# -*- 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 # # 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()