148 lines
4.2 KiB
Python
148 lines
4.2 KiB
Python
|
"""Provides functions for computing the efficiency of nodes and graphs."""
|
||
|
|
|
||
|
|
import networkx as nx
|
||
|
|
from networkx.exception import NetworkXNoPath
|
||
|
|
|
||
|
|
from ..utils import not_implemented_for
|
||
|
|
|
||
|
|
__all__ = ["efficiency", "local_efficiency", "global_efficiency"]
|
||
|
|
|
||
|
|
|
||
|
|
@not_implemented_for("directed")
|
||
|
|
def efficiency(G, u, v):
|
||
|
|
"""Returns the efficiency of a pair of nodes in a graph.
|
||
|
|
|
||
|
|
The *efficiency* of a pair of nodes is the multiplicative inverse of the
|
||
|
|
shortest path distance between the nodes [1]_. Returns 0 if no path
|
||
|
|
between nodes.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
G : :class:`networkx.Graph`
|
||
|
|
An undirected graph for which to compute the average local efficiency.
|
||
|
|
u, v : node
|
||
|
|
Nodes in the graph ``G``.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
float
|
||
|
|
Multiplicative inverse of the shortest path distance between the nodes.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Edge weights are ignored when computing the shortest path distances.
|
||
|
|
|
||
|
|
See also
|
||
|
|
--------
|
||
|
|
local_efficiency
|
||
|
|
global_efficiency
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Latora, Vito, and Massimo Marchiori.
|
||
|
|
"Efficient behavior of small-world networks."
|
||
|
|
*Physical Review Letters* 87.19 (2001): 198701.
|
||
|
|
<https://doi.org/10.1103/PhysRevLett.87.198701>
|
||
|
|
|
||
|
|
"""
|
||
|
|
try:
|
||
|
|
eff = 1 / nx.shortest_path_length(G, u, v)
|
||
|
|
except NetworkXNoPath:
|
||
|
|
eff = 0
|
||
|
|
return eff
|
||
|
|
|
||
|
|
|
||
|
|
@not_implemented_for("directed")
|
||
|
|
def global_efficiency(G):
|
||
|
|
"""Returns the average global efficiency of the graph.
|
||
|
|
|
||
|
|
The *efficiency* of a pair of nodes in a graph is the multiplicative
|
||
|
|
inverse of the shortest path distance between the nodes. The *average
|
||
|
|
global efficiency* of a graph is the average efficiency of all pairs of
|
||
|
|
nodes [1]_.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
G : :class:`networkx.Graph`
|
||
|
|
An undirected graph for which to compute the average global efficiency.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
float
|
||
|
|
The average global efficiency of the graph.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Edge weights are ignored when computing the shortest path distances.
|
||
|
|
|
||
|
|
See also
|
||
|
|
--------
|
||
|
|
local_efficiency
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Latora, Vito, and Massimo Marchiori.
|
||
|
|
"Efficient behavior of small-world networks."
|
||
|
|
*Physical Review Letters* 87.19 (2001): 198701.
|
||
|
|
<https://doi.org/10.1103/PhysRevLett.87.198701>
|
||
|
|
|
||
|
|
"""
|
||
|
|
n = len(G)
|
||
|
|
denom = n * (n - 1)
|
||
|
|
if denom != 0:
|
||
|
|
lengths = nx.all_pairs_shortest_path_length(G)
|
||
|
|
g_eff = 0
|
||
|
|
for source, targets in lengths:
|
||
|
|
for target, distance in targets.items():
|
||
|
|
if distance > 0:
|
||
|
|
g_eff += 1 / distance
|
||
|
|
g_eff /= denom
|
||
|
|
# g_eff = sum(1 / d for s, tgts in lengths
|
||
|
|
# for t, d in tgts.items() if d > 0) / denom
|
||
|
|
else:
|
||
|
|
g_eff = 0
|
||
|
|
# TODO This can be made more efficient by computing all pairs shortest
|
||
|
|
# path lengths in parallel.
|
||
|
|
return g_eff
|
||
|
|
|
||
|
|
|
||
|
|
@not_implemented_for("directed")
|
||
|
|
def local_efficiency(G):
|
||
|
|
"""Returns the average local efficiency of the graph.
|
||
|
|
|
||
|
|
The *efficiency* of a pair of nodes in a graph is the multiplicative
|
||
|
|
inverse of the shortest path distance between the nodes. The *local
|
||
|
|
efficiency* of a node in the graph is the average global efficiency of the
|
||
|
|
subgraph induced by the neighbors of the node. The *average local
|
||
|
|
efficiency* is the average of the local efficiencies of each node [1]_.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
G : :class:`networkx.Graph`
|
||
|
|
An undirected graph for which to compute the average local efficiency.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
float
|
||
|
|
The average local efficiency of the graph.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Edge weights are ignored when computing the shortest path distances.
|
||
|
|
|
||
|
|
See also
|
||
|
|
--------
|
||
|
|
global_efficiency
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Latora, Vito, and Massimo Marchiori.
|
||
|
|
"Efficient behavior of small-world networks."
|
||
|
|
*Physical Review Letters* 87.19 (2001): 198701.
|
||
|
|
<https://doi.org/10.1103/PhysRevLett.87.198701>
|
||
|
|
|
||
|
|
"""
|
||
|
|
# TODO This summation can be trivially parallelized.
|
||
|
|
efficiency_list = (global_efficiency(G.subgraph(G[v])) for v in G)
|
||
|
|
return sum(efficiency_list) / len(G)
|