Isotypical representation - Revision history

en>Mark viking: No longer an orphan

2014-10-27T18:18:15Z

No longer an orphan

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	~~{{Orphan\|date=November 2013}}~~		Crop Farmers Rave from Stevensville, spends time with interests like butterflies, property developers [http://turkishwiki.com/Savills_Singapore house in singapore] singapore and cross-stitch. Recently took some time to visit Keoladeo National Park.

	~~'''Price's model''' (named after the physicist [[Derek J. de Solla Price]]) was the first model which generalized the [[Simon model]]<ref name="simon">Simon~~, H. A., ''On a class of skew distribution functions'', Biometrika '''42''', 425-440 (1955).</ref> to be used for networks, especially for growing networks. Price's model belongs to the broader class of network growing models (together with the highly influential [[Barabási–Albert model]]) whose primary target is to explain the origination of networks with ~~strongly skewed degree distributions. The model picked up the ideas of the [[Simon model]] reflecting the concept of [[rich get richer]]~~, ~~also known as the~~ [[Matthew effect]]. [[Price]] took the example of a network of citations between scientific papers and expressed its properties. His idea was that the way how an old vertex (existing paper) gets new edges (new citations) should be proportional to the number of existing edges (existing citations) the vertex already has. This was referred to as ''cumulative advantage'', now also known as [[preferential attachment]]. Price's work is also significant in providing the first known example of a [[scale-free network]] (although it was named later). His ideas were used to describe many real-world networks such as the [[Web]].

	~~== The model ==~~

	~~===Basics===~~

	Let us take a directed graph with ''n'' nodes. Let <math>p_k</math> denote the fraction of nodes with ''k'' degrees with <math>\sum_{k}{p_k}=1</math>. Each new node has a given out-degree (namely those papers it cites) and it is fixed in the long run. This does not mean that the out-degrees can not vary across nodes, simply we assume that the mean out-degree ''m'' is fixed over time. It is clear, that <math>\sum_{k}{kp_k}=m</math>, consequently ''m'' is not restricted to integers. The most trivial form of preferential attachment means that a new node connects to an existing node proportionally to its in-degrees. In other words, a new paper cites an existing paper in proportional to its in-degrees. The caveat of such idea is that no new paper is cited when it is joined to the network so it is going to have zero probability of being cited in the future (which necessarily is not how it happens). To overcome this, [[Derek J. de Solla Price\|Price]] proposed that an attachment should be proportional to some <math>k+k_0</math> with <math>k_0</math> constant. In general <math>k_0</~~math> can be arbitrary, yet Price proposes a <math>k_0=1<~~/~~math>, in that way an initial citation is associated with the paper itself (so the proportionality factor is now ''k'' + 1 instead of ''k'')~~. ~~The probability of a new edge connecting to any node with a degree ''k'' is~~

	~~: <math>\frac{(k+1)p_k}{\sum_k (k+1)p_k}=\frac{(k+1)p_k}{m+1}<~~/~~math>~~

	~~===Evolution of the network===~~

	~~The next question is the net change~~ in the number of nodes with degree ''k'' when we add new nodes to the network. Naturally, this number is decreasing, as some ''k''-degree nodes have new edges, hence becoming (''k'' + 1)-degree nodes; but on the other hand this number is also increasing, as some (''k'' − 1)-degree nodes might get new edges, becoming ''k'' degree nodes. To express this net change formally, let us denote the fraction of ''k''-degree nodes at a network of ''n'' vertices with <math>p_{k,n}</math>:

	~~: <math>(n+1)p_{k,n+1}-np_{k,n}=[kp_{k-1,n}-(k+1)p_{k,n}~~]~~\frac{m}{m+1}\text{ for }k\geq 1,</math>~~

	and

	~~: <math>(n+1)p_{0,n+1}~~-~~np_{0,n}=1-p_{0,n}\frac{m}{m+1}\text{ for }k=0~~.~~</math>~~

	~~To obtain a stationary solution for <math>p_{k,n+1}=p_{k,n}=p_k</math>, first let us express <math>p_{k}</math> using the well-known [[master equation]] method, as~~

	~~: <math>p_k=\begin{cases}[kp_{k-1}-(k+1)p_k]\frac{m}{m+1} & \text{for } k\geq 1 \\1-p_0\frac{m}{m+1} & \text{for } k=0\end{cases}</math>~~

	~~After~~ some ~~manipulation, the expression above yields~~ to

	~~: <math>p_0=\frac{m+1}{2m+1}</math>~~

	~~and~~

	~~: <math>p_0=\frac{k!}{(k+2+1/m)\cdots(3+1/m)}=(1+1/m)\mathbf{B}(k+1,2+1/m),</math>~~

	with <math>\mathbf{B}(a,b)</math> being the [[beta function\|Beta-function]]. As a consequence, <math>p_k\sim k^{-(2+1/m)}</math>. This is identical to saying that <math>p_k</math> follows a [[power law\|power-law distribution]] with exponent <math>\alpha=2+1/m</math>. Typically, this places the exponent between 2 and 3, which is the case for many real world networks. [[Derek J. de Solla Price\|Price]] tested his model by comparing to the citation network data and concluded that the resulting ''m'' is feasible to produce a sufficiently good [[power law\|power-law distribution]].

	~~===Generalization===~~

	~~It is straightforward how to generalize the above results to the case when <math>k_0\neq 1</math>. Basic calculations show that~~

	~~: <math>p_k=\frac{m+1}{m(k_0+1)+1}\frac{\mathbf{B}(k+k_0,2+1/m)}{\mathbf{B}(k_0,2+1/m)},</math>~~

	~~which once more yields to a power law distribution of <math>p_k</math> with the same exponent <math>\alpha=2+1/m</math> for large ''k'' and fixed <math>k_0</math>.~~

	~~==Notes==~~

	For further discussion, please see,<ref name="drog">Dorogovtsev, S. N., Mendes, J. F. F., and Samukhin, A. N., ''Structure of growing networks with preferential linking'', Phys. Rev. Lett. '''85''', 4633–4636 (2000).</ref><ref name="krap">Krapivsky, P. L. and Redner, S., ''Organization of growing random networks'', Phys. Rev. E '''63''', 066123 (2001).</ref> and.<ref name="drog2">Dorogovtsev, S. N. and Mendes, J. F. F., ''Evolution of networks'', Advances in Physics '''51''', 1079–1187 (2002).</ref><ref name="mendes">Krapivsky, P. L. and Redner, S., ''Rate equation approach for growing networks'', in R. Pastor-Satorras and J. Rubi (eds.), Proceedings of the XVIII Sitges Conference on Statistical Mechanics, Lecture Notes in Physics, Springer, Berlin (2003).</ref> [[Derek J. de Solla Price\|Price]] was able to derive these results but this was how far he could get with it, without the provision of computational resources. Fortunately, many work dedicated to preferential attachment and network growth had been enabled by recent technological progress. Without question, the most remarkable contribution to this topic so far is the [[Barabási–Albert model]].<ref name="BA">Barabási, Albert-László and Réka Albert, [http://arxiv.org/abs/cond-mat/9910332 ''Emergence of scaling in rendom networks''], Science, '''286''', 509–512, (1999)</ref>

	~~==Sources==~~
	*Newman, M. E. J., The Structure and Function of Complex Networks, SIAM Review, '''45'''(2), 167–256 (2003)

	~~==References==~~

	~~<references />~~

	~~[[Category:Mathematical modeling]]~~
	~~[[Category:Networks]]~~

en>Noix07: /* References */

2014-02-03T17:49:48Z

References

New page

{{Orphan|date=November 2013}}

'''Price's model''' (named after the physicist [[Derek J. de Solla Price]]) was the first model which generalized the [[Simon model]]<ref name="simon">Simon, H. A., ''On a class of skew distribution functions'', Biometrika '''42''', 425-440 (1955).</ref> to be used for networks, especially for growing networks. Price's model belongs to the broader class of network growing models (together with the highly influential [[Barabási–Albert model]]) whose primary target is to explain the origination of networks with strongly skewed degree distributions. The model picked up the ideas of the [[Simon model]] reflecting the concept of [[rich get richer]], also known as the [[Matthew effect]]. [[Price]] took the example of a network of citations between scientific papers and expressed its properties. His idea was that the way how an old vertex (existing paper) gets new edges (new citations) should be proportional to the number of existing edges (existing citations) the vertex already has. This was referred to as ''cumulative advantage'', now also known as [[preferential attachment]]. Price's work is also significant in providing the first known example of a [[scale-free network]] (although it was named later). His ideas were used to describe many real-world networks such as the [[Web]].

== The model ==

===Basics===

Let us take a directed graph with ''n'' nodes. Let <math>p_k</math> denote the fraction of nodes with ''k'' degrees with <math>\sum_{k}{p_k}=1</math>. Each new node has a given out-degree (namely those papers it cites) and it is fixed in the long run. This does not mean that the out-degrees can not vary across nodes, simply we assume that the mean out-degree ''m'' is fixed over time. It is clear, that <math>\sum_{k}{kp_k}=m</math>, consequently ''m'' is not restricted to integers. The most trivial form of preferential attachment means that a new node connects to an existing node proportionally to its in-degrees. In other words, a new paper cites an existing paper in proportional to its in-degrees. The caveat of such idea is that no new paper is cited when it is joined to the network so it is going to have zero probability of being cited in the future (which necessarily is not how it happens). To overcome this, [[Derek J. de Solla Price|Price]] proposed that an attachment should be proportional to some <math>k+k_0</math> with <math>k_0</math> constant. In general <math>k_0</math> can be arbitrary, yet Price proposes a <math>k_0=1</math>, in that way an initial citation is associated with the paper itself (so the proportionality factor is now ''k'' + 1 instead of ''k''). The probability of a new edge connecting to any node with a degree ''k'' is

: <math>\frac{(k+1)p_k}{\sum_k (k+1)p_k}=\frac{(k+1)p_k}{m+1}</math>

===Evolution of the network===

The next question is the net change in the number of nodes with degree ''k'' when we add new nodes to the network. Naturally, this number is decreasing, as some ''k''-degree nodes have new edges, hence becoming (''k'' + 1)-degree nodes; but on the other hand this number is also increasing, as some (''k'' − 1)-degree nodes might get new edges, becoming ''k'' degree nodes. To express this net change formally, let us denote the fraction of ''k''-degree nodes at a network of ''n'' vertices with <math>p_{k,n}</math>:

: <math>(n+1)p_{k,n+1}-np_{k,n}=[kp_{k-1,n}-(k+1)p_{k,n}]\frac{m}{m+1}\text{ for }k\geq 1,</math>

and

: <math>(n+1)p_{0,n+1}-np_{0,n}=1-p_{0,n}\frac{m}{m+1}\text{ for }k=0.</math>

To obtain a stationary solution for <math>p_{k,n+1}=p_{k,n}=p_k</math>, first let us express <math>p_{k}</math> using the well-known [[master equation]] method, as

: <math>p_k=\begin{cases}[kp_{k-1}-(k+1)p_k]\frac{m}{m+1} & \text{for } k\geq 1 \\1-p_0\frac{m}{m+1} & \text{for } k=0\end{cases}</math>

After some manipulation, the expression above yields to

: <math>p_0=\frac{m+1}{2m+1}</math>

and

: <math>p_0=\frac{k!}{(k+2+1/m)\cdots(3+1/m)}=(1+1/m)\mathbf{B}(k+1,2+1/m),</math>

with <math>\mathbf{B}(a,b)</math> being the [[beta function|Beta-function]]. As a consequence, <math>p_k\sim k^{-(2+1/m)}</math>. This is identical to saying that <math>p_k</math> follows a [[power law|power-law distribution]] with exponent <math>\alpha=2+1/m</math>. Typically, this places the exponent between 2 and 3, which is the case for many real world networks. [[Derek J. de Solla Price|Price]] tested his model by comparing to the citation network data and concluded that the resulting ''m'' is feasible to produce a sufficiently good [[power law|power-law distribution]].

===Generalization===

It is straightforward how to generalize the above results to the case when <math>k_0\neq 1</math>. Basic calculations show that

: <math>p_k=\frac{m+1}{m(k_0+1)+1}\frac{\mathbf{B}(k+k_0,2+1/m)}{\mathbf{B}(k_0,2+1/m)},</math>

which once more yields to a power law distribution of <math>p_k</math> with the same exponent <math>\alpha=2+1/m</math> for large ''k'' and fixed <math>k_0</math>.

==Notes==

For further discussion, please see,<ref name="drog">Dorogovtsev, S. N., Mendes, J. F. F., and Samukhin, A. N., ''Structure of growing networks with preferential linking'', Phys. Rev. Lett. '''85''', 4633–4636 (2000).</ref><ref name="krap">Krapivsky, P. L. and Redner, S., ''Organization of growing random networks'', Phys. Rev. E '''63''', 066123 (2001).</ref> and.<ref name="drog2">Dorogovtsev, S. N. and Mendes, J. F. F., ''Evolution of networks'', Advances in Physics '''51''', 1079–1187 (2002).</ref><ref name="mendes">Krapivsky, P. L. and Redner, S., ''Rate equation approach for growing networks'', in R. Pastor-Satorras and J. Rubi (eds.), Proceedings of the XVIII Sitges Conference on Statistical Mechanics, Lecture Notes in Physics, Springer, Berlin (2003).</ref> [[Derek J. de Solla Price|Price]] was able to derive these results but this was how far he could get with it, without the provision of computational resources. Fortunately, many work dedicated to preferential attachment and network growth had been enabled by recent technological progress. Without question, the most remarkable contribution to this topic so far is the [[Barabási–Albert model]].<ref name="BA">Barabási, Albert-László and Réka Albert, [http://arxiv.org/abs/cond-mat/9910332 ''Emergence of scaling in rendom networks''], Science, '''286''', 509–512, (1999)</ref>

==Sources==
*Newman, M. E. J., The Structure and Function of Complex Networks, SIAM Review, '''45'''(2), 167–256 (2003)

==References==

<references />

[[Category:Mathematical modeling]]
[[Category:Networks]]