geographical_threshold_graph

geographical_threshold_graph(n, theta, alpha=2, dim=2, pos=None, weight=None, metric=None)[source]

Returns a geographical threshold graph.

The geographical threshold graph model places n nodes uniformly at random in a rectangular domain. Each node u is assigned a weight \(w_u\). Two nodes u and v are joined by an edge if

\[w_u + w_v \ge \theta r^{\alpha}\]

where r is the distance between u and v, and \(\theta\), \(\alpha\) are parameters.

Parameters:

  • n (int or iterable) – Number of nodes or iterable of nodes

  • theta (float) – Threshold value

  • alpha (float, optional) – Exponent of distance function

  • dim (int, optional) – Dimension of graph

  • pos (dict) – Node positions as a dictionary of tuples keyed by node.

  • weight (dict) – Node weights as a dictionary of numbers keyed by node.

  • metric (function) – A metric on vectors of numbers (represented as lists or tuples). This must be a function that accepts two lists (or tuples) as input and yields a number as output. The function must also satisfy the four requirements of a metric. Specifically, if d is the function and x, y, and z are vectors in the graph, then d must satisfy

    1. d*(*x, y) ≥ 0,
    2. d*(*x, y) = 0 if and only if x = y,
    3. d*(*x, y) = d*(*y, x),
    4. d*(*x, z) ≤ d*(*x, y) + d*(*y, z).

    If this argument is not specified, the Euclidean distance metric is used.
Returns:

A random geographic threshold graph, undirected and without self-loops.

Each node has a node attribute 'pos' that stores the position of that node in Euclidean space as provided by the pos keyword argument or, if pos was not provided, as generated by this function. Similarly, each node has a node attribute 'weight' that stores the weight of that node as provided or as generated.
Return type:

Graph

Examples

Specify an alternate distance metric using the metric keyword argument. For example, to use the “taxicab metric” instead of the default Euclidean metric:

>>> dist = lambda x, y: sum(abs(a - b) for a, b in zip(x, y))
>>> G = nx.geographical_threshold_graph(10, 0.1, metric=dist)

Notes

If weights are not specified they are assigned to nodes by drawing randomly from the exponential distribution with rate parameter \(\lambda=1\). To specify weights from a different distribution, use the weight keyword argument:

>>> import random
>>> n = 20
>>> w = {i: random.expovariate(5.0) for i in range(n)}
>>> G = nx.geographical_threshold_graph(20, 50, weight=w)

If node positions are not specified they are randomly assigned from the uniform distribution.

References

[1]Masuda, N., Miwa, H., Konno, N.: Geographical threshold graphs with small-world and scale-free properties. Physical Review E 71, 036108 (2005)
[2]Milan Bradonjić, Aric Hagberg and Allon G. Percus, Giant component and connectivity in geographical threshold graphs, in Algorithms and Models for the Web-Graph (WAW 2007), Antony Bonato and Fan Chung (Eds), pp. 209–216, 2007