networkx.generators.geometric.geographical_threshold_graph¶

geographical_threshold_graph
(n, theta, dim=2, pos=None, weight=None, metric=None, p_dist=None, seed=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
\[(w_u + w_v)h(r) \ge \theta\]where
r
is the distance betweenu
andv
, h(r) is a probability of connection as a function ofr
, and \(\theta\) as the threshold parameter. h(r) corresponds to the p_dist parameter. Parameters
n (int or iterable) – Number of nodes or iterable of nodes
theta (float) – Threshold value
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
\(d(x, y) \ge 0\),
\(d(x, y) = 0\) if and only if \(x = y\),
\(d(x, y) = d(y, x)\),
\(d(x, z) \le d(x, y) + d(y, z)\).
If this argument is not specified, the Euclidean distance metric is used.
p_dist (function, optional) – A probability density function computing the probability of connecting two nodes that are of distance, r, computed by metric. The probability density function,
p_dist
, must be any function that takes the metric value as input and outputs a single probability value between 01. The scipy.stats package has many probability distribution functions implemented and tools for custom probability distribution definitions [2], and passing the .pdf method of scipy.stats distributions can be used here. If the probability function,p_dist
, is not supplied, the default exponential function :math:r^{2}
is used.seed (integer, random_state, or None (default)) – Indicator of random number generation state. See Randomness.
 Returns
A random geographic threshold graph, undirected and without selfloops.
Each node has a node attribute
pos
that stores the position of that node in Euclidean space as provided by thepos
keyword argument or, ifpos
was not provided, as generated by this function. Similarly, each node has a node attributeweight
that stores the weight of that node as provided or as generated. Return type
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 smallworld and scalefree 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 WebGraph (WAW 2007), Antony Bonato and Fan Chung (Eds), pp. 209–216, 2007