Source code for networkx.algorithms.distance_measures

"""Graph diameter, radius, eccentricity and other properties."""

import networkx as nx
from networkx.utils import not_implemented_for

__all__ = [
    "extrema_bounding",
    "eccentricity",
    "diameter",
    "radius",
    "periphery",
    "center",
    "barycenter",
    "resistance_distance",
]


[docs]def extrema_bounding(G, compute="diameter"): """Compute requested extreme distance metric of undirected graph G Computation is based on smart lower and upper bounds, and in practice linear in the number of nodes, rather than quadratic (except for some border cases such as complete graphs or circle shaped graphs). Parameters ---------- G : NetworkX graph An undirected graph compute : string denoting the requesting metric "diameter" for the maximal eccentricity value, "radius" for the minimal eccentricity value, "periphery" for the set of nodes with eccentricity equal to the diameter "center" for the set of nodes with eccentricity equal to the radius Returns ------- value : value of the requested metric int for "diameter" and "radius" or list of nodes for "center" and "periphery" Raises ------ NetworkXError If the graph consists of multiple components Notes ----- This algorithm was proposed in the following papers: F.W. Takes and W.A. Kosters, Determining the Diameter of Small World Networks, in Proceedings of the 20th ACM International Conference on Information and Knowledge Management (CIKM 2011), pp. 1191-1196, 2011. doi: https://doi.org/10.1145/2063576.2063748 F.W. Takes and W.A. Kosters, Computing the Eccentricity Distribution of Large Graphs, Algorithms 6(1): 100-118, 2013. doi: https://doi.org/10.3390/a6010100 M. Borassi, P. Crescenzi, M. Habib, W.A. Kosters, A. Marino and F.W. Takes, Fast Graph Diameter and Radius BFS-Based Computation in (Weakly Connected) Real-World Graphs, Theoretical Computer Science 586: 59-80, 2015. doi: https://doi.org/10.1016/j.tcs.2015.02.033 """ # init variables degrees = dict(G.degree()) # start with the highest degree node minlowernode = max(degrees, key=degrees.get) N = len(degrees) # number of nodes # alternate between smallest lower and largest upper bound high = False # status variables ecc_lower = dict.fromkeys(G, 0) ecc_upper = dict.fromkeys(G, N) candidates = set(G) # (re)set bound extremes minlower = N maxlower = 0 minupper = N maxupper = 0 # repeat the following until there are no more candidates while candidates: if high: current = maxuppernode # select node with largest upper bound else: current = minlowernode # select node with smallest lower bound high = not high # get distances from/to current node and derive eccentricity dist = dict(nx.single_source_shortest_path_length(G, current)) if len(dist) != N: msg = "Cannot compute metric because graph is not connected." raise nx.NetworkXError(msg) current_ecc = max(dist.values()) # print status update # print ("ecc of " + str(current) + " (" + str(ecc_lower[current]) + "/" # + str(ecc_upper[current]) + ", deg: " + str(dist[current]) + ") is " # + str(current_ecc)) # print(ecc_upper) # (re)set bound extremes maxuppernode = None minlowernode = None # update node bounds for i in candidates: # update eccentricity bounds d = dist[i] ecc_lower[i] = low = max(ecc_lower[i], max(d, (current_ecc - d))) ecc_upper[i] = upp = min(ecc_upper[i], current_ecc + d) # update min/max values of lower and upper bounds minlower = min(ecc_lower[i], minlower) maxlower = max(ecc_lower[i], maxlower) minupper = min(ecc_upper[i], minupper) maxupper = max(ecc_upper[i], maxupper) # update candidate set if compute == "diameter": ruled_out = { i for i in candidates if ecc_upper[i] <= maxlower and 2 * ecc_lower[i] >= maxupper } elif compute == "radius": ruled_out = { i for i in candidates if ecc_lower[i] >= minupper and ecc_upper[i] + 1 <= 2 * minlower } elif compute == "periphery": ruled_out = { i for i in candidates if ecc_upper[i] < maxlower and (maxlower == maxupper or ecc_lower[i] > maxupper) } elif compute == "center": ruled_out = { i for i in candidates if ecc_lower[i] > minupper and (minlower == minupper or ecc_upper[i] + 1 < 2 * minlower) } elif compute == "eccentricities": ruled_out = {} ruled_out.update(i for i in candidates if ecc_lower[i] == ecc_upper[i]) candidates -= ruled_out # for i in ruled_out: # print("removing %g: ecc_u: %g maxl: %g ecc_l: %g maxu: %g"% # (i,ecc_upper[i],maxlower,ecc_lower[i],maxupper)) # print("node %g: ecc_u: %g maxl: %g ecc_l: %g maxu: %g"% # (4,ecc_upper[4],maxlower,ecc_lower[4],maxupper)) # print("NODE 4: %g"%(ecc_upper[4] <= maxlower)) # print("NODE 4: %g"%(2 * ecc_lower[4] >= maxupper)) # print("NODE 4: %g"%(ecc_upper[4] <= maxlower # and 2 * ecc_lower[4] >= maxupper)) # updating maxuppernode and minlowernode for selection in next round for i in candidates: if ( minlowernode is None or ( ecc_lower[i] == ecc_lower[minlowernode] and degrees[i] > degrees[minlowernode] ) or (ecc_lower[i] < ecc_lower[minlowernode]) ): minlowernode = i if ( maxuppernode is None or ( ecc_upper[i] == ecc_upper[maxuppernode] and degrees[i] > degrees[maxuppernode] ) or (ecc_upper[i] > ecc_upper[maxuppernode]) ): maxuppernode = i # print status update # print (" min=" + str(minlower) + "/" + str(minupper) + # " max=" + str(maxlower) + "/" + str(maxupper) + # " candidates: " + str(len(candidates))) # print("cand:",candidates) # print("ecc_l",ecc_lower) # print("ecc_u",ecc_upper) # wait = input("press Enter to continue") # return the correct value of the requested metric if compute == "diameter": return maxlower elif compute == "radius": return minupper elif compute == "periphery": p = [v for v in G if ecc_lower[v] == maxlower] return p elif compute == "center": c = [v for v in G if ecc_upper[v] == minupper] return c elif compute == "eccentricities": return ecc_lower return None
[docs]def eccentricity(G, v=None, sp=None): """Returns the eccentricity of nodes in G. The eccentricity of a node v is the maximum distance from v to all other nodes in G. Parameters ---------- G : NetworkX graph A graph v : node, optional Return value of specified node sp : dict of dicts, optional All pairs shortest path lengths as a dictionary of dictionaries Returns ------- ecc : dictionary A dictionary of eccentricity values keyed by node. """ # if v is None: # none, use entire graph # nodes=G.nodes() # elif v in G: # is v a single node # nodes=[v] # else: # assume v is a container of nodes # nodes=v order = G.order() e = {} for n in G.nbunch_iter(v): if sp is None: length = nx.single_source_shortest_path_length(G, n) L = len(length) else: try: length = sp[n] L = len(length) except TypeError as e: raise nx.NetworkXError('Format of "sp" is invalid.') from e if L != order: if G.is_directed(): msg = ( "Found infinite path length because the digraph is not" " strongly connected" ) else: msg = "Found infinite path length because the graph is not" " connected" raise nx.NetworkXError(msg) e[n] = max(length.values()) if v in G: return e[v] # return single value else: return e
[docs]def diameter(G, e=None, usebounds=False): """Returns the diameter of the graph G. The diameter is the maximum eccentricity. Parameters ---------- G : NetworkX graph A graph e : eccentricity dictionary, optional A precomputed dictionary of eccentricities. Returns ------- d : integer Diameter of graph See Also -------- eccentricity """ if usebounds is True and e is None and not G.is_directed(): return extrema_bounding(G, compute="diameter") if e is None: e = eccentricity(G) return max(e.values())
[docs]def periphery(G, e=None, usebounds=False): """Returns the periphery of the graph G. The periphery is the set of nodes with eccentricity equal to the diameter. Parameters ---------- G : NetworkX graph A graph e : eccentricity dictionary, optional A precomputed dictionary of eccentricities. Returns ------- p : list List of nodes in periphery See Also -------- barycenter center """ if usebounds is True and e is None and not G.is_directed(): return extrema_bounding(G, compute="periphery") if e is None: e = eccentricity(G) diameter = max(e.values()) p = [v for v in e if e[v] == diameter] return p
[docs]def radius(G, e=None, usebounds=False): """Returns the radius of the graph G. The radius is the minimum eccentricity. Parameters ---------- G : NetworkX graph A graph e : eccentricity dictionary, optional A precomputed dictionary of eccentricities. Returns ------- r : integer Radius of graph """ if usebounds is True and e is None and not G.is_directed(): return extrema_bounding(G, compute="radius") if e is None: e = eccentricity(G) return min(e.values())
[docs]def center(G, e=None, usebounds=False): """Returns the center of the graph G. The center is the set of nodes with eccentricity equal to radius. Parameters ---------- G : NetworkX graph A graph e : eccentricity dictionary, optional A precomputed dictionary of eccentricities. Returns ------- c : list List of nodes in center See Also -------- barycenter periphery """ if usebounds is True and e is None and not G.is_directed(): return extrema_bounding(G, compute="center") if e is None: e = eccentricity(G) radius = min(e.values()) p = [v for v in e if e[v] == radius] return p
[docs]def barycenter(G, weight=None, attr=None, sp=None): r"""Calculate barycenter of a connected graph, optionally with edge weights. The :dfn:`barycenter` a :func:`connected <networkx.algorithms.components.is_connected>` graph :math:`G` is the subgraph induced by the set of its nodes :math:`v` minimizing the objective function .. math:: \sum_{u \in V(G)} d_G(u, v), where :math:`d_G` is the (possibly weighted) :func:`path length <networkx.algorithms.shortest_paths.generic.shortest_path_length>`. The barycenter is also called the :dfn:`median`. See [West01]_, p. 78. Parameters ---------- G : :class:`networkx.Graph` The connected graph :math:`G`. weight : :class:`str`, optional Passed through to :func:`~networkx.algorithms.shortest_paths.generic.shortest_path_length`. attr : :class:`str`, optional If given, write the value of the objective function to each node's `attr` attribute. Otherwise do not store the value. sp : dict of dicts, optional All pairs shortest path lengths as a dictionary of dictionaries Returns ------- list Nodes of `G` that induce the barycenter of `G`. Raises ------ NetworkXNoPath If `G` is disconnected. `G` may appear disconnected to :func:`barycenter` if `sp` is given but is missing shortest path lengths for any pairs. ValueError If `sp` and `weight` are both given. See Also -------- center periphery """ if sp is None: sp = nx.shortest_path_length(G, weight=weight) else: sp = sp.items() if weight is not None: raise ValueError("Cannot use both sp, weight arguments together") smallest, barycenter_vertices, n = float("inf"), [], len(G) for v, dists in sp: if len(dists) < n: raise nx.NetworkXNoPath( f"Input graph {G} is disconnected, so every induced subgraph " "has infinite barycentricity." ) barycentricity = sum(dists.values()) if attr is not None: G.nodes[v][attr] = barycentricity if barycentricity < smallest: smallest = barycentricity barycenter_vertices = [v] elif barycentricity == smallest: barycenter_vertices.append(v) return barycenter_vertices
def _laplacian_submatrix(node, mat, node_list): """Removes row/col from a sparse matrix and returns the submatrix""" j = node_list.index(node) n = list(range(len(node_list))) n.pop(j) if mat.shape[0] != mat.shape[1]: raise nx.NetworkXError("Matrix must be square") elif len(node_list) != mat.shape[0]: msg = "Node list length does not match matrix dimentions" raise nx.NetworkXError(msg) mat = mat.tocsr() mat = mat[n, :] mat = mat.tocsc() mat = mat[:, n] node_list.pop(j) return mat, node_list def _count_lu_permutations(perm_array): """Counts the number of permutations in SuperLU perm_c or perm_r""" perm_cnt = 0 arr = perm_array.tolist() for i in range(len(arr)): if i != arr[i]: perm_cnt += 1 n = arr.index(i) arr[n] = arr[i] arr[i] = i return perm_cnt
[docs]@not_implemented_for("directed") def resistance_distance(G, nodeA, nodeB, weight=None, invert_weight=True): """Returns the resistance distance between node A and node B on graph G. The resistance distance between two nodes of a graph is akin to treating the graph as a grid of resistorses with a resistance equal to the provided weight. If weight is not provided, then a weight of 1 is used for all edges. Parameters ---------- G : NetworkX graph A graph nodeA : node A node within graph G. nodeB : node A node within graph G, exclusive of Node A. weight : string or None, optional (default=None) The edge data key used to compute the resistance distance. If None, then each edge has weight 1. invert_weight : boolean (default=True) Proper calculation of resistance distance requires building the Laplacian matrix with the reciprocal of the weight. Not required if the weight is already inverted. Weight cannot be zero. Returns ------- rd : float Value of effective resistance distance Notes ----- Overview discussion: * https://en.wikipedia.org/wiki/Resistance_distance * http://mathworld.wolfram.com/ResistanceDistance.html Additional details: Vaya Sapobi Samui Vos, “Methods for determining the effective resistance,” M.S., Mathematisch Instituut, Universiteit Leiden, Leiden, Netherlands, 2016 Available: `Link to thesis <https://www.universiteitleiden.nl/binaries/content/assets/science/mi/scripties/master/vos_vaya_master.pdf>`_ """ import numpy as np import scipy as sp import scipy.sparse.linalg # call as sp.sparse.linalg if not nx.is_connected(G): msg = "Graph G must be strongly connected." raise nx.NetworkXError(msg) elif nodeA not in G: msg = "Node A is not in graph G." raise nx.NetworkXError(msg) elif nodeB not in G: msg = "Node B is not in graph G." raise nx.NetworkXError(msg) elif nodeA == nodeB: msg = "Node A and Node B cannot be the same." raise nx.NetworkXError(msg) G = G.copy() node_list = list(G) if invert_weight and weight is not None: if G.is_multigraph(): for (u, v, k, d) in G.edges(keys=True, data=True): d[weight] = 1 / d[weight] else: for (u, v, d) in G.edges(data=True): d[weight] = 1 / d[weight] # Replace with collapsing topology or approximated zero? # Using determinants to compute the effective resistance is more memory # efficent than directly calculating the psuedo-inverse L = nx.laplacian_matrix(G, node_list, weight=weight) Lsub_a, node_list_a = _laplacian_submatrix(nodeA, L.copy(), node_list[:]) Lsub_ab, node_list_ab = _laplacian_submatrix(nodeB, Lsub_a.copy(), node_list_a[:]) # Factorize Laplacian submatrixes and extract diagonals # Order the diagonals to minimize the likelihood over overflows # during computing the determinant lu_a = sp.sparse.linalg.splu(Lsub_a, options=dict(SymmetricMode=True)) LdiagA = lu_a.U.diagonal() LdiagA_s = np.product(np.sign(LdiagA)) * np.product(lu_a.L.diagonal()) LdiagA_s *= (-1) ** _count_lu_permutations(lu_a.perm_r) LdiagA_s *= (-1) ** _count_lu_permutations(lu_a.perm_c) LdiagA = np.absolute(LdiagA) LdiagA = np.sort(LdiagA) lu_ab = sp.sparse.linalg.splu(Lsub_ab, options=dict(SymmetricMode=True)) LdiagAB = lu_ab.U.diagonal() LdiagAB_s = np.product(np.sign(LdiagAB)) * np.product(lu_ab.L.diagonal()) LdiagAB_s *= (-1) ** _count_lu_permutations(lu_ab.perm_r) LdiagAB_s *= (-1) ** _count_lu_permutations(lu_ab.perm_c) LdiagAB = np.absolute(LdiagAB) LdiagAB = np.sort(LdiagAB) # Calculate the ratio of determinant, rd = det(Lsub_ab)/det(Lsub_a) Ldet = np.product(np.divide(np.append(LdiagAB, [1]), LdiagA)) rd = Ldet * LdiagAB_s / LdiagA_s return rd