path: root/examples/gromov/plot_entropic_semirelaxed_fgw.py

# -*- coding: utf-8 -*-
"""
==========================
Entropic-regularized semi-relaxed (Fused) Gromov-Wasserstein example
==========================

This example is designed to show how to use the entropic semi-relaxed Gromov-Wasserstein
and the entropic semi-relaxed Fused Gromov-Wasserstein divergences.

Entropic-regularized sr(F)GW between two graphs G1 and G2 searches for a reweighing of the nodes of
G2 at a minimal entropic-regularized (F)GW distance from G1.

First, we generate two graphs following Stochastic Block Models, then show
how to compute their srGW matchings and illustrate them. These graphs are then
endowed with node features and we follow the same process with srFGW.

[48] Cédric Vincent-Cuaz, Rémi Flamary, Marco Corneli, Titouan Vayer, Nicolas Courty.
"Semi-relaxed Gromov-Wasserstein divergence and applications on graphs"
International Conference on Learning Representations (ICLR), 2021.
"""

# Author: Cédric Vincent-Cuaz <cedvincentcuaz@gmail.com>
#
# License: MIT License

# sphinx_gallery_thumbnail_number = 1

import numpy as np
import matplotlib.pylab as pl
from ot.gromov import entropic_semirelaxed_gromov_wasserstein, entropic_semirelaxed_fused_gromov_wasserstein, gromov_wasserstein, fused_gromov_wasserstein
import networkx
from networkx.generators.community import stochastic_block_model as sbm

#############################################################################
#
# Generate two graphs following Stochastic Block models of 2 and 3 clusters.
# ---------------------------------------------


N2 = 20  # 2 communities
N3 = 30  # 3 communities
p2 = [[1., 0.1],
      [0.1, 0.9]]
p3 = [[1., 0.1, 0.],
      [0.1, 0.95, 0.1],
      [0., 0.1, 0.9]]
G2 = sbm(seed=0, sizes=[N2 // 2, N2 // 2], p=p2)
G3 = sbm(seed=0, sizes=[N3 // 3, N3 // 3, N3 // 3], p=p3)


C2 = networkx.to_numpy_array(G2)
C3 = networkx.to_numpy_array(G3)

h2 = np.ones(C2.shape[0]) / C2.shape[0]
h3 = np.ones(C3.shape[0]) / C3.shape[0]

# Add weights on the edges for visualization later on
weight_intra_G2 = 5
weight_inter_G2 = 0.5
weight_intra_G3 = 1.
weight_inter_G3 = 1.5

weightedG2 = networkx.Graph()
part_G2 = [G2.nodes[i]['block'] for i in range(N2)]

for node in G2.nodes():
    weightedG2.add_node(node)
for i, j in G2.edges():
    if part_G2[i] == part_G2[j]:
        weightedG2.add_edge(i, j, weight=weight_intra_G2)
    else:
        weightedG2.add_edge(i, j, weight=weight_inter_G2)

weightedG3 = networkx.Graph()
part_G3 = [G3.nodes[i]['block'] for i in range(N3)]

for node in G3.nodes():
    weightedG3.add_node(node)
for i, j in G3.edges():
    if part_G3[i] == part_G3[j]:
        weightedG3.add_edge(i, j, weight=weight_intra_G3)
    else:
        weightedG3.add_edge(i, j, weight=weight_inter_G3)

#############################################################################
#
# Compute their entropic-regularized semi-relaxed Gromov-Wasserstein divergences
# ---------------------------------------------

# 0) GW(C2, h2, C3, h3) for reference
OT, log = gromov_wasserstein(C2, C3, h2, h3, symmetric=True, log=True)
gw = log['gw_dist']

# 1) srGW_e(C2, h2, C3)
OT_23, log_23 = entropic_semirelaxed_gromov_wasserstein(
    C2, C3, h2, symmetric=True, epsilon=1., G0=None, log=True)
srgw_23 = log_23['srgw_dist']

# 2) srGW_e(C3, h3, C2)

OT_32, log_32 = entropic_semirelaxed_gromov_wasserstein(
    C3, C2, h3, symmetric=None, epsilon=1., G0=None, log=True)
srgw_32 = log_32['srgw_dist']

print('GW(C2, C3) = ', gw)
print('srGW_e(C2, h2, C3) = ', srgw_23)
print('srGW_e(C3, h3, C2) = ', srgw_32)


#############################################################################
#
# Visualization of the entropic-regularized semi-relaxed Gromov-Wasserstein matchings
# ---------------------------------------------
#
# We color nodes of the graph on the right - then project its node colors
# based on the optimal transport plan from the entropic srGW matching.
# We adjust the intensity of links across domains proportionaly to the mass
# sent, adding a minimal intensity of 0.1 if mass sent is not zero.


def draw_graph(G, C, nodes_color_part, Gweights=None,
               pos=None, edge_color='black', node_size=None,
               shiftx=0, seed=0):

    if (pos is None):
        pos = networkx.spring_layout(G, scale=1., seed=seed)

    if shiftx != 0:
        for k, v in pos.items():
            v[0] = v[0] + shiftx

    alpha_edge = 0.7
    width_edge = 1.8
    if Gweights is None:
        networkx.draw_networkx_edges(G, pos, width=width_edge, alpha=alpha_edge, edge_color=edge_color)
    else:
        # We make more visible connections between activated nodes
        n = len(Gweights)
        edgelist_activated = []
        edgelist_deactivated = []
        for i in range(n):
            for j in range(n):
                if Gweights[i] * Gweights[j] * C[i, j] > 0:
                    edgelist_activated.append((i, j))
                elif C[i, j] > 0:
                    edgelist_deactivated.append((i, j))

        networkx.draw_networkx_edges(G, pos, edgelist=edgelist_activated,
                                     width=width_edge, alpha=alpha_edge,
                                     edge_color=edge_color)
        networkx.draw_networkx_edges(G, pos, edgelist=edgelist_deactivated,
                                     width=width_edge, alpha=0.1,
                                     edge_color=edge_color)

    if Gweights is None:
        for node, node_color in enumerate(nodes_color_part):
            networkx.draw_networkx_nodes(G, pos, nodelist=[node],
                                         node_size=node_size, alpha=1,
                                         node_color=node_color)
    else:
        scaled_Gweights = Gweights / (0.5 * Gweights.max())
        nodes_size = node_size * scaled_Gweights
        for node, node_color in enumerate(nodes_color_part):
            networkx.draw_networkx_nodes(G, pos, nodelist=[node],
                                         node_size=nodes_size[node], alpha=1,
                                         node_color=node_color)
    return pos


def draw_transp_colored_srGW(G1, C1, G2, C2, part_G1,
                             p1, p2, T, pos1=None, pos2=None,
                             shiftx=4, switchx=False, node_size=70,
                             seed_G1=0, seed_G2=0):
    starting_color = 0
    # get graphs partition and their coloring
    part1 = part_G1.copy()
    unique_colors = ['C%s' % (starting_color + i) for i in np.unique(part1)]
    nodes_color_part1 = []
    for cluster in part1:
        nodes_color_part1.append(unique_colors[cluster])

    nodes_color_part2 = []
    # T: getting colors assignment from argmin of columns
    for i in range(len(G2.nodes())):
        j = np.argmax(T[:, i])
        nodes_color_part2.append(nodes_color_part1[j])
    pos1 = draw_graph(G1, C1, nodes_color_part1, Gweights=p1,
                      pos=pos1, node_size=node_size, shiftx=0, seed=seed_G1)
    pos2 = draw_graph(G2, C2, nodes_color_part2, Gweights=p2, pos=pos2,
                      node_size=node_size, shiftx=shiftx, seed=seed_G2)
    for k1, v1 in pos1.items():
        max_Tk1 = np.max(T[k1, :])
        for k2, v2 in pos2.items():
            if (T[k1, k2] > 0):
                pl.plot([pos1[k1][0], pos2[k2][0]],
                        [pos1[k1][1], pos2[k2][1]],
                        '-', lw=0.6, alpha=min(T[k1, k2] / max_Tk1 + 0.1, 1.),
                        color=nodes_color_part1[k1])
    return pos1, pos2


node_size = 40
fontsize = 10
seed_G2 = 0
seed_G3 = 4

pl.figure(1, figsize=(8, 2.5))
pl.clf()
pl.subplot(121)
pl.axis('off')
pl.axis
pl.title(r'$srGW_e(\mathbf{C_2},\mathbf{h_2},\mathbf{C_3}) =%s$' % (np.round(srgw_23, 3)), fontsize=fontsize)

hbar2 = OT_23.sum(axis=0)
pos1, pos2 = draw_transp_colored_srGW(
    weightedG2, C2, weightedG3, C3, part_G2, p1=None, p2=hbar2, T=OT_23,
    shiftx=1.5, node_size=node_size, seed_G1=seed_G2, seed_G2=seed_G3)
pl.subplot(122)
pl.axis('off')
hbar3 = OT_32.sum(axis=0)
pl.title(r'$srGW_e(\mathbf{C_3}, \mathbf{h_3},\mathbf{C_2}) =%s$' % (np.round(srgw_32, 3)), fontsize=fontsize)
pos1, pos2 = draw_transp_colored_srGW(
    weightedG3, C3, weightedG2, C2, part_G3, p1=None, p2=hbar3, T=OT_32,
    pos1=pos2, pos2=pos1, shiftx=3., node_size=node_size, seed_G1=0, seed_G2=0)
pl.tight_layout()

pl.show()

#############################################################################
#
# Add node features
# ---------------------------------------------

# We add node features with given mean - by clusters
# and inversely proportional to clusters' intra-connectivity

F2 = np.zeros((N2, 1))
for i, c in enumerate(part_G2):
    F2[i, 0] = np.random.normal(loc=c, scale=0.01)

F3 = np.zeros((N3, 1))
for i, c in enumerate(part_G3):
    F3[i, 0] = np.random.normal(loc=2. - c, scale=0.01)

#############################################################################
#
# Compute their semi-relaxed Fused Gromov-Wasserstein divergences
# ---------------------------------------------

alpha = 0.5
# Compute pairwise euclidean distance between node features
M = (F2 ** 2).dot(np.ones((1, N3))) + np.ones((N2, 1)).dot((F3 ** 2).T) - 2 * F2.dot(F3.T)

# 0) FGW_alpha(C2, F2, h2, C3, F3, h3) for reference

OT, log = fused_gromov_wasserstein(
    M, C2, C3, h2, h3, symmetric=True, alpha=alpha, log=True)
fgw = log['fgw_dist']

# 1) srFGW_e(C2, F2, h2, C3, F3)
OT_23, log_23 = entropic_semirelaxed_fused_gromov_wasserstein(
    M, C2, C3, h2, symmetric=True, epsilon=1., alpha=0.5, log=True, G0=None)
srfgw_23 = log_23['srfgw_dist']

# 2) srFGW(C3, F3, h3, C2, F2)

OT_32, log_32 = entropic_semirelaxed_fused_gromov_wasserstein(
    M.T, C3, C2, h3, symmetric=None, epsilon=1., alpha=alpha, log=True, G0=None)
srfgw_32 = log_32['srfgw_dist']

print('FGW(C2, F2, C3, F3) = ', fgw)
print(r'$srGW_e$(C2, F2, h2, C3, F3) = ', srfgw_23)
print(r'$srGW_e$(C3, F3, h3, C2, F2) = ', srfgw_32)

#############################################################################
#
# Visualization of the entropic semi-relaxed Fused Gromov-Wasserstein matchings
# ---------------------------------------------
#
# We color nodes of the graph on the right - then project its node colors
# based on the optimal transport plan from the srFGW matching
# NB: colors refer to clusters - not to node features

pl.figure(2, figsize=(8, 2.5))
pl.clf()
pl.subplot(121)
pl.axis('off')
pl.axis
pl.title(r'$srFGW_e(\mathbf{C_2},\mathbf{F_2},\mathbf{h_2},\mathbf{C_3},\mathbf{F_3}) =%s$' % (np.round(srfgw_23, 3)), fontsize=fontsize)

hbar2 = OT_23.sum(axis=0)
pos1, pos2 = draw_transp_colored_srGW(
    weightedG2, C2, weightedG3, C3, part_G2, p1=None, p2=hbar2, T=OT_23,
    shiftx=1.5, node_size=node_size, seed_G1=seed_G2, seed_G2=seed_G3)
pl.subplot(122)
pl.axis('off')
hbar3 = OT_32.sum(axis=0)
pl.title(r'$srFGW_e(\mathbf{C_3}, \mathbf{F_3}, \mathbf{h_3}, \mathbf{C_2}, \mathbf{F_2}) =%s$' % (np.round(srfgw_32, 3)), fontsize=fontsize)
pos1, pos2 = draw_transp_colored_srGW(
    weightedG3, C3, weightedG2, C2, part_G3, p1=None, p2=hbar3, T=OT_32,
    pos1=pos2, pos2=pos1, shiftx=3., node_size=node_size, seed_G1=0, seed_G2=0)
pl.tight_layout()

pl.show()