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1048 lines
39 KiB
1048 lines
39 KiB
"""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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"geometric_soft_configuration_graph",
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]
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@nx._dispatchable(node_attrs="pos_name")
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def geometric_edges(G, radius, p=2, *, pos_name="pos"):
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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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pos_name : string, default="pos"
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The name of the node attribute which represents the position of each
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node in 2D coordinates. Every node in the Graph must have this attribute.
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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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... [
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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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... )
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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_name):
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if pos is None:
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raise nx.NetworkXError(
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f"Node {n} (and all nodes) must have a '{pos_name}' attribute."
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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, pos_name)
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def _geometric_edges(G, radius, p, pos_name):
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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_name)
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try:
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import scipy as sp
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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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@nx._dispatchable(graphs=None, returns_graph=True)
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def random_geometric_graph(
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n, radius, dim=2, pos=None, p=2, seed=None, *, pos_name="pos"
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):
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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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pos_name : string, default="pos"
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The name of the node attribute which represents the position
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in 2D coordinates of the node in the returned graph.
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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_name)
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G.add_edges_from(_geometric_edges(G, radius, p, pos_name))
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return G
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@py_random_state(6)
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@nx._dispatchable(graphs=None, returns_graph=True)
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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, *, pos_name="pos"
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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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pos_name : string, default="pos"
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The name of the node attribute which represents the position
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in 2D coordinates of the node in the returned graph.
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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_name)
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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, pos_name)))
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return G
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@py_random_state(7)
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@nx._dispatchable(graphs=None, returns_graph=True)
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def geographical_threshold_graph(
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n,
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theta,
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dim=2,
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pos=None,
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weight=None,
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metric=None,
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p_dist=None,
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seed=None,
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*,
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pos_name="pos",
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weight_name="weight",
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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
|
|
probability density definitions, and passing the ``.pdf`` method of
|
|
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.
|
|
seed : integer, random_state, or None (default)
|
|
Indicator of random number generation state.
|
|
See :ref:`Randomness<randomness>`.
|
|
pos_name : string, default="pos"
|
|
The name of the node attribute which represents the position
|
|
in 2D coordinates of the node in the returned graph.
|
|
weight_name : string, default="weight"
|
|
The name of the node attribute which represents the weight
|
|
of the node in the returned graph.
|
|
|
|
Returns
|
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-------
|
|
Graph
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|
A random geographic threshold graph, undirected and without
|
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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.
|
|
|
|
Examples
|
|
--------
|
|
Specify an alternate distance metric using the ``metric`` keyword
|
|
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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|
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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
|
|
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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|
.. [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.
|
|
if metric is None:
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metric = math.dist
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nx.set_node_attributes(G, weight, weight_name)
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nx.set_node_attributes(G, pos, pos_name)
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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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|
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# Returns ``True`` if and only if the nodes whose attributes are
|
|
# ``du`` and ``dv`` should be joined, according to the threshold
|
|
# condition.
|
|
def should_join(pair):
|
|
u, v = pair
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u_pos, v_pos = pos[u], pos[v]
|
|
u_weight, v_weight = weight[u], weight[v]
|
|
return (u_weight + v_weight) * p_dist(metric(u_pos, v_pos)) >= theta
|
|
|
|
G.add_edges_from(filter(should_join, combinations(G, 2)))
|
|
return G
|
|
|
|
|
|
@py_random_state(6)
|
|
@nx._dispatchable(graphs=None, returns_graph=True)
|
|
def waxman_graph(
|
|
n,
|
|
beta=0.4,
|
|
alpha=0.1,
|
|
L=None,
|
|
domain=(0, 0, 1, 1),
|
|
metric=None,
|
|
seed=None,
|
|
*,
|
|
pos_name="pos",
|
|
):
|
|
r"""Returns a Waxman random graph.
|
|
|
|
The Waxman random graph model places `n` nodes uniformly at random
|
|
in a rectangular domain. Each pair of nodes at distance `d` is
|
|
joined by an edge with probability
|
|
|
|
.. math::
|
|
p = \beta \exp(-d / \alpha L).
|
|
|
|
This function implements both Waxman models, using the `L` keyword
|
|
argument.
|
|
|
|
* Waxman-1: if `L` is not specified, it is set to be the maximum distance
|
|
between any pair of nodes.
|
|
* Waxman-2: if `L` is specified, the distance between a pair of nodes is
|
|
chosen uniformly at random from the interval `[0, L]`.
|
|
|
|
Parameters
|
|
----------
|
|
n : int or iterable
|
|
Number of nodes or iterable of nodes
|
|
beta: float
|
|
Model parameter
|
|
alpha: float
|
|
Model parameter
|
|
L : float, optional
|
|
Maximum distance between nodes. If not specified, the actual distance
|
|
is calculated.
|
|
domain : four-tuple of numbers, optional
|
|
Domain size, given as a tuple of the form `(x_min, y_min, x_max,
|
|
y_max)`.
|
|
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. $d(x, y) \ge 0$,
|
|
2. $d(x, y) = 0$ if and only if $x = y$,
|
|
3. $d(x, y) = d(y, x)$,
|
|
4. $d(x, z) \le d(x, y) + d(y, z)$.
|
|
|
|
If this argument is not specified, the Euclidean distance metric is
|
|
used.
|
|
|
|
.. _metric: https://en.wikipedia.org/wiki/Metric_%28mathematics%29
|
|
|
|
seed : integer, random_state, or None (default)
|
|
Indicator of random number generation state.
|
|
See :ref:`Randomness<randomness>`.
|
|
pos_name : string, default="pos"
|
|
The name of the node attribute which represents the position
|
|
in 2D coordinates of the node in the returned graph.
|
|
|
|
Returns
|
|
-------
|
|
Graph
|
|
A random Waxman graph, undirected and without self-loops. Each
|
|
node has a node attribute ``'pos'`` that stores the position of
|
|
that node in Euclidean space as generated by this function.
|
|
|
|
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.waxman_graph(10, 0.5, 0.1, metric=dist)
|
|
|
|
.. _taxicab metric: https://en.wikipedia.org/wiki/Taxicab_geometry
|
|
.. _Euclidean metric: https://en.wikipedia.org/wiki/Euclidean_distance
|
|
|
|
Notes
|
|
-----
|
|
Starting in NetworkX 2.0 the parameters alpha and beta align with their
|
|
usual roles in the probability distribution. In earlier versions their
|
|
positions in the expression were reversed. Their position in the calling
|
|
sequence reversed as well to minimize backward incompatibility.
|
|
|
|
References
|
|
----------
|
|
.. [1] B. M. Waxman, *Routing of multipoint connections*.
|
|
IEEE J. Select. Areas Commun. 6(9),(1988) 1617--1622.
|
|
"""
|
|
G = nx.empty_graph(n)
|
|
(xmin, ymin, xmax, ymax) = domain
|
|
# Each node gets a uniformly random position in the given rectangle.
|
|
pos = {v: (seed.uniform(xmin, xmax), seed.uniform(ymin, ymax)) for v in G}
|
|
nx.set_node_attributes(G, pos, pos_name)
|
|
# If no distance metric is provided, use Euclidean distance.
|
|
if metric is None:
|
|
metric = math.dist
|
|
# If the maximum distance L is not specified (that is, we are in the
|
|
# Waxman-1 model), then find the maximum distance between any pair
|
|
# of nodes.
|
|
#
|
|
# In the Waxman-1 model, join nodes randomly based on distance. In
|
|
# the Waxman-2 model, join randomly based on random l.
|
|
if L is None:
|
|
L = max(metric(x, y) for x, y in combinations(pos.values(), 2))
|
|
|
|
def dist(u, v):
|
|
return metric(pos[u], pos[v])
|
|
|
|
else:
|
|
|
|
def dist(u, v):
|
|
return seed.random() * L
|
|
|
|
# `pair` is the pair of nodes to decide whether to join.
|
|
def should_join(pair):
|
|
return seed.random() < beta * math.exp(-dist(*pair) / (alpha * L))
|
|
|
|
G.add_edges_from(filter(should_join, combinations(G, 2)))
|
|
return G
|
|
|
|
|
|
@py_random_state(5)
|
|
@nx._dispatchable(graphs=None, returns_graph=True)
|
|
def navigable_small_world_graph(n, p=1, q=1, r=2, dim=2, seed=None):
|
|
r"""Returns a navigable small-world graph.
|
|
|
|
A navigable small-world graph is a directed grid with additional long-range
|
|
connections that are chosen randomly.
|
|
|
|
[...] we begin with a set of nodes [...] that are identified with the set
|
|
of lattice points in an $n \times n$ square,
|
|
$\{(i, j): i \in \{1, 2, \ldots, n\}, j \in \{1, 2, \ldots, n\}\}$,
|
|
and we define the *lattice distance* between two nodes $(i, j)$ and
|
|
$(k, l)$ to be the number of "lattice steps" separating them:
|
|
$d((i, j), (k, l)) = |k - i| + |l - j|$.
|
|
|
|
For a universal constant $p >= 1$, the node $u$ has a directed edge to
|
|
every other node within lattice distance $p$---these are its *local
|
|
contacts*. For universal constants $q >= 0$ and $r >= 0$ we also
|
|
construct directed edges from $u$ to $q$ other nodes (the *long-range
|
|
contacts*) using independent random trials; the $i$th directed edge from
|
|
$u$ has endpoint $v$ with probability proportional to $[d(u,v)]^{-r}$.
|
|
|
|
-- [1]_
|
|
|
|
Parameters
|
|
----------
|
|
n : int
|
|
The length of one side of the lattice; the number of nodes in
|
|
the graph is therefore $n^2$.
|
|
p : int
|
|
The diameter of short range connections. Each node is joined with every
|
|
other node within this lattice distance.
|
|
q : int
|
|
The number of long-range connections for each node.
|
|
r : float
|
|
Exponent for decaying probability of connections. The probability of
|
|
connecting to a node at lattice distance $d$ is $1/d^r$.
|
|
dim : int
|
|
Dimension of grid
|
|
seed : integer, random_state, or None (default)
|
|
Indicator of random number generation state.
|
|
See :ref:`Randomness<randomness>`.
|
|
|
|
References
|
|
----------
|
|
.. [1] J. Kleinberg. The small-world phenomenon: An algorithmic
|
|
perspective. Proc. 32nd ACM Symposium on Theory of Computing, 2000.
|
|
"""
|
|
if p < 1:
|
|
raise nx.NetworkXException("p must be >= 1")
|
|
if q < 0:
|
|
raise nx.NetworkXException("q must be >= 0")
|
|
if r < 0:
|
|
raise nx.NetworkXException("r must be >= 0")
|
|
|
|
G = nx.DiGraph()
|
|
nodes = list(product(range(n), repeat=dim))
|
|
for p1 in nodes:
|
|
probs = [0]
|
|
for p2 in nodes:
|
|
if p1 == p2:
|
|
continue
|
|
d = sum((abs(b - a) for a, b in zip(p1, p2)))
|
|
if d <= p:
|
|
G.add_edge(p1, p2)
|
|
probs.append(d**-r)
|
|
cdf = list(accumulate(probs))
|
|
for _ in range(q):
|
|
target = nodes[bisect_left(cdf, seed.uniform(0, cdf[-1]))]
|
|
G.add_edge(p1, target)
|
|
return G
|
|
|
|
|
|
@py_random_state(7)
|
|
@nx._dispatchable(graphs=None, returns_graph=True)
|
|
def thresholded_random_geometric_graph(
|
|
n,
|
|
radius,
|
|
theta,
|
|
dim=2,
|
|
pos=None,
|
|
weight=None,
|
|
p=2,
|
|
seed=None,
|
|
*,
|
|
pos_name="pos",
|
|
weight_name="weight",
|
|
):
|
|
r"""Returns a thresholded random geometric graph in the unit cube.
|
|
|
|
The thresholded random geometric graph [1] model places `n` nodes
|
|
uniformly at random in the unit cube of dimensions `dim`. Each node
|
|
`u` is assigned a weight :math:`w_u`. Two nodes `u` and `v` are
|
|
joined by an edge if they are within the maximum connection distance,
|
|
`radius` computed by the `p`-Minkowski distance and the summation of
|
|
weights :math:`w_u` + :math:`w_v` is greater than or equal
|
|
to the threshold parameter `theta`.
|
|
|
|
Edges within `radius` of each other are determined using a KDTree when
|
|
SciPy is available. This reduces the time complexity from :math:`O(n^2)`
|
|
to :math:`O(n)`.
|
|
|
|
Parameters
|
|
----------
|
|
n : int or iterable
|
|
Number of nodes or iterable of nodes
|
|
radius: float
|
|
Distance threshold value
|
|
theta: float
|
|
Threshold value
|
|
dim : int, optional
|
|
Dimension of graph
|
|
pos : dict, optional
|
|
A dictionary keyed by node with node positions as values.
|
|
weight : dict, optional
|
|
Node weights as a dictionary of numbers keyed by node.
|
|
p : float, optional (default 2)
|
|
Which Minkowski distance metric to use. `p` has to meet the condition
|
|
``1 <= p <= infinity``.
|
|
|
|
If this argument is not specified, the :math:`L^2` metric
|
|
(the Euclidean distance metric), p = 2 is used.
|
|
|
|
This should not be confused with the `p` of an Erdős-Rényi random
|
|
graph, which represents probability.
|
|
seed : integer, random_state, or None (default)
|
|
Indicator of random number generation state.
|
|
See :ref:`Randomness<randomness>`.
|
|
pos_name : string, default="pos"
|
|
The name of the node attribute which represents the position
|
|
in 2D coordinates of the node in the returned graph.
|
|
weight_name : string, default="weight"
|
|
The name of the node attribute which represents the weight
|
|
of the node in the returned graph.
|
|
|
|
Returns
|
|
-------
|
|
Graph
|
|
A thresholded random geographic 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 nodethre
|
|
attribute ``'weight'`` that stores the weight of that node as
|
|
provided or as generated.
|
|
|
|
Examples
|
|
--------
|
|
Default Graph:
|
|
|
|
G = nx.thresholded_random_geometric_graph(50, 0.2, 0.1)
|
|
|
|
Custom Graph:
|
|
|
|
Create a thresholded random geometric graph on 50 uniformly distributed
|
|
nodes where nodes are joined by an edge if their sum weights drawn from
|
|
a exponential distribution with rate = 5 are >= theta = 0.1 and their
|
|
Euclidean distance is at most 0.2.
|
|
|
|
Notes
|
|
-----
|
|
This uses a *k*-d tree to build the graph.
|
|
|
|
The `pos` keyword argument can be used to specify node positions so you
|
|
can create an arbitrary distribution and domain for positions.
|
|
|
|
For example, to use a 2D Gaussian distribution of node positions with mean
|
|
(0, 0) and standard deviation 2
|
|
|
|
If weights are not specified they are assigned to nodes by drawing randomly
|
|
from the exponential distribution with rate parameter :math:`\lambda=1`.
|
|
To specify weights from a different distribution, use the `weight` keyword
|
|
argument::
|
|
|
|
::
|
|
|
|
>>> import random
|
|
>>> import math
|
|
>>> n = 50
|
|
>>> pos = {i: (random.gauss(0, 2), random.gauss(0, 2)) for i in range(n)}
|
|
>>> w = {i: random.expovariate(5.0) for i in range(n)}
|
|
>>> G = nx.thresholded_random_geometric_graph(n, 0.2, 0.1, 2, pos, w)
|
|
|
|
References
|
|
----------
|
|
.. [1] http://cole-maclean.github.io/blog/files/thesis.pdf
|
|
|
|
"""
|
|
G = nx.empty_graph(n)
|
|
G.name = f"thresholded_random_geometric_graph({n}, {radius}, {theta}, {dim})"
|
|
# If no weights are provided, choose them from an exponential
|
|
# distribution.
|
|
if weight is None:
|
|
weight = {v: seed.expovariate(1) for v in G}
|
|
# If no positions are provided, choose uniformly random vectors in
|
|
# Euclidean space of the specified dimension.
|
|
if pos is None:
|
|
pos = {v: [seed.random() for i in range(dim)] for v in G}
|
|
# If no distance metric is provided, use Euclidean distance.
|
|
nx.set_node_attributes(G, weight, weight_name)
|
|
nx.set_node_attributes(G, pos, pos_name)
|
|
|
|
edges = (
|
|
(u, v)
|
|
for u, v in _geometric_edges(G, radius, p, pos_name)
|
|
if weight[u] + weight[v] >= theta
|
|
)
|
|
G.add_edges_from(edges)
|
|
return G
|
|
|
|
|
|
@py_random_state(5)
|
|
@nx._dispatchable(graphs=None, returns_graph=True)
|
|
def geometric_soft_configuration_graph(
|
|
*, beta, n=None, gamma=None, mean_degree=None, kappas=None, seed=None
|
|
):
|
|
r"""Returns a random graph from the geometric soft configuration model.
|
|
|
|
The $\mathbb{S}^1$ model [1]_ is the geometric soft configuration model
|
|
which is able to explain many fundamental features of real networks such as
|
|
small-world property, heteregenous degree distributions, high level of
|
|
clustering, and self-similarity.
|
|
|
|
In the geometric soft configuration model, a node $i$ is assigned two hidden
|
|
variables: a hidden degree $\kappa_i$, quantifying its popularity, influence,
|
|
or importance, and an angular position $\theta_i$ in a circle abstracting the
|
|
similarity space, where angular distances between nodes are a proxy for their
|
|
similarity. Focusing on the angular position, this model is often called
|
|
the $\mathbb{S}^1$ model (a one-dimensional sphere). The circle's radius is
|
|
adjusted to $R = N/2\pi$, where $N$ is the number of nodes, so that the density
|
|
is set to 1 without loss of generality.
|
|
|
|
The connection probability between any pair of nodes increases with
|
|
the product of their hidden degrees (i.e., their combined popularities),
|
|
and decreases with the angular distance between the two nodes.
|
|
Specifically, nodes $i$ and $j$ are connected with the probability
|
|
|
|
$p_{ij} = \frac{1}{1 + \frac{d_{ij}^\beta}{\left(\mu \kappa_i \kappa_j\right)^{\max(1, \beta)}}}$
|
|
|
|
where $d_{ij} = R\Delta\theta_{ij}$ is the arc length of the circle between
|
|
nodes $i$ and $j$ separated by an angular distance $\Delta\theta_{ij}$.
|
|
Parameters $\mu$ and $\beta$ (also called inverse temperature) control the
|
|
average degree and the clustering coefficient, respectively.
|
|
|
|
It can be shown [2]_ that the model undergoes a structural phase transition
|
|
at $\beta=1$ so that for $\beta<1$ networks are unclustered in the thermodynamic
|
|
limit (when $N\to \infty$) whereas for $\beta>1$ the ensemble generates
|
|
networks with finite clustering coefficient.
|
|
|
|
The $\mathbb{S}^1$ model can be expressed as a purely geometric model
|
|
$\mathbb{H}^2$ in the hyperbolic plane [3]_ by mapping the hidden degree of
|
|
each node into a radial coordinate as
|
|
|
|
$r_i = \hat{R} - \frac{2 \max(1, \beta)}{\beta \zeta} \ln \left(\frac{\kappa_i}{\kappa_0}\right)$
|
|
|
|
where $\hat{R}$ is the radius of the hyperbolic disk and $\zeta$ is the curvature,
|
|
|
|
$\hat{R} = \frac{2}{\zeta} \ln \left(\frac{N}{\pi}\right)
|
|
- \frac{2\max(1, \beta)}{\beta \zeta} \ln (\mu \kappa_0^2)$
|
|
|
|
The connection probability then reads
|
|
|
|
$p_{ij} = \frac{1}{1 + \exp\left({\frac{\beta\zeta}{2} (x_{ij} - \hat{R})}\right)}$
|
|
|
|
where
|
|
|
|
$x_{ij} = r_i + r_j + \frac{2}{\zeta} \ln \frac{\Delta\theta_{ij}}{2}$
|
|
|
|
is a good approximation of the hyperbolic distance between two nodes separated
|
|
by an angular distance $\Delta\theta_{ij}$ with radial coordinates $r_i$ and $r_j$.
|
|
For $\beta > 1$, the curvature $\zeta = 1$, for $\beta < 1$, $\zeta = \beta^{-1}$.
|
|
|
|
|
|
Parameters
|
|
----------
|
|
Either `n`, `gamma`, `mean_degree` are provided or `kappas`. The values of
|
|
`n`, `gamma`, `mean_degree` (if provided) are used to construct a random
|
|
kappa-dict keyed by node with values sampled from a power-law distribution.
|
|
|
|
beta : positive number
|
|
Inverse temperature, controlling the clustering coefficient.
|
|
n : int (default: None)
|
|
Size of the network (number of nodes).
|
|
If not provided, `kappas` must be provided and holds the nodes.
|
|
gamma : float (default: None)
|
|
Exponent of the power-law distribution for hidden degrees `kappas`.
|
|
If not provided, `kappas` must be provided directly.
|
|
mean_degree : float (default: None)
|
|
The mean degree in the network.
|
|
If not provided, `kappas` must be provided directly.
|
|
kappas : dict (default: None)
|
|
A dict keyed by node to its hidden degree value.
|
|
If not provided, random values are computed based on a power-law
|
|
distribution using `n`, `gamma` and `mean_degree`.
|
|
seed : int, random_state, or None (default)
|
|
Indicator of random number generation state.
|
|
See :ref:`Randomness<randomness>`.
|
|
|
|
Returns
|
|
-------
|
|
Graph
|
|
A random geometric soft configuration graph (undirected with no self-loops).
|
|
Each node has three node-attributes:
|
|
|
|
- ``kappa`` that represents the hidden degree.
|
|
|
|
- ``theta`` the position in the similarity space ($\mathbb{S}^1$) which is
|
|
also the angular position in the hyperbolic plane.
|
|
|
|
- ``radius`` the radial position in the hyperbolic plane
|
|
(based on the hidden degree).
|
|
|
|
|
|
Examples
|
|
--------
|
|
Generate a network with specified parameters:
|
|
|
|
>>> G = nx.geometric_soft_configuration_graph(beta=1.5, n=100, gamma=2.7, mean_degree=5)
|
|
|
|
Create a geometric soft configuration graph with 100 nodes. The $\beta$ parameter
|
|
is set to 1.5 and the exponent of the powerlaw distribution of the hidden
|
|
degrees is 2.7 with mean value of 5.
|
|
|
|
Generate a network with predefined hidden degrees:
|
|
|
|
>>> kappas = {i: 10 for i in range(100)}
|
|
>>> G = nx.geometric_soft_configuration_graph(beta=2.5, kappas=kappas)
|
|
|
|
Create a geometric soft configuration graph with 100 nodes. The $\beta$ parameter
|
|
is set to 2.5 and all nodes with hidden degree $\kappa=10$.
|
|
|
|
|
|
References
|
|
----------
|
|
.. [1] Serrano, M. Á., Krioukov, D., & Boguñá, M. (2008). Self-similarity
|
|
of complex networks and hidden metric spaces. Physical review letters, 100(7), 078701.
|
|
|
|
.. [2] van der Kolk, J., Serrano, M. Á., & Boguñá, M. (2022). An anomalous
|
|
topological phase transition in spatial random graphs. Communications Physics, 5(1), 245.
|
|
|
|
.. [3] Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A., & Boguná, M. (2010).
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Hyperbolic geometry of complex networks. Physical Review E, 82(3), 036106.
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"""
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if beta <= 0:
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raise nx.NetworkXError("The parameter beta cannot be smaller or equal to 0.")
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if kappas is not None:
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if not all((n is None, gamma is None, mean_degree is None)):
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raise nx.NetworkXError(
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"When kappas is input, n, gamma and mean_degree must not be."
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)
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n = len(kappas)
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mean_degree = sum(kappas) / len(kappas)
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else:
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if any((n is None, gamma is None, mean_degree is None)):
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raise nx.NetworkXError(
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"Please provide either kappas, or all 3 of: n, gamma and mean_degree."
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)
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# Generate `n` hidden degrees from a powerlaw distribution
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# with given exponent `gamma` and mean value `mean_degree`
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gam_ratio = (gamma - 2) / (gamma - 1)
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kappa_0 = mean_degree * gam_ratio * (1 - 1 / n) / (1 - 1 / n**gam_ratio)
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base = 1 - 1 / n
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power = 1 / (1 - gamma)
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kappas = {i: kappa_0 * (1 - seed.random() * base) ** power for i in range(n)}
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G = nx.Graph()
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R = n / (2 * math.pi)
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# Approximate values for mu in the thermodynamic limit (when n -> infinity)
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if beta > 1:
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mu = beta * math.sin(math.pi / beta) / (2 * math.pi * mean_degree)
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elif beta == 1:
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mu = 1 / (2 * mean_degree * math.log(n))
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else:
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mu = (1 - beta) / (2**beta * mean_degree * n ** (1 - beta))
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# Generate random positions on a circle
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thetas = {k: seed.uniform(0, 2 * math.pi) for k in kappas}
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for u in kappas:
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for v in list(G):
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angle = math.pi - math.fabs(math.pi - math.fabs(thetas[u] - thetas[v]))
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dij = math.pow(R * angle, beta)
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mu_kappas = math.pow(mu * kappas[u] * kappas[v], max(1, beta))
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p_ij = 1 / (1 + dij / mu_kappas)
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# Create an edge with a certain connection probability
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if seed.random() < p_ij:
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G.add_edge(u, v)
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G.add_node(u)
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nx.set_node_attributes(G, thetas, "theta")
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nx.set_node_attributes(G, kappas, "kappa")
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# Map hidden degrees into the radial coordiantes
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zeta = 1 if beta > 1 else 1 / beta
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kappa_min = min(kappas.values())
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R_c = 2 * max(1, beta) / (beta * zeta)
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R_hat = (2 / zeta) * math.log(n / math.pi) - R_c * math.log(mu * kappa_min)
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radii = {node: R_hat - R_c * math.log(kappa) for node, kappa in kappas.items()}
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nx.set_node_attributes(G, radii, "radius")
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return G
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