805 lines
29 KiB
Python
805 lines
29 KiB
Python
"""Generators for geometric graphs.
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"""
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import math
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from bisect import bisect_left
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from itertools import accumulate, combinations, product
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import networkx as nx
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from networkx.utils import py_random_state
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__all__ = [
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"geometric_edges",
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"geographical_threshold_graph",
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"navigable_small_world_graph",
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"random_geometric_graph",
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"soft_random_geometric_graph",
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"thresholded_random_geometric_graph",
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"waxman_graph",
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]
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def euclidean(x, y):
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"""Returns the Euclidean distance between the vectors ``x`` and ``y``.
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Each of ``x`` and ``y`` can be any iterable of numbers. The
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iterables must be of the same length.
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.. deprecated:: 2.7
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"""
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import warnings
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msg = (
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"euclidean is deprecated and will be removed in 3.0."
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"Use math.dist(x, y) instead."
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)
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warnings.warn(msg, DeprecationWarning, stacklevel=2)
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return math.dist(x, y)
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def geometric_edges(G, radius, p=2):
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"""Returns edge list of node pairs within `radius` of each other.
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Parameters
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----------
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G : networkx graph
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The graph from which to generate the edge list. The nodes in `G` should
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have an attribute ``pos`` corresponding to the node position, which is
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used to compute the distance to other nodes.
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radius : scalar
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The distance threshold. Edges are included in the edge list if the
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distance between the two nodes is less than `radius`.
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p : scalar, default=2
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The `Minkowski distance metric
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<https://en.wikipedia.org/wiki/Minkowski_distance>`_ used to compute
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distances. The default value is 2, i.e. Euclidean distance.
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Returns
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-------
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edges : list
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List of edges whose distances are less than `radius`
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Notes
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-----
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Radius uses Minkowski distance metric `p`.
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If scipy is available, `scipy.spatial.cKDTree` is used to speed computation.
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Examples
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--------
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Create a graph with nodes that have a "pos" attribute representing 2D
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coordinates.
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>>> G = nx.Graph()
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>>> G.add_nodes_from([
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... (0, {"pos": (0, 0)}),
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... (1, {"pos": (3, 0)}),
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... (2, {"pos": (8, 0)}),
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... ])
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>>> nx.geometric_edges(G, radius=1)
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[]
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>>> nx.geometric_edges(G, radius=4)
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[(0, 1)]
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>>> nx.geometric_edges(G, radius=6)
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[(0, 1), (1, 2)]
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>>> nx.geometric_edges(G, radius=9)
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[(0, 1), (0, 2), (1, 2)]
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"""
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# Input validation - every node must have a "pos" attribute
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for n, pos in G.nodes(data="pos"):
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if pos is None:
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raise nx.NetworkXError(
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f"All nodes in `G` must have a 'pos' attribute. Check node {n}"
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)
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# NOTE: See _geometric_edges for the actual implementation. The reason this
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# is split into two functions is to avoid the overhead of input validation
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# every time the function is called internally in one of the other
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# geometric generators
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return _geometric_edges(G, radius, p)
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def _geometric_edges(G, radius, p=2):
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"""
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Implements `geometric_edges` without input validation. See `geometric_edges`
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for complete docstring.
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"""
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nodes_pos = G.nodes(data="pos")
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try:
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import scipy as sp
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import scipy.spatial # call as sp.spatial
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except ImportError:
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# no scipy KDTree so compute by for-loop
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radius_p = radius**p
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edges = [
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(u, v)
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for (u, pu), (v, pv) in combinations(nodes_pos, 2)
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if sum(abs(a - b) ** p for a, b in zip(pu, pv)) <= radius_p
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]
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return edges
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# scipy KDTree is available
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nodes, coords = list(zip(*nodes_pos))
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kdtree = sp.spatial.cKDTree(coords) # Cannot provide generator.
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edge_indexes = kdtree.query_pairs(radius, p)
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edges = [(nodes[u], nodes[v]) for u, v in sorted(edge_indexes)]
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return edges
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@py_random_state(5)
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def random_geometric_graph(n, radius, dim=2, pos=None, p=2, seed=None):
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"""Returns a random geometric graph in the unit cube of dimensions `dim`.
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The random geometric graph model places `n` nodes uniformly at
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random in the unit cube. Two nodes are joined by an edge if the
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distance between the nodes is at most `radius`.
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Edges are determined using a KDTree when SciPy is available.
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This reduces the time complexity from $O(n^2)$ to $O(n)$.
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Parameters
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----------
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n : int or iterable
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Number of nodes or iterable of nodes
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radius: float
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Distance threshold value
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dim : int, optional
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Dimension of graph
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pos : dict, optional
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A dictionary keyed by node with node positions as values.
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p : float, optional
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Which Minkowski distance metric to use. `p` has to meet the condition
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``1 <= p <= infinity``.
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If this argument is not specified, the :math:`L^2` metric
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(the Euclidean distance metric), p = 2 is used.
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This should not be confused with the `p` of an Erdős-Rényi random
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graph, which represents probability.
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seed : integer, random_state, or None (default)
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Indicator of random number generation state.
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See :ref:`Randomness<randomness>`.
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Returns
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-------
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Graph
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A random geometric graph, undirected and without self-loops.
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Each node has a node attribute ``'pos'`` that stores the
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position of that node in Euclidean space as provided by the
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``pos`` keyword argument or, if ``pos`` was not provided, as
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generated by this function.
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Examples
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--------
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Create a random geometric graph on twenty nodes where nodes are joined by
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an edge if their distance is at most 0.1::
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>>> G = nx.random_geometric_graph(20, 0.1)
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Notes
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-----
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This uses a *k*-d tree to build the graph.
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The `pos` keyword argument can be used to specify node positions so you
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can create an arbitrary distribution and domain for positions.
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For example, to use a 2D Gaussian distribution of node positions with mean
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(0, 0) and standard deviation 2::
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>>> import random
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>>> n = 20
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>>> pos = {i: (random.gauss(0, 2), random.gauss(0, 2)) for i in range(n)}
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>>> G = nx.random_geometric_graph(n, 0.2, pos=pos)
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References
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----------
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.. [1] Penrose, Mathew, *Random Geometric Graphs*,
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Oxford Studies in Probability, 5, 2003.
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"""
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# TODO Is this function just a special case of the geographical
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# threshold graph?
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#
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# half_radius = {v: radius / 2 for v in n}
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# return geographical_threshold_graph(nodes, theta=1, alpha=1,
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# weight=half_radius)
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#
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G = nx.empty_graph(n)
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# If no positions are provided, choose uniformly random vectors in
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# Euclidean space of the specified dimension.
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if pos is None:
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pos = {v: [seed.random() for i in range(dim)] for v in G}
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nx.set_node_attributes(G, pos, "pos")
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G.add_edges_from(_geometric_edges(G, radius, p))
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return G
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@py_random_state(6)
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def soft_random_geometric_graph(
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n, radius, dim=2, pos=None, p=2, p_dist=None, seed=None
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):
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r"""Returns a soft random geometric graph in the unit cube.
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The soft random geometric graph [1] model places `n` nodes uniformly at
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random in the unit cube in dimension `dim`. Two nodes of distance, `dist`,
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computed by the `p`-Minkowski distance metric are joined by an edge with
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probability `p_dist` if the computed distance metric value of the nodes
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is at most `radius`, otherwise they are not joined.
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Edges within `radius` of each other are determined using a KDTree when
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SciPy is available. This reduces the time complexity from :math:`O(n^2)`
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to :math:`O(n)`.
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Parameters
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----------
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n : int or iterable
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Number of nodes or iterable of nodes
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radius: float
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Distance threshold value
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dim : int, optional
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Dimension of graph
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pos : dict, optional
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A dictionary keyed by node with node positions as values.
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p : float, optional
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Which Minkowski distance metric to use.
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`p` has to meet the condition ``1 <= p <= infinity``.
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If this argument is not specified, the :math:`L^2` metric
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(the Euclidean distance metric), p = 2 is used.
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This should not be confused with the `p` of an Erdős-Rényi random
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graph, which represents probability.
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p_dist : function, optional
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A probability density function computing the probability of
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connecting two nodes that are of distance, dist, computed by the
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Minkowski distance metric. The probability density function, `p_dist`,
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must be any function that takes the metric value as input
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and outputs a single probability value between 0-1. The scipy.stats
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package has many probability distribution functions implemented and
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tools for custom probability distribution definitions [2], and passing
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the .pdf method of scipy.stats distributions can be used here. If the
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probability function, `p_dist`, is not supplied, the default function
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is an exponential distribution with rate parameter :math:`\lambda=1`.
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seed : integer, random_state, or None (default)
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Indicator of random number generation state.
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See :ref:`Randomness<randomness>`.
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Returns
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-------
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Graph
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A soft random geometric graph, undirected and without self-loops.
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Each node has a node attribute ``'pos'`` that stores the
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position of that node in Euclidean space as provided by the
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``pos`` keyword argument or, if ``pos`` was not provided, as
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generated by this function.
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Examples
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--------
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Default Graph:
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G = nx.soft_random_geometric_graph(50, 0.2)
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Custom Graph:
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Create a soft random geometric graph on 100 uniformly distributed nodes
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where nodes are joined by an edge with probability computed from an
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exponential distribution with rate parameter :math:`\lambda=1` if their
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Euclidean distance is at most 0.2.
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Notes
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-----
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This uses a *k*-d tree to build the graph.
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The `pos` keyword argument can be used to specify node positions so you
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can create an arbitrary distribution and domain for positions.
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For example, to use a 2D Gaussian distribution of node positions with mean
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(0, 0) and standard deviation 2
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The scipy.stats package can be used to define the probability distribution
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with the .pdf method used as `p_dist`.
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::
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>>> import random
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>>> import math
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>>> n = 100
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>>> pos = {i: (random.gauss(0, 2), random.gauss(0, 2)) for i in range(n)}
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>>> p_dist = lambda dist: math.exp(-dist)
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>>> G = nx.soft_random_geometric_graph(n, 0.2, pos=pos, p_dist=p_dist)
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References
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----------
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.. [1] Penrose, Mathew D. "Connectivity of soft random geometric graphs."
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The Annals of Applied Probability 26.2 (2016): 986-1028.
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.. [2] scipy.stats -
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https://docs.scipy.org/doc/scipy/reference/tutorial/stats.html
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"""
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G = nx.empty_graph(n)
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G.name = f"soft_random_geometric_graph({n}, {radius}, {dim})"
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# If no positions are provided, choose uniformly random vectors in
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# Euclidean space of the specified dimension.
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if pos is None:
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pos = {v: [seed.random() for i in range(dim)] for v in G}
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nx.set_node_attributes(G, pos, "pos")
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# if p_dist function not supplied the default function is an exponential
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# distribution with rate parameter :math:`\lambda=1`.
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if p_dist is None:
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def p_dist(dist):
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return math.exp(-dist)
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def should_join(edge):
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u, v = edge
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dist = (sum(abs(a - b) ** p for a, b in zip(pos[u], pos[v]))) ** (1 / p)
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return seed.random() < p_dist(dist)
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G.add_edges_from(filter(should_join, _geometric_edges(G, radius, p)))
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return G
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@py_random_state(7)
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def geographical_threshold_graph(
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n, theta, dim=2, pos=None, weight=None, metric=None, p_dist=None, seed=None
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):
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r"""Returns a geographical threshold graph.
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The geographical threshold graph model places $n$ nodes uniformly at
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random in a rectangular domain. Each node $u$ is assigned a weight
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$w_u$. Two nodes $u$ and $v$ are joined by an edge if
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.. math::
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(w_u + w_v)p_{dist}(r) \ge \theta
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where `r` is the distance between `u` and `v`, `p_dist` is any function of
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`r`, and :math:`\theta` as the threshold parameter. `p_dist` is used to
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give weight to the distance between nodes when deciding whether or not
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they should be connected. The larger `p_dist` is, the more prone nodes
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separated by `r` are to be connected, and vice versa.
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Parameters
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----------
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n : int or iterable
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Number of nodes or iterable of nodes
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theta: float
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Threshold value
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dim : int, optional
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Dimension of graph
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pos : dict
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Node positions as a dictionary of tuples keyed by node.
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weight : dict
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Node weights as a dictionary of numbers keyed by node.
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metric : function
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A metric on vectors of numbers (represented as lists or
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tuples). This must be a function that accepts two lists (or
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tuples) as input and yields a number as output. The function
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must also satisfy the four requirements of a `metric`_.
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Specifically, if $d$ is the function and $x$, $y$,
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and $z$ are vectors in the graph, then $d$ must satisfy
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1. $d(x, y) \ge 0$,
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2. $d(x, y) = 0$ if and only if $x = y$,
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3. $d(x, y) = d(y, x)$,
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4. $d(x, z) \le d(x, y) + d(y, z)$.
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If this argument is not specified, the Euclidean distance metric is
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used.
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.. _metric: https://en.wikipedia.org/wiki/Metric_%28mathematics%29
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p_dist : function, optional
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Any function used to give weight to the distance between nodes when
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deciding whether or not they should be connected. `p_dist` was
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originally conceived as a probability density function giving the
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probability of connecting two nodes that are of metric distance `r`
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apart. The implementation here allows for more arbitrary definitions
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of `p_dist` that do not need to correspond to valid probability
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density functions. The :mod:`scipy.stats` package has many
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probability density functions implemented and tools for custom
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probability density definitions, and passing the ``.pdf`` method of
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scipy.stats distributions can be used here. If ``p_dist=None``
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(the default), the exponential function :math:`r^{-2}` is used.
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seed : integer, random_state, or None (default)
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Indicator of random number generation state.
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See :ref:`Randomness<randomness>`.
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Returns
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-------
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Graph
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A random geographic threshold graph, undirected and without
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self-loops.
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Each node has a node attribute ``pos`` that stores the
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position of that node in Euclidean space as provided by the
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``pos`` keyword argument or, if ``pos`` was not provided, as
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generated by this function. Similarly, each node has a node
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attribute ``weight`` that stores the weight of that node as
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provided or as generated.
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Examples
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--------
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Specify an alternate distance metric using the ``metric`` keyword
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argument. For example, to use the `taxicab metric`_ instead of the
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default `Euclidean metric`_::
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>>> dist = lambda x, y: sum(abs(a - b) for a, b in zip(x, y))
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>>> G = nx.geographical_threshold_graph(10, 0.1, metric=dist)
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.. _taxicab metric: https://en.wikipedia.org/wiki/Taxicab_geometry
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.. _Euclidean metric: https://en.wikipedia.org/wiki/Euclidean_distance
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Notes
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-----
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If weights are not specified they are assigned to nodes by drawing randomly
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from the exponential distribution with rate parameter $\lambda=1$.
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To specify weights from a different distribution, use the `weight` keyword
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argument::
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>>> import random
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>>> n = 20
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>>> w = {i: random.expovariate(5.0) for i in range(n)}
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>>> G = nx.geographical_threshold_graph(20, 50, weight=w)
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If node positions are not specified they are randomly assigned from the
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uniform distribution.
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References
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----------
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.. [1] Masuda, N., Miwa, H., Konno, N.:
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Geographical threshold graphs with small-world and scale-free
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properties.
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Physical Review E 71, 036108 (2005)
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.. [2] Milan Bradonjić, Aric Hagberg and Allon G. Percus,
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Giant component and connectivity in geographical threshold graphs,
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in Algorithms and Models for the Web-Graph (WAW 2007),
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Antony Bonato and Fan Chung (Eds), pp. 209--216, 2007
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"""
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G = nx.empty_graph(n)
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# If no weights are provided, choose them from an exponential
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# distribution.
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if weight is None:
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weight = {v: seed.expovariate(1) for v in G}
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# If no positions are provided, choose uniformly random vectors in
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# Euclidean space of the specified dimension.
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if pos is None:
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pos = {v: [seed.random() for i in range(dim)] for v in G}
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# If no distance metric is provided, use Euclidean distance.
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if metric is None:
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metric = math.dist
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nx.set_node_attributes(G, weight, "weight")
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nx.set_node_attributes(G, pos, "pos")
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# if p_dist is not supplied, use default r^-2
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if p_dist is None:
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def p_dist(r):
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return r**-2
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# Returns ``True`` if and only if the nodes whose attributes are
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# ``du`` and ``dv`` should be joined, according to the threshold
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# condition.
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def should_join(pair):
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u, v = pair
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u_pos, v_pos = pos[u], pos[v]
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u_weight, v_weight = weight[u], weight[v]
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return (u_weight + v_weight) * p_dist(metric(u_pos, v_pos)) >= theta
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G.add_edges_from(filter(should_join, combinations(G, 2)))
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return G
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@py_random_state(6)
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def waxman_graph(
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n, beta=0.4, alpha=0.1, L=None, domain=(0, 0, 1, 1), metric=None, seed=None
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):
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r"""Returns a Waxman random graph.
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The Waxman random graph model places `n` nodes uniformly at random
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in a rectangular domain. Each pair of nodes at distance `d` is
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joined by an edge with probability
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.. math::
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p = \beta \exp(-d / \alpha L).
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This function implements both Waxman models, using the `L` keyword
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argument.
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* Waxman-1: if `L` is not specified, it is set to be the maximum distance
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between any pair of nodes.
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* Waxman-2: if `L` is specified, the distance between a pair of nodes is
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chosen uniformly at random from the interval `[0, L]`.
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Parameters
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----------
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n : int or iterable
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Number of nodes or iterable of nodes
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beta: float
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Model parameter
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alpha: float
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Model parameter
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L : float, optional
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Maximum distance between nodes. If not specified, the actual distance
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is calculated.
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domain : four-tuple of numbers, optional
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Domain size, given as a tuple of the form `(x_min, y_min, x_max,
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y_max)`.
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metric : function
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A metric on vectors of numbers (represented as lists or
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tuples). This must be a function that accepts two lists (or
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tuples) as input and yields a number as output. The function
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must also satisfy the four requirements of a `metric`_.
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Specifically, if $d$ is the function and $x$, $y$,
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and $z$ are vectors in the graph, then $d$ must satisfy
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1. $d(x, y) \ge 0$,
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2. $d(x, y) = 0$ if and only if $x = y$,
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3. $d(x, y) = d(y, x)$,
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4. $d(x, z) \le d(x, y) + d(y, z)$.
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If this argument is not specified, the Euclidean distance metric is
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used.
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.. _metric: https://en.wikipedia.org/wiki/Metric_%28mathematics%29
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seed : integer, random_state, or None (default)
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Indicator of random number generation state.
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See :ref:`Randomness<randomness>`.
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Returns
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-------
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Graph
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A random Waxman graph, undirected and without self-loops. Each
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node has a node attribute ``'pos'`` that stores the position of
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that node in Euclidean space as generated by this function.
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Examples
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--------
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Specify an alternate distance metric using the ``metric`` keyword
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argument. For example, to use the "`taxicab metric`_" instead of the
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default `Euclidean metric`_::
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>>> dist = lambda x, y: sum(abs(a - b) for a, b in zip(x, y))
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>>> G = nx.waxman_graph(10, 0.5, 0.1, metric=dist)
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.. _taxicab metric: https://en.wikipedia.org/wiki/Taxicab_geometry
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.. _Euclidean metric: https://en.wikipedia.org/wiki/Euclidean_distance
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Notes
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-----
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Starting in NetworkX 2.0 the parameters alpha and beta align with their
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usual roles in the probability distribution. In earlier versions their
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positions in the expression were reversed. Their position in the calling
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sequence reversed as well to minimize backward incompatibility.
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References
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----------
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.. [1] B. M. Waxman, *Routing of multipoint connections*.
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IEEE J. Select. Areas Commun. 6(9),(1988) 1617--1622.
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"""
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G = nx.empty_graph(n)
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(xmin, ymin, xmax, ymax) = domain
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# Each node gets a uniformly random position in the given rectangle.
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pos = {v: (seed.uniform(xmin, xmax), seed.uniform(ymin, ymax)) for v in G}
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nx.set_node_attributes(G, pos, "pos")
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# If no distance metric is provided, use Euclidean distance.
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if metric is None:
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metric = math.dist
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# If the maximum distance L is not specified (that is, we are in the
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# Waxman-1 model), then find the maximum distance between any pair
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# of nodes.
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#
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# In the Waxman-1 model, join nodes randomly based on distance. In
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# the Waxman-2 model, join randomly based on random l.
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if L is None:
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L = max(metric(x, y) for x, y in combinations(pos.values(), 2))
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def dist(u, v):
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return metric(pos[u], pos[v])
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else:
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def dist(u, v):
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return seed.random() * L
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# `pair` is the pair of nodes to decide whether to join.
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def should_join(pair):
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return seed.random() < beta * math.exp(-dist(*pair) / (alpha * L))
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G.add_edges_from(filter(should_join, combinations(G, 2)))
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return G
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@py_random_state(5)
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def navigable_small_world_graph(n, p=1, q=1, r=2, dim=2, seed=None):
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r"""Returns a navigable small-world graph.
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A navigable small-world graph is a directed grid with additional long-range
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connections that are chosen randomly.
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[...] we begin with a set of nodes [...] that are identified with the set
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of lattice points in an $n \times n$ square,
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$\{(i, j): i \in \{1, 2, \ldots, n\}, j \in \{1, 2, \ldots, n\}\}$,
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and we define the *lattice distance* between two nodes $(i, j)$ and
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$(k, l)$ to be the number of "lattice steps" separating them:
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$d((i, j), (k, l)) = |k - i| + |l - j|$.
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For a universal constant $p >= 1$, the node $u$ has a directed edge to
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every other node within lattice distance $p$---these are its *local
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contacts*. For universal constants $q >= 0$ and $r >= 0$ we also
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construct directed edges from $u$ to $q$ other nodes (the *long-range
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contacts*) using independent random trials; the $i$th directed edge from
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$u$ has endpoint $v$ with probability proportional to $[d(u,v)]^{-r}$.
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-- [1]_
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Parameters
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----------
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n : int
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The length of one side of the lattice; the number of nodes in
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the graph is therefore $n^2$.
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p : int
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The diameter of short range connections. Each node is joined with every
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other node within this lattice distance.
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q : int
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The number of long-range connections for each node.
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r : float
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Exponent for decaying probability of connections. The probability of
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connecting to a node at lattice distance $d$ is $1/d^r$.
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dim : int
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Dimension of grid
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seed : integer, random_state, or None (default)
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Indicator of random number generation state.
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See :ref:`Randomness<randomness>`.
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References
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----------
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.. [1] J. Kleinberg. The small-world phenomenon: An algorithmic
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perspective. Proc. 32nd ACM Symposium on Theory of Computing, 2000.
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"""
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if p < 1:
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raise nx.NetworkXException("p must be >= 1")
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if q < 0:
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raise nx.NetworkXException("q must be >= 0")
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if r < 0:
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raise nx.NetworkXException("r must be >= 1")
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G = nx.DiGraph()
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nodes = list(product(range(n), repeat=dim))
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for p1 in nodes:
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probs = [0]
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for p2 in nodes:
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if p1 == p2:
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continue
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d = sum((abs(b - a) for a, b in zip(p1, p2)))
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if d <= p:
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G.add_edge(p1, p2)
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probs.append(d**-r)
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cdf = list(accumulate(probs))
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for _ in range(q):
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target = nodes[bisect_left(cdf, seed.uniform(0, cdf[-1]))]
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G.add_edge(p1, target)
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return G
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@py_random_state(7)
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def thresholded_random_geometric_graph(
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n, radius, theta, dim=2, pos=None, weight=None, p=2, seed=None
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):
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r"""Returns a thresholded random geometric graph in the unit cube.
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The thresholded random geometric graph [1] model places `n` nodes
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uniformly at random in the unit cube of dimensions `dim`. Each node
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`u` is assigned a weight :math:`w_u`. Two nodes `u` and `v` are
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joined by an edge if they are within the maximum connection distance,
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`radius` computed by the `p`-Minkowski distance and the summation of
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weights :math:`w_u` + :math:`w_v` is greater than or equal
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to the threshold parameter `theta`.
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Edges within `radius` of each other are determined using a KDTree when
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SciPy is available. This reduces the time complexity from :math:`O(n^2)`
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to :math:`O(n)`.
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Parameters
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----------
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n : int or iterable
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Number of nodes or iterable of nodes
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radius: float
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Distance threshold value
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theta: float
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Threshold value
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dim : int, optional
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Dimension of graph
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pos : dict, optional
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A dictionary keyed by node with node positions as values.
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weight : dict, optional
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Node weights as a dictionary of numbers keyed by node.
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p : float, optional (default 2)
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Which Minkowski distance metric to use. `p` has to meet the condition
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``1 <= p <= infinity``.
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If this argument is not specified, the :math:`L^2` metric
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(the Euclidean distance metric), p = 2 is used.
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This should not be confused with the `p` of an Erdős-Rényi random
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graph, which represents probability.
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seed : integer, random_state, or None (default)
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Indicator of random number generation state.
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See :ref:`Randomness<randomness>`.
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Returns
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-------
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Graph
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A thresholded random geographic graph, undirected and without
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self-loops.
|
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|
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Each node has a node attribute ``'pos'`` that stores the
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position of that node in Euclidean space as provided by the
|
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``pos`` keyword argument or, if ``pos`` was not provided, as
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generated by this function. Similarly, each node has a nodethre
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attribute ``'weight'`` that stores the weight of that node as
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provided or as generated.
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Examples
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--------
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Default Graph:
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G = nx.thresholded_random_geometric_graph(50, 0.2, 0.1)
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Custom Graph:
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Create a thresholded random geometric graph on 50 uniformly distributed
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nodes where nodes are joined by an edge if their sum weights drawn from
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a exponential distribution with rate = 5 are >= theta = 0.1 and their
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Euclidean distance is at most 0.2.
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Notes
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-----
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This uses a *k*-d tree to build the graph.
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The `pos` keyword argument can be used to specify node positions so you
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can create an arbitrary distribution and domain for positions.
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For example, to use a 2D Gaussian distribution of node positions with mean
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(0, 0) and standard deviation 2
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If weights are not specified they are assigned to nodes by drawing randomly
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from the exponential distribution with rate parameter :math:`\lambda=1`.
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To specify weights from a different distribution, use the `weight` keyword
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argument::
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::
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>>> import random
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>>> import math
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>>> n = 50
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>>> pos = {i: (random.gauss(0, 2), random.gauss(0, 2)) for i in range(n)}
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>>> w = {i: random.expovariate(5.0) for i in range(n)}
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>>> G = nx.thresholded_random_geometric_graph(n, 0.2, 0.1, 2, pos, w)
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|
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References
|
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----------
|
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.. [1] http://cole-maclean.github.io/blog/files/thesis.pdf
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|
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"""
|
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G = nx.empty_graph(n)
|
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G.name = f"thresholded_random_geometric_graph({n}, {radius}, {theta}, {dim})"
|
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# If no weights are provided, choose them from an exponential
|
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# distribution.
|
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if weight is None:
|
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weight = {v: seed.expovariate(1) for v in G}
|
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# If no positions are provided, choose uniformly random vectors in
|
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# Euclidean space of the specified dimension.
|
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if pos is None:
|
|
pos = {v: [seed.random() for i in range(dim)] for v in G}
|
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# If no distance metric is provided, use Euclidean distance.
|
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nx.set_node_attributes(G, weight, "weight")
|
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nx.set_node_attributes(G, pos, "pos")
|
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|
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edges = (
|
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(u, v)
|
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for u, v in _geometric_edges(G, radius, p)
|
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if weight[u] + weight[v] >= theta
|
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)
|
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G.add_edges_from(edges)
|
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return G
|