J. Arrow（1972），Milton Friedman（1976），Herbert A. Simon（1978），Gerard Debreu（1983），John Forbes Nash，Jr.（1994），James Mirrlees（1996），Daniel McFadden（2000），Daniel Kahneman（2002），Robert J.Aumann（2005），Leonid Hurwicz（2007），Roger Myerson（2007），Alvin E.Roth（2012）和Lloyd S. Shapley（2012）和Jean Tirole（2014）。

Some investment firms have been founded by mathematicians with low Erdős numbers, among them James B. Ax of Axcom Technologies, and James H. Simons of Renaissance Technologies, both with an Erdős number of 3.

一些埃尔德什数低的数学家创立了投资公司，其中包括Axcom Technologies的James B. Ax和Renaissance Technologies的James H. Simons，两者的埃尔德什数均为3。

Philosophy 哲学领域

Since the more formal versions of philosophy share reasoning with the basics of mathematics, these fields overlap considerably, and Erdős numbers are available for many philosophers. Philosopher John P. Burgess has an Erdős number of 2. Jon Barwise and Joel David Hamkins, both with Erdős number 2, have also contributed extensively to philosophy, but are primarily described as mathematicians.

由于哲学的本质与数学基础缘由互通，因此它们有很多重叠的地方，许多哲学家都可以使用埃尔德什数。哲学家John P. Burgess的埃尔德什数为2。Barwise和Joel David Hamkins埃尔德什数都为2，他们为哲学做出了大量贡献，但通常被称为数学家。

Law 法律领域

Judge Richard Posner, having coauthored with Alvin E. Roth, has an Erdős number of at most 4. Roberto Mangabeira Unger, a politician, philosopher and legal theorist who teaches at Harvard Law School, has an Erdős number of at most 4, having coauthored with Lee Smolin.

与Alvin E. Roth合作的法官Richard Posner的埃尔德什数最多为4。政治家，哲学家和法律理论家Roberto Mangabeira Unger与Lee Smolin曾经合作过，其埃尔德什数最多为4。

Politics 政治领域

Angela Merkel, Chancellor of Germany from 2005 to the present, has an Erdős number of at most 5.

从2005年至今的德国总理安格拉·默克尔Angela Merkel的埃尔德什数最多为5。

Engineering 工程领域

Some fields of engineering, in particular communication theory and cryptography, make direct use of the discrete mathematics championed by Erdős. It is therefore not surprising that practitioners in these fields have low Erdős numbers. For example, Robert McEliece, a professor of electrical engineering at Caltech, had an Erdős number of 1, having collaborated with Erdős himself. Cryptographers Ron Rivest, Adi Shamir, and Leonard Adleman, inventors of the RSA cryptosystem, all have Erdős number 2.

工程的某些领域，尤其是通信理论和密码学，直接利用了埃尔德什数主要涉及的离散数学。因此，这些领域的从业人员的埃尔德什数低就不足为奇了。例如，加州理工学院电气工程学教授Robert McEliece与埃尔德什本人合作，其埃尔德什数为1。RSA密码系统的发明者，密码学家Ron Rivest，Adi Shamir和Leonard Adleman的埃尔德什数均为2。

Social network analysis 社交网络分析领域

Anthropologist Douglas R. White has an Erdős number of 2 via graph theorist Frank Harary. Sociologist Barry Wellman has an Erdős number of 3 via social network analyst and statistician Ove Frank, another collaborator of Harary's.

人类学家道格拉斯·怀特Douglas R. White通过与图论家弗兰克·哈拉里Frank Harary合作得到埃尔德什数为2。社会学家巴里·韦尔曼Barry Wellman通过与社交网络分析师和统计学家Ove Frank（Harve's的另一位合作者）合作得到了埃尔德什数为3。

Linguistics 语言学领域

The Romanian mathematician and computational linguist Solomon Marcus had an Erdős number of 1 for a paper in Acta Mathematica Hungarica that he co-authored with Erdős in 1957.

The Romanian mathematician and computational linguist Solomon Marcus had an Erdős number of 1 for a paper in Acta Mathematica Hungarica that he co-authored with Erdős in 1957.

罗马尼亚数学家和计算语言学家Solomon Marcus在1957年与埃尔德什合作了《 Acta Mathematica Hungarica》中的一篇论文，因此他的埃尔德什数为1。

Impact 影响

1985年，保罗·埃尔德什在阿德莱德大学任教，他的学生陶哲轩（Terence Tao）当时只有10岁。陶后来成为加州大学洛杉矶分校的数学教授，于2006年获得费尔兹奖，并于2007年当选为皇家学会会员。他的埃尔德什数为2。

Erdős numbers have been a part of the folklore of mathematicians throughout the world for many years. Among all working mathematicians at the turn of the millennium who have a finite Erdős number, the numbers range up to 15, the median is 5, and the mean is 4.65; almost everyone with a finite Erdős number has a number less than 8. Due to the very high frequency of interdisciplinary collaboration in science today, very large numbers of non-mathematicians in many other fields of science also have finite Erdős numbers. For example, political scientist Steven Brams has an Erdős number of 2. In biomedical research, it is common for statisticians to be among the authors of publications, and many statisticians can be linked to Erdős via John Tukey, who has an Erdős number of 2. Similarly, the prominent geneticist Eric Lander and the mathematician Daniel Kleitman have collaborated on papers, and since Kleitman has an Erdős number of 1, a large fraction of the genetics and genomics community can be linked via Lander and his numerous collaborators. Similarly, collaboration with Gustavus Simmons opened the door for Erdős numbers within the cryptographic research community, and many linguists have finite Erdős numbers, many due to chains of collaboration with such notable scholars as Noam Chomsky (Erdős number 4), William Labov (3), Mark Liberman (3), Geoffrey Pullum (3), or Ivan Sag (4). There are also connections with arts fields.

Erdős numbers have been a part of the folklore of mathematicians throughout the world for many years. Among all working mathematicians at the turn of the millennium who have a finite Erdős number, the numbers range up to 15, the median is 5, and the mean is 4.65; For example, political scientist Steven Brams has an Erdős number of 2. In biomedical research, it is common for statisticians to be among the authors of publications, and many statisticians can be linked to Erdős via John Tukey, who has an Erdős number of 2. Similarly, the prominent geneticist Eric Lander and the mathematician Daniel Kleitman have collaborated on papers, and since Kleitman has an Erdős number of 1, a large fraction of the genetics and genomics community can be linked via Lander and his numerous collaborators. Similarly, collaboration with Gustavus Simmons opened the door for Erdős numbers within the cryptographic research community, and many linguists have finite Erdős numbers, many due to chains of collaboration with such notable scholars as Noam Chomsky (Erdős number 4), William Labov (3), Mark Liberman (3), Geoffrey Pullum (3), or Ivan Sag (4). There are also connections with arts fields.

多年以来，埃尔德什数在数学家之间一直盛行。在千年之交的所有在职数学家中，都伴随着一个有限埃尔德什数，数字范围最大为15，中位数为5，平均值为4.65。几乎每个具有有限埃尔德什数的人其数字都小于8。由于当今科学领域跨学科合作的频率很高，因此许多其他科学领域的大量非数学家也具有有限的埃尔德什数。例如，政治学家Steven Brams的埃尔德什数为2。在生物医学研究中，统计学家通常是出版物的作者，许多统计学家可以通过John Tukey（其埃尔德什数为2）与埃尔德什链接。同样，著名的遗传学家Eric Lander和数学家Daniel Kleitman在论文上进行了合作，由于Kleitman的埃尔德什数为1，因此可以通过Lander及其众多合作者将遗传学和基因组学领域的大部分联系起来。另外，与Gustavus Simmons的合作为密码研究界内的埃尔德什数打开了大门，许多语言学家拥有有限的埃尔德什数，这许多是由于与Noam Chomsky（埃尔德什数为4），William Labov（埃尔德什数为3）等著名学者的合作产生，类似有Mark Liberman（3），Geoffrey Pullum（3）或Ivan Sag（4）。同时与艺术领域也有联系。

According to Alex Lopez-Ortiz, all the Fields and Nevanlinna prize winners during the three cycles in 1986 to 1994 have Erdős numbers of at most 9.

根据亚历克斯·洛佩兹·奥尔蒂斯Alex Lopez-Ortiz的说法，在1986年至1994年的三个周期中，所有费尔兹奖Fields和内凡琳娜奖Nevanlinna prize得主的埃尔德什数最多为9。

Earlier mathematicians published fewer papers than modern ones, and more rarely published jointly written papers. The earliest person known to have a finite Erdős number is either Antoine Lavoisier (born 1743, Erdős number 13), Richard Dedekind (born 1831, Erdős number 7), or Ferdinand Georg Frobenius (born 1849, Erdős number 3), depending on the standard of publication eligibility.

较早的数学家发表的论文通常少于现代的，而且很少发表联合论文。已知拥有有限埃尔德什数的最早学者是Antoine Lavoisier（生于1743年，埃尔德什数为13），Richard Dedekind（生于1831年，埃尔德什数为7）或Ferdinand Georg Frobenius（生于1849年，埃尔德什数为3），具体取决于出版物资格标准。

Martin Tompa proposed a directed graph version of the Erdős number problem, by orienting edges of the collaboration graph from the alphabetically earlier author to the alphabetically later author and defining the monotone Erdős number of an author to be the length of a longest path from Erdős to the author in this directed graph. He finds a path of this type of length 12.

Martin Tompa proposed a directed graph version of the Erdős number problem, by orienting edges of the collaboration graph from the alphabetically earlier author to the alphabetically later author and defining the monotone Erdős number of an author to be the length of a longest path from Erdős to the author in this directed graph. He finds a path of this type of length 12.

马丁·汤帕Martin Tompa提出了埃尔德什数问题的有向图版本，通过定向协作图，将字母顺序更早的作者到字母顺序更晚的作者进行排列，并将作者的单调埃尔德什数定义为该有向图中从埃尔德什到作者的最长路径的长度。他发现这种路径长度为12。

Also, Michael Barr suggests "rational Erdős numbers", generalizing the idea that a person who has written p joint papers with Erdős should be assigned Erdős number 1/p. From the collaboration multigraph of the second kind (although he also has a way to deal with the case of the first kind)—with one edge between two mathematicians for each joint paper they have produced—form an electrical network with a one-ohm resistor on each edge. The total resistance between two nodes tells how "close" these two nodes are.

Also, Michael Barr suggests "rational Erdős numbers", generalizing the idea that a person who has written p joint papers with Erdős should be assigned Erdős number 1/p. From the collaboration multigraph of the second kind (although he also has a way to deal with the case of the first kind)—with one edge between two mathematicians for each joint paper they have produced—form an electrical network with a one-ohm resistor on each edge. The total resistance between two nodes tells how "close" these two nodes are.

另外，迈克尔·巴尔Michael Barr曾建议使用“合理的埃尔德什数”，通俗的说就是与埃尔德共同撰写过p篇论文的人应被分配埃尔德什数的1/p。根据第二种的协作多重图（尽管他也有办法处理第一种情况），即在他们所合著的每篇联合论文中，两个数学家之间都有一条边，这个边缘上都有一个1欧姆电阻器的电网。两个节点之间的总电阻表明这两个节点有多“相近”。

It has been argued that "for an individual researcher, a measure such as Erdős number captures the structural properties of [the] network whereas the h-index captures the citation impact of the publications," and that "One can be easily convinced that ranking in coauthorship networks should take into account both measures to generate a realistic and acceptable ranking."

有人争辩说：“对于独立研究人员而言，诸如埃尔德什数之类的量度可以捕获网络的结构特性，而h指数则可以捕获出版物的引文影响。” 并且“可以很容易地使人相信，共同作者网络中的排名应该同时考虑到两种方法，以产生现实且可接受的排名。”

In 2004 William Tozier, a mathematician with an Erdős number of 4, auctioned off a co-authorship on eBay, hence providing the buyer with an Erdős number of 5. The winning bid of $1031 was posted by a Spanish mathematician, who however did not intend to pay but just placed the bid to stop what he considered a mockery.

2004年，数学家埃尔德什数为4的William Tozier在eBay上拍卖了合著者，因此为买家提供了埃尔德什数为5的机会。一位西班牙数学家发布了1031美元的中标价格，不过他并不打算付款，而只是提出了中标要求，以阻止他认为是嘲弄的行为。

Variations 演变

A number of variations on the concept have been proposed to apply to other fields.

目前出现了很多对该概念进行变型的提议以应用于其他领域。

The best known is the Bacon number (as in the game Six Degrees of Kevin Bacon), connecting actors that appeared in a film together to the actor Kevin Bacon. It was created in 1994, 25 years after Goffman's article on the Erdős number.

最著名的是游戏《与凯文·培根的六度分隔》中的培根数，将电影中出现的演员与演员凯文·培根联系在一起。它开始于1994年，距高夫曼关于埃尔德什数的文章发表25年。

A small number of people are connected to both Erdős and Bacon and thus have an Erdős–Bacon number, which combines the two numbers by taking their sum. One example is the actress-mathematician Danica McKellar, best known for playing Winnie Cooper on the TV series The Wonder Years. Her Erdős number is 4, and her Bacon number is 2.

A small number of people are connected to both Erdős and Bacon and thus have an Erdős–Bacon number, which combines the two numbers by taking their sum. One example is the actress-mathematician Danica McKellar, best known for playing Winnie Cooper on the TV series The Wonder Years. Her Erdős number is 4, and her Bacon number is 2.

很少一部分人同时与埃尔德什和培根相连，因此有一个埃尔德什-培根数，该数通过求和将两个数相加。一个例子是女演员兼数学家丹妮卡·麦凯拉Danica McKellar，她在电视连续剧《纯真年代》中扮演温妮·库珀而闻名。她的埃尔德什数是4，她的培根数是2。

Further extension is possible. For example, the "Erdős–Bacon–Sabbath number" is the sum of the Erdős–Bacon number and the collaborative distance to the band Black Sabbath in terms of singing in public. Physicist Stephen Hawking had an Erdős–Bacon–Sabbath number of 8, and actress Natalie Portman has one of 11 (her Erdős number is 5).

以此类推可以进一步扩展，例如，“埃尔德什-培根–萨巴什数”是“埃尔德什-培根数”在大众音乐领域与黑色安息日Black Sabbath乐队的协作距离总和。物理学家斯蒂芬·霍金Stephen Hawking的埃尔德什–培根–萨巴什数为8，女演员娜塔莉·波特曼Natalie Portman的埃德斯–培根–萨巴什数为11（她的埃尔德什数为5）。

References 参考文献

External links 相关链接

Jerry Grossman, The Erdős Number Project. Contains statistics and a complete list of all mathematicians with an Erdős number less than or equal to 2.

"On a Portion of the Well-Known Collaboration Graph", Jerrold W. Grossman and Patrick D. F. Ion.

"Some Analyses of Erdős Collaboration Graph", Vladimir Batagelj and Andrej Mrvar.

American Mathematical Society, [1]. A search engine for Erdős numbers and collaboration distance between other authors. As of 18 November 2011 no special access is required.

Numberphile video. Ron Graham on imaginary Erdős numbers.

Jerry Grossman,《埃尔德什数项目》.统计数据以及埃尔德什数小于或等于2的所有数学家的完整列表.

"著名协作图的分配", Jerrold W. Grossman and Patrick D. F. Ion.

"埃尔德什协作图的部分分析", Vladimir Batagelj and Andrej Mrvar.

American Mathematical Society, [1]. 计算埃尔德什数和其他作者之间协作距离的搜索引擎. As of 18 November 2011 no special access is required.

数字视频. Ron Graham on imaginary Erdős numbers.

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