Plot graphs’ barycenter using FGW

This example illustrates the computation barycenter of labeled graphs using FGW

Requires networkx >=2

Vayer Titouan, Chapel Laetitia, Flamary R{‘e}mi, Tavenard Romain

and Courty Nicolas

“Optimal Transport for structured data with application on graphs” International Conference on Machine Learning (ICML). 2019.

# Author: Titouan Vayer <>
# License: MIT License
import numpy as np
import matplotlib.pyplot as plt
import networkx as nx
import math
from scipy.sparse.csgraph import shortest_path
import matplotlib.colors as mcol
from matplotlib import cm
from ot.gromov import fgw_barycenters
def find_thresh(C, inf=0.5, sup=3, step=10):
    """ Trick to find the adequate thresholds from where value of the C matrix are considered close enough to say that nodes are connected
        Tthe threshold is found by a linesearch between values "inf" and "sup" with "step" thresholds tested.
        The optimal threshold is the one which minimizes the reconstruction error between the shortest_path matrix coming from the thresholded adjency matrix
        and the original matrix.
    C : ndarray, shape (n_nodes,n_nodes)
            The structure matrix to threshold
    inf : float
          The beginning of the linesearch
    sup : float
          The end of the linesearch
    step : integer
            Number of thresholds tested
    dist = []
    search = np.linspace(inf, sup, step)
    for thresh in search:
        Cprime = sp_to_adjency(C, 0, thresh)
        SC = shortest_path(Cprime, method='D')
        SC[SC == float('inf')] = 100
        dist.append(np.linalg.norm(SC - C))
    return search[np.argmin(dist)], dist

def sp_to_adjency(C, threshinf=0.2, threshsup=1.8):
    """ Thresholds the structure matrix in order to compute an adjency matrix.
    All values between threshinf and threshsup are considered representing connected nodes and set to 1. Else are set to 0
    C : ndarray, shape (n_nodes,n_nodes)
        The structure matrix to threshold
    threshinf : float
        The minimum value of distance from which the new value is set to 1
    threshsup : float
        The maximum value of distance from which the new value is set to 1
    C : ndarray, shape (n_nodes,n_nodes)
        The threshold matrix. Each element is in {0,1}
    H = np.zeros_like(C)
    np.fill_diagonal(H, np.diagonal(C))
    C = C - H
    C = np.minimum(np.maximum(C, threshinf), threshsup)
    C[C == threshsup] = 0
    C[C != 0] = 1

    return C

def build_noisy_circular_graph(N=20, mu=0, sigma=0.3, with_noise=False, structure_noise=False, p=None):
    """ Create a noisy circular graph
    g = nx.Graph()
    for i in range(N):
        noise = float(np.random.normal(mu, sigma, 1))
        if with_noise:
            g.add_node(i, attr_name=math.sin((2 * i * math.pi / N)) + noise)
            g.add_node(i, attr_name=math.sin(2 * i * math.pi / N))
        g.add_edge(i, i + 1)
        if structure_noise:
            randomint = np.random.randint(0, p)
            if randomint == 0:
                if i <= N - 3:
                    g.add_edge(i, i + 2)
                if i == N - 2:
                    g.add_edge(i, 0)
                if i == N - 1:
                    g.add_edge(i, 1)
    g.add_edge(N, 0)
    noise = float(np.random.normal(mu, sigma, 1))
    if with_noise:
        g.add_node(N, attr_name=math.sin((2 * N * math.pi / N)) + noise)
        g.add_node(N, attr_name=math.sin(2 * N * math.pi / N))
    return g

def graph_colors(nx_graph, vmin=0, vmax=7):
    cnorm = mcol.Normalize(vmin=vmin, vmax=vmax)
    cpick = cm.ScalarMappable(norm=cnorm, cmap='viridis')
    val_map = {}
    for k, v in nx.get_node_attributes(nx_graph, 'attr_name').items():
        val_map[k] = cpick.to_rgba(v)
    colors = []
    for node in nx_graph.nodes():
    return colors

Generate data

We build a dataset of noisy circular graphs. Noise is added on the structures by random connections and on the features by gaussian noise.

X0 = []
for k in range(9):
    X0.append(build_noisy_circular_graph(np.random.randint(15, 25), with_noise=True, structure_noise=True, p=3))

Plot data

plt.figure(figsize=(8, 10))
for i in range(len(X0)):
    plt.subplot(3, 3, i + 1)
    g = X0[i]
    pos = nx.kamada_kawai_layout(g)
    nx.draw(g, pos=pos, node_color=graph_colors(g, vmin=-1, vmax=1), with_labels=False, node_size=100)
plt.suptitle('Dataset of noisy graphs. Color indicates the label', fontsize=20)


/home/circleci/project/examples/ UserWarning: Matplotlib is currently using agg, which is a non-GUI backend, so cannot show the figure.

Barycenter computation

Features distances are the euclidean distances

Cs = [shortest_path(nx.adjacency_matrix(x)) for x in X0]
ps = [np.ones(len(x.nodes())) / len(x.nodes()) for x in X0]
Ys = [np.array([v for (k, v) in nx.get_node_attributes(x, 'attr_name').items()]).reshape(-1, 1) for x in X0]
lambdas = np.array([np.ones(len(Ys)) / len(Ys)]).ravel()
sizebary = 15  # we choose a barycenter with 15 nodes

A, C, log = fgw_barycenters(sizebary, Ys, Cs, ps, lambdas, alpha=0.95, log=True)

Plot Barycenter

bary = nx.from_numpy_matrix(sp_to_adjency(C, threshinf=0, threshsup=find_thresh(C, sup=100, step=100)[0]))
for i, v in enumerate(A.ravel()):
    bary.add_node(i, attr_name=v)
pos = nx.kamada_kawai_layout(bary)
nx.draw(bary, pos=pos, node_color=graph_colors(bary, vmin=-1, vmax=1), with_labels=False)
plt.suptitle('Barycenter', fontsize=20)


/home/circleci/project/examples/ UserWarning: Matplotlib is currently using agg, which is a non-GUI backend, so cannot show the figure.

Total running time of the script: ( 0 minutes 2.140 seconds)

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