Editing Branching process (section)

== Mathematical formulation ==

The most common formulation of a branching process is that of the [[Galton&ndash;Watson process]].  Let ''Z''<sub>''n''</sub> denote the state in period ''n'' (often interpreted as the size of generation&nbsp;''n''), and let ''X''<sub>''n,i''</sub> be a random variable denoting the number of direct successors of member ''i'' in period ''n'', where ''X''<sub>''n,i''</sub> are [[independent and identically distributed random variables]] over all ''n''&nbsp;∈{&nbsp;0,&nbsp;1,&nbsp;2,&nbsp;...} and ''i''&nbsp;∈&nbsp;{1,&nbsp;...,&nbsp;''Z''<sub>''n''</sub>}.  Then the recurrence equation is

:<math>Z_{n+1} = \sum_{i=1}^{Z_n} X_{n,i}</math>

with ''Z''<sub>0</sub> = 1.

Alternatively, the branching process can be formulated as a [[random walk]].  Let ''S''<sub>''i''</sub> denote the state in period ''i'', and let ''X''<sub>''i''</sub> be a random variable that is [[iid]] over all ''i''.  Then the recurrence equation is

:<math>S_{i+1} = S_i+X_{i+1}-1 = \sum_{j=1}^{i+1} X_j-i</math>

with ''S''<sub>0</sub> = 1.  To gain some intuition for this formulation, imagine a walk where the goal is to visit every node, but every time a previously unvisited node is visited, additional nodes are revealed that must also be visited.  Let ''S''<sub>''i''</sub> represent the number of revealed but unvisited nodes in period ''i'', and let ''X''<sub>''i''</sub> represent the number of new nodes that are revealed when node ''i'' is visited.  Then in each period, the number of revealed but unvisited nodes equals the number of such nodes in the previous period, plus the new nodes that are revealed when visiting a node, minus the node that is visited.  The process ends once all revealed nodes have been visited.

=== Continuous-time branching processes ===
For discrete-time branching processes, the "branching time" is fixed to be ''1'' for all individuals. For continuous-time branching processes, each individual waits for a random time (which is a continuous random variable), and then divides according to the given distribution. The waiting time for different individuals are independent, and are independent with the number of children. In general, the waiting time is an exponential variable with parameter ''λ'' for all individuals, so that the process is Markovian.