Source code for graspy.simulations.simulations

# Copyright 2019 NeuroData (http://neurodata.io)
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import numpy as np

from ..utils import symmetrize, cartprod
import warnings


def _n_to_labels(n):
    n_cumsum = n.cumsum()
    labels = np.zeros(n.sum(), dtype=np.int64)
    for i in range(1, len(n)):
        labels[n_cumsum[i - 1] : n_cumsum[i]] = i
    return labels


def sample_edges(P, directed=False, loops=False):
    """
    Gemerates a binary random graph based on the P matrix provided

    Each element in P represents the probability of a connection between 
    a vertex indexed by the row i and the column j. 

    Parameters
    ----------
    P: np.ndarray, shape (n_vertices, n_vertices)
        Matrix of probabilities (between 0 and 1) for a random graph
    directed: boolean, optional (default=False)
        If False, output adjacency matrix will be symmetric. Otherwise, output adjacency
        matrix will be asymmetric.
    loops: boolean, optional (default=False)
        If False, no edges will be sampled in the diagonal. Otherwise, edges
        are sampled in the diagonal.

    Returns
    -------
    A: ndarray (n_vertices, n_vertices)
        Binary adjacency matrix the same size as P representing a random
        graph

    References
    ----------
    .. [1] Sussman, D.L., Tang, M., Fishkind, D.E., Priebe, C.E.  "A
       Consistent Adjacency Spectral Embedding for Stochastic Blockmodel Graphs,"
       Journal of the American Statistical Association, Vol. 107(499), 2012
    """
    if type(P) is not np.ndarray:
        raise TypeError("P must be numpy.ndarray")
    if len(P.shape) != 2:
        raise ValueError("P must have dimension 2 (n_vertices, n_dimensions)")
    if P.shape[0] != P.shape[1]:
        raise ValueError("P must be a square matrix")
    if not directed:
        # can cut down on sampling by ~half
        triu_inds = np.triu_indices(P.shape[0])
        samples = np.random.binomial(1, P[triu_inds])
        A = np.zeros_like(P)
        A[triu_inds] = samples
        A = symmetrize(A, method="triu")
    else:
        A = np.random.binomial(1, P)

    if loops:
        return A
    else:
        return A - np.diag(np.diag(A))


[docs]def er_np(n, p, directed=False, loops=False, wt=1, wtargs=None, dc=None, dc_kws={}): r""" Samples a Erdos Renyi (n, p) graph with specified edge probability. Erdos Renyi (n, p) graph is a simple graph with n vertices and a probability p of edges being connected. Read more in the :ref:`tutorials <simulations_tutorials>` Parameters ---------- n: int Number of vertices p: float Probability of an edge existing between two vertices, between 0 and 1. directed: boolean, optional (default=False) If False, output adjacency matrix will be symmetric. Otherwise, output adjacency matrix will be asymmetric. loops: boolean, optional (default=False) If False, no edges will be sampled in the diagonal. Otherwise, edges are sampled in the diagonal. wt: object, optional (default=1) Weight function for each of the edges, taking only a size argument. This weight function will be randomly assigned for selected edges. If 1, graph produced is binary. wtargs: dictionary, optional (default=None) Optional arguments for parameters that can be passed to weight function ``wt``. dc: function or array-like, shape (n_vertices) `dc` is used to generate a degree-corrected Erdos Renyi Model in which each node in the graph has a parameter to specify its expected degree relative to other nodes. - function: should generate a non-negative number to be used as a degree correction to create a heterogenous degree distribution. A weight will be generated for each vertex, normalized so that the sum of weights is 1. - array-like of scalars, shape (n_vertices): The weights should sum to 1; otherwise, they will be normalized and a warning will be thrown. The scalar associated with each vertex is the node's relative expected degree. dc_kws: dictionary Ignored if `dc` is none or array of scalar. If `dc` is a function, `dc_kws` corresponds to its named arguments. If not specified, in either case all functions will assume their default parameters. Returns ------- A : ndarray, shape (n, n) Sampled adjacency matrix Examples -------- >>> np.random.seed(1) >>> n = 4 >>> p = 0.25 To sample a binary Erdos Renyi (n, p) graph: >>> er_np(n, p) array([[0., 0., 1., 0.], [0., 0., 1., 0.], [1., 1., 0., 0.], [0., 0., 0., 0.]]) To sample a weighted Erdos Renyi (n, p) graph with Uniform(0, 1) distribution: >>> wt = np.random.uniform >>> wtargs = dict(low=0, high=1) >>> er_np(n, p, wt=wt, wtargs=wtargs) array([[0. , 0. , 0.95788953, 0.53316528], [0. , 0. , 0. , 0. ], [0.95788953, 0. , 0. , 0.31551563], [0.53316528, 0. , 0.31551563, 0. ]]) """ if isinstance(dc, (list, np.ndarray)) and all(callable(f) for f in dc): raise TypeError("dc is not of type function or array-like of scalars") if not np.issubdtype(type(n), np.integer): raise TypeError("n is not of type int.") if not np.issubdtype(type(p), np.floating): raise TypeError("p is not of type float.") if type(loops) is not bool: raise TypeError("loops is not of type bool.") if type(directed) is not bool: raise TypeError("directed is not of type bool.") n_sbm = np.array([n]) p_sbm = np.array([[p]]) g = sbm(n_sbm, p_sbm, directed, loops, wt, wtargs, dc, dc_kws) return g
[docs]def er_nm(n, m, directed=False, loops=False, wt=1, wtargs=None): r""" Samples an Erdos Renyi (n, m) graph with specified number of edges. Erdos Renyi (n, m) graph is a simple graph with n vertices and exactly m number of total edges. Read more in the :ref:`tutorials <simulations_tutorials>` Parameters ---------- n: int Number of vertices m: int Number of edges, a value between 1 and :math:`n^2`. directed: boolean, optional (default=False) If False, output adjacency matrix will be symmetric. Otherwise, output adjacency matrix will be asymmetric. loops: boolean, optional (default=False) If False, no edges will be sampled in the diagonal. Otherwise, edges are sampled in the diagonal. wt: object, optional (default=1) Weight function for each of the edges, taking only a size argument. This weight function will be randomly assigned for selected edges. If 1, graph produced is binary. wtargs: dictionary, optional (default=None) Optional arguments for parameters that can be passed to weight function ``wt``. Returns ------- A: ndarray, shape (n, n) Sampled adjacency matrix Examples -------- >>> np.random.seed(1) >>> n = 4 >>> m = 4 To sample a binary Erdos Renyi (n, m) graph: >>> er_nm(n, m) array([[0., 1., 1., 1.], [1., 0., 0., 1.], [1., 0., 0., 0.], [1., 1., 0., 0.]]) To sample a weighted Erdos Renyi (n, m) graph with Uniform(0, 1) distribution: >>> wt = np.random.uniform >>> wtargs = dict(low=0, high=1) >>> er_nm(n, m, wt=wt, wtargs=wtargs) array([[0. , 0.66974604, 0. , 0.38791074], [0.66974604, 0. , 0. , 0.39658073], [0. , 0. , 0. , 0.93553907], [0.38791074, 0.39658073, 0.93553907, 0. ]]) """ if not np.issubdtype(type(m), np.integer): raise TypeError("m is not of type int.") elif m <= 0: msg = "m must be > 0." raise ValueError(msg) if not np.issubdtype(type(n), np.integer): raise TypeError("n is not of type int.") elif n <= 0: msg = "n must be > 0." raise ValueError(msg) if type(directed) is not bool: raise TypeError("directed is not of type bool.") if type(loops) is not bool: raise TypeError("loops is not of type bool.") # check weight function if not np.issubdtype(type(wt), np.integer): if not callable(wt): raise TypeError("You have not passed a function for wt.") # compute max number of edges to sample if loops: if directed: max_edges = n ** 2 msg = "n^2" else: max_edges = n * (n + 1) // 2 msg = "n(n+1)/2" else: if directed: max_edges = n * (n - 1) msg = "n(n-1)" else: max_edges = n * (n - 1) // 2 msg = "n(n-1)/2" if m > max_edges: msg = "You have passed a number of edges, {}, exceeding {}, {}." msg = msg.format(m, msg, max_edges) raise ValueError(msg) A = np.zeros((n, n)) # check if directedness is desired if directed: if loops: # use all of the indices idx = np.where(np.logical_not(A)) else: # use only the off-diagonal indices idx = np.where(~np.eye(n, dtype=bool)) else: # use upper-triangle indices, and ignore diagonal according # to loops argument idx = np.triu_indices(n, k=int(loops is False)) # get idx in 1d coordinates by ravelling triu = np.ravel_multi_index(idx, A.shape) # choose M of them triu = np.random.choice(triu, size=m, replace=False) # unravel back triu = np.unravel_index(triu, A.shape) # check weight function if not np.issubdtype(type(wt), np.number): wt = wt(size=m, **wtargs) A[triu] = wt if not directed: A = symmetrize(A, method="triu") return A
[docs]def sbm( n, p, directed=False, loops=False, wt=1, wtargs=None, dc=None, dc_kws={}, return_labels=False, ): """ Samples a graph from the stochastic block model (SBM). SBM produces a graph with specified communities, in which each community can have different sizes and edge probabilities. Read more in the :ref:`tutorials <simulations_tutorials>` Parameters ---------- n: list of int, shape (n_communities) Number of vertices in each community. Communities are assigned n[0], n[1], ... p: array-like, shape (n_communities, n_communities) Probability of an edge between each of the communities, where p[i, j] indicates the probability of a connection between edges in communities [i, j]. 0 < p[i, j] < 1 for all i, j. directed: boolean, optional (default=False) If False, output adjacency matrix will be symmetric. Otherwise, output adjacency matrix will be asymmetric. loops: boolean, optional (default=False) If False, no edges will be sampled in the diagonal. Otherwise, edges are sampled in the diagonal. wt: object or array-like, shape (n_communities, n_communities) if Wt is an object, a weight function to use globally over the sbm for assigning weights. 1 indicates to produce a binary graph. If Wt is an array-like, a weight function for each of the edge communities. Wt[i, j] corresponds to the weight function between communities i and j. If the entry is a function, should accept an argument for size. An entry of Wt[i, j] = 1 will produce a binary subgraph over the i, j community. wtargs: dictionary or array-like, shape (n_communities, n_communities) if Wt is an object, Wtargs corresponds to the trailing arguments to pass to the weight function. If Wt is an array-like, Wtargs[i, j] corresponds to trailing arguments to pass to Wt[i, j]. dc: function or array-like, shape (n_vertices) or (n_communities), optional `dc` is used to generate a degree-corrected stochastic block model [1] in which each node in the graph has a parameter to specify its expected degree relative to other nodes within its community. - function: should generate a non-negative number to be used as a degree correction to create a heterogenous degree distribution. A weight will be generated for each vertex, normalized so that the sum of weights in each block is 1. - array-like of functions, shape (n_communities): Each function will generate the degree distribution for its respective community. - array-like of scalars, shape (n_vertices): The weights in each block should sum to 1; otherwise, they will be normalized and a warning will be thrown. The scalar associated with each vertex is the node's relative expected degree within its community. dc_kws: dictionary or array-like, shape (n_communities), optional Ignored if `dc` is none or array of scalar. If `dc` is a function, `dc_kws` corresponds to its named arguments. If `dc` is an array-like of functions, `dc_kws` should be an array-like, shape (n_communities), of dictionary. Each dictionary is the named arguments for the corresponding function for that community. If not specified, in either case all functions will assume their default parameters. return_labels: boolean, optional (default=False) If False, only output is adjacency matrix. Otherwise, an additional output will be an array with length equal to the number of vertices in the graph, where each entry in the array labels which block a vertex in the graph is in. References ---------- .. [1] Tai Qin and Karl Rohe. "Regularized spectral clustering under the Degree-Corrected Stochastic Blockmodel," Advances in Neural Information Processing Systems 26, 2013 Returns ------- A: ndarray, shape (sum(n), sum(n)) Sampled adjacency matrix labels: ndarray, shape (sum(n)) Label vector Examples -------- >>> np.random.seed(1) >>> n = [3, 3] >>> p = [[0.5, 0.1], [0.1, 0.5]] To sample a binary 2-block SBM graph: >>> sbm(n, p) array([[0., 0., 1., 0., 0., 0.], [0., 0., 1., 0., 0., 1.], [1., 1., 0., 0., 0., 0.], [0., 0., 0., 0., 1., 0.], [0., 0., 0., 1., 0., 0.], [0., 1., 0., 0., 0., 0.]]) To sample a weighted 2-block SBM graph with Poisson(2) distribution: >>> wt = np.random.poisson >>> wtargs = dict(lam=2) >>> sbm(n, p, wt=wt, wtargs=wtargs) array([[0., 4., 0., 1., 0., 0.], [4., 0., 0., 0., 0., 2.], [0., 0., 0., 0., 0., 0.], [1., 0., 0., 0., 0., 0.], [0., 0., 0., 0., 0., 0.], [0., 2., 0., 0., 0., 0.]]) """ # Check n if not isinstance(n, (list, np.ndarray)): msg = "n must be a list or np.array, not {}.".format(type(n)) raise TypeError(msg) else: n = np.array(n) if not np.issubdtype(n.dtype, np.integer): msg = "There are non-integer elements in n" raise ValueError(msg) # Check p if not isinstance(p, (list, np.ndarray)): msg = "p must be a list or np.array, not {}.".format(type(p)) raise TypeError(msg) else: p = np.array(p) if not np.issubdtype(p.dtype, np.number): msg = "There are non-numeric elements in p" raise ValueError(msg) elif p.shape != (n.size, n.size): msg = "p is must have shape len(n) x len(n), not {}".format(p.shape) raise ValueError(msg) elif np.any(p < 0) or np.any(p > 1): msg = "Values in p must be in between 0 and 1." raise ValueError(msg) # Check wt and wtargs if not np.issubdtype(type(wt), np.number) and not callable(wt): if not isinstance(wt, (list, np.ndarray)): msg = "wt must be a numeric, list, or np.array, not {}".format(type(wt)) raise TypeError(msg) if not isinstance(wtargs, (list, np.ndarray)): msg = "wtargs must be a numeric, list, or np.array, not {}".format( type(wtargs) ) raise TypeError(msg) wt = np.array(wt, dtype=object) wtargs = np.array(wtargs, dtype=object) # if not number, check dimensions if wt.shape != (n.size, n.size): msg = "wt must have size len(n) x len(n), not {}".format(wt.shape) raise ValueError(msg) if wtargs.shape != (n.size, n.size): msg = "wtargs must have size len(n) x len(n), not {}".format(wtargs.shape) raise ValueError(msg) # check if each element is a function for element in wt.ravel(): if not callable(element): msg = "{} is not a callable function.".format(element) raise TypeError(msg) else: wt = np.full(p.shape, wt, dtype=object) wtargs = np.full(p.shape, wtargs, dtype=object) # Check directed if not directed: if np.any(p != p.T): raise ValueError("Specified undirected, but P is directed.") if np.any(wt != wt.T): raise ValueError("Specified undirected, but Wt is directed.") if np.any(wtargs != wtargs.T): raise ValueError("Specified undirected, but Wtargs is directed.") K = len(n) # the number of communities counter = 0 # get a list of community indices cmties = [] for i in range(0, K): cmties.append(range(counter, counter + n[i])) counter += n[i] # Check degree-corrected input parameters if callable(dc): # Check that the paramters are a dict if not isinstance(dc_kws, dict): msg = "dc_kws must be of type dict not{}".format(type(dc_kws)) raise TypeError(msg) # Create the probability matrix for each vertex dcProbs = np.array([dc(**dc_kws) for _ in range(0, sum(n))], dtype="float") for indices in cmties: dcProbs[indices] /= sum(dcProbs[indices]) elif isinstance(dc, (list, np.ndarray)) and np.issubdtype( np.array(dc).dtype, np.number ): dcProbs = np.array(dc, dtype=float) # Check size and element types if not np.issubdtype(dcProbs.dtype, np.number): msg = "There are non-numeric elements in dc, {}".format(dcProbs.dtype) raise ValueError(msg) elif dcProbs.shape != (sum(n),): msg = "dc must have size equal to the number of" msg += " vertices {0}, not {1}".format(sum(n), dcProbs.shape) raise ValueError(msg) elif np.any(dcProbs < 0): msg = "Values in dc cannot be negative." raise ValueError(msg) # Check that probabilities sum to 1 in each block for i in range(0, K): if not np.isclose(sum(dcProbs[cmties[i]]), 1, atol=1.0e-8): msg = "Block {} probabilities should sum to 1, normalizing...".format(i) warnings.warn(msg, UserWarning) dcProbs[cmties[i]] /= sum(dcProbs[cmties[i]]) elif isinstance(dc, (list, np.ndarray)) and all(callable(f) for f in dc): dcFuncs = np.array(dc) if dcFuncs.shape != (len(n),): msg = "dc must have size equal to the number of blocks {0}, not {1}".format( len(n), dcFuncs.shape ) raise ValueError(msg) # Check that the parameters type, length, and type if not isinstance(dc_kws, (list, np.ndarray)): # Allows for nonspecification of default parameters for all functions if dc_kws == {}: dc_kws = [{} for _ in range(0, len(n))] else: msg = "dc_kws must be of type list or np.ndarray, not {}".format( type(dc_kws) ) raise TypeError(msg) elif not len(dc_kws) == len(n): msg = "dc_kws must have size equal to" msg += " the number of blocks {0}, not {1}".format(len(n), len(dc_kws)) raise ValueError(msg) elif not all(type(kw) == dict for kw in dc_kws): msg = "dc_kws elements must all be of type dict" raise TypeError(msg) # Create the probability matrix for each vertex dcProbs = np.array( [ dcFunc(**kws) for dcFunc, kws, size in zip(dcFuncs, dc_kws, n) for _ in range(0, size) ], dtype="float", ) # dcProbs = dcProbs.astype(float) for indices in cmties: dcProbs[indices] /= sum(dcProbs[indices]) # dcProbs[indices] = dcProbs / dcProbs[indices].sum() elif dc is not None: msg = "dc must be a function or a list or np.array of numbers or callable" msg += " functions, not {}".format(type(dc)) raise ValueError(msg) # End Checks, begin simulation A = np.zeros((sum(n), sum(n))) for i in range(0, K): if directed: jrange = range(0, K) else: jrange = range(i, K) for j in jrange: block_wt = wt[i, j] block_wtargs = wtargs[i, j] block_p = p[i, j] # identify submatrix for community i, j # cartesian product to identify edges for community i,j pair cprod = cartprod(cmties[i], cmties[j]) # get idx in 1d coordinates by ravelling triu = np.ravel_multi_index((cprod[:, 0], cprod[:, 1]), A.shape) pchoice = np.random.uniform(size=len(triu)) if dc is not None: # (v1,v2) connected with probability p*k_i*k_j*dcP[v1]*dcP[v2] num_edges = sum(pchoice < block_p) edge_dist = dcProbs[cprod[:, 0]] * dcProbs[cprod[:, 1]] # If n_edges greater than support of dc distribution, pick fewer edges if num_edges > sum(edge_dist > 0): msg = "More edges sampled than nonzero pairwise dc entries." msg += " Picking fewer edges" warnings.warn(msg, UserWarning) num_edges = sum(edge_dist > 0) triu = np.random.choice( triu, size=num_edges, replace=False, p=edge_dist ) else: # connected with probability p triu = triu[pchoice < block_p] if type(block_wt) is not int: block_wt = block_wt(size=len(triu), **block_wtargs) triu = np.unravel_index(triu, A.shape) A[triu] = block_wt if not loops: A = A - np.diag(np.diag(A)) if not directed: A = symmetrize(A, method="triu") if return_labels: labels = _n_to_labels(n) return A, labels return A
[docs]def rdpg(X, Y=None, rescale=False, directed=False, loops=False, wt=1, wtargs=None): r""" Samples a random graph based on the latent positions in X (and optionally in Y) If only X :math:`\in\mathbb{R}^{n\times d}` is given, the P matrix is calculated as :math:`P = XX^T`. If X, Y :math:`\in\mathbb{R}^{n\times d}` is given, then :math:`P = XY^T`. These operations correspond to the dot products between a set of latent positions, so each row in X or Y represents the latent positions in :math:`\mathbb{R}^{d}` for a single vertex in the random graph Note that this function may also rescale or clip the resulting P matrix to get probabilities between 0 and 1, or remove loops. A binary random graph is then sampled from the P matrix described by X (and possibly Y). Read more in the :ref:`tutorials <simulations_tutorials>` Parameters ---------- X: np.ndarray, shape (n_vertices, n_dimensions) latent position from which to generate a P matrix if Y is given, interpreted as the left latent position Y: np.ndarray, shape (n_vertices, n_dimensions) or None, optional right latent position from which to generate a P matrix rescale: boolean, optional (default=False) when rescale is True, will subtract the minimum value in P (if it is below 0) and divide by the maximum (if it is above 1) to ensure that P has entries between 0 and 1. If False, elements of P outside of [0, 1] will be clipped directed: boolean, optional (default=False) If False, output adjacency matrix will be symmetric. Otherwise, output adjacency matrix will be asymmetric. loops: boolean, optional (default=False) If False, no edges will be sampled in the diagonal. Diagonal elements in P matrix are removed prior to rescaling (see above) which may affect behavior. Otherwise, edges are sampled in the diagonal. wt: object, optional (default=1) Weight function for each of the edges, taking only a size argument. This weight function will be randomly assigned for selected edges. If 1, graph produced is binary. wtargs: dictionary, optional (default=None) Optional arguments for parameters that can be passed to weight function ``wt``. Returns ------- A: ndarray (n_vertices, n_vertices) A matrix representing the probabilities of connections between vertices in a random graph based on their latent positions References ---------- .. [1] Sussman, D.L., Tang, M., Fishkind, D.E., Priebe, C.E. "A Consistent Adjacency Spectral Embedding for Stochastic Blockmodel Graphs," Journal of the American Statistical Association, Vol. 107(499), 2012 Examples -------- >>> np.random.seed(1) Generate random latent positions using 2-dimensional Dirichlet distribution. >>> X = np.random.dirichlet([1, 1], size=5) Sample a binary RDPG using sampled latent positions. >>> rdpg(X, loops=False) array([[0., 1., 0., 0., 1.], [1., 0., 0., 1., 1.], [0., 0., 0., 1., 1.], [0., 1., 1., 0., 0.], [1., 1., 1., 0., 0.]]) Sample a weighted RDPG with Poisson(2) weight distribution >>> wt = np.random.poisson >>> wtargs = dict(lam=2) >>> rdpg(X, loops=False, wt=wt, wtargs=wtargs) array([[0., 4., 0., 2., 0.], [1., 0., 0., 0., 0.], [0., 0., 0., 0., 2.], [1., 0., 0., 0., 1.], [0., 2., 2., 0., 0.]]) """ P = p_from_latent(X, Y, rescale=rescale, loops=loops) A = sample_edges(P, directed=directed, loops=loops) # check weight function if (not np.issubdtype(type(wt), np.integer)) and ( not np.issubdtype(type(wt), np.floating) ): if not callable(wt): raise TypeError("You have not passed a function for wt.") if not np.issubdtype(type(wt), np.number): wts = wt(size=(np.count_nonzero(A)), **wtargs) A[A > 0] = wts else: A *= wt return A
def p_from_latent(X, Y=None, rescale=False, loops=True): r""" Gemerates a matrix of connection probabilities for a random graph based on a set of latent positions If only X is given, the P matrix is calculated as :math:`P = XX^T` If X and Y is given, then :math:`P = XY^T` These operations correspond to the dot products between a set of latent positions, so each row in X or Y represents the latent positions in :math:`\mathbb{R}^{num-columns}` for a single vertex in the random graph Note that this function may also rescale or clip the resulting P matrix to get probabilities between 0 and 1, or remove loops Parameters ---------- X: np.ndarray, shape (n_vertices, n_dimensions) latent position from which to generate a P matrix if Y is given, interpreted as the left latent position Y: np.ndarray, shape (n_vertices, n_dimensions) or None, optional right latent position from which to generate a P matrix rescale: boolean, optional (default=False) when rescale is True, will subtract the minimum value in P (if it is below 0) and divide by the maximum (if it is above 1) to ensure that P has entries between 0 and 1. If False, elements of P outside of [0, 1] will be clipped loops: boolean, optional (default=True) whether to allow elements on the diagonal (corresponding to self connections in a graph) in the returned P matrix. If loops is False, these elements are removed prior to rescaling (see above) which may affect behavior Returns ------- P: ndarray (n_vertices, n_vertices) A matrix representing the probabilities of connections between vertices in a random graph based on their latent positions References ---------- .. [1] Sussman, D.L., Tang, M., Fishkind, D.E., Priebe, C.E. "A Consistent Adjacency Spectral Embedding for Stochastic Blockmodel Graphs," Journal of the American Statistical Association, Vol. 107(499), 2012 """ if Y is None: Y = X if type(X) is not np.ndarray or type(Y) is not np.ndarray: raise TypeError("Latent positions must be numpy.ndarray") if X.ndim != 2 or Y.ndim != 2: raise ValueError( "Latent positions must have dimension 2 (n_vertices, n_dimensions)" ) if X.shape != Y.shape: raise ValueError("Dimensions of latent positions X and Y must be the same") P = X @ Y.T # should this be before or after the rescaling, could give diff answers if not loops: P = P - np.diag(np.diag(P)) if rescale: if P.min() < 0: P = P - P.min() if P.max() > 1: P = P / P.max() else: P[P < 0] = 0 P[P > 1] = 1 return P