Example MCMC inference

As an example, we are going to find the partition that gives the minimum description length on a small graph in dataset/southernWomen.edgelist.

To begin, let’s first import the functions that we need.

from biSBM.ioutils import get_edgelist, get_types
from biSBM.optimalks import *

Here, we will fit the biSBM via the Markov Chain Monte Carlo algorithm. We have provided a wrapper to the binary for this task. Let’s import it.:

from engines.mcmc import *
mcmc = MCMC(f_engine="engines/bipartiteSBM-MCMC/bin/mcmc")

The engines.mcmc is a wrapper class for the C++ engine. It generates the commandline input that runs as a spawned process in the background. In this example, it will generate a string that tells the C++ program to do graph partition of the southern women dataset at \(K_a=3\) and \(K_b=2\).

mcmc.prepare_engine("dataset/test/southernWomen.edgelist", 18, 14, 3, 2)
# Out[*]: 'engines/bipartiteSBM-MCMC/bin/mcmc -e dataset/test/southernWomen.edgelist -n 6 6 6 7 7 -t 1000000 -x 100000 -c abrupt_cool -a 100000.0 -y 18 14 -z 3 2 -E 0.001 -g

In addition, we have to tell the program which are type-a nodes and which are type-b. We assume that the node indices in the dataset is placed in a specific order; that is, nodes of type-a are indexed first and then followed by nodes of type-b.

For example, if we have a sample network whose number (i.e., size) of type-a nodes is \(n_a=18\) and the number of type-b nodes is \(n_b=14\).

This means that the nodes which indexed with \(0 \dots 17\) are type-a nodes and those indexed with \(18 \dots 31\) are type-b nodes. Once we have specified the engine, it’s time to bake the dataset!

edgelist = get_edgelist("dataset/test/southernWomen.edgelist")
types = get_types("dataset/test/southernWomen.types")

For types, we can also use,

types = mcmc.gen_types(n1, n2)  # n1=18 & n2=14

We can then feed these three variables into the main OptimalKs Class.

oks = OptimalKs(mcmc, edgelist, types)

Now, we can start the heuristic search!

# Out[*]:
# OrderedDict([((1, 1), 191.72536162138402),
#              ((6, 7), 227.47573446636372),
#              ((2, 1), 199.25454713207995),
#              ((1, 2), 197.11255878689138),
#              ((1, 3), 202.30070134785217),
#              ((2, 2), 199.5005556514051),
#              ((2, 3), 196.01966191156208),
#              ((3, 1), 203.69566837215007),
#              ((3, 2), 204.73016933010257),
#              ((3, 3), 201.34320240110824)])

We should expect to wait for a minute or two for the program to complete (depending on the size of the network).

For the sourthernWomen dataset, we will see that there are no statistically significant communities other than the fact of being a bipartite network. That is, we reached a trivial conclusion that \(K_a=1\) and \(K_b=1\).

If you are interested to run the heuristic again, just to check the consistency of the result, you can re-initiate the biSBM.optimalks.OptimalKs class and then re-do the minimize_bisbm_dl().

Once the algorithm stops, it will output the trace and entropy (a.k.a, description length) at the \((K_a, K_b)\) point whose description length is minimal. This is the best network parameterization after running the algorithm once, we can use it as a approach to Bayesian inference and compare the description length with other models.

To see the algorithmic outcome, we may simply run:

# Out[*]:
# {'algm_args': {'init_ka': 6, 'init_kb': 7, 'i_0': 0.058872511354072427},
#  'na': 18,
#  'nb': 14,
#  'e': 89,
#  'avg_k': 5.5625,
#  'ka': 1,
#  'kb': 1,
#  'mdl': 191.72536162138402,
#  'dl': {'adjacency': 108.99607644928233,
#   'partition': 5.5294290875114225,
#   'degree': 77.19985608459028,
#   'edges': 0.0}}

We conclude that the partition for the southern women dataset is trivial. There is no structure other than bipartite that support the blockmodeling of the dataset.