Structure and Dynamics on Graphs

# 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)

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  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.

import igraph as ig
import numpy as np

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

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() #### Trace Functions through Stochastic Lanczos Quadrature (SLQ)

##### 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")


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).

## Project details

This version 1.0.4 1.0.3 1.0.2 1.0.1

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