Generating a graph with certain degree distributio

Question

I am trying to generate a random graph that has small-world properties (exhibits a power law distribution). I just started using the networkx package and discovered that it offers a variety of random graph generation. Can someone tell me if it possible to generate a graph where a given node's degree follows a gamma distribution (either in R or using python's networkx package)?

Answer 1

Including the above mentioned, networkx provides 4 algorithms that receives the degree_distribution as an input:

• configuration_model: explain by @eric
• expected_degree_graph: use a probabilistic approach based on the expected degree of each node. It won't give you the exact degrees but an approximation.
• havel_hakimi_graph: this one tries to connect the nodes with highest degree first
• random_degree_sequence_graph: as far as I can see, this is similar to what @JonC suggested; it has a trials parameter as there is no guarantee of finding a suitable configuration.

The full list (including some versions of the algorithms for directed graphs) is here.

I also found a couple of papers:

• Efficient and exact sampling of simple graphs with given arbitrary degree sequence
• A Sequential Importance Sampling Algorithm for Generating Random Graphs with Prescribed Degrees

Answer 2

If you want to use the configuration model something like this should work in NetworkX:

import random 
import networkx as nx
z=[int(random.gammavariate(alpha=9.0,beta=2.0)) for i in range(100)]
G=nx.configuration_model(z)

You might need to adjust the mean of the sequence z depending on parameters in the gamma distribution. Also z doesn't need to be graphical (you'll get a multigraph), but it does need an even sum so you might have to try a few random sequences (or add 1)...

The NetworkX documentation notes for configuration_model give another example, a reference and how to remove parallel edges and self loops:

Notes
-----
As described by Newman [1]_.

A non-graphical degree sequence (not realizable by some simple
graph) is allowed since this function returns graphs with self
loops and parallel edges.  An exception is raised if the degree
sequence does not have an even sum.

This configuration model construction process can lead to
duplicate edges and loops.  You can remove the self-loops and
parallel edges (see below) which will likely result in a graph
that doesn't have the exact degree sequence specified.  This
"finite-size effect" decreases as the size of the graph increases.

References
----------
.. [1] M.E.J. Newman, "The structure and function
       of complex networks", SIAM REVIEW 45-2, pp 167-256, 2003.

Examples
--------
>>> from networkx.utils import powerlaw_sequence
>>> z=nx.create_degree_sequence(100,powerlaw_sequence)
>>> G=nx.configuration_model(z)

To remove parallel edges:

>>> G=nx.Graph(G)

To remove self loops:

>>> G.remove_edges_from(G.selfloop_edges())

Here is an example similar to the one at http://networkx.lanl.gov/examples/drawing/degree_histogram.html that makes a drawing including a graph layout of the largest connected component:

#!/usr/bin/env python
import random
import matplotlib.pyplot as plt
import networkx as nx

def seq(n):
    return [random.gammavariate(alpha=2.0,beta=1.0) for i in range(100)]    
z=nx.create_degree_sequence(100,seq)
nx.is_valid_degree_sequence(z)
G=nx.configuration_model(z)  # configuration model

degree_sequence=sorted(nx.degree(G).values(),reverse=True) # degree sequence
print "Degree sequence", degree_sequence
dmax=max(degree_sequence)

plt.hist(degree_sequence,bins=dmax)
plt.title("Degree histogram")
plt.ylabel("count")
plt.xlabel("degree")

# draw graph in inset 
plt.axes([0.45,0.45,0.45,0.45])
Gcc=nx.connected_component_subgraphs(G)[0]
pos=nx.spring_layout(Gcc)
plt.axis('off')
nx.draw_networkx_nodes(Gcc,pos,node_size=20)
nx.draw_networkx_edges(Gcc,pos,alpha=0.4)

plt.savefig("degree_histogram.png")
plt.show()

Answer 3

I did this a while ago in base Python... IIRC, I used the following method. From memory, so this may not be entirely accurate, but hopefully it's worth something:

1. Chose the number of nodes, N, in your graph, and the density (existing edges over possible edges), D. This implies the number of edges, E.
2. For each node, assign its degree by first choosing a random positive number x and finding P(x), where P is your pdf. The node's degree is (P(x)*E/2) -1.
3. Chose a node at random, and connect it to another random node. If either node has realized its assigned degree, eliminate it from further selection. Repeat E times.

N.B. that this doesn't create a connected graph in general.

Answer 4

I know this is very late, but you can do the same thing, albeit a little more straightforward, with mathematica.

RandomGraph[DegreeGraphDistribution[{3, 3, 3, 3, 3, 3, 3, 3}], 4]

This will generate 4 random graphs, with each node having a prescribed degree.