def sphere():
n_samples = 2000
n = 3
edges_percent = 0.2
X = np.ndarray(shape=(n_samples, n))
for i in range(n_samples):
x = np.random.multivariate_normal(np.zeros(n), np.eye(n))
X[i] = x / np.linalg.norm(x)
plot_sphere(X, 'sphere')
inner_prods = X @ X.T
dists = np.arccos(np.clip(inner_prods, -1, 1))
print('Finished computing distances')
dists_vec = dists[np.triu_indices(n_samples, 1)]
n_edges = int((edges_percent * 0.5 * n_samples * (n_samples - 1)) / 100)
neigh_threshold = np.partition(dists_vec, n_edges)[n_edges]
print('Number of edges: {}, neigh threshold: {}'.format(
n_edges, neigh_threshold))
graph = nx.Graph()
graph.add_edges_from(np.argwhere(dists < neigh_threshold))
graph.remove_edges_from(graph.selfloop_edges())
graph = ricciCurvature(graph, alpha=0.99, method='OTD')
graph = formanCurvature(graph)
o_curvatures, f_curvatures = get_edge_curvatures(graph)
plot_curvatures(o_curvatures, 'sphere_ollivier')
plot_curvatures(f_curvatures, 'sphere_forman')
def grid():
g = nx.grid_graph([5, 5, 5], periodic=True)
g = ricciCurvature(g, alpha=0.99, method='OTD')
g = formanCurvature(g)
o_curvs, f_curvs = get_edge_curvatures(g)
plot_curvatures(o_curvs, 'o_grid')
plot_curvatures(f_curvs, 'f_grid')
print(o_curvs, f_curvs)
def full_graph():
graph = nx.complete_graph(10)
graph = ricciCurvature(graph, alpha=0.5, method='ATD')
graph = formanCurvature(graph)
o_curvatures, f_curvatures = get_edge_curvatures(graph)
plot_curvatures(o_curvatures, 'full_ollivier')
plot_curvatures(f_curvatures, 'full_forman')
def regular_sphere(toy=False):
if toy:
# yapf: disable
X = [
[0, 0, 1], # north pole
[1, 0, 0], [-1, 0, 0], [0, 1, 0], [0, -1, 0], # equator
[0, 0, -1], # south pole
]
# yapf: enable
else:
# Algorithm from https://www.cmu.edu/biolphys/deserno/pdf/sphere_equi.pdf
n_samples = 100
a = 4 * np.pi / n_samples
d = np.sqrt(a)
m_nu = int(np.pi / d)
d_nu = np.pi / m_nu
d_phi = a / d_nu
X = []
for m in range(m_nu):
nu = np.pi * (m + 0.5) / m_nu
m_phi = int(2 * np.pi * np.sin(nu) / d_phi)
for n in range(m_phi):
phi = 2 * np.pi * n / m_phi
x = np.array([
np.sin(nu) * np.cos(phi),
np.sin(nu) * np.sin(phi),
np.cos(nu)
])
X.append(x)
print('Number of points: ', len(X))
# plot it to make sure it looks 'regular'
X = np.array(X)
plot_sphere(X, 'regular_sphere')
# compute distances
inner_prods = X @ X.T
dists = np.arccos(np.clip(inner_prods, -1, 1))
print('Finished computing distances')
# search for the smallest distance which gives a single connected component
# graph = search_smallest_dist_for_connected_graph(dists)
# even better: use the largest-smallest connecting distance
graph = largest_smallest_connecting_distance(dists)
# degree distribution; should be small
plot_degree_distribution(graph, 'regular_sphere')
# curvature
graph = ricciCurvature(graph, alpha=0.99, method='OTD')
graph = formanCurvature(graph)
o_curvatures, f_curvatures = get_edge_curvatures(graph)
if toy:
print(o_curvatures, f_curvatures)
else:
plot_curvatures(o_curvatures, 'regular_sphere_ollivier')
plot_curvatures(f_curvatures, 'regular_sphere_forman')
def small_sphere():
# This yields positive curvatures, as expected, so the conclusion is that
# the other ones working with the sphere yield a lot of 0 curvatures because
# of the dense sampling which makes the neighbourhood of each node to look
# flat.
g = nx.Graph()
g.add_edges_from([(0, 1), (0, 2), (0, 3), (0, 4), (1, 2), (1, 3), (1, 5),
(2, 4), (2, 5), (3, 4), (3, 5), (4, 5)])
g = ricciCurvature(g, alpha=0.5, method='OTD')
g = formanCurvature(g)
o_curvs, f_curvs = get_edge_curvatures(g)
print(o_curvs, f_curvs)
def erdos_renyi():
n = 1000
p = 0.01
g = nx.Graph()
for i in range(n):
for j in range(i + 1, n):
if np.random.rand() < p:
g.add_edge(i, j)
print('Number of edges: ', g.number_of_edges())
print('Number of connected components: ', nx.number_connected_components(g))
g = ricciCurvature(g, alpha=0.5, method='OTD')
g = formanCurvature(g)
o_curvs, f_curvs = get_edge_curvatures(g)
plot_curvatures(o_curvs, 'o_gnp')
plot_curvatures(f_curvs, 'f_gnp')
def balanced_tree():
branching = 3
depth = 5
g = nx.balanced_tree(branching, depth)
print('Number of edges: ', g.number_of_edges())
g = ricciCurvature(g, alpha=0.99, method='OTD')
g = formanCurvature(g)
o_curvatures, f_curvatures = get_edge_curvatures(g)
plot_curvatures(o_curvatures, 'balanced_tree_ollivier')
plot_curvatures(f_curvatures, 'balanced_tree_forman')
# this shows that the positively curved edges are on the last layer
for limit in range(1, depth + 1):
curvatures = []
for u, v in nx.bfs_edges(g, 0, depth_limit=limit):
curvatures.append(g[u][v]['ricciCurvature'])
plot_curvatures(np.array(curvatures), '{}_balanced'.format(limit))
def tree():
graph = nx.Graph()
graph.add_edge(0, 1)
for i in range(2, 2000):
j = i - np.random.geometric(0.01)
while j < 0:
j = j + i
graph.add_edge(i, j)
nx.nx_pydot.write_dot(graph, 'output/random_tree.dot')
graph = ricciCurvature(graph, alpha=0.99, method='OTD')
graph = formanCurvature(graph)
o_curvatures, f_curvatures = get_edge_curvatures(graph)
plot_curvatures(o_curvatures, 'tree_ollivier')
plot_curvatures(f_curvatures, 'tree_forman')
# skip the last layer
curvatures = []
for _, v, attrs in graph.edges(data=True):
if graph.degree[v] > 1:
curvatures.append(attrs['ricciCurvature'])
plot_curvatures(curvatures, 'tree_no_leaves_ollivier')
def hypercube():
g = nx.hypercube_graph(5)
g = ricciCurvature(g, alpha=0.5, method='OTD')
g = formanCurvature(g)
o_curvs, f_curvs = get_edge_curvatures(g)
print(o_curvs, f_curvs)
def cycle():
g = nx.cycle_graph(10)
g = ricciCurvature(g, alpha=0.5, method='OTD')
g = formanCurvature(g)
o_curvatures, f_curvatures = get_edge_curvatures(g)
print(o_curvatures, f_curvatures)
import networkx as nx
from GraphRicciCurvature.FormanRicci import formanCurvature
from GraphRicciCurvature.OllivierRicci import ricciCurvature
from GraphRicciCurvature.RicciFlow import compute_ricciFlow
# Import an example NetworkX karate club graph
G = nx.karate_club_graph()
# Compute the Ollivier-Ricci curvature of the given graph G
G = ricciCurvature(G, alpha=0.5, weight=None, verbose=False)
print("Karate Club Graph: The Ollivier-Ricci curvature of edge (0,1) is %f" %
G[0][1]["ricciCurvature"])
# Compute the Forman-Ricci curvature of the given graph G
G = formanCurvature(G, verbose=False)
print("Karate Club Graph: The Forman-Ricci curvature of edge (0,1) is %f" %
G[0][1]["formanCurvature"])
#-----------------------------------
# Construct a directed graph example
Gd = nx.DiGraph()
Gd.add_edges_from([(1, 2), (2, 3), (3, 4), (2, 4), (4, 2)])
# Compute the Ollivier-Ricci curvature of the given directed graph Gd
Gd = ricciCurvature(Gd)
for n1, n2 in Gd.edges():
print("Directed Graph: The Ollivier-Ricci curvature of edge(%d,%d) id %f" %
(n1, n2, Gd[n1][n2]["ricciCurvature"]))
# Compute the Forman-Ricci curvature of the given directed graph Gd