matrix-count
Estimating integer matrix counting problems
Maximilian Jerdee
We provide analytic estimates and sampling-based algorithms for a variety of counting problems defined over integer matrices.
For example, we may count the number of non-negative symmetric integer matrices with even diagonal entries and a given row sum. This is the number of (multi-)graphs with a given degree sequence. We can also estimate the number of such matrices under combinations of the further conditions:
- Binary matrices (only simple graphs).
- Fixed total sum of diagonal entries (number of self edges).
- Fixed sum of entries in blocks of matrix. (number of edges between prescribed groups) [Not yet implemented]
We also include methods for estimating the number of non-negative integer matrices with a given row sum and column sum as described in Jerdee, Kirkley, Newman (2022). [Not yet implemented]
These problems can also be generalized as sums over matrices $A$ weighted by a Dirichlet-multinomial factor on their entries
$$w(A) = \prod_{i < j}\binom{A_{ij} + \alpha - 1}{\alpha - 1} \prod_i \binom{A_{ii}/2 + \alpha - 1}{\alpha - 1}.$$
Note that $\alpha = 1$ corresponds to the uniform count. This more general estimate acts as the partition function of a generalized random multigraph model.
Installation
matrix-count may be installed through pip:
pip install matrix-count
or be built locally by cloning this repository and running
pip install .
in the base directory.
Typical usage
Once installed, the package can be imported as
import matrix_count
Note that this is not import matrix-count.
The package can then be used to evaluate rapid analytic estimates of these counting problems, to sample from the space of such matrices, and to converge to the exact number of these matrices.
# Margin of a 8x8 symmetric non-negative integer matrix with even diagonal entries
margin = [10, 9, 8, 7, 6, 5, 4, 3]
# Estimate the logarithm of the number of symmetric matrices with given margin sum
# (number of multigraphs with given degree sequence)
estimate = matrix_count.estimate_log_symmetric_matrices(margin, alpha=1)
print("Estimated log count of symmetric matrices:", estimate)
# Count the number of such matrices
count, count_err = matrix_count.count_log_symmetric_matrices(margin, alpha=1)
print("Log count of symmetric matrices:", count, "+/-", count_err)
# Sample from the space of such matrices
num_samples = 3
for _t in range(num_samples):
sample, entropy = matrix_count.sample_symmetric_matrix(margin)
print("Sampled matrix:")
print(sample)
print("Minus log probability of sampled matrix:", entropy)
Further usage examples can be found in the examples directory of the
repository and the package documentation.
1 maintainer
Metadata
1 maintainer
Release files for matrix-count 0.1.6
For a detailed explanation of source distributions (sdists) and built distributions (wheels), please see the package formats documentation.
Source distribution (sdist)
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| matrix_count-0.1.6.tar.gz | 803.8 kB | Without provenance | Details |
Built distributions (wheels)
Total release size: 19.5 MB
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