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Source code for networkx.generators.community

#    Copyright(C) 2011-2019 by
#    Ben Edwards <bedwards@cs.unm.edu>
#    Aric Hagberg <hagberg@lanl.gov>
#    Konstantinos Karakatsanis <dinoskarakas@gmail.com>
#    All rights reserved.
#    BSD license.
#
# Authors:  Ben Edwards (bedwards@cs.unm.edu)
#           Aric Hagberg (hagberg@lanl.gov)
#           Konstantinos Karakatsanis (dinoskarakas@gmail.com)
#           Jean-Gabriel Young (jean.gabriel.young@gmail.com)
"""Generators for classes of graphs used in studying social networks."""
from __future__ import division
import itertools
import math
import networkx as nx
from networkx.utils import py_random_state

__all__ = ['caveman_graph', 'connected_caveman_graph',
           'relaxed_caveman_graph', 'random_partition_graph',
           'planted_partition_graph', 'gaussian_random_partition_graph',
           'ring_of_cliques', 'windmill_graph', 'stochastic_block_model']


[docs]def caveman_graph(l, k): """Returns a caveman graph of `l` cliques of size `k`. Parameters ---------- l : int Number of cliques k : int Size of cliques Returns ------- G : NetworkX Graph caveman graph Notes ----- This returns an undirected graph, it can be converted to a directed graph using :func:`nx.to_directed`, or a multigraph using ``nx.MultiGraph(nx.caveman_graph(l, k))``. Only the undirected version is described in [1]_ and it is unclear which of the directed generalizations is most useful. Examples -------- >>> G = nx.caveman_graph(3, 3) See also -------- connected_caveman_graph References ---------- .. [1] Watts, D. J. 'Networks, Dynamics, and the Small-World Phenomenon.' Amer. J. Soc. 105, 493-527, 1999. """ # l disjoint cliques of size k G = nx.empty_graph(l * k) if k > 1: for start in range(0, l * k, k): edges = itertools.combinations(range(start, start + k), 2) G.add_edges_from(edges) return G
[docs]def connected_caveman_graph(l, k): """Returns a connected caveman graph of `l` cliques of size `k`. The connected caveman graph is formed by creating `n` cliques of size `k`, then a single edge in each clique is rewired to a node in an adjacent clique. Parameters ---------- l : int number of cliques k : int size of cliques Returns ------- G : NetworkX Graph connected caveman graph Notes ----- This returns an undirected graph, it can be converted to a directed graph using :func:`nx.to_directed`, or a multigraph using ``nx.MultiGraph(nx.caveman_graph(l, k))``. Only the undirected version is described in [1]_ and it is unclear which of the directed generalizations is most useful. Examples -------- >>> G = nx.connected_caveman_graph(3, 3) References ---------- .. [1] Watts, D. J. 'Networks, Dynamics, and the Small-World Phenomenon.' Amer. J. Soc. 105, 493-527, 1999. """ G = nx.caveman_graph(l, k) for start in range(0, l * k, k): G.remove_edge(start, start + 1) G.add_edge(start, (start - 1) % (l * k)) return G
[docs]@py_random_state(3) def relaxed_caveman_graph(l, k, p, seed=None): """Returns a relaxed caveman graph. A relaxed caveman graph starts with `l` cliques of size `k`. Edges are then randomly rewired with probability `p` to link different cliques. Parameters ---------- l : int Number of groups k : int Size of cliques p : float Probabilty of rewiring each edge. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. Returns ------- G : NetworkX Graph Relaxed Caveman Graph Raises ------ NetworkXError: If p is not in [0,1] Examples -------- >>> G = nx.relaxed_caveman_graph(2, 3, 0.1, seed=42) References ---------- .. [1] Santo Fortunato, Community Detection in Graphs, Physics Reports Volume 486, Issues 3-5, February 2010, Pages 75-174. https://arxiv.org/abs/0906.0612 """ G = nx.caveman_graph(l, k) nodes = list(G) for (u, v) in G.edges(): if seed.random() < p: # rewire the edge x = seed.choice(nodes) if G.has_edge(u, x): continue G.remove_edge(u, v) G.add_edge(u, x) return G
[docs]@py_random_state(3) def random_partition_graph(sizes, p_in, p_out, seed=None, directed=False): """Returns the random partition graph with a partition of sizes. A partition graph is a graph of communities with sizes defined by s in sizes. Nodes in the same group are connected with probability p_in and nodes of different groups are connected with probability p_out. Parameters ---------- sizes : list of ints Sizes of groups p_in : float probability of edges with in groups p_out : float probability of edges between groups directed : boolean optional, default=False Whether to create a directed graph seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. Returns ------- G : NetworkX Graph or DiGraph random partition graph of size sum(gs) Raises ------ NetworkXError If p_in or p_out is not in [0,1] Examples -------- >>> G = nx.random_partition_graph([10,10,10],.25,.01) >>> len(G) 30 >>> partition = G.graph['partition'] >>> len(partition) 3 Notes ----- This is a generalization of the planted-l-partition described in [1]_. It allows for the creation of groups of any size. The partition is store as a graph attribute 'partition'. References ---------- .. [1] Santo Fortunato 'Community Detection in Graphs' Physical Reports Volume 486, Issue 3-5 p. 75-174. https://arxiv.org/abs/0906.0612 """ # Use geometric method for O(n+m) complexity algorithm # partition = nx.community_sets(nx.get_node_attributes(G, 'affiliation')) if not 0.0 <= p_in <= 1.0: raise nx.NetworkXError("p_in must be in [0,1]") if not 0.0 <= p_out <= 1.0: raise nx.NetworkXError("p_out must be in [0,1]") # create connection matrix num_blocks = len(sizes) p = [[p_out for s in range(num_blocks)] for r in range(num_blocks)] for r in range(num_blocks): p[r][r] = p_in return stochastic_block_model(sizes, p, nodelist=None, seed=seed, directed=directed, selfloops=False, sparse=True)
[docs]@py_random_state(4) def planted_partition_graph(l, k, p_in, p_out, seed=None, directed=False): """Returns the planted l-partition graph. This model partitions a graph with n=l*k vertices in l groups with k vertices each. Vertices of the same group are linked with a probability p_in, and vertices of different groups are linked with probability p_out. Parameters ---------- l : int Number of groups k : int Number of vertices in each group p_in : float probability of connecting vertices within a group p_out : float probability of connected vertices between groups seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. directed : bool,optional (default=False) If True return a directed graph Returns ------- G : NetworkX Graph or DiGraph planted l-partition graph Raises ------ NetworkXError: If p_in,p_out are not in [0,1] or Examples -------- >>> G = nx.planted_partition_graph(4, 3, 0.5, 0.1, seed=42) See Also -------- random_partition_model References ---------- .. [1] A. Condon, R.M. Karp, Algorithms for graph partitioning on the planted partition model, Random Struct. Algor. 18 (2001) 116-140. .. [2] Santo Fortunato 'Community Detection in Graphs' Physical Reports Volume 486, Issue 3-5 p. 75-174. https://arxiv.org/abs/0906.0612 """ return random_partition_graph([k] * l, p_in, p_out, seed, directed)
[docs]@py_random_state(6) def gaussian_random_partition_graph(n, s, v, p_in, p_out, directed=False, seed=None): """Generate a Gaussian random partition graph. A Gaussian random partition graph is created by creating k partitions each with a size drawn from a normal distribution with mean s and variance s/v. Nodes are connected within clusters with probability p_in and between clusters with probability p_out[1] Parameters ---------- n : int Number of nodes in the graph s : float Mean cluster size v : float Shape parameter. The variance of cluster size distribution is s/v. p_in : float Probabilty of intra cluster connection. p_out : float Probability of inter cluster connection. directed : boolean, optional default=False Whether to create a directed graph or not seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. Returns ------- G : NetworkX Graph or DiGraph gaussian random partition graph Raises ------ NetworkXError If s is > n If p_in or p_out is not in [0,1] Notes ----- Note the number of partitions is dependent on s,v and n, and that the last partition may be considerably smaller, as it is sized to simply fill out the nodes [1] See Also -------- random_partition_graph Examples -------- >>> G = nx.gaussian_random_partition_graph(100,10,10,.25,.1) >>> len(G) 100 References ---------- .. [1] Ulrik Brandes, Marco Gaertler, Dorothea Wagner, Experiments on Graph Clustering Algorithms, In the proceedings of the 11th Europ. Symp. Algorithms, 2003. """ if s > n: raise nx.NetworkXError("s must be <= n") assigned = 0 sizes = [] while True: size = int(seed.gauss(s, float(s) / v + 0.5)) if size < 1: # how to handle 0 or negative sizes? continue if assigned + size >= n: sizes.append(n - assigned) break assigned += size sizes.append(size) return random_partition_graph(sizes, p_in, p_out, directed, seed)
[docs]def ring_of_cliques(num_cliques, clique_size): """Defines a "ring of cliques" graph. A ring of cliques graph is consisting of cliques, connected through single links. Each clique is a complete graph. Parameters ---------- num_cliques : int Number of cliques clique_size : int Size of cliques Returns ------- G : NetworkX Graph ring of cliques graph Raises ------ NetworkXError If the number of cliques is lower than 2 or if the size of cliques is smaller than 2. Examples -------- >>> G = nx.ring_of_cliques(8, 4) See Also -------- connected_caveman_graph Notes ----- The `connected_caveman_graph` graph removes a link from each clique to connect it with the next clique. Instead, the `ring_of_cliques` graph simply adds the link without removing any link from the cliques. """ if num_cliques < 2: raise nx.NetworkXError('A ring of cliques must have at least ' 'two cliques') if clique_size < 2: raise nx.NetworkXError('The cliques must have at least two nodes') G = nx.Graph() for i in range(num_cliques): edges = itertools.combinations(range(i * clique_size, i * clique_size + clique_size), 2) G.add_edges_from(edges) G.add_edge(i * clique_size + 1, (i + 1) * clique_size % (num_cliques * clique_size)) return G
[docs]def windmill_graph(n, k): """Generate a windmill graph. A windmill graph is a graph of `n` cliques each of size `k` that are all joined at one node. It can be thought of as taking a disjoint union of `n` cliques of size `k`, selecting one point from each, and contracting all of the selected points. Alternatively, one could generate `n` cliques of size `k-1` and one node that is connected to all other nodes in the graph. Parameters ---------- n : int Number of cliques k : int Size of cliques Returns ------- G : NetworkX Graph windmill graph with n cliques of size k Raises ------ NetworkXError If the number of cliques is less than two If the size of the cliques are less than two Examples -------- >>> G = nx.windmill_graph(4, 5) Notes ----- The node labeled `0` will be the node connected to all other nodes. Note that windmill graphs are usually denoted `Wd(k,n)`, so the parameters are in the opposite order as the parameters of this method. """ if n < 2: msg = 'A windmill graph must have at least two cliques' raise nx.NetworkXError(msg) if k < 2: raise nx.NetworkXError('The cliques must have at least two nodes') G = nx.disjoint_union_all(itertools.chain([nx.complete_graph(k)], (nx.complete_graph(k - 1) for _ in range(n - 1)))) G.add_edges_from((0, i) for i in range(k, G.number_of_nodes())) return G
[docs]@py_random_state(3) def stochastic_block_model(sizes, p, nodelist=None, seed=None, directed=False, selfloops=False, sparse=True): """Returns a stochastic block model graph. This model partitions the nodes in blocks of arbitrary sizes, and places edges between pairs of nodes independently, with a probability that depends on the blocks. Parameters ---------- sizes : list of ints Sizes of blocks p : list of list of floats Element (r,s) gives the density of edges going from the nodes of group r to nodes of group s. p must match the number of groups (len(sizes) == len(p)), and it must be symmetric if the graph is undirected. nodelist : list, optional The block tags are assigned according to the node identifiers in nodelist. If nodelist is None, then the ordering is the range [0,sum(sizes)-1]. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. directed : boolean optional, default=False Whether to create a directed graph or not. selfloops : boolean optional, default=False Whether to include self-loops or not. sparse: boolean optional, default=True Use the sparse heuristic to speed up the generator. Returns ------- g : NetworkX Graph or DiGraph Stochastic block model graph of size sum(sizes) Raises ------ NetworkXError If probabilities are not in [0,1]. If the probability matrix is not square (directed case). If the probability matrix is not symmetric (undirected case). If the sizes list does not match nodelist or the probability matrix. If nodelist contains duplicate. Examples -------- >>> sizes = [75, 75, 300] >>> probs = [[0.25, 0.05, 0.02], ... [0.05, 0.35, 0.07], ... [0.02, 0.07, 0.40]] >>> g = nx.stochastic_block_model(sizes, probs, seed=0) >>> len(g) 450 >>> H = nx.quotient_graph(g, g.graph['partition'], relabel=True) >>> for v in H.nodes(data=True): ... print(round(v[1]['density'], 3)) ... 0.245 0.348 0.405 >>> for v in H.edges(data=True): ... print(round(1.0 * v[2]['weight'] / (sizes[v[0]] * sizes[v[1]]), 3)) ... 0.051 0.022 0.07 See Also -------- random_partition_graph planted_partition_graph gaussian_random_partition_graph gnp_random_graph References ---------- .. [1] Holland, P. W., Laskey, K. B., & Leinhardt, S., "Stochastic blockmodels: First steps", Social networks, 5(2), 109-137, 1983. """ # Check if dimensions match if len(sizes) != len(p): raise nx.NetworkXException("'sizes' and 'p' do not match.") # Check for probability symmetry (undirected) and shape (directed) for row in p: if len(p) != len(row): raise nx.NetworkXException("'p' must be a square matrix.") if not directed: p_transpose = [list(i) for i in zip(*p)] for i in zip(p, p_transpose): for j in zip(i[0], i[1]): if abs(j[0] - j[1]) > 1e-08: raise nx.NetworkXException("'p' must be symmetric.") # Check for probability range for row in p: for prob in row: if prob < 0 or prob > 1: raise nx.NetworkXException("Entries of 'p' not in [0,1].") # Check for nodelist consistency if nodelist is not None: if len(nodelist) != sum(sizes): raise nx.NetworkXException("'nodelist' and 'sizes' do not match.") if len(nodelist) != len(set(nodelist)): raise nx.NetworkXException("nodelist contains duplicate.") else: nodelist = range(0, sum(sizes)) # Setup the graph conditionally to the directed switch. block_range = range(len(sizes)) if directed: g = nx.DiGraph() block_iter = itertools.product(block_range, block_range) else: g = nx.Graph() block_iter = itertools.combinations_with_replacement(block_range, 2) # Split nodelist in a partition (list of sets). size_cumsum = [sum(sizes[0:x]) for x in range(0, len(sizes) + 1)] g.graph['partition'] = [set(nodelist[size_cumsum[x]:size_cumsum[x + 1]]) for x in range(0, len(size_cumsum) - 1)] # Setup nodes and graph name for block_id, nodes in enumerate(g.graph['partition']): for node in nodes: g.add_node(node, block=block_id) g.name = "stochastic_block_model" # Test for edge existence parts = g.graph['partition'] for i, j in block_iter: if i == j: if directed: if selfloops: edges = itertools.product(parts[i], parts[i]) else: edges = itertools.permutations(parts[i], 2) else: edges = itertools.combinations(parts[i], 2) if selfloops: edges = itertools.chain(edges, zip(parts[i], parts[i])) for e in edges: if seed.random() < p[i][j]: g.add_edge(*e) else: edges = itertools.product(parts[i], parts[j]) if sparse: if p[i][j] == 1: # Test edges cases p_ij = 0 or 1 for e in edges: g.add_edge(*e) elif p[i][j] > 0: while True: try: logrand = math.log(seed.random()) skip = math.floor(logrand / math.log(1 - p[i][j])) # consume "skip" edges next(itertools.islice(edges, skip, skip), None) e = next(edges) g.add_edge(*e) # __safe except StopIteration: break else: for e in edges: if seed.random() < p[i][j]: g.add_edge(*e) # __safe return g