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→The master equation approach
===The master equation approach===
===The master equation approach===
A [[master equation]] can express the behaviour of an undirected growing network where, at each time step, a new node is added to the network, linked to an old node (randomly chosen and without preference). The initial network is formed by two nodes and two links between them at time <math>t = 2</math>, this configuration is necessary only to simplify further calculations, so at time <math>t = n</math> the network have <math>n</math> nodes and <math>n</math> links.
The master equation for this network is:
: <math>p(k,s,t+1) = \frac 1 t p(k-1,s,t) + \left(1 - \frac 1 t \right)p(k,s,t),</math>
where <math>p(k,s,t)</math> is the probability to have the node <math>s</math> with degree <math>k</math> at time <math>t+1</math>, and <math>s</math> is the time step when this node was added to the network. Note that there are only two ways for an old node <math>s</math> to have <math>k</math> links at time <math>t+1</math>:
* The node <math>s</math> have degree <math>k-1</math> at time <math>t</math> and will be linked by the new node with probability <math>1/t</math>
* Already has degree <math>k</math> at time <math>t</math> and will not be linked by the new node.
After simplifying this model, the degree distribution is <math>P(k) = 2^{-k}. </math><ref name="dorogovtsev-mendes">{{cite book|last1=Dorogovtsev|first1=S N|last2=Mendes|first2=J F F|title=Evolution of Networks: From Biological Nets to the Internet and WWW|date=2003|publisher=Oxford University Press, Inc.|location=New York, NY, USA|isbn=978-0198515906}}</ref>
Based on this growing network, an epidemic model is developed following a simple rule: Each time the new node is added and after choosing the old node to link, a decision is made: whether or not this new node will be infected. The master equation for this epidemic model is:
: <math>p_r(k,s,t) = r_t \frac 1 t p_r(k-1,s,t) + \left(1 - \frac 1 t \right) p_r(k,s,t),</math>
where <math>r_t</math> represents the decision to infect (<math>r_t = 1</math>) or not (<math>r_t = 0</math>). Solving this master equation, the following solution is obtained: <math>\tilde{P}_r(k) = \left(\frac r 2 \right)^k. </math><ref name="cotacallapa-hase">{{cite journal|last1=Cotacallapa|first1=M|last2=Hase|first2=M O|title=Epidemics in networks: a master equation approach|journal=Journal of Physics A|date=2016|volume=49|issue=6|page=065001|doi=10.1088/1751-8113/49/6/065001|bibcode=2016JPhA...49f5001C|arxiv=1604.01049}}</ref>
===主方程法===
A [[master equation]] can express the behaviour of an undirected growing network where, at each time step, a new node is added to the network, linked to an old node (randomly chosen and without preference). The initial network is formed by two nodes and two links between them at time <math>t = 2</math>, this configuration is necessary only to simplify further calculations, so at time <math>t = n</math> the network have <math>n</math> nodes and <math>n</math> links.
A [[master equation]] can express the behaviour of an undirected growing network where, at each time step, a new node is added to the network, linked to an old node (randomly chosen and without preference). The initial network is formed by two nodes and two links between them at time <math>t = 2</math>, this configuration is necessary only to simplify further calculations, so at time <math>t = n</math> the network have <math>n</math> nodes and <math>n</math> links.