Animating Network Growth with NetworkX and Matplotlib

Upon review, that code wasn't nearly as relevant to this problem as I'd thought. However, I was able to use this SO answer and this SO answer to cobble together an answer for you. Here's code that will create a graph, add 50 random nodes and 50 random edges to it, and display an image of the graph after every node and edge is added. A few of the key changes from your code:

1. This code is based on using pylab.ion() (see here for more info).
2. Your code tries to use one figure and change the image within it. This code creates a new figure for each frame.
3. This code doesn't write out to .mp4. If you really need to do that, I would suggest writing the figures out to .png file while rendering them, and converting to mp4 after the fact.
4. Note that this code uses matplotlib.pyplot.pause() instead of time.sleep(). The figures won't render if you use time.sleep().

Good luck!

import random
import pylab
from matplotlib.pyplot import pause
import networkx as nx
pylab.ion()

graph = nx.Graph()
node_number = 0
graph.add_node(node_number, Position=(random.randrange(0, 100), random.randrange(0, 100)))

def get_fig():
    global node_number
    node_number += 1
    graph.add_node(node_number, Position=(random.randrange(0, 100), random.randrange(0, 100)))
    graph.add_edge(node_number, random.choice(graph.nodes()))
    fig = pylab.figure()
    nx.draw(graph, pos=nx.get_node_attributes(graph,'Position'))
    return fig

num_plots = 50;
pylab.show()

for i in range(num_plots):

    fig = get_fig()
    fig.canvas.draw()
    pylab.draw()
    pause(2)
    pylab.close(fig)

As the existing answers are quite old, here is a version of how the code could look like with current versions of matplotlib and networkx.

import random
import networkx as nx
import matplotlib.pyplot as plt

graph = nx.Graph()
num_plots = 50
for node_number in range(num_plots):
    graph.add_node(node_number, Position=(random.randrange(0, 100), random.randrange(0, 100)))
    graph.add_edge(node_number, random.choice(list(graph.nodes())))
    nx.draw(graph, pos=nx.get_node_attributes(graph,'Position'))
    plt.pause(0.5)

An improved version of bretlance's. Hope it will be helpful. It will show animations but not picture after picture.

Still don't know how the owner of the question Animate drawing networkx edges made use of matplotlib's animation

#!/usr/bin/env python
import random
import pylab
from matplotlib.pyplot import pause
import networkx as nx
pylab.ion()

graph = nx.Graph()
node_number = 0
graph.add_node(node_number, Position=(random.randrange(0, 100), random.randrange(0, 100)))

def get_fig():
    global node_number
    node_number += 1
    graph.add_node(node_number, Position=(random.randrange(0, 100), random.randrange(0, 100)))
    graph.add_edge(node_number, random.choice(graph.nodes()))
    nx.draw(graph, pos=nx.get_node_attributes(graph,'Position'))

num_plots = 50;
pylab.show()

for i in range(num_plots):

    get_fig()
    pylab.draw()
    pause(2)