eMaTe it is a python package which the main goal is to provide methods capable of estimating the spectral densities and trace functions of large sparse matrices. eMaTe can run in both CPU and GPU and can estimate the spectral density and related trace functions, such as entropy and Estrada index, even in directed or undirected networks with million of nodes.

CITE

Characterization and comparison of large directed graphs through the spectra of the magnetic Laplacian

@article{FdeResende2020,
  doi = {10.1063/5.0006891},
  url = {https://doi.org/10.1063/5.0006891},
  year = {2020},
  month = jul,
  publisher = {{AIP} Publishing},
  volume = {30},
  number = {7},
  pages = {073141},
  author = {Bruno Messias F. de Resende and Luciano da F. Costa},
  title = {Characterization and comparison of large directed networks through the spectra of the magnetic Laplacian},
  journal = {Chaos: An Interdisciplinary Journal of Nonlinear Science}
}

Install

pip install emate

If you a have a GPU you should also install cupy.

Kernel Polynomial Method (KPM)

The Kernel Polynomial Method can estimate the spectral density of large sparse Hermitan matrices with a computational cost almost linear. This method combines three key ingredients: the Chebyshev expansion + the stochastic trace estimator + kernel smoothing.

Example

import networkx as nx
import numpy as np

n = 3000
g = nx.erdos_renyi_graph(n , 3/n)
W = nx.adjacency_matrix(g)

vals  = np.linalg.eigvals(W.todense()).real

from emate.hermitian import tfkpm


num_moments = 40
num_vecs = 40
extra_points = 10
ek, rho = tfkpm(W, num_moments, num_vecs, extra_points)

import matplotlib.pyplot as plt
plt.hist(vals, density=True, bins=100, alpha=.9, color="steelblue")
plt.scatter(ek, rho, c="tomato", zorder=999, alpha=0.9, marker="d")

If the CUPY package it is available in your machine, you can also use the cupy implementation. When compared to tf-kpm, the Cupy-kpm is slower for median matrices (100k) and faster for larger matrices (> 10^6). The main reason it's because the tf-kpm was implemented in order to calc all te moments in a single step.

import matplotlib.pyplot as plt
from emate.hermitian import cupykpm

num_moments = 40
num_vecs = 40
extra_points = 10
ek, rho = cupykpm(W.tocsr(), num_moments, num_vecs, extra_points)
plt.hist(vals, density=True, bins=100, alpha=.9, color="steelblue")
plt.scatter(ek.get(), rho.get(), c="tomato", zorder=999, alpha=0.9, marker="d")

Stochastic Lanczos Quadrature (SLQ)

The problem of estimating the trace of matrix functions appears in applications ranging from machine learning and scientific computing, to computational biology.[2]

Example

Computing the Estrada index

from emate.symmetric.slq import pyslq
import tensorflow as tf

def trace_function(eig_vals):
    return tf.exp(eig_vals)

num_vecs = 100
num_steps = 50
approximated_estrada_index, _ = pyslq(L_sparse, num_vecs, num_steps,  trace_function)
exact_estrada_index =  np.sum(np.exp(vals_laplacian))
approximated_estrada_index, exact_estrada_index

The above code returns

(3058.012, 3063.16457163222)

Entropy

import scipy
import scipy.sparse

def entropy(eig_vals):
  s = 0.
  for val in eig_vals:
    if val > 0:
      s += -val*np.log(val)
  return s

L = np.array(G.laplacian(normalized=True), dtype=np.float64)
vals_laplacian = np.linalg.eigvalsh(L).real

exact_entropy =  entropy(vals_laplacian)


def trace_function(eig_vals):
  def entropy(val):
    return tf.cond(val>0, lambda:-val*tf.log(val), lambda: 0.)
  
  return tf.map_fn(entropy, eig_vals)
 
L_sparse = scipy.sparse.coo_matrix(L)
    
num_vecs = 100
num_steps = 50
approximated_entropy, _ = pyslq(L_sparse, num_vecs, num_steps,  trace_function)

approximated_entropy, exact_entropy

(-509.46283, -512.5283224633046)

[1] Hutchinson, M. F. (1990). A stochastic estimator of the trace of the influence matrix for laplacian smoothing splines. Communications in Statistics-Simulation and Computation, 19(2), 433-450.

[2] Ubaru, S., Chen, J., & Saad, Y. (2017). Fast Estimation of tr(f(A)) via Stochastic Lanczos Quadrature. SIAM Journal on Matrix Analysis and Applications, 38(4), 1075-1099.

[3] The Kernel Polynomial Method applied to tight binding systems with time-dependence