hSrSSKrSSKJr SSKJrJrJr SSKr SSK J r /SQr \ R"SS9SS S .S jj5rS r\ "S 5\ R"SSS9SS S .Sjj55r\ "S5\ R"SSS9SS S .Sjj55r\ "S5\ R"SSS9SS SS.Sjj55r\ "S5\ R"SSS9S S S .Sjj55r\ "S 5\ R"SSS9S!Sj55r\ "S5\ R"SSS9S"S SS.Sjj55r\ "S 5\ R"SSS9SSSSSS.Sj55rg)#z Generators for geometric graphs.N) bisect_left) accumulate combinationsproduct)py_random_state)geometric_edgesgeographical_threshold_graphnavigable_small_world_graphrandom_geometric_graphsoft_random_geometric_graph"thresholded_random_geometric_graph waxman_graph"geometric_soft_configuration_graphpos_name) node_attrspos)rcURUS9H%upEUbM [R"SUSUS35e [XX#5$)aReturns edge list of node pairs within `radius` of each other. Parameters ---------- G : networkx graph The graph from which to generate the edge list. The nodes in `G` should have an attribute ``pos`` corresponding to the node position, which is used to compute the distance to other nodes. radius : scalar The distance threshold. Edges are included in the edge list if the distance between the two nodes is less than `radius`. pos_name : string, default="pos" The name of the node attribute which represents the position of each node in 2D coordinates. Every node in the Graph must have this attribute. p : scalar, default=2 The `Minkowski distance metric `_ used to compute distances. The default value is 2, i.e. Euclidean distance. Returns ------- edges : list List of edges whose distances are less than `radius` Notes ----- Radius uses Minkowski distance metric `p`. If scipy is available, `scipy.spatial.cKDTree` is used to speed computation. Examples -------- Create a graph with nodes that have a "pos" attribute representing 2D coordinates. >>> G = nx.Graph() >>> G.add_nodes_from( ... [ ... (0, {"pos": (0, 0)}), ... (1, {"pos": (3, 0)}), ... (2, {"pos": (8, 0)}), ... ] ... ) >>> nx.geometric_edges(G, radius=1) [] >>> nx.geometric_edges(G, radius=4) [(0, 1)] >>> nx.geometric_edges(G, radius=6) [(0, 1), (1, 2)] >>> nx.geometric_edges(G, radius=9) [(0, 1), (0, 2), (1, 2)] datazNode z (and all nodes) must have a 'z ' attribute.)nodesnx NetworkXError_geometric_edges)Gradiusprnrs O/var/www/html/env/lib/python3.13/site-packages/networkx/generators/geometric.pyrrsVl''x'( ;""s8 ,O ) Aq 33c^URUS9nSSKn[ [ U65upURRU 5nURUT5n[U5VV s/sH upyXX4PM n nn U $![aa UT-n[US5VVV V s/sH1uupxup[ U4Sj[ X555U::dM.Xy4PM3 Os sn n nnfn n nnn U s$f=fs sn nf)zf Implements `geometric_edges` without input validation. See `geometric_edges` for complete docstring. rrNc3H># UHup[X- 5T-v M g7fNabs.0abrs r #_geometric_edges..gs;{tq3qu:?{") rscipy ImportErrorrsumziplistspatialcKDTree query_pairssorted)rrrr nodes_posspradius_pupuvpvedgesrcoordskdtree edge_indexess ` rrrYs X&I i)ME ZZ   'F%%fa0L.4\.B C.Bdaeh !.BE C L 19%1A$> $> !;s2{;;xG QF$>    Ds)A=$C+= C(-C C C('C(T)graphs returns_graphc ([R"U5nUc;UVV s0sH,o[U5V s/sHoR5PM sn _M. nnn [R"XsU5 UR [ XqXF55 U$s sn fs sn nf)u 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`. 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 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. )r empty_graphrangerandomset_node_attributesadd_edges_fromr) rrdimrrseedrrr;is rr r rs{f qA {?@Aq!%*5*Q;;=*55qA18,%a=> H 6AsBB  B Bc ^^^^[R"U5nSUSUSUS3UlTc=UV V s0sH-o[U5V s/sHn TR 5PM sn _M/ sn n m[R "UTU5 TcSmUUUU4Sjn UR [U [XTU555 U$s sn fs sn n f)uReturns 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`. 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 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 zsoft_random_geometric_graph(, )c0[R"U*5$r#)mathexp)dists rp_dist+soft_random_geometric_graph..p_distFs88TE? "rc>Uup[U4Sj[TUTU555ST- -nTR5T"U5:$)Nc3H># UHup[X- 5T-v M g7fr#r$r&s rr*Csoft_random_geometric_graph..should_join..KsD0CCJ!O0Cr,)r/r0rG)edger9r;rTrrUrrKs r should_join0soft_random_geometric_graph..should_joinIsGDCFCF0CDD!a%P{{}vd|++r) rrEnamerFrGrHrIfilterr) rrrJrrrUrKrrr;rLr\s ```` rr r sT qA+A3b3%q AAF {?@Aq!%*5*Q4;;=*55qA1c8,~ #,, VK)9!Q)QRS H!6AsCB;!C;Cweight)r weight_namec ^^^^^[R"U5n Tc"U V s0sHoURS5_M sn mTc<U V V s0sH,o[U5V s/sHoR 5PM sn _M. sn n mTc[ R m[R"U TU 5 [R"U TU5 TcSmUUUUU4Sjn U R[U [U S555 U $s sn fs sn fs sn n f)uReturns 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)p_{dist}(r) \ge \theta where `r` is the distance between `u` and `v`, `p_dist` is any function of `r`, and :math:`\theta` as the threshold parameter. `p_dist` is used to give weight to the distance between nodes when deciding whether or not they should be connected. The larger `p_dist` is, the more prone nodes separated by `r` are to be connected, and vice versa. 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 Any function used to give weight to the distance between nodes when deciding whether or not they should be connected. `p_dist` was originally conceived as a probability density function giving the probability of connecting two nodes that are of metric distance `r` apart. The implementation here allows for more arbitrary definitions of `p_dist` that do not need to correspond to valid probability density functions. The :mod:`scipy.stats` package has many 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`` (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`. 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 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 rZc US-$)N)rs rrU,geographical_threshold_graph..p_dists b5Lrc\>UupT UT UpCT UT UpeXV-T"T"X455-T :$r#rf) pairr9r;u_posv_posu_weightv_weightmetricrUrthetaras rr\1geographical_threshold_graph..should_joinsE1vs1vu#AYq (#vfU.B'CCuLLrr!) rrE expovariaterFrGrRrTrHrIr_r)rrprJrrarorUrKrrbrr;rLr\s ` ```` rr r RsJ qA~234!QT%%a((!4 {?@Aq!%*5*Q;;=*55qA ~1fk21c8,~  MM VKa);<= H756AsC: DC?5D?Dc ^^^^^^^[R"U5nUuppUV s0sH%oTRX5TRX54_M' sn m[R"UTU5 Tc[R mTc5[ U4Sj[TR5S555mUU4SjmOUU4SjmUUUUU4SjnUR[U[US555 U$s sn f)a 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`. 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. c38># UHupT"X5v M g7fr#rf)r'xyros rr*waxman_graph..lsG)Fq )Fsr!c">T"TUTU5$r#rf)r9r;rors rrTwaxman_graph..distns#a&#a&) )rc*>TR5T-$r#)rG)r9r;LrKs rrTryss;;=1$ $rcl>TR5T[R"T"U6*TT-- 5-:$r#)rGrRrS)rjr{alphabetarTrKs rr\!waxman_graph..should_joinws1{{}tdhhd |uqy/I&JJJJr) rrEuniformrHrRrTmaxrvaluesrIr_)rr~r}r{domainrorKrrxminyminxmaxymaxr;r\rTrs ``` `` @@rrrsH qA%TLM NAqt||D'd)AB BA NC1c8, ~ y GcjjlA)FG G *  %KKVKa);<= H7 Os,C1c US:a[R"S5eUS:a[R"S5eUS:a[R"S5e[R"5n[[ [ U5US95nUHnS/n UHOn X:XaM [ S[X555n X::aURX5 U RX*-5 MQ [[U 55n [ U5H5n U[XRSU S55nURX5 M7 M U$) aReturns 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`. References ---------- .. [1] J. Kleinberg. The small-world phenomenon: An algorithmic perspective. Proc. 32nd ACM Symposium on Theory of Computing, 2000. rZzp must be >= 1rzq must be >= 0zr must be >= 0)repeatc3@# UHup[X!- 5v M g7fr#r$)r'r(r)s rr*.navigable_small_world_graph..s8KDASZZKs) rNetworkXExceptionDiGraphr1rrFr/r0add_edgeappendrrr)rrqrgrJrKrrp1probsp2dcdf_targets rr r ~s` 1u""#3441u""#3441u""#344 A q#. /EBx8CK89Av 2" LLB  :e$%qA;sLLCG,DEFF JJr " Hrc ^^[R"U5n SUSUSTSUS3 U lTc"U V s0sHoURS5_M sn mUc;U V V s0sH,o[ U5V s/sHoR 5PM sn _M. nn n [R "U TU 5 [R "XU5 UU4Sj[XXh55n U RU 5 U $s sn fs sn fs sn n f)u 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`. 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 z#thresholded_random_geometric_graph(rOrPrZc3P># UHupTUTU-T:dMX4v M g7fr#rf)r'r9r;rpras rr*5thresholded_random_geometric_graph..Js3