Editing Percolation theory (section)

== Computation of the critical parameter ==

For most infinite lattice graphs, {{math|''p''<sub>c</sub>}} cannot be calculated exactly, though in some cases {{math|''p''<sub>c</sub>}} there is an exact value. For example:

*for the [[square lattice]] {{math|'''ℤ'''<sup>2</sup>}} in two dimensions, {{math|''p''<sub>c</sub> {{=}} {{sfrac|1|2}}}} for bond percolation, a fact which was an open question for more than 20 years and was finally resolved by [[Harry Kesten]] in the early 1980s,<ref name="BollobásRiordan2006">{{cite journal|last1=Bollobás|first1=Béla|last2=Riordan|first2=Oliver|title=Sharp thresholds and percolation in the plane|journal=Random Structures and Algorithms |volume=29|issue=4|year=2006|pages=524–548|issn=1042-9832|doi=10.1002/rsa.20134|arxiv=math/0412510|s2cid=7342807}}</ref> see {{harvtxt|Kesten|1982}}. For site percolation on the square lattice, the value of {{math|''p''<sub>c</sub>}} is not known from analytic derivation but only via simulations of large lattices which provide the estimate {{math|''p''<sub>c</sub> {{=}} }} 0.59274621 ± 0.00000013.<ref>{{cite journal |author=MEJ Newman |author2=RM Ziff|year=2000|title=Efficient Monte Carlo algorithm and high-precision results for percolation |journal=Physical Review Letters |issue=19|volume=85|pages=4104–4107|doi=10.1103/physrevlett.85.4104 |pmid=11056635|arxiv=cond-mat/0005264|bibcode=2000PhRvL..85.4104N|s2cid=747665}}</ref>  
*A limit case for lattices in high dimensions is given by the [[Bethe lattice]], whose threshold is at {{math|''p''<sub>c</sub> {{=}} {{sfrac|1|''z'' − 1}}}} for a [[coordination number]]&nbsp;{{math|''z''}}. In other words: for the regular [[Tree (graph theory)|tree]] of degree <math>z</math>, <math>p_c</math> is equal to <math>1/(z-1)</math>. 
[[File:Front de percolation.png|thumb|Percolation front]]

* For a random [[Tree (graph theory)|tree-like]] network without degree-degree correlation, it can be shown that such network can have a [[giant component]], and the [[percolation threshold]] (transmission probability) is given by <math>p_c = \frac{1}{g_1'(1)}</math>,  where <math>g_1(z)</math> is the [[Degree distribution#Generating functions method|generating function]] corresponding to the [[Degree distribution#Generating functions method|excess degree distribution]]. So, for random [[Erdős–Rényi model|Erdős–Rényi networks]] of average degree <math>\langle k\rangle</math>, {{math|''p''<sub>c</sub> {{=}} {{sfrac|1|&lang;k&rang;}}}}.<ref>{{cite journal|author=Erdős, P.|author2=Rényi, A.|name-list-style=amp|year=1959|title=On random graphs I.|journal=Publ. Math.|issue=6|pages=290–297}}</ref><ref>{{cite journal|author=Erdős, P.|author2=Rényi, A.|name-list-style=amp|year=1960|title=The evolution of random graphs|journal=Publ. Math. Inst. Hung. Acad. Sci.|issue=5|pages=17–61}}</ref><ref>{{cite journal|author=Bolloba's, B.|year=1985|title=Random Graphs|journal=Academic}}</ref>
* In networks with low [[Clustering coefficient|clustering]], <math>
0 < C \ll 1
</math>, the critical point gets scaled by <math>
(1-C)^{-1}
</math> such that:

<math>p_c = \frac{1}{1-C}\frac{1}{g_1'(1)}.</math><ref>{{Cite journal|last1=Berchenko|first1=Yakir|last2=Artzy-Randrup|first2=Yael|last3=Teicher|first3=Mina|last4=Stone|first4=Lewi|date=2009-03-30|title=Emergence and Size of the Giant Component in Clustered Random Graphs with a Given Degree Distribution|url=https://link.aps.org/doi/10.1103/PhysRevLett.102.138701|journal=Physical Review Letters|language=en|volume=102|issue=13|pages=138701|doi=10.1103/PhysRevLett.102.138701|pmid=19392410|bibcode=2009PhRvL.102m8701B|issn=0031-9007|access-date=2022-02-24|archive-date=2023-02-04|archive-url=https://web.archive.org/web/20230204143725/https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.102.138701|url-status=live}}</ref>  

This indicates that for a given degree distribution, the clustering leads to a larger percolation threshold, mainly because for a fixed number of links, the clustering structure reinforces the core of the network with the price of diluting the global connections. For networks with high clustering, strong clustering could induce the core–periphery structure, in which the core and periphery might percolate at different critical points, and the above approximate treatment is not applicable.<ref>{{Cite journal|last1=Li|first1=Ming|last2=Liu|first2=Run-Ran|last3=Lü|first3=Linyuan|last4=Hu|first4=Mao-Bin|last5=Xu|first5=Shuqi|last6=Zhang|first6=Yi-Cheng|date=2021-04-25|title=Percolation on complex networks: Theory and application|url=https://www.sciencedirect.com/science/article/pii/S0370157320304269|journal=Physics Reports|series=Percolation on complex networks: Theory and application|language=en|volume=907|pages=1–68|doi=10.1016/j.physrep.2020.12.003|arxiv=2101.11761 |bibcode=2021PhR...907....1L |s2cid=231719831 |issn=0370-1573}}</ref>