Working with Betweenness Centrality#

The betweenness centrality of a graph is a measure of centrality based on shortest paths. For every pair of nodes in a connected graph, there is at least a single shortest path between the nodes such that the number of edges the path passes through is minimized. The betweenness centrality for a given graph node is the number of these shortest paths that pass through the node.

This is defined as:

\[c_B(v) =\sum_{s,t \in V} \frac{\sigma(s, t|v)}{\sigma(s, t)}\]

where \(V\) is the set of nodes, \(\sigma(s, t)\) is the number of shortest \((s, t)\) paths, and \(\sigma(s, t|v)\) is the number of those paths passing through some node \(v\) other than \(s, t\). If \(s = t\), \(\sigma(s, t) = 1\), and if \(v \in {s, t}\), \(\sigma(s, t|v) = 0\)

This tutorial will take you through the process of calculating the betweenness centrality of a graph and visualizing it.

Generate a Graph#

To start we need to generate a graph:

import rustworkx as rx
from rustworkx.visualization import mpl_draw

graph = rx.generators.hexagonal_lattice_graph(4, 4)

Calculate the Betweeness Centrality#

The betweenness_centrality() function can be used to calculate the betweenness centrality for each node in the graph.

import pprint

centrality = rx.betweenness_centrality(graph)
# Print the centrality value for the first 5 nodes in the graph
pprint.pprint({x: centrality[x] for x in range(5)})
{0: 0.007277212457600987,
 1: 0.02047046385621779,
 2: 0.07491079688119466,
 3: 0.04242324126690451,
 4: 0.09205321351482312}

The output of betweenness_centrality() is a CentralityMapping which is a custom mapping type that maps the node index to the centrality value as a float. This is a mapping and not a sequence because node indices (and edge indices too, which is not relevant here) are not guaranteed to be a contiguous sequence if there are removals.

Visualize the Betweenness Centrality#

Now that we’ve found the betweenness centrality for graph lets visualize it. We’ll color each node in the output visualization using its calculated centrality:

import matplotlib.pyplot as plt

# Generate a color list
colors = []
for node in graph.node_indices():
# Generate a visualization with a colorbar
plt.rcParams['figure.figsize'] = [15, 10]
ax = plt.gca()
sm =
plt.colorbar(sm, ax=ax)
plt.title("Betweenness Centrality of a 4 x 4 Hexagonal Lattice Graph")
mpl_draw(graph, node_color=colors, ax=ax)

Alternatively, you can use graphviz_draw():

from rustworkx.visualization import graphviz_draw
import matplotlib

# For graphviz visualization we need to assign the data payload for each
# node to its centrality value so that we can color based on this
for node, btw in centrality.items():
    graph[node] = btw

# Leverage matplotlib for color map
colormap = matplotlib.colormaps["magma"]
norm = matplotlib.colors.Normalize(

def color_node(node):
    rgba = matplotlib.colors.to_hex(colormap(norm(node)), keep_alpha=True)
    return {
        "color": f"\"{rgba}\"",
        "fillcolor": f"\"{rgba}\"",
        "style": "filled",
        "shape": "circle",
        "label": "%.2f" % node,

graphviz_draw(graph, node_attr_fn=color_node, method="neato")