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"""Generators for geometric graphs. """ from bisect import bisect_left from itertools import accumulate, combinations, product from math import sqrt import math try: from scipy.spatial import cKDTree as KDTree except ImportError: _is_scipy_available = False else: _is_scipy_available = True import networkx as nx from networkx.utils import nodes_or_number, py_random_state __all__ = [ "geographical_threshold_graph", "waxman_graph", "navigable_small_world_graph", "random_geometric_graph", "soft_random_geometric_graph", "thresholded_random_geometric_graph", ] def euclidean(x, y): """Returns the Euclidean distance between the vectors ``x`` and ``y``. Each of ``x`` and ``y`` can be any iterable of numbers. The iterables must be of the same length. """ return sqrt(sum((a - b) ** 2 for a, b in zip(x, y))) def _fast_edges(G, radius, p): """Returns edge list of node pairs within `radius` of each other using scipy KDTree and Minkowski distance metric `p` Requires scipy to be installed. """ pos = nx.get_node_attributes(G, "pos") nodes, coords = list(zip(*pos.items())) kdtree = KDTree(coords) # Cannot provide generator. edge_indexes = kdtree.query_pairs(radius, p) edges = ((nodes[u], nodes[v]) for u, v in edge_indexes) return edges def _slow_edges(G, radius, p): """Returns edge list of node pairs within `radius` of each other using Minkowski distance metric `p` Works without scipy, but in `O(n^2)` time. """ # TODO This can be parallelized. edges = [] for (u, pu), (v, pv) in combinations(G.nodes(data="pos"), 2): if sum(abs(a - b) ** p for a, b in zip(pu, pv)) <= radius ** p: edges.append((u, v)) return edges @py_random_state(5) @nodes_or_number(0) def random_geometric_graph(n, radius, dim=2, pos=None, p=2, seed=None): """Returns a random geometric graph in the unit cube of dimensions `dim`. The random geometric graph model places `n` nodes uniformly at random in the unit cube. Two nodes are joined by an edge if the distance between the nodes is at most `radius`. Edges are determined using a KDTree when SciPy is available. This reduces the time complexity from $O(n^2)$ to $O(n)$. Parameters ---------- n : int or iterable Number of nodes or iterable of nodes radius: float Distance threshold value dim : int, optional Dimension of graph pos : dict, optional A dictionary keyed by node with node positions as values. p : float, optional 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>`. Returns ------- Graph A random geometric 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. Examples -------- Create a random geometric graph on twenty nodes where nodes are joined by an edge if their distance is at most 0.1:: >>> G = nx.random_geometric_graph(20, 0.1) 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:: >>> import random >>> n = 20 >>> pos = {i: (random.gauss(0, 2), random.gauss(0, 2)) for i in range(n)} >>> G = nx.random_geometric_graph(n, 0.2, pos=pos) References ---------- .. [1] Penrose, Mathew, *Random Geometric Graphs*, Oxford Studies in Probability, 5, 2003. """ # TODO Is this function just a special case of the geographical # threshold graph? # # n_name, nodes = n # half_radius = {v: radius / 2 for v in nodes} # return geographical_threshold_graph(nodes, theta=1, alpha=1, # weight=half_radius) # n_name, nodes = n G = nx.Graph() G.add_nodes_from(nodes) # 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 nodes} nx.set_node_attributes(G, pos, "pos") if _is_scipy_available: edges = _fast_edges(G, radius, p) else: edges = _slow_edges(G, radius, p) G.add_edges_from(edges) return G @py_random_state(6) @nodes_or_number(0) def soft_random_geometric_graph( n, radius, dim=2, pos=None, p=2, p_dist=None, seed=None ): r"""Returns a soft random geometric graph in the unit cube. The soft random geometric graph [1] model places `n` nodes uniformly at random in the unit cube in dimension `dim`. Two nodes of distance, `dist`, computed by the `p`-Minkowski distance metric are joined by an edge with probability `p_dist` if the computed distance metric value of the nodes is at most `radius`, otherwise they are not joined. 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 dim : int, optional Dimension of graph pos : dict, optional A dictionary keyed by node with node positions as values. p : float, optional 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. p_dist : function, optional A probability density function computing the probability of connecting two nodes that are of distance, dist, computed by the Minkowski distance metric. The probability density function, `p_dist`, must be any function that takes the metric value as input and outputs a single probability value between 0-1. The scipy.stats package has many probability distribution functions implemented and tools for custom probability distribution definitions [2], and passing the .pdf method of scipy.stats distributions can be used here. If the probability function, `p_dist`, is not supplied, the default function is an exponential distribution with rate parameter :math:`\lambda=1`. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. Returns ------- Graph A soft random geometric 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. Examples -------- Default Graph: G = nx.soft_random_geometric_graph(50, 0.2) Custom Graph: Create a soft random geometric graph on 100 uniformly distributed nodes where nodes are joined by an edge with probability computed from an exponential distribution with rate parameter :math:`\lambda=1` if 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 The scipy.stats package can be used to define the probability distribution with the .pdf method used as `p_dist`. :: >>> import random >>> import math >>> n = 100 >>> pos = {i: (random.gauss(0, 2), random.gauss(0, 2)) for i in range(n)} >>> p_dist = lambda dist: math.exp(-dist) >>> G = nx.soft_random_geometric_graph(n, 0.2, pos=pos, p_dist=p_dist) References ---------- .. [1] Penrose, Mathew D. "Connectivity of soft random geometric graphs." The Annals of Applied Probability 26.2 (2016): 986-1028. [2] scipy.stats - https://docs.scipy.org/doc/scipy/reference/tutorial/stats.html """ n_name, nodes = n G = nx.Graph() G.name = f"soft_random_geometric_graph({n}, {radius}, {dim})" G.add_nodes_from(nodes) # 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 nodes} nx.set_node_attributes(G, pos, "pos") # if p_dist function not supplied the default function is an exponential # distribution with rate parameter :math:`\lambda=1`. if p_dist is None: def p_dist(dist): return math.exp(-dist) def should_join(pair): u, v = pair u_pos, v_pos = pos[u], pos[v] dist = (sum(abs(a - b) ** p for a, b in zip(u_pos, v_pos))) ** (1 / p) # Check if dist <= radius parameter. This check is redundant if scipy # is available and _fast_edges routine is used, but provides the # check in case scipy is not available and all edge combinations # need to be checked if dist <= radius: return seed.random() < p_dist(dist) else: return False if _is_scipy_available: edges = _fast_edges(G, radius, p) G.add_edges_from(filter(should_join, edges)) else: G.add_edges_from(filter(should_join, combinations(G, 2))) return G @py_random_state(7) @nodes_or_number(0) def geographical_threshold_graph( n, theta, dim=2, pos=None, weight=None, metric=None, p_dist=None, seed=None ): r"""Returns a geographical threshold graph. The geographical threshold graph model places $n$ nodes uniformly at random in a rectangular domain. Each node $u$ is assigned a weight $w_u$. Two nodes $u$ and $v$ are joined by an edge if .. math:: (w_u + w_v)h(r) \ge \theta where `r` is the distance between `u` and `v`, h(r) is a probability of connection as a function of `r`, and :math:`\theta` as the threshold parameter. h(r) corresponds to the p_dist parameter. Parameters ---------- n : int or iterable Number of nodes or iterable of nodes theta: float Threshold value dim : int, optional Dimension of graph pos : dict Node positions as a dictionary of tuples keyed by node. weight : dict Node weights as a dictionary of numbers keyed by node. metric : function A metric on vectors of numbers (represented as lists or tuples). This must be a function that accepts two lists (or tuples) as input and yields a number as output. The function must also satisfy the four requirements of a `metric`_. Specifically, if $d$ is the function and $x$, $y$, and $z$ are vectors in the graph, then $d$ must satisfy 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 p_dist : function, optional A probability density function computing the probability of connecting two nodes that are of distance, r, computed by metric. The probability density function, `p_dist`, must be any function that takes the metric value as input and outputs a single probability value between 0-1. The scipy.stats package has many probability distribution functions implemented and tools for custom probability distribution definitions [2], and passing the .pdf method of scipy.stats distributions can be used here. If the probability function, `p_dist`, is not supplied, the default exponential function :math: `r^{-2}` is used. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. Returns ------- Graph A random geographic threshold graph, undirected and without self-loops. Each node has a node attribute ``pos`` that stores the position of that node in Euclidean space as provided by the ``pos`` keyword argument or, if ``pos`` was not provided, as generated by this function. Similarly, each node has a node attribute ``weight`` that stores the weight of that node as provided or as generated. Examples -------- Specify an alternate distance metric using the ``metric`` keyword argument. For example, to use the `taxicab metric`_ instead of the default `Euclidean metric`_:: >>> dist = lambda x, y: sum(abs(a - b) for a, b in zip(x, y)) >>> G = nx.geographical_threshold_graph(10, 0.1, metric=dist) .. _taxicab metric: https://en.wikipedia.org/wiki/Taxicab_geometry .. _Euclidean metric: https://en.wikipedia.org/wiki/Euclidean_distance Notes ----- If weights are not specified they are assigned to nodes by drawing randomly from the exponential distribution with rate parameter $\lambda=1$. To specify weights from a different distribution, use the `weight` keyword argument:: >>> import random >>> n = 20 >>> w = {i: random.expovariate(5.0) for i in range(n)} >>> G = nx.geographical_threshold_graph(20, 50, weight=w) If node positions are not specified they are randomly assigned from the uniform distribution. References ---------- .. [1] Masuda, N., Miwa, H., Konno, N.: Geographical threshold graphs with small-world and scale-free properties. Physical Review E 71, 036108 (2005) .. [2] Milan Bradonjić, Aric Hagberg and Allon G. Percus, Giant component and connectivity in geographical threshold graphs, in Algorithms and Models for the Web-Graph (WAW 2007), Antony Bonato and Fan Chung (Eds), pp. 209--216, 2007 """ n_name, nodes = n G = nx.Graph() G.add_nodes_from(nodes) # 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 nodes} # If no distance metric is provided, use Euclidean distance. if metric is None: metric = euclidean nx.set_node_attributes(G, weight, "weight") nx.set_node_attributes(G, pos, "pos") # if p_dist is not supplied, use default r^-2 if p_dist is None: def p_dist(r): return r ** -2 # 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 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) @nodes_or_number(0) def waxman_graph( n, beta=0.4, alpha=0.1, L=None, domain=(0, 0, 1, 1), metric=None, seed=None ): 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>`. 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. """ n_name, nodes = n G = nx.Graph() G.add_nodes_from(nodes) (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") # If no distance metric is provided, use Euclidean distance. if metric is None: metric = euclidean # 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) 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 >= 1") 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) @nodes_or_number(0) def thresholded_random_geometric_graph( n, radius, theta, dim=2, pos=None, weight=None, p=2, seed=None ): 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 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>`. 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 """ n_name, nodes = n G = nx.Graph() G.name = f"thresholded_random_geometric_graph({n}, {radius}, {theta}, {dim})" G.add_nodes_from(nodes) # 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 nodes} # If no distance metric is provided, use Euclidean distance. nx.set_node_attributes(G, weight, "weight") nx.set_node_attributes(G, pos, "pos") # Returns ``True`` if and only if the nodes whose attributes are # ``du`` and ``dv`` should be joined, according to the threshold # condition and node pairs are within the maximum connection # distance, ``radius``. def should_join(pair): u, v = pair u_weight, v_weight = weight[u], weight[v] u_pos, v_pos = pos[u], pos[v] dist = (sum(abs(a - b) ** p for a, b in zip(u_pos, v_pos))) ** (1 / p) # Check if dist is <= radius parameter. This check is redundant if # scipy is available and _fast_edges routine is used, but provides # the check in case scipy is not available and all edge combinations # need to be checked if dist <= radius: return theta <= u_weight + v_weight else: return False if _is_scipy_available: edges = _fast_edges(G, radius, p) G.add_edges_from(filter(should_join, edges)) else: G.add_edges_from(filter(should_join, combinations(G, 2))) return G