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A classic example of a preferential attachment process is the growth in the number of [[species]] per [[genus]] in some higher [[taxon]] of biotic organisms.<ref name=YulePhilTrans>{{cite journal | last=Yule | first=G. U. | title=A Mathematical Theory of Evolution, based on the Conclusions of Dr. J. C. Willis, F.R.S | journal=[[Philosophical Transactions of the Royal Society B]] | volume=213 | pages=21–87 | year=1925 | doi=10.1098/rstb.1925.0002 | issue=402–410| doi-access=free }}</ref>  New genera ("urns") are added to a taxon whenever a newly appearing species is considered sufficiently different from its predecessors that it does not belong in any of the current genera.  New species ("balls") are added as old ones [[speciation|speciate]] (i.e., split in two) and, assuming that new species belong to the same genus as their parent (except for those that start new genera), the probability that a new species is added to a genus will be proportional to the number of species the genus already has.  This process, first studied by [[Udny Yule|Yule]], is a ''[[linear]]'' preferential attachment process, since the rate at which genera accrue new species is linear in the number they already have.

A classic example of a preferential attachment process is the growth in the number of [[species]] per [[genus]] in some higher [[taxon]] of biotic organisms. New genera ("urns") are added to a taxon whenever a newly appearing species is considered sufficiently different from its predecessors that it does not belong in any of the current genera.  New species ("balls") are added as old ones [[speciation|speciate]] (i.e., split in two) and, assuming that new species belong to the same genus as their parent (except for those that start new genera), the probability that a new species is added to a genus will be proportional to the number of species the genus already has.  This process, first studied by [[Udny Yule|Yule]], is a ''[[linear]]'' preferential attachment process, since the rate at which genera accrue new species is linear in the number they already have.

A classic example of a preferential attachment process is the growth in the number of species per genus in some higher taxon of biotic organisms.  New genera ("urns") are added to a taxon whenever a newly appearing species is considered sufficiently different from its predecessors that it does not belong in any of the current genera.  New species ("balls") are added as old ones speciate (i.e., split in two) and, assuming that new species belong to the same genus as their parent (except for those that start new genera), the probability that a new species is added to a genus will be proportional to the number of species the genus already has.  This process, first studied by Yule, is a linear preferential attachment process, since the rate at which genera accrue new species is linear in the number they already have.

A classic example of a preferential attachment process is the growth in the number of species per genus in some higher taxon of biotic organisms.  New genera ("urns") are added to a taxon whenever a newly appearing species is considered sufficiently different from its predecessors that it does not belong in any of the current genera.  New species ("balls") are added as old ones speciate (i.e., split in two) and, assuming that new species belong to the same genus as their parent (except for those that start new genera), the probability that a new species is added to a genus will be proportional to the number of species the genus already has.  This process, first studied by Yule, is a linear preferential attachment process, since the rate at which genera accrue new species is linear in the number they already have.

Linear preferential attachment processes in which the number of urns increases are known to produce a distribution of balls over the urns following the so-called [[Yule distribution]].  In the most general form of the process, balls are added to the system at an overall rate of ''m'' new balls for each new urn.  Each newly created urn starts out with ''k''₀ balls and further balls are added to urns at a rate proportional to the number ''k'' that they already have plus a constant ''a''&nbsp;>&nbsp;−''k''₀.  With these definitions, the fraction ''P''(''k'') of urns having ''k'' balls in the limit of long time is given by<ref>{{cite journal | last=Newman | first=M. E. J. | title=Power laws, Pareto distributions and Zipf's law | journal=Contemporary Physics | volume=46 | pages=323–351 | year=2005 | arxiv=cond-mat/0412004 | doi=10.1080/00107510500052444 | issue=5 | bibcode=2005ConPh..46..323N}}</ref>

Linear preferential attachment processes in which the number of urns increases are known to produce a distribution of balls over the urns following the so-called [[Yule distribution]].  In the most general form of the process, balls are added to the system at an overall rate of ''m'' new balls for each new urn.  Each newly created urn starts out with ''k''₀ balls and further balls are added to urns at a rate proportional to the number ''k'' that they already have plus a constant ''a''&nbsp;>&nbsp;−''k''₀.  With these definitions, the fraction ''P''(''k'') of urns having ''k'' balls in the limit of long time is given by

Linear preferential attachment processes in which the number of urns increases are known to produce a distribution of balls over the urns following the so-called Yule distribution.  In the most general form of the process, balls are added to the system at an overall rate of m new balls for each new urn.  Each newly created urn starts out with k₀ balls and further balls are added to urns at a rate proportional to the number k that they already have plus a constant a&nbsp;>&nbsp;−k₀.  With these definitions, the fraction P(k) of urns having k balls in the limit of long time is given by

Linear preferential attachment processes in which the number of urns increases are known to produce a distribution of balls over the urns following the so-called Yule distribution.  In the most general form of the process, balls are added to the system at an overall rate of m new balls for each new urn.  Each newly created urn starts out with k₀ balls and further balls are added to urns at a rate proportional to the number k that they already have plus a constant a&nbsp;>&nbsp;−k₀.  With these definitions, the fraction P(k) of urns having k balls in the limit of long time is given by

In other words, the preferential attachment process generates a "[[Long tail|long-tailed]]" distribution following a [[Pareto distribution]] or [[power law]] in its tail.  This is the primary reason for the historical interest in preferential attachment: the species distribution and many other phenomena are observed empirically to follow power laws and the preferential attachment process is a leading candidate mechanism to explain this behavior.  Preferential attachment is considered a possible candidate for, among other things, the distribution of the sizes of cities,<ref name=SimonBiomet>{{cite journal | last=Simon | first=H. A. | title=On a class of skew distribution functions | journal=Biometrika | volume=42 | pages=425–440  | year=1955 | doi=10.1093/biomet/42.3-4.425  | issue=3–4 }}</ref> the wealth of extremely wealthy individuals,<ref name=SimonBiomet /> the number of citations received by learned publications,<ref name=PriceJASIS>{{cite journal | last=Price | first=D. J. de S.  | title=A general theory of bibliometric and other cumulative advantage processes | journal=J. Amer. Soc. Inform. Sci. | volume=27 | pages=292–306 | year=1976 | url=http://garfield.library.upenn.edu/price/pricetheory1976.pdf | doi=10.1002/asi.4630270505 | issue=5}}</ref> and the number of links to pages on the World Wide Web.<ref name=BAScience>{{cite journal | last=Barabási | first=A.-L. |author2=R. Albert | title=Emergence of scaling in random networks | journal=Science | volume=286 | pages=509–512 | year=1999 | arxiv=cond-mat/9910332 | doi=10.1126/science.286.5439.509 | issue=5439 | pmid=10521342| bibcode=1999Sci...286..509B }}</ref>

In other words, the preferential attachment process generates a "[[Long tail|long-tailed]]" distribution following a [[Pareto distribution]] or [[power law]] in its tail.  This is the primary reason for the historical interest in preferential attachment: the species distribution and many other phenomena are observed empirically to follow power laws and the preferential attachment process is a leading candidate mechanism to explain this behavior.  Preferential attachment is considered a possible candidate for, among other things, the distribution of the sizes of cities, the wealth of extremely wealthy individuals, the number of citations received by learned publications, and the number of links to pages on the World Wide Web.

In other words, the preferential attachment process generates a "long-tailed" distribution following a Pareto distribution or power law in its tail.  This is the primary reason for the historical interest in preferential attachment: the species distribution and many other phenomena are observed empirically to follow power laws and the preferential attachment process is a leading candidate mechanism to explain this behavior.  Preferential attachment is considered a possible candidate for, among other things, the distribution of the sizes of cities, the wealth of extremely wealthy individuals, and the number of links to pages on the World Wide Web.

In other words, the preferential attachment process generates a "long-tailed" distribution following a Pareto distribution or power law in its tail.  This is the primary reason for the historical interest in preferential attachment: the species distribution and many other phenomena are observed empirically to follow power laws and the preferential attachment process is a leading candidate mechanism to explain this behavior.  Preferential attachment is considered a possible candidate for, among other things, the distribution of the sizes of cities, the wealth of extremely wealthy individuals, and the number of links to pages on the World Wide Web.

The general model described here includes many other specific models as special cases.  In the species/genus example above, for instance, each genus starts out with a single species (''k''₀&nbsp;=&nbsp;1) and gains new species in direct proportion to the number it already has (''a''&nbsp;=&nbsp;0), and hence ''P''(''k'')&nbsp;=&nbsp;B(''k'',&nbsp;''γ'')/B(''k''₀,&nbsp;''γ''&nbsp;−&nbsp;1) with ''γ''=2&nbsp;+&nbsp;1/''m''.  Similarly the Price model for scientific citations<ref name=PriceJASIS /> corresponds to the case ''k''₀&nbsp;=&nbsp;0, ''a''&nbsp; =&nbsp;1 and the widely studied [[Barabási-Albert model]]<ref name=BAScience /> corresponds to ''k''₀&nbsp;=&nbsp;''m'', ''a''&nbsp;=&nbsp;0.

The general model described here includes many other specific models as special cases.  In the species/genus example above, for instance, each genus starts out with a single species (''k''₀&nbsp;=&nbsp;1) and gains new species in direct proportion to the number it already has (''a''&nbsp;=&nbsp;0), and hence ''P''(''k'')&nbsp;=&nbsp;B(''k'',&nbsp;''γ'')/B(''k''₀,&nbsp;''γ''&nbsp;−&nbsp;1) with ''γ''=2&nbsp;+&nbsp;1/''m''.  Similarly the Price model for scientific citations corresponds to the case ''k''₀&nbsp;=&nbsp;0, ''a''&nbsp; =&nbsp;1 and the widely studied [[Barabási-Albert model]] corresponds to ''k''₀&nbsp;=&nbsp;''m'', ''a''&nbsp;=&nbsp;0.

The general model described here includes many other specific models as special cases.  In the species/genus example above, for instance, each genus starts out with a single species (k₀&nbsp;=&nbsp;1) and gains new species in direct proportion to the number it already has (a&nbsp;=&nbsp;0), and hence P(k)&nbsp;=&nbsp;B(k,&nbsp;γ)/B(k₀,&nbsp;γ&nbsp;−&nbsp;1) with γ=2&nbsp;+&nbsp;1/m.  Similarly the Price model for scientific citations corresponds to the case k₀&nbsp;=&nbsp;0, a&nbsp; =&nbsp;1 and the widely studied Barabási-Albert model corresponds to k₀&nbsp;=&nbsp;m, a&nbsp;=&nbsp;0.

The general model described here includes many other specific models as special cases.  In the species/genus example above, for instance, each genus starts out with a single species (k₀&nbsp;=&nbsp;1) and gains new species in direct proportion to the number it already has (a&nbsp;=&nbsp;0), and hence P(k)&nbsp;=&nbsp;B(k,&nbsp;γ)/B(k₀,&nbsp;γ&nbsp;−&nbsp;1) with γ=2&nbsp;+&nbsp;1/m.  Similarly the Price model for scientific citations corresponds to the case k₀&nbsp;=&nbsp;0, a&nbsp; =&nbsp;1 and the widely studied Barabási-Albert model corresponds to k₀&nbsp;=&nbsp;m, a&nbsp;=&nbsp;0.

Preferential attachment is sometimes referred to as the [[Matthew effect]], but the two are not precisely equivalent.  The Matthew effect, first discussed by [[Robert K. Merton]],<ref>{{cite journal | last=Merton | first=Robert K. | authorlink=Robert K. Merton | title=The Matthew effect in science | journal=Science | volume=159 | pages=56–63 | year=1968 | doi=10.1126/science.159.3810.56 | pmid=17737466 | issue=3810| bibcode=1968Sci...159...56M }}</ref> is named for a passage in the [[bible|biblical]] [[Gospel of Matthew]]: "For everyone who has will be given more, and he will have an abundance. Whoever does not have, even what he has will be taken from him." ([[Gospel of Matthew|Matthew]] [[s:Bible (New International Version)/Matthew#25:29|25:29]], [[New International Version]].)  The preferential attachment process does not incorporate the taking away part. This point may be moot, however, since the scientific insight behind the Matthew effect is in any case entirely different.  Qualitatively it is intended to describe not a mechanical multiplicative effect like preferential attachment but a specific human behavior in which people are more likely to give credit to the famous than to the little known.  The classic example of the Matthew effect is a scientific discovery made simultaneously by two different people, one well known and the other little known.  It is claimed that under these circumstances people tend more often to credit the discovery to the well-known scientist.  Thus the real-world phenomenon the Matthew effect is intended to describe is quite distinct from (though certainly related to) preferential attachment.

Preferential attachment is sometimes referred to as the [[Matthew effect]], but the two are not precisely equivalent.  The Matthew effect, first discussed by [[Robert K. Merton]], is named for a passage in the [[bible|biblical]] [[Gospel of Matthew]]: "For everyone who has will be given more, and he will have an abundance. Whoever does not have, even what he has will be taken from him." ([[Gospel of Matthew|Matthew]] [[s:Bible (New International Version)/Matthew#25:29|25:29]], [[New International Version]].)  The preferential attachment process does not incorporate the taking away part. This point may be moot, however, since the scientific insight behind the Matthew effect is in any case entirely different.  Qualitatively it is intended to describe not a mechanical multiplicative effect like preferential attachment but a specific human behavior in which people are more likely to give credit to the famous than to the little known.  The classic example of the Matthew effect is a scientific discovery made simultaneously by two different people, one well known and the other little known.  It is claimed that under these circumstances people tend more often to credit the discovery to the well-known scientist.  Thus the real-world phenomenon the Matthew effect is intended to describe is quite distinct from (though certainly related to) preferential attachment.

Preferential attachment is sometimes referred to as the Matthew effect, but the two are not precisely equivalent.  The Matthew effect, first discussed by Robert K. Merton, is named for a passage in the biblical Gospel of Matthew: "For everyone who has will be given more, and he will have an abundance. Whoever does not have, even what he has will be taken from him." (Matthew 25:29, New International Version.)  The preferential attachment process does not incorporate the taking away part. This point may be moot, however, since the scientific insight behind the Matthew effect is in any case entirely different.  Qualitatively it is intended to describe not a mechanical multiplicative effect like preferential attachment but a specific human behavior in which people are more likely to give credit to the famous than to the little known.  The classic example of the Matthew effect is a scientific discovery made simultaneously by two different people, one well known and the other little known.  It is claimed that under these circumstances people tend more often to credit the discovery to the well-known scientist.  Thus the real-world phenomenon the Matthew effect is intended to describe is quite distinct from (though certainly related to) preferential attachment.

Preferential attachment is sometimes referred to as the Matthew effect, but the two are not precisely equivalent.  The Matthew effect, first discussed by Robert K. Merton, is named for a passage in the biblical Gospel of Matthew: "For everyone who has will be given more, and he will have an abundance. Whoever does not have, even what he has will be taken from him." (Matthew 25:29, New International Version.)  The preferential attachment process does not incorporate the taking away part. This point may be moot, however, since the scientific insight behind the Matthew effect is in any case entirely different.  Qualitatively it is intended to describe not a mechanical multiplicative effect like preferential attachment but a specific human behavior in which people are more likely to give credit to the famous than to the little known.  The classic example of the Matthew effect is a scientific discovery made simultaneously by two different people, one well known and the other little known.  It is claimed that under these circumstances people tend more often to credit the discovery to the well-known scientist.  Thus the real-world phenomenon the Matthew effect is intended to describe is quite distinct from (though certainly related to) preferential attachment.

==History 历史==

==History 历史==

The first rigorous consideration of preferential attachment seems to be that of [[Udny Yule]] in 1925, who used it to explain the power-law distribution of the number of species per genus of flowering plants.<ref name=YulePhilTrans />  The process is sometimes called a "Yule process" in his honor.  Yule was able to show that the process gave rise to a distribution with a power-law tail, but the details of his proof are, by today's standards, contorted and difficult, since the modern tools of stochastic process theory did not yet exist and he was forced to use more cumbersome methods of proof.

The first rigorous consideration of preferential attachment seems to be that of [[Udny Yule]] in 1925, who used it to explain the power-law distribution of the number of species per genus of flowering plants. The process is sometimes called a "Yule process" in his honor.  Yule was able to show that the process gave rise to a distribution with a power-law tail, but the details of his proof are, by today's standards, contorted and difficult, since the modern tools of stochastic process theory did not yet exist and he was forced to use more cumbersome methods of proof.

The first rigorous consideration of preferential attachment seems to be that of Udny Yule in 1925, who used it to explain the power-law distribution of the number of species per genus of flowering plants.  The process is sometimes called a "Yule process" in his honor.  Yule was able to show that the process gave rise to a distribution with a power-law tail, but the details of his proof are, by today's standards, contorted and difficult, since the modern tools of stochastic process theory did not yet exist and he was forced to use more cumbersome methods of proof.

The first rigorous consideration of preferential attachment seems to be that of Udny Yule in 1925, who used it to explain the power-law distribution of the number of species per genus of flowering plants.  The process is sometimes called a "Yule process" in his honor.  Yule was able to show that the process gave rise to a distribution with a power-law tail, but the details of his proof are, by today's standards, contorted and difficult, since the modern tools of stochastic process theory did not yet exist and he was forced to use more cumbersome methods of proof.

Most modern treatments of preferential attachment make use of the [[master equation]] method, whose use in this context was pioneered by [[Herbert A. Simon|Simon]] in 1955, in work on the distribution of sizes of cities and other phenomena.<ref name=SimonBiomet />

Most modern treatments of preferential attachment make use of the [[master equation]] method, whose use in this context was pioneered by [[Herbert A. Simon|Simon]] in 1955, in work on the distribution of sizes of cities and other phenomena.

Most modern treatments of preferential attachment make use of the master equation method, whose use in this context was pioneered by Simon in 1955, in work on the distribution of sizes of cities and other phenomena.

Most modern treatments of preferential attachment make use of the master equation method, whose use in this context was pioneered by Simon in 1955, in work on the distribution of sizes of cities and other phenomena.

The first application of preferential attachment to learned citations was given by [[Derek J. de Solla Price|Price]] in 1976.<ref name=PriceJASIS />  (He referred to the process as a "cumulative advantage" process.)  His was also the first application of the process to the growth of a network, producing what would now be called a [[scale-free network]].  It is in the context of network growth that the process is most frequently studied today.  Price also promoted preferential attachment as a possible explanation for power laws in many other phenomena, including [[Lotka's law]] of scientific productivity and [[Bradford's law]] of journal use.

The first application of preferential attachment to learned citations was given by [[Derek J. de Solla Price|Price]] in 1976. (He referred to the process as a "cumulative advantage" process.)  His was also the first application of the process to the growth of a network, producing what would now be called a [[scale-free network]].  It is in the context of network growth that the process is most frequently studied today.  Price also promoted preferential attachment as a possible explanation for power laws in many other phenomena, including [[Lotka's law]] of scientific productivity and [[Bradford's law]] of journal use.

The first application of preferential attachment to learned citations was given by Price in 1976.  (He referred to the process as a "cumulative advantage" process.)  His was also the first application of the process to the growth of a network, producing what would now be called a scale-free network.  It is in the context of network growth that the process is most frequently studied today.  Price also promoted preferential attachment as a possible explanation for power laws in many other phenomena, including Lotka's law of scientific productivity and Bradford's law of journal use.

The first application of preferential attachment to learned citations was given by Price in 1976.  (He referred to the process as a "cumulative advantage" process.)  His was also the first application of the process to the growth of a network, producing what would now be called a scale-free network.  It is in the context of network growth that the process is most frequently studied today.  Price also promoted preferential attachment as a possible explanation for power laws in many other phenomena, including Lotka's law of scientific productivity and Bradford's law of journal use.

The application of preferential attachment to the growth of the World Wide Web was proposed by [[BA model|Barabási and Albert]] in 1999.<ref name=BAScience /> Barabási and Albert also coined the name "preferential attachment" by which the process is best known today and suggested that the process might apply to the growth of other networks as well. For growing networks, the precise functional form of preferential attachment can be estimated by [[maximum likelihood estimation]].<ref>{{cite journal |last1=Pham |first1=Thong |last2=Sheridan |first2=Paul |last3=Shimodaira |first3=Hidetoshi |title=PAFit: A Statistical Method for Measuring Preferential Attachment in Temporal Complex Networks |journal=PLoS ONE |date=September 17, 2015 |volume=10 |issue=9 |doi=10.1371/journal.pone.0137796 |url=http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0137796 |pages=e0137796 |pmid=26378457 |pmc=4574777|bibcode=2015PLoSO..1037796P }}</ref>

The application of preferential attachment to the growth of the World Wide Web was proposed by [[BA model|Barabási and Albert]] in 1999.  Barabási and Albert also coined the name "preferential attachment" by which the process is best known today and suggested that the process might apply to the growth of other networks as well. For growing networks, the precise functional form of preferential attachment can be estimated by [[maximum likelihood estimation]].<ref>{{cite journal |last1=Pham |first1=Thong |last2=Sheridan |first2=Paul |last3=Shimodaira |first3=Hidetoshi |title=PAFit: A Statistical Method for Measuring Preferential Attachment in Temporal Complex Networks |journal=PLoS ONE |date=September 17, 2015 |volume=10 |issue=9 |doi=10.1371/journal.pone.0137796 |url=http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0137796 |pages=e0137796 |pmid=26378457 |pmc=4574777|bibcode=2015PLoSO..1037796P }}</ref>

The application of preferential attachment to the growth of the World Wide Web was proposed by Barabási and Albert in 1999.

The application of preferential attachment to the growth of the World Wide Web was proposed by Barabási and Albert in 1999.