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‎_downloads/059a63fc6cb655dcd2f1c9a61bb9d7ec/plot_OT_2D_samples.ipynb‎

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"name": "python",
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"nbconvert_exporter": "python",
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"version": "3.9.12"
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"version": "3.9.16"
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"nbformat": 4,

‎_downloads/07fcc19ba03226cd3d83d4e40ec44385/auto_examples_python.zip‎

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‎_downloads/0e15d87ff8ae3d0dd037bf7ddc80d8a7/plot_variance.ipynb‎

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"name": "python",
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"version": "3.9.16"
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‎_downloads/0e77b32207676919c68b7f28c6029b1b/plot_unmix_optim_torch.ipynb‎

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"name": "python",
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"version": "3.9.16"
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‎_downloads/0eb518a5cb2af5a1870fc1f120d1483e/plot_partial_wass_and_gromov.ipynb‎

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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.9.12"
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"version": "3.9.16"
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}
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‎_downloads/12ea2f8b4bdc97ebeef0837212c6d0c8/plot_semirelaxed_fgw.py‎

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# -*- coding: utf-8 -*-
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"""
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==========================
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Semi-relaxed (Fused) Gromov-Wasserstein example
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==========================
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This example is designed to show how to use the semi-relaxed Gromov-Wasserstein
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and the semi-relaxed Fused Gromov-Wasserstein divergences.
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sr(F)GW between two graphs G1 and G2 searches for a reweighing of the nodes of
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G2 at a minimal (F)GW distance from G1.
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First, we generate two graphs following Stochastic Block Models, then show
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how to compute their srGW matchings and illustrate them. These graphs are then
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endowed with node features and we follow the same process with srFGW.
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[48] Cédric Vincent-Cuaz, Rémi Flamary, Marco Corneli, Titouan Vayer, Nicolas Courty.
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"Semi-relaxed Gromov-Wasserstein divergence and applications on graphs"
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International Conference on Learning Representations (ICLR), 2021.
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"""
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# Author: Cédric Vincent-Cuaz <cedvincentcuaz@gmail.com>
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#
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# License: MIT License
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# sphinx_gallery_thumbnail_number = 1
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import numpy as np
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import matplotlib.pylab as pl
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from ot.gromov import semirelaxed_gromov_wasserstein, semirelaxed_fused_gromov_wasserstein, gromov_wasserstein, fused_gromov_wasserstein
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import networkx
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from networkx.generators.community import stochastic_block_model as sbm
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#############################################################################
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#
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# Generate two graphs following Stochastic Block models of 2 and 3 clusters.
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# ---------------------------------------------
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N2 = 20 # 2 communities
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N3 = 30 # 3 communities
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p2 = [[1., 0.1],
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[0.1, 0.9]]
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p3 = [[1., 0.1, 0.],
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[0.1, 0.95, 0.1],
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[0., 0.1, 0.9]]
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G2 = sbm(seed=0, sizes=[N2 // 2, N2 // 2], p=p2)
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G3 = sbm(seed=0, sizes=[N3 // 3, N3 // 3, N3 // 3], p=p3)
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C2 = networkx.to_numpy_array(G2)
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C3 = networkx.to_numpy_array(G3)
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h2 = np.ones(C2.shape[0]) / C2.shape[0]
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h3 = np.ones(C3.shape[0]) / C3.shape[0]
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# Add weights on the edges for visualization later on
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weight_intra_G2 = 5
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weight_inter_G2 = 0.5
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weight_intra_G3 = 1.
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weight_inter_G3 = 1.5
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weightedG2 = networkx.Graph()
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part_G2 = [G2.nodes[i]['block'] for i in range(N2)]
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for node in G2.nodes():
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weightedG2.add_node(node)
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for i, j in G2.edges():
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if part_G2[i] == part_G2[j]:
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weightedG2.add_edge(i, j, weight=weight_intra_G2)
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else:
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weightedG2.add_edge(i, j, weight=weight_inter_G2)
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weightedG3 = networkx.Graph()
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part_G3 = [G3.nodes[i]['block'] for i in range(N3)]
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for node in G3.nodes():
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weightedG3.add_node(node)
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for i, j in G3.edges():
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if part_G3[i] == part_G3[j]:
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weightedG3.add_edge(i, j, weight=weight_intra_G3)
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else:
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weightedG3.add_edge(i, j, weight=weight_inter_G3)
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#############################################################################
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#
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# Compute their semi-relaxed Gromov-Wasserstein divergences
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# ---------------------------------------------
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# 0) GW(C2, h2, C3, h3) for reference
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OT, log = gromov_wasserstein(C2, C3, h2, h3, symmetric=True, log=True)
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gw = log['gw_dist']
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# 1) srGW(C2, h2, C3)
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OT_23, log_23 = semirelaxed_gromov_wasserstein(C2, C3, h2, symmetric=True,
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log=True, G0=None)
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srgw_23 = log_23['srgw_dist']
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# 2) srGW(C3, h3, C2)
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OT_32, log_32 = semirelaxed_gromov_wasserstein(C3, C2, h3, symmetric=None,
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log=True, G0=OT.T)
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srgw_32 = log_32['srgw_dist']
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print('GW(C2, C3) = ', gw)
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print('srGW(C2, h2, C3) = ', srgw_23)
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print('srGW(C3, h3, C2) = ', srgw_32)
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#############################################################################
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#
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# Visualization of the semi-relaxed Gromov-Wasserstein matchings
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# ---------------------------------------------
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#
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# We color nodes of the graph on the right - then project its node colors
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# based on the optimal transport plan from the srGW matching
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def draw_graph(G, C, nodes_color_part, Gweights=None,
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pos=None, edge_color='black', node_size=None,
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shiftx=0, seed=0):
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if (pos is None):
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pos = networkx.spring_layout(G, scale=1., seed=seed)
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if shiftx != 0:
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for k, v in pos.items():
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v[0] = v[0] + shiftx
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alpha_edge = 0.7
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width_edge = 1.8
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if Gweights is None:
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networkx.draw_networkx_edges(G, pos, width=width_edge, alpha=alpha_edge, edge_color=edge_color)
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else:
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# We make more visible connections between activated nodes
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n = len(Gweights)
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edgelist_activated = []
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edgelist_deactivated = []
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for i in range(n):
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for j in range(n):
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if Gweights[i] * Gweights[j] * C[i, j] > 0:
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edgelist_activated.append((i, j))
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elif C[i, j] > 0:
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edgelist_deactivated.append((i, j))
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networkx.draw_networkx_edges(G, pos, edgelist=edgelist_activated,
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width=width_edge, alpha=alpha_edge,
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edge_color=edge_color)
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networkx.draw_networkx_edges(G, pos, edgelist=edgelist_deactivated,
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width=width_edge, alpha=0.1,
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edge_color=edge_color)
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if Gweights is None:
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for node, node_color in enumerate(nodes_color_part):
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networkx.draw_networkx_nodes(G, pos, nodelist=[node],
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node_size=node_size, alpha=1,
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node_color=node_color)
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else:
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scaled_Gweights = Gweights / (0.5 * Gweights.max())
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nodes_size = node_size * scaled_Gweights
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for node, node_color in enumerate(nodes_color_part):
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networkx.draw_networkx_nodes(G, pos, nodelist=[node],
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node_size=nodes_size[node], alpha=1,
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node_color=node_color)
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return pos
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def draw_transp_colored_srGW(G1, C1, G2, C2, part_G1,
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p1, p2, T, pos1=None, pos2=None,
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shiftx=4, switchx=False, node_size=70,
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seed_G1=0, seed_G2=0):
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starting_color = 0
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# get graphs partition and their coloring
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part1 = part_G1.copy()
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unique_colors = ['C%s' % (starting_color + i) for i in np.unique(part1)]
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nodes_color_part1 = []
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for cluster in part1:
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nodes_color_part1.append(unique_colors[cluster])
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nodes_color_part2 = []
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# T: getting colors assignment from argmin of columns
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for i in range(len(G2.nodes())):
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j = np.argmax(T[:, i])
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nodes_color_part2.append(nodes_color_part1[j])
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pos1 = draw_graph(G1, C1, nodes_color_part1, Gweights=p1,
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pos=pos1, node_size=node_size, shiftx=0, seed=seed_G1)
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pos2 = draw_graph(G2, C2, nodes_color_part2, Gweights=p2, pos=pos2,
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node_size=node_size, shiftx=shiftx, seed=seed_G2)
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for k1, v1 in pos1.items():
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for k2, v2 in pos2.items():
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if (T[k1, k2] > 0):
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pl.plot([pos1[k1][0], pos2[k2][0]],
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[pos1[k1][1], pos2[k2][1]],
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'-', lw=0.8, alpha=0.5,
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color=nodes_color_part1[k1])
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return pos1, pos2
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node_size = 40
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fontsize = 10
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seed_G2 = 0
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seed_G3 = 4
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pl.figure(1, figsize=(8, 2.5))
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pl.clf()
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pl.subplot(121)
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pl.axis('off')
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pl.axis
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pl.title(r'srGW$(\mathbf{C_2},\mathbf{h_2},\mathbf{C_3}) =%s$' % (np.round(srgw_23, 3)), fontsize=fontsize)
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hbar2 = OT_23.sum(axis=0)
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pos1, pos2 = draw_transp_colored_srGW(
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weightedG2, C2, weightedG3, C3, part_G2, p1=None, p2=hbar2, T=OT_23,
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shiftx=1.5, node_size=node_size, seed_G1=seed_G2, seed_G2=seed_G3)
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pl.subplot(122)
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pl.axis('off')
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hbar3 = OT_32.sum(axis=0)
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pl.title(r'srGW$(\mathbf{C_3}, \mathbf{h_3},\mathbf{C_2}) =%s$' % (np.round(srgw_32, 3)), fontsize=fontsize)
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pos1, pos2 = draw_transp_colored_srGW(
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weightedG3, C3, weightedG2, C2, part_G3, p1=None, p2=hbar3, T=OT_32,
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pos1=pos2, pos2=pos1, shiftx=3., node_size=node_size, seed_G1=0, seed_G2=0)
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pl.tight_layout()
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pl.show()
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#############################################################################
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#
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# Add node features
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# ---------------------------------------------
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# We add node features with given mean - by clusters
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# and inversely proportional to clusters' intra-connectivity
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F2 = np.zeros((N2, 1))
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for i, c in enumerate(part_G2):
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F2[i, 0] = np.random.normal(loc=c, scale=0.01)
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F3 = np.zeros((N3, 1))
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for i, c in enumerate(part_G3):
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F3[i, 0] = np.random.normal(loc=2. - c, scale=0.01)
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#############################################################################
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#
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# Compute their semi-relaxed Fused Gromov-Wasserstein divergences
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# ---------------------------------------------
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alpha = 0.5
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# Compute pairwise euclidean distance between node features
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M = (F2 ** 2).dot(np.ones((1, N3))) + np.ones((N2, 1)).dot((F3 ** 2).T) - 2 * F2.dot(F3.T)
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# 0) FGW_alpha(C2, F2, h2, C3, F3, h3) for reference
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OT, log = fused_gromov_wasserstein(
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M, C2, C3, h2, h3, symmetric=True, alpha=alpha, log=True)
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fgw = log['fgw_dist']
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# 1) srFGW(C2, F2, h2, C3, F3)
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OT_23, log_23 = semirelaxed_fused_gromov_wasserstein(
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M, C2, C3, h2, symmetric=True, alpha=0.5, log=True, G0=None)
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srfgw_23 = log_23['srfgw_dist']
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# 2) srFGW(C3, F3, h3, C2, F2)
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OT_32, log_32 = semirelaxed_fused_gromov_wasserstein(
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M.T, C3, C2, h3, symmetric=None, alpha=alpha, log=True, G0=None)
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srfgw_32 = log_32['srfgw_dist']
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print('FGW(C2, F2, C3, F3) = ', fgw)
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print('srGW(C2, F2, h2, C3, F3) = ', srfgw_23)
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print('srGW(C3, F3, h3, C2, F2) = ', srfgw_32)
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#############################################################################
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#
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# Visualization of the semi-relaxed Fused Gromov-Wasserstein matchings
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# ---------------------------------------------
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#
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# We color nodes of the graph on the right - then project its node colors
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# based on the optimal transport plan from the srFGW matching
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# NB: colors refer to clusters - not to node features
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pl.figure(2, figsize=(8, 2.5))
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pl.clf()
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pl.subplot(121)
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pl.axis('off')
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pl.axis
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pl.title(r'srFGW$(\mathbf{C_2},\mathbf{F_2},\mathbf{h_2},\mathbf{C_3},\mathbf{F_3}) =%s$' % (np.round(srfgw_23, 3)), fontsize=fontsize)
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hbar2 = OT_23.sum(axis=0)
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pos1, pos2 = draw_transp_colored_srGW(
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weightedG2, C2, weightedG3, C3, part_G2, p1=None, p2=hbar2, T=OT_23,
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shiftx=1.5, node_size=node_size, seed_G1=seed_G2, seed_G2=seed_G3)
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pl.subplot(122)
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pl.axis('off')
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hbar3 = OT_32.sum(axis=0)
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pl.title(r'srFGW$(\mathbf{C_3}, \mathbf{F_3}, \mathbf{h_3}, \mathbf{C_2}, \mathbf{F_2}) =%s$' % (np.round(srfgw_32, 3)), fontsize=fontsize)
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pos1, pos2 = draw_transp_colored_srGW(
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weightedG3, C3, weightedG2, C2, part_G3, p1=None, p2=hbar3, T=OT_32,
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pos1=pos2, pos2=pos1, shiftx=3., node_size=node_size, seed_G1=0, seed_G2=0)
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pl.tight_layout()
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pl.show()

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