Skip to main content

Structure and Dynamics on Graphs

Project description

Structure and Dynamics on Graphs (Beta)

The main goal of StDoG is to provide a package which can be used to study dynamical and structural properties (like spectra) on graphs with a large number of vertices. The modules of StDoG are being built by combining codes written in Tensorflow + CUDA and C++.

1 - Install

pip install stdog

2 - Examples

2.1 - Dynamics

2.1.1 - Kuramoto

Tensorflow
import numpy as np
import igraph as ig
from stdog.utils.misc import ig2sparse  #Function to convert igraph format to sparse matrix


num_couplings = 40
N = 20480

G = ig.Graph.Erdos_Renyi(N, 3/N)
adj = ig2sparse(G)

omegas = np.random.normal(size= N).astype("float32")
couplings = np.linspace(0.0,4.,num_couplings)
phases =  np.array([
    np.random.uniform(-np.pi,np.pi,N)
    for i_l in range(num_couplings)

],dtype=np.float32)


precision =32

dt = 0.01
num_temps = 50000
total_time = dt*num_temps
total_time_transient = total_time
transient = False
from stdog.dynamics.kuramoto import Heuns

heuns_0 = Heuns(adj, phases, omegas, couplings, total_time, dt,         
    device="/gpu:0", # or /cpu:
    precision=precision, transient=transient)

heuns_0.run()
heuns_0.transient = True
heuns_0.total_time = total_time_transient
heuns_0.run()
order_parameter_list = heuns_0.order_parameter_list # (num_couplings, total_time//dt)
import matplotlib.pyplot as plt

r = np.mean(order_parameter_list, axis=1)
stdr = np.std(order_parameter_list, axis=1)

plt.ion()
fig, ax1 = plt.subplots()
ax1.plot(couplings,r,'.-')
ax2 = ax1.twinx()
ax2.plot(couplings,stdr,'r.-')
plt.show()

CUDA - Faster than Tensorflow implementation

If CUDA is available. You can install our another package, stdogpkg/cukuramoto (C)

pip install cukuramoto
from stdog.dynamics.kuramoto.cuheuns import CUHeuns as cuHeuns

heuns_0 = cuHeuns(adj, phases, omegas,  couplings,
    total_time, dt, block_size = 1024, transient = False)

heuns_0.run()

heuns_0.transient = True
heuns_0.total_time = total_time_transient
heuns_0.run()
order_parameter_list = heuns_0.order_parameter_list #

2.2 Spectral

Spectral Density

The Kernel Polynomial Method [1] 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 [2] + kernel smoothing.

import igraph as ig
import numpy as np

N = 3000
G = ig.Graph.Erdos_Renyi(N, 3/N)

W = np.array(G.get_adjacency().data, dtype=np.float64)
vals = np.linalg.eigvalsh(W).real
import stdog.spectra as spectra
from stdog.utils.misc import ig2sparse 

W = ig2sparse(G)
num_moments = 300
num_vecs = 200
extra_points = 10
ek, rho = spectra.dos.kpm(W, num_moments, num_vecs, extra_points, device="/gpu:0")
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")
plt.ylim(0, 1)
plt.show()

kpm

Trace Functions through Stochastic Lanczos Quadrature (SLQ)[3]

Computing custom trace functions
from stdog.spectra.trace_function import slq
import tensorflow as tf

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

num_vecs = 100
num_steps = 50
approximated_estrada_index, _ = slq(L_sparse, num_vecs, num_steps,  trace_function, device="/gpu:0")
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
from stdog.spectra.trace_function import entropy as slq_entropy

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)

L_sparse = scipy.sparse.coo_matrix(L)
    
num_vecs = 100
num_steps = 50
approximated_entropy = slq_entropy(
    L_sparse, num_vecs, num_steps, device="/cpu:0")

approximated_entropy, exact_entropy
(-509.46283, -512.5283224633046)

References

1 - Wang, L.W., 1994. Calculating the density of states and optical-absorption spectra of large quantum systems by the plane-wave moments method. Physical Review B, 49(15), p.10154.

2 - 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), pp.433-450.

3 - 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 - How to cite

Thomas Peron, Bruno Messias, Angélica S. Mata, Francisco A. Rodrigues, and Yamir Moreno. On the onset of synchronization of Kuramoto oscillators in scale-free networks. arXiv:1905.02256 (2019).

4 - Acknowledgements

This work has been supported also by FAPESP grants 11/50761-2 and 2015/22308-2. Research carriedout using the computational resources of the Center forMathematical Sciences Applied to Industry (CeMEAI)funded by FAPESP (grant 2013/07375-0).

Responsible authors

@devmessias, @tkdmperon

Project details


Download files

Download the file for your platform. If you're not sure which to choose, learn more about installing packages.

Files for stdog, version 1.0.4
Filename, size File type Python version Upload date Hashes
Filename, size stdog-1.0.4.tar.gz (35.6 kB) File type Source Python version None Upload date Hashes View hashes

Supported by

Elastic Elastic Search Pingdom Pingdom Monitoring Google Google BigQuery Sentry Sentry Error logging AWS AWS Cloud computing DataDog DataDog Monitoring Fastly Fastly CDN DigiCert DigiCert EV certificate StatusPage StatusPage Status page