Generating Cluster Graphs

• Generating Cluster Graphs

Generating Cluster Graphs

This example shows how to find the communities in a graph, then contract each community into a single node using cluster_graph(). For this tutorial, we’ll use the Donald Knuth’s Les Miserables Network, which shows the coapperances of characters in the novel Les Miserables. The network can be obtained here.

import igraph as ig
import matplotlib.pyplot as plt

# Load the graph
g = ig.load("./lesmis/lesmis.gml")

First, let’s visualise the original communities, using community_edge_betweenness() to separate out vertices into clusters. (For a more focused tutorial on just visualising communities, check out Communities).

# Generate communities
communities = g.community_edge_betweenness()
communities = communities.as_clustering() # Convert into a VertexClustering for plotting

# Print them out
for i, community in enumerate(communities):
    print(f"Community {i}:")
    for v in community:
        print(f"\t{g.vs[v]['label']}")

# Set community colors
num_communities = len(communities)
palette1 = ig.RainbowPalette(n=num_communities)
for i, community in enumerate(communities):
    g.vs[community]["color"] = i
    community_edges = g.es.select(_within=community)
    community_edges["color"] = i

g.vs["label"] = ["\n\n" + label for label in g.vs["label"]] # Move the labels below the vertices

# Plot the communities
fig1, ax1 = plt.subplots()
ig.plot(
    communities,
    target=ax1,
    mark_groups=True,
    palette=palette1,
    vertex_size=0.1,
    edge_width=0.5,
)
fig1.set_size_inches(20, 20)
fig1.savefig("../figures/communities.png", dpi=200)

Now let’s try and contract the information down, using only a single vertex to represent each community. We start by defining attribute values for each of the nodes in the original graph.

# Assign x, y, and sizes for each node
layout = g.layout_kamada_kawai()
g.vs["x"], g.vs["y"] = list(zip(*layout))
g.vs["size"] = 1
g.es["size"] = 1

This is so we can define how each of these attributes get combined together when we call cluster_graph().

# Generate cluster graph
cluster_graph = communities.cluster_graph(
    combine_vertices={
        "x": "mean",
        "y": "mean",
        "color": "first",
        "size": "sum",
    },
    combine_edges={
        "size": "sum",
    },
)

Here, we take the mean of x and y values so that the nodes in the cluster graph are placed at the center of the original cluster’s position.

Note

mean, first, and sum are all built-in collapsing functions, along with prod, median, max, min, last, random. You can also define your own custom collapsing functions, which take in a list and return a single element representing the combined attribute value. For more details on igraph contraction, see contract_vertices()

Finally we plot out the graph using our calculated attributes:

# Plot the cluster graph
palette2 = ig.GradientPalette("gainsboro", "black")
g.es["color"] = [palette2.get(int(i)) for i in ig.rescale(cluster_graph.es["size"], (0, 255), clamp=True)]

fig2, ax2 = plt.subplots()
ig.plot(
    cluster_graph,
    target=ax2,
    palette=palette1,
    # set a minimum size on vertex_size, otherwise vertices are too small
    vertex_size=[max(0.2, size / 20) for size in cluster_graph.vs["size"]],
    edge_color=g.es["color"],
    edge_width=0.8,
)

# Add a legend
legend_handles = []
for i in range(num_communities):
    handle = ax2.scatter(
        [], [],
        s=100,
        facecolor=palette1.get(i),
        edgecolor="k",
        label=i,
    )
    legend_handles.append(handle)

ax2.legend(
    handles=legend_handles,
    title='Community:',
    bbox_to_anchor=(0, 1.0),
    bbox_transform=ax2.transAxes,
)

fig2.set_size_inches(10, 10)
fig2.savefig("../figures/cluster_graph.png", dpi=150)

The two figures are shown side by side here:

… and the final output of the communities we printed out are displayed below:

Community 0:
    Myriel
    Napoleon
    MlleBaptistine
    MmeMagloire
    CountessDeLo
    Geborand
    Champtercier
    Cravatte
    Count
    OldMan
Community 1:
    Labarre
    Valjean
    MmeDeR
    Isabeau
    Gervais
    Bamatabois
    Simplice
    Scaufflaire
    Woman1
    Judge
    Champmathieu
    Brevet
    Chenildieu
    Cochepaille
Community 2:
    Marguerite
    Tholomyes
    Listolier
    Fameuil
    Blacheville
    Favourite
    Dahlia
    Zephine
    Fantine
    Perpetue
Community 3:
    MmeThenardier
    Thenardier
    Javert
    Pontmercy
    Eponine
    Anzelma
    Gueulemer
    Babet
    Claquesous
    Montparnasse
    Brujon
Community 4:
    Cosette
    Woman2
    Gillenormand
    Magnon
    MlleGillenormand
    MmePontmercy
    MlleVaubois
    LtGillenormand
    BaronessT
    Toussaint
Community 5:
    Fauchelevent
    MotherInnocent
    Gribier
Community 6:
    Boulatruelle
Community 7:
    Jondrette
    MmeBurgon
Community 8:
    Gavroche
    Marius
    Mabeuf
    Enjolras
    Combeferre
    Prouvaire
    Feuilly
    Courfeyrac
    Bahorel
    Bossuet
    Joly
    Grantaire
    MmeHucheloup
Community 9:
    MotherPlutarch
Community 10:
    Child1
    Child2