Source code for networkx.algorithms.smallworld

"""Functions for estimating the small-world-ness of graphs.

A small world network is characterized by a small average shortest path length,
and a large clustering coefficient.

Small-worldness is commonly measured with the coefficient sigma or omega.

Both coefficients compare the average clustering coefficient and shortest path
length of a given graph against the same quantities for an equivalent random
or lattice graph.

For more information, see the Wikipedia article on small-world network [1]_.

.. [1] Small-world network:: https://en.wikipedia.org/wiki/Small-world_network

"""
import networkx as nx
from networkx.utils import not_implemented_for, py_random_state

__all__ = ["random_reference", "lattice_reference", "sigma", "omega"]


[docs] @not_implemented_for("directed") @not_implemented_for("multigraph") @py_random_state(3) @nx._dispatchable def random_reference(G, niter=1, connectivity=True, seed=None): """Compute a random graph by swapping edges of a given graph. Parameters ---------- G : graph An undirected graph with 4 or more nodes. niter : integer (optional, default=1) An edge is rewired approximately `niter` times. connectivity : boolean (optional, default=True) When True, ensure connectivity for the randomized graph. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. Returns ------- G : graph The randomized graph. Raises ------ NetworkXError If there are fewer than 4 nodes or 2 edges in `G` Notes ----- The implementation is adapted from the algorithm by Maslov and Sneppen (2002) [1]_. References ---------- .. [1] Maslov, Sergei, and Kim Sneppen. "Specificity and stability in topology of protein networks." Science 296.5569 (2002): 910-913. """ if len(G) < 4: raise nx.NetworkXError("Graph has fewer than four nodes.") if len(G.edges) < 2: raise nx.NetworkXError("Graph has fewer that 2 edges") from networkx.utils import cumulative_distribution, discrete_sequence local_conn = nx.connectivity.local_edge_connectivity G = G.copy() keys, degrees = zip(*G.degree()) # keys, degree cdf = cumulative_distribution(degrees) # cdf of degree nnodes = len(G) nedges = nx.number_of_edges(G) niter = niter * nedges ntries = int(nnodes * nedges / (nnodes * (nnodes - 1) / 2)) swapcount = 0 for i in range(niter): n = 0 while n < ntries: # pick two random edges without creating edge list # choose source node indices from discrete distribution (ai, ci) = discrete_sequence(2, cdistribution=cdf, seed=seed) if ai == ci: continue # same source, skip a = keys[ai] # convert index to label c = keys[ci] # choose target uniformly from neighbors b = seed.choice(list(G.neighbors(a))) d = seed.choice(list(G.neighbors(c))) if b in [a, c, d] or d in [a, b, c]: continue # all vertices should be different # don't create parallel edges if (d not in G[a]) and (b not in G[c]): G.add_edge(a, d) G.add_edge(c, b) G.remove_edge(a, b) G.remove_edge(c, d) # Check if the graph is still connected if connectivity and local_conn(G, a, b) == 0: # Not connected, revert the swap G.remove_edge(a, d) G.remove_edge(c, b) G.add_edge(a, b) G.add_edge(c, d) else: swapcount += 1 break n += 1 return G
[docs] @not_implemented_for("directed") @not_implemented_for("multigraph") @py_random_state(4) @nx._dispatchable def lattice_reference(G, niter=5, D=None, connectivity=True, seed=None): """Latticize the given graph by swapping edges. Parameters ---------- G : graph An undirected graph. niter : integer (optional, default=1) An edge is rewired approximately niter times. D : numpy.array (optional, default=None) Distance to the diagonal matrix. connectivity : boolean (optional, default=True) Ensure connectivity for the latticized graph when set to True. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. Returns ------- G : graph The latticized graph. Raises ------ NetworkXError If there are fewer than 4 nodes or 2 edges in `G` Notes ----- The implementation is adapted from the algorithm by Sporns et al. [1]_. which is inspired from the original work by Maslov and Sneppen(2002) [2]_. References ---------- .. [1] Sporns, Olaf, and Jonathan D. Zwi. "The small world of the cerebral cortex." Neuroinformatics 2.2 (2004): 145-162. .. [2] Maslov, Sergei, and Kim Sneppen. "Specificity and stability in topology of protein networks." Science 296.5569 (2002): 910-913. """ import numpy as np from networkx.utils import cumulative_distribution, discrete_sequence local_conn = nx.connectivity.local_edge_connectivity if len(G) < 4: raise nx.NetworkXError("Graph has fewer than four nodes.") if len(G.edges) < 2: raise nx.NetworkXError("Graph has fewer that 2 edges") # Instead of choosing uniformly at random from a generated edge list, # this algorithm chooses nonuniformly from the set of nodes with # probability weighted by degree. G = G.copy() keys, degrees = zip(*G.degree()) # keys, degree cdf = cumulative_distribution(degrees) # cdf of degree nnodes = len(G) nedges = nx.number_of_edges(G) if D is None: D = np.zeros((nnodes, nnodes)) un = np.arange(1, nnodes) um = np.arange(nnodes - 1, 0, -1) u = np.append((0,), np.where(un < um, un, um)) for v in range(int(np.ceil(nnodes / 2))): D[nnodes - v - 1, :] = np.append(u[v + 1 :], u[: v + 1]) D[v, :] = D[nnodes - v - 1, :][::-1] niter = niter * nedges # maximal number of rewiring attempts per 'niter' max_attempts = int(nnodes * nedges / (nnodes * (nnodes - 1) / 2)) for _ in range(niter): n = 0 while n < max_attempts: # pick two random edges without creating edge list # choose source node indices from discrete distribution (ai, ci) = discrete_sequence(2, cdistribution=cdf, seed=seed) if ai == ci: continue # same source, skip a = keys[ai] # convert index to label c = keys[ci] # choose target uniformly from neighbors b = seed.choice(list(G.neighbors(a))) d = seed.choice(list(G.neighbors(c))) bi = keys.index(b) di = keys.index(d) if b in [a, c, d] or d in [a, b, c]: continue # all vertices should be different # don't create parallel edges if (d not in G[a]) and (b not in G[c]): if D[ai, bi] + D[ci, di] >= D[ai, ci] + D[bi, di]: # only swap if we get closer to the diagonal G.add_edge(a, d) G.add_edge(c, b) G.remove_edge(a, b) G.remove_edge(c, d) # Check if the graph is still connected if connectivity and local_conn(G, a, b) == 0: # Not connected, revert the swap G.remove_edge(a, d) G.remove_edge(c, b) G.add_edge(a, b) G.add_edge(c, d) else: break n += 1 return G
[docs] @not_implemented_for("directed") @not_implemented_for("multigraph") @py_random_state(3) @nx._dispatchable def sigma(G, niter=100, nrand=10, seed=None): """Returns the small-world coefficient (sigma) of the given graph. The small-world coefficient is defined as: sigma = C/Cr / L/Lr where C and L are respectively the average clustering coefficient and average shortest path length of G. Cr and Lr are respectively the average clustering coefficient and average shortest path length of an equivalent random graph. A graph is commonly classified as small-world if sigma>1. Parameters ---------- G : NetworkX graph An undirected graph. niter : integer (optional, default=100) Approximate number of rewiring per edge to compute the equivalent random graph. nrand : integer (optional, default=10) Number of random graphs generated to compute the average clustering coefficient (Cr) and average shortest path length (Lr). seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. Returns ------- sigma : float The small-world coefficient of G. Notes ----- The implementation is adapted from Humphries et al. [1]_ [2]_. References ---------- .. [1] The brainstem reticular formation is a small-world, not scale-free, network M. D. Humphries, K. Gurney and T. J. Prescott, Proc. Roy. Soc. B 2006 273, 503-511, doi:10.1098/rspb.2005.3354. .. [2] Humphries and Gurney (2008). "Network 'Small-World-Ness': A Quantitative Method for Determining Canonical Network Equivalence". PLoS One. 3 (4). PMID 18446219. doi:10.1371/journal.pone.0002051. """ import numpy as np # Compute the mean clustering coefficient and average shortest path length # for an equivalent random graph randMetrics = {"C": [], "L": []} for i in range(nrand): Gr = random_reference(G, niter=niter, seed=seed) randMetrics["C"].append(nx.transitivity(Gr)) randMetrics["L"].append(nx.average_shortest_path_length(Gr)) C = nx.transitivity(G) L = nx.average_shortest_path_length(G) Cr = np.mean(randMetrics["C"]) Lr = np.mean(randMetrics["L"]) sigma = (C / Cr) / (L / Lr) return sigma
[docs] @not_implemented_for("directed") @not_implemented_for("multigraph") @py_random_state(3) @nx._dispatchable def omega(G, niter=5, nrand=10, seed=None): """Returns the small-world coefficient (omega) of a graph The small-world coefficient of a graph G is: omega = Lr/L - C/Cl where C and L are respectively the average clustering coefficient and average shortest path length of G. Lr is the average shortest path length of an equivalent random graph and Cl is the average clustering coefficient of an equivalent lattice graph. The small-world coefficient (omega) measures how much G is like a lattice or a random graph. Negative values mean G is similar to a lattice whereas positive values mean G is a random graph. Values close to 0 mean that G has small-world characteristics. Parameters ---------- G : NetworkX graph An undirected graph. niter: integer (optional, default=5) Approximate number of rewiring per edge to compute the equivalent random graph. nrand: integer (optional, default=10) Number of random graphs generated to compute the maximal clustering coefficient (Cr) and average shortest path length (Lr). seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. Returns ------- omega : float The small-world coefficient (omega) Notes ----- The implementation is adapted from the algorithm by Telesford et al. [1]_. References ---------- .. [1] Telesford, Joyce, Hayasaka, Burdette, and Laurienti (2011). "The Ubiquity of Small-World Networks". Brain Connectivity. 1 (0038): 367-75. PMC 3604768. PMID 22432451. doi:10.1089/brain.2011.0038. """ import numpy as np # Compute the mean clustering coefficient and average shortest path length # for an equivalent random graph randMetrics = {"C": [], "L": []} # Calculate initial average clustering coefficient which potentially will # get replaced by higher clustering coefficients from generated lattice # reference graphs Cl = nx.average_clustering(G) niter_lattice_reference = niter niter_random_reference = niter * 2 for _ in range(nrand): # Generate random graph Gr = random_reference(G, niter=niter_random_reference, seed=seed) randMetrics["L"].append(nx.average_shortest_path_length(Gr)) # Generate lattice graph Gl = lattice_reference(G, niter=niter_lattice_reference, seed=seed) # Replace old clustering coefficient, if clustering is higher in # generated lattice reference Cl_temp = nx.average_clustering(Gl) if Cl_temp > Cl: Cl = Cl_temp C = nx.average_clustering(G) L = nx.average_shortest_path_length(G) Lr = np.mean(randMetrics["L"]) omega = (Lr / L) - (C / Cl) return omega