non_randomness

non_randomness(G, k=None, weight='weight')

Compute the non-randomness of a graph.

The first value \(R_G\) is the sum of non-randomness values of all edges within the graph (where the non-randomness of an edge tends to be small when the two nodes linked by that edge are from two different communities).

The second value \(R_G^*\) is a relative measure that indicates to what extent G is different from a random graph in terms of probability. The closer it is to 0, the higher the likelihood the graph was generated by an Erdős–Rényi model.

Parameters:

GNetworkX graph

Graph must be undirected, connected, and without self-loops.

kint or None, optional (default=None)

The number of communities in G. If k is not set, the function uses a default community detection algorithm (label_propagation_communities()) to set it.

weightstring or None, optional (default=”weight”)

The name of an edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1, i.e., the graph is binary.

Returns:

(float, float) tuple

The first value is \(R_G\), the non-randomness of the graph, the second is \(R_G^*\), the relative non-randomness w.r.t. the Erdős–Rényi model.

Raises:

NetworkXNotImplemented

If the input graph is directed or a multigraph.

NetworkXException

If the input graph is not connected.

NetworkXError

If the input graph contains self-loops or has no edges.

ValueError

If k is not in \(\{1, \dots, n-1\}\), where \(n\) is the number of nodes, or if k is such that the computed edge probability \(p = \frac{2km}{n(n-k)}\) does not satisfy \(0 < p < 1\).

Notes

If a weight argument is passed, this algorithm will use the eigenvalues of the weighted adjacency matrix instead.

The output of this function corresponds to (4.4) and (4.5) in [1]. A lower value of \(R^*_G\) indicates a more random graph; one can think of \(1 - \Phi(R_G^*)\) as the similarity between the graph and a random graph, where \(\Phi(x)\) is the cumulative distribution function of the standard normal distribution.

Theorem 2 in [2] states that for any graph \(G\) with \(n\) nodes, \(m\) edges, and \(k\) communities, its non-randomness is bounded below by the non-randomness of an \(r\)-regular graph (a graph where each node has degree \(r\)), and bounded above by the non-randomness of an \(l\)-complete graph (a graph where each community is a clique of \(l\) nodes).

References

[1]

Xiaowei Ying and Xintao Wu, On Randomness Measures for Social Networks, SIAM International Conference on Data Mining. 2009 https://doi.org/10.1137/1.9781611972795.61

[2]

Ying, Xiaowei & Wu, Leting & Wu, Xintao. (2012). A Spectrum-Based Framework for Quantifying Randomness of Social Networks. IEEE Transactions on Knowledge and Data Engineering 23(12):1842–1856. https://dl.acm.org/doi/abs/10.1109/TKDE.2010.218

Examples

>>> G = nx.karate_club_graph()
>>> nr, nr_rd = nx.non_randomness(G, 2)
>>> nr, nr_rd = nx.non_randomness(G, 2, "weight")

When the number of communities k is not specified, label_propagation_communities() is used to compute it. This algorithm can give different results depending on the order of nodes and edges in the graph. For example, while the following graphs are identical, computing the non-randomness of each of them yields different results:

>>> G1, G2 = nx.Graph(), nx.Graph()
>>> G1.add_edges_from([(0, 1), (1, 2), (1, 3), (3, 4)])
>>> G2.add_edges_from([(0, 1), (1, 3), (1, 2), (3, 4)])
>>> [round(r, 6) for r in nx.non_randomness(G1)]
[-1.847759, -5.842437]
>>> [round(r, 6) for r in nx.non_randomness(G2)]
Traceback (most recent call last):
 ...
ValueError: invalid number of communities for graph with 5 nodes and 4 edges: 2

This is because the community detection algorithm finds 1 community in G1 and 2 communities in G2. This can be resolved by specifying the number of communities k:

>>> [round(r, 6) for r in nx.non_randomness(G2, k=1)]
[-1.847759, -5.842437]