.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/gromov/plot_entropic_semirelaxed_fgw.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_gromov_plot_entropic_semirelaxed_fgw.py: ========================== 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. .. GENERATED FROM PYTHON SOURCE LINES 21-34 .. code-block:: Python # Author: Cédric Vincent-Cuaz # # 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 .. GENERATED FROM PYTHON SOURCE LINES 35-37 Generate two graphs following Stochastic Block models of 2 and 3 clusters. --------------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 38-85 .. code-block:: Python 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) .. GENERATED FROM PYTHON SOURCE LINES 86-88 Compute their entropic-regularized semi-relaxed Gromov-Wasserstein divergences --------------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 89-110 .. code-block:: Python # 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) .. rst-class:: sphx-glr-script-out .. code-block:: none GW(C2, C3) = 0.255 srGW_e(C2, h2, C3) = 0.06000000014822844 srGW_e(C3, h3, C2) = 0.1577777782120945 .. GENERATED FROM PYTHON SOURCE LINES 111-118 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. .. GENERATED FROM PYTHON SOURCE LINES 119-229 .. code-block:: Python 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() .. image-sg:: /auto_examples/gromov/images/sphx_glr_plot_entropic_semirelaxed_fgw_001.png :alt: $srGW_e(\mathbf{C_2},\mathbf{h_2},\mathbf{C_3}) =0.06$, $srGW_e(\mathbf{C_3}, \mathbf{h_3},\mathbf{C_2}) =0.158$ :srcset: /auto_examples/gromov/images/sphx_glr_plot_entropic_semirelaxed_fgw_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 230-232 Add node features --------------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 233-245 .. code-block:: Python # 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) .. GENERATED FROM PYTHON SOURCE LINES 246-248 Compute their semi-relaxed Fused Gromov-Wasserstein divergences --------------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 249-275 .. code-block:: Python 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) .. rst-class:: sphx-glr-script-out .. code-block:: none FGW(C2, F2, C3, F3) = 0.38089508056745364 $srGW_e$(C2, F2, h2, C3, F3) = 0.0325729126416439 $srGW_e$(C3, F3, h3, C2, F2) = 0.24125701598915783 .. GENERATED FROM PYTHON SOURCE LINES 276-282 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 .. GENERATED FROM PYTHON SOURCE LINES 283-305 .. code-block:: Python 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() .. image-sg:: /auto_examples/gromov/images/sphx_glr_plot_entropic_semirelaxed_fgw_002.png :alt: $srFGW_e(\mathbf{C_2},\mathbf{F_2},\mathbf{h_2},\mathbf{C_3},\mathbf{F_3}) =0.033$, $srFGW_e(\mathbf{C_3}, \mathbf{F_3}, \mathbf{h_3}, \mathbf{C_2}, \mathbf{F_2}) =0.241$ :srcset: /auto_examples/gromov/images/sphx_glr_plot_entropic_semirelaxed_fgw_002.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 4.998 seconds) .. _sphx_glr_download_auto_examples_gromov_plot_entropic_semirelaxed_fgw.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_entropic_semirelaxed_fgw.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_entropic_semirelaxed_fgw.py ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_