betweenness_centrality

betweenness_centrality(G, k=None, normalized=True, weight=None, endpoints=False, seed=None)

Compute the shortest-path betweenness centrality for nodes.

Betweenness centrality of a node \(v\) is the sum of the fraction of all-pairs shortest paths that pass through \(v\).

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

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 \(s\) and \(t\). If \(s = t\), \(\sigma(s, t) = 1\), and if \(v \in \{s, t\}\), \(\sigma(s, t | v) = 0\) [2]. The denominator \(\sigma(s, t)\) is a normalization factor that can be turned off to get the raw path counts.

Parameters:

Ggraph

A NetworkX graph.

kint, optional (default=None)

If k is not None, use k sampled nodes as sources for the considered paths. The resulting sampled counts are then inflated to approximate betweenness. Higher values of k give better approximation. Must have k <= len(G).

normalizedbool, optional (default=True)

If True, the betweenness values are rescaled by dividing by the number of possible \((s, t)\)-pairs in the graph.

weightNone or string, optional (default=None)

If None, all edge weights are 1. Otherwise holds the name of the edge attribute used as weight. Weights are used to calculate weighted shortest paths, so they are interpreted as distances.

endpointsbool, optional (default=False)

If True, include the endpoints \(s\) and \(t\) in the shortest path counts. This is taken into account when rescaling the values.

seedinteger, random_state, or None (default)

Indicator of random number generation state. See Randomness. Note that this is only used if k is not None.

Returns:

nodesdict

Dictionary of nodes with betweenness centrality as the value.

See also

betweenness_centrality_subset

edge_betweenness_centrality

load_centrality

Notes

The algorithm is from Ulrik Brandes [1]. See [4] for the original first published version and [2] for details on algorithms for variations and related metrics.

For approximate betweenness calculations, set k to the number of sampled nodes (“pivots”) used as sources to estimate the betweenness values. The formula then sums over \(s\) is in these pivots, instead of over all nodes. The resulting sum is then inflated to approximate the full sum. For a discussion of how to choose k for efficiency, see [3].

For weighted graphs the edge weights must be greater than zero. Zero edge weights can produce an infinite number of equal length paths between pairs of nodes.

Directed graphs and undirected graphs count paths differently. In directed graphs, each pair of source-target nodes is considered separately in each direction, as the shortest paths can differ by direction. However, in undirected graphs, each pair of nodes is considered only once, as the shortest paths are symmetric. This means the normalization factor to divide by is \(N(N-1)\) for directed graphs and \(N(N-1)/2\) for undirected graphs, where \(N = n\) (the number of nodes) if endpoints are included and \(N = n-1\) otherwise.

This algorithm is not guaranteed to be correct if edge weights are floating point numbers. As a workaround you can use integer numbers by multiplying the relevant edge attributes by a convenient constant factor (e.g. 100) and converting to integers.

References

[1]

Ulrik Brandes: A Faster Algorithm for Betweenness Centrality. Journal of Mathematical Sociology 25(2):163–177, 2001. https://doi.org/10.1080/0022250X.2001.9990249

[2] (1,2)

Ulrik Brandes: On Variants of Shortest-Path Betweenness Centrality and their Generic Computation. Social Networks 30(2):136–145, 2008. https://doi.org/10.1016/j.socnet.2007.11.001

[3]

Ulrik Brandes and Christian Pich: Centrality Estimation in Large Networks. International Journal of Bifurcation and Chaos 17(7):2303–2318, 2007. https://dx.doi.org/10.1142/S0218127407018403

[4]

Linton C. Freeman: A set of measures of centrality based on betweenness. Sociometry 40: 35–41, 1977 https://doi.org/10.2307/3033543

Examples

Consider an undirected 3-path. Each pair of nodes has exactly one shortest path between them. Since the graph is undirected, only ordered pairs are counted. Of these (and when endpoints is False), none of the shortest paths pass through 0 and 2, and only the shortest path between 0 and 2 passes through 1. As such, the counts should be {0: 0, 1: 1, 2: 0}.

>>> G = nx.path_graph(3)
>>> nx.betweenness_centrality(G, normalized=False, endpoints=False)
{0: 0.0, 1: 1.0, 2: 0.0}

If endpoints is True, we also need to count endpoints as being on the path: \(\sigma(s, t | s) = \sigma(s, t | t) = \sigma(s, t)\). In our example, 0 is then part of two shortest paths (0 to 1 and 0 to 2); similarly, 2 is part of two shortest paths (0 to 2 and 1 to 2). 1 is part of all three shortest paths. This makes the new raw counts {0: 2, 1: 3, 2: 2}.

>>> nx.betweenness_centrality(G, normalized=False, endpoints=True)
{0: 2.0, 1: 3.0, 2: 2.0}

With normalization, the values are divided by the number of ordered \((s, t)\)-pairs. If we are not counting endpoints, there are \(n - 1\) possible choices for \(s\) (all except the node we are computing betweenness centrality for), which in turn leaves \(n - 2\) possible choices for \(t\) as \(s \ne t\). The total number of ordered pairs when endpoints is False is \((n - 1)(n - 2)/2 = 1\). If endpoints is True, there are \(n(n - 1)/2 = 3\) ordered \((s, t)\)-pairs to divide by.

>>> nx.betweenness_centrality(G, normalized=True, endpoints=False)
{0: 0.0, 1: 1.0, 2: 0.0}
>>> nx.betweenness_centrality(G, normalized=True, endpoints=True)
{0: 0.6666666666666666, 1: 1.0, 2: 0.6666666666666666}

If the graph is directed instead, we now need to consider \((s, t)\)-pairs in both directions. Our example becomes a directed 3-path. Without counting endpoints, we only have one path through 1 (0 to 2). This means the raw counts are {0: 0, 1: 1, 2: 0}.

>>> DG = nx.path_graph(3, create_using=nx.DiGraph)
>>> nx.betweenness_centrality(DG, normalized=False, endpoints=False)
{0: 0.0, 1: 1.0, 2: 0.0}

If we do include endpoints, the raw counts are {0: 2, 1: 3, 2: 2}.

>>> nx.betweenness_centrality(DG, normalized=False, endpoints=True)
{0: 2.0, 1: 3.0, 2: 2.0}

If we want to normalize directed betweenness centrality, the raw counts are normalized by the number of \((s, t)\)-pairs. There are \(n(n - 1)\) possible paths with endpoints and \((n - 1)(n - 2)\) without endpoints. In our example, that’s 6 with endpoints and 2 without endpoints.

>>> nx.betweenness_centrality(DG, normalized=True, endpoints=True)
{0: 0.3333333333333333, 1: 0.5, 2: 0.3333333333333333}
>>> nx.betweenness_centrality(DG, normalized=True, endpoints=False)
{0: 0.0, 1: 0.5, 2: 0.0}

Computing the full betweenness centrality can be costly. This function can also be used to compute approximate betweenness centrality by setting k. This only determines the number of source nodes to sample; all nodes are targets.

For simplicity, we only consider the case where endpoints are included in the counts. Since the partial sums only include k terms, instead of n, we multiply them by n / k, to approximate the full sum. As the sets of sources and targets are not the same anymore, paths have to be counted in a directed way. We thus count each as half a path. This ensures that the results approximate the standard betweenness for k == n.

For instance, in the undirected 3-path graph case, setting k = 2 (with seed=42) selects nodes 0 and 2 as sources. This means only shortest paths starting at these nodes are considered. The raw counts with endpoints are {0: 3, 1: 4, 2: 3}. Accounting for the partial sum and applying the undirectedness half-path correction, we get

>>> nx.betweenness_centrality(G, k=2, normalized=False, endpoints=True, seed=42)
{0: 2.25, 1: 3.0, 2: 2.25}

When normalizing, we instead want to divide by the total number of \((s, t)\)-pairs. This is \(k(n - 1)\) with endpoints.

>>> nx.betweenness_centrality(G, k=2, normalized=True, endpoints=True, seed=42)
{0: 0.75, 1: 1.0, 2: 0.75}
----

Additional backends implement this function

cugraphGPU-accelerated backend.

weight parameter is not yet supported, and RNG with seed may be different. Normalization when using k and endpoints=False does not currently match NetworkX. nx-cugraph was updated in 25.04 to match networkx/networkx#7908, but does not yet match networkx/networkx#7949. These changes were introduced in NetworkX 3.5. The next release of nx-cugraph, 25.06, will match NetworkX 3.5.

parallelA networkx backend that uses joblib to run graph algorithms in parallel. Find the nx-parallel’s configuration guide here

The parallel computation is implemented by dividing the nodes into chunks and computing betweenness centrality for each chunk concurrently.

Additional parameters:

get_chunksstr, function (default = “chunks”)

A function that takes in a list of all the nodes as input and returns an iterable node_chunks. The default chunking is done by slicing the nodes into n_jobs number of chunks.

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