Generating functions¶
Network scientists often use generating functions as a means
of computing with entire probability distributions at once. epydemic
includes a small generating functions library suitable for using this
approach in conjunction with simulations if desired, for example to
check whether a given theoretical prediction is upheld numerically.
Importing the library¶
Generating functions aren’t imported along with the rest of
epydemic. Instead, import the epydemic.gf package:
import epydemic.gf
Alternatively (and better), import only the functions you need:
from epydemic.gf import gf_from_network, gf_er
Construction¶
epydemic’s generating functions can be constructed in four
different ways: from an explicit series, from a sequence of
coefficients, from a function that returns the coefficient of a given
term, or from a network.
- epydemic.gf.gf_from_series(f)¶
Create a generating function from the function that defines the complete series.
- Parameters
f (
Callable[[float],float]) – the series function- Return type
GF- Returns
the generating function
- epydemic.gf.gf_from_coefficients(cs)¶
Create a generating function from a list of coefficients.
- Parameters
cs (
List[float]) – the list of coefficients- Return type
GF- Returns
a generating function
- epydemic.gf.gf_from_coefficient_function(f)¶
Create a generating function from a function that computes the coefficient for the given term.
- Parameters
f (
Callable[[int],float]) – the coefficients function- Return type
GF- Returns
a generating function
- epydemic.gf.gf_from_network(g)¶
Create a generating function representing the actual distribution of degrees in the given network.
- Parameters
g (
Graph) – the network- Return type
GF- Returns
a generating function
There are also standard constructors for common generating functions corresponding to common degree distributions.
Warning
When implementing a generating function using a series with
gf_from_series() it’s important that the series can be
manipulated and computed in certain specific ways. See
Providing a custom series for details. (The other construction methods
have no similar traps for the unwary.)
In use¶
Generating functions expose three main functions. Firstly, a generating function can be evaluated by applying it to a value:
isolated_fraction = gf(0)
Secondly, a generating function can be queried to extract the coefficient attached to a particular term:
nodes_of_degree_5 = gf[5]
Thirdly, a generating function can be differentiated one or more times:
G0 = gf_er(50000, 20)
# extract the mean degree
G0prime = G0.dx()
mean_degree = G0prime(1)
# extract the degree-5 nodes (the hard way)
nodes_of_degree_5 = (G0.dx(5))(0) / math.factorial(5)