"""Functions to generate random topologies according to a number of models.
The generated topologies are either Topology or DirectedTopology objects.
"""
import math
import random
import networkx as nx
from fnss.util import random_from_pdf
from fnss.topologies.topology import Topology
__all__ = [
'erdos_renyi_topology',
'waxman_1_topology',
'waxman_2_topology',
'barabasi_albert_topology',
'extended_barabasi_albert_topology',
'glp_topology'
]
[docs]def erdos_renyi_topology(n, p, seed=None, fast=False):
r"""Return a random graph :math:`G_{n,p}` (Erdos-Renyi graph, binomial
graph).
Chooses each of the possible edges with probability p.
Parameters
----------
n : int
The number of nodes.
p : float
Probability for edge creation.
seed : int, optional
Seed for random number generator (default=None).
fast : boolean, optional
Uses the algorithm proposed by [3]_, which is faster for small p
References
----------
.. [1] P. Erdos and A. Renyi, On Random Graphs, Publ. Math. 6, 290 (1959).
.. [2] E. N. Gilbert, Random Graphs, Ann. Math. Stat., 30, 1141 (1959).
.. [3] Vladimir Batagelj and Ulrik Brandes,
"Efficient generation of large random networks",
Phys. Rev. E, 71, 036113, 2005.
"""
# validate input parameters
if not isinstance(n, int) or n < 0:
raise ValueError('n must be a positive integer')
if p > 1 or p < 0:
raise ValueError('p must be a value in (0,1)')
if fast:
G = Topology(nx.fast_gnp_random_graph(n, p, seed=seed))
else:
G = Topology(nx.gnp_random_graph(n, p, seed=seed))
G.name = "erdos_renyi_topology(%s, %s)" % (n, p)
G.graph['type'] = 'er'
return G
[docs]def waxman_1_topology(n, alpha=0.4, beta=0.1, L=1.0,
distance_unit='Km', seed=None):
r"""
Return a Waxman-1 random topology.
The Waxman-1 random topology models assigns link between nodes with
probability
.. math::
p = \alpha*exp(-d/(\beta*L)).
where the distance *d* is chosen randomly in *[0,L]*.
Parameters
----------
n : int
Number of nodes
alpha : float
Model parameter chosen in *(0,1]* (higher alpha increases link density)
beta : float
Model parameter chosen in *(0,1]* (higher beta increases difference
between density of short and long links)
L : float
Maximum distance between nodes.
seed : int, optional
Seed for random number generator (default=None).
Returns
-------
G : Topology
Notes
-----
Each node of G has the attributes *latitude* and *longitude*. These
attributes are not expressed in degrees but in *distance_unit*.
Each edge of G has the attribute *length*, which is also expressed in
*distance_unit*.
References
----------
.. [1] B. M. Waxman, Routing of multipoint connections.
IEEE J. Select. Areas Commun. 6(9),(1988) 1617-1622.
"""
# validate input parameters
if not isinstance(n, int) or n <= 0:
raise ValueError('n must be a positive integer')
if alpha > 1 or alpha <= 0 or beta > 1 or beta <= 0:
raise ValueError('alpha and beta must be float values in (0,1]')
if L <= 0:
raise ValueError('L must be a positive number')
if seed is not None:
random.seed(seed)
G = Topology(type='waxman_1', distance_unit=distance_unit)
G.name = "waxman_1_topology(%s, %s, %s, %s)" % (n, alpha, beta, L)
G.add_nodes_from(range(n))
nodes = list(G.nodes())
while nodes:
u = nodes.pop()
for v in nodes:
d = L * random.random()
if random.random() < alpha * math.exp(-d / (beta * L)):
G.add_edge(u, v, length=d)
return G
[docs]def waxman_2_topology(n, alpha=0.4, beta=0.1, domain=(0, 0, 1, 1),
distance_unit='Km', seed=None):
r"""Return a Waxman-2 random topology.
The Waxman-2 random topology models place n nodes uniformly at random
in a rectangular domain. Two nodes u, v are connected with a link
with probability
.. math::
p = \alpha*exp(-d/(\beta*L)).
where the distance *d* is the Euclidean distance between the nodes u and v.
and *L* is the maximum distance between all nodes in the graph.
Parameters
----------
n : int
Number of nodes
alpha : float
Model parameter chosen in *(0,1]* (higher alpha increases link density)
beta : float
Model parameter chosen in *(0,1]* (higher beta increases difference
between density of short and long links)
domain : tuple of numbers, optional
Domain size (xmin, ymin, xmax, ymax)
seed : int, optional
Seed for random number generator (default=None).
Returns
-------
G : Topology
Notes
-----
Each edge of G has the attribute *length*
References
----------
.. [1] B. M. Waxman, Routing of multipoint connections.
IEEE J. Select. Areas Commun. 6(9),(1988) 1617-1622.
"""
# validate input parameters
if not isinstance(n, int) or n <= 0:
raise ValueError('n must be a positive integer')
if alpha > 1 or alpha <= 0 or beta > 1 or beta <= 0:
raise ValueError('alpha and beta must be float values in (0,1]')
if not isinstance(domain, tuple) or len(domain) != 4:
raise ValueError('domain must be a tuple of 4 number')
(xmin, ymin, xmax, ymax) = domain
if xmin > xmax:
raise ValueError('In domain, xmin cannot be greater than xmax')
if ymin > ymax:
raise ValueError('In domain, ymin cannot be greater than ymax')
if seed is not None:
random.seed(seed)
G = Topology(type='waxman_2', distance_unit=distance_unit)
G.name = "waxman_2_topology(%s, %s, %s)" % (n, alpha, beta)
G.add_nodes_from(range(n))
for v in G.nodes():
G.node[v]['latitude'] = (ymin + (ymax - ymin)) * random.random()
G.node[v]['longitude'] = (xmin + (xmax - xmin)) * random.random()
l = {}
nodes = list(G.nodes())
while nodes:
u = nodes.pop()
for v in nodes:
x_u = G.node[u]['longitude']
x_v = G.node[v]['longitude']
y_u = G.node[u]['latitude']
y_v = G.node[v]['latitude']
l[(u, v)] = math.sqrt((x_u - x_v) ** 2 + (y_u - y_v) ** 2)
L = max(l.values())
for (u, v), d in l.items():
if random.random() < alpha * math.exp(-d / (beta * L)):
G.add_edge(u, v, length=d)
return G
# This is the classical BA model, without rewiring and add
[docs]def barabasi_albert_topology(n, m, m0, seed=None):
r"""
Return a random topology using Barabasi-Albert preferential attachment
model.
A topology of n nodes is grown by attaching new nodes each with m links
that are preferentially attached to existing nodes with high degree.
More precisely, the Barabasi-Albert topology is built as follows. First, a
line topology with m0 nodes is created. Then at each step, one node is
added and connected to m existing nodes. These nodes are selected randomly
with probability
.. math::
\Pi(i) = \frac{deg(i)}{sum_{v \in V} deg V}.
Where i is the selected node and V is the set of nodes of the graph.
Parameters
----------
n : int
Number of nodes
m : int
Number of edges to attach from a new node to existing nodes
m0 : int
Number of nodes initially attached to the network
seed : int, optional
Seed for random number generator (default=None).
Returns
-------
G : Topology
Notes
-----
The initialization is a graph with with m nodes connected by :math:`m -1`
edges.
It does not use the Barabasi-Albert method provided by NetworkX because it
does not allow to specify *m0* parameter.
There are no disconnected subgraphs in the topology.
References
----------
.. [1] A. L. Barabasi and R. Albert "Emergence of scaling in
random networks", Science 286, pp 509-512, 1999.
"""
def calc_pi(G):
"""Calculate BA Pi function for all nodes of the graph"""
degree = dict(G.degree())
den = float(sum(degree.values()))
return {node: degree[node] / den for node in G.nodes()}
# input parameters
if n < 1 or m < 1 or m0 < 1:
raise ValueError('n, m and m0 must be positive integers')
if m >= m0:
raise ValueError('m must be <= m0')
if n < m0:
raise ValueError('n must be > m0')
if seed is not None:
random.seed(seed)
# Step 1: Add m0 nodes. These nodes are interconnected together
# because otherwise they will end up isolated at the end
G = Topology(nx.path_graph(m0))
G.name = "ba_topology(%d,%d,%d)" % (n, m, m0)
G.graph['type'] = 'ba'
# Step 2: Add one node and connect it with m links
while G.number_of_nodes() < n:
pi = calc_pi(G)
u = G.number_of_nodes()
G.add_node(u)
new_links = 0
while new_links < m:
v = random_from_pdf(pi)
if not G.has_edge(u, v):
G.add_edge(u, v)
new_links += 1
return G
# This is the extended BA model, with rewiring and add
[docs]def extended_barabasi_albert_topology(n, m, m0, p, q, seed=None):
r"""
Return a random topology using the extended Barabasi-Albert preferential
attachment model.
Differently from the original Barabasi-Albert model, this model takes into
account the presence of local events, such as the addition of new links or
the rewiring of existing links.
More precisely, the Barabasi-Albert topology is built as follows. First, a
topology with *m0* isolated nodes is created. Then, at each step:
with probability *p* add *m* new links between existing nodes, selected
with probability:
.. math::
\Pi(i) = \frac{deg(i) + 1}{\sum_{v \in V} (deg(v) + 1)}
with probability *q* rewire *m* links. Each link to be rewired is selected as
follows: a node i is randomly selected and a link is randomly removed from
it. The node i is then connected to a new node randomly selected with
probability :math:`\Pi(i)`,
with probability :math:`1-p-q` add a new node and attach it to m nodes of
the existing topology selected with probability :math:`\Pi(i)`
Repeat the previous step until the topology comprises n nodes in total.
Parameters
----------
n : int
Number of nodes
m : int
Number of edges to attach from a new node to existing nodes
m0 : int
Number of edges initially attached to the network
p : float
The probability that new links are added
q : float
The probability that existing links are rewired
seed : int, optional
Seed for random number generator (default=None).
Returns
-------
G : Topology
References
----------
.. [1] A. L. Barabasi and R. Albert "Topology of evolving networks: local
events and universality", Physical Review Letters 85(24), 2000.
"""
def calc_pi(G):
"""Calculate extended-BA Pi function for all nodes of the graph"""
degree = dict(G.degree())
den = float(sum(degree.values()) + G.number_of_nodes())
return {node: (degree[node] + 1) / den for node in G.nodes()}
# input parameters
if n < 1 or m < 1 or m0 < 1:
raise ValueError('n, m and m0 must be a positive integer')
if m >= m0:
raise ValueError('m must be <= m0')
if n < m0:
raise ValueError('n must be > m0')
if p > 1 or p < 0:
raise ValueError('p must be included between 0 and 1')
if q > 1 or q < 0:
raise ValueError('q must be included between 0 and 1')
if p + q > 1:
raise ValueError('p + q must be <= 1')
if seed is not None:
random.seed(seed)
G = Topology(type='extended_ba')
G.name = "ext_ba_topology(%d, %d, %d, %f, %f)" % (n, m, m0, p, q)
# Step 1: Add m0 isolated nodes
G.add_nodes_from(range(m0))
while G.number_of_nodes() < n:
pi = calc_pi(G)
r = random.random()
if r <= p:
# add m new links with probability p
n_nodes = G.number_of_nodes()
n_edges = G.number_of_edges()
max_n_edges = (n_nodes * (n_nodes - 1)) / 2
if n_edges + m > max_n_edges: # cannot add m links
continue # rewire or add nodes
new_links = 0
while new_links < m:
u = random_from_pdf(pi)
v = random_from_pdf(pi)
if u is not v and not G.has_edge(u, v):
G.add_edge(u, v)
new_links += 1
elif r > p and r <= p + q:
# rewire m links with probability q
rewired_links = 0
while rewired_links < m:
i = random.choice(list(G.nodes())) # pick up node randomly (uniform)
if len(G.adj[i]) is 0: # if i has no edges, I cannot rewire
break
j = random.choice(list(G.adj[i].keys())) # node to be disconnected
k = random_from_pdf(pi) # new node to be connected
if i is not k and j is not k and not G.has_edge(i, k):
G.remove_edge(i, j)
G.add_edge(i, k)
rewired_links += 1
else:
# add a new node with probability 1 - p - q
new_node = G.number_of_nodes()
G.add_node(new_node)
new_links = 0
while new_links < m:
existing_node = random_from_pdf(pi)
if not G.has_edge(new_node, existing_node):
G.add_edge(new_node, existing_node)
new_links += 1
return G
[docs]def glp_topology(n, m, m0, p, beta, seed=None):
r"""
Return a random topology using the Generalized Linear Preference (GLP)
preferential attachment model.
It differs from the extended Barabasi-Albert model in that there is link
rewiring and a beta parameter is introduced to fine-tune preferential
attachment.
More precisely, the GLP topology is built as follows. First, a
line topology with *m0* nodes is created. Then, at each step:
with probability *p*, add *m* new links between existing nodes, selected
with probability:
.. math::
\Pi(i) = \frac{deg(i) - \beta 1}{\sum_{v \in V} (deg(v) - \beta)}
with probability :math:`1-p`, add a new node and attach it to m nodes of
the existing topology selected with probability :math:`\Pi(i)`
Repeat the previous step until the topology comprises n nodes in total.
Parameters
----------
n : int
Number of nodes
m : int
Number of edges to attach from a new node to existing nodes
m0 : int
Number of edges initially attached to the network
p : float
The probability that new links are added
beta : float
Parameter to fine-tune preferntial attachment: beta < 1
seed : int, optional
Seed for random number generator (default=None).
Returns
-------
G : Topology
References
----------
.. [1] T. Bu and D. Towsey "On distinguishing between Internet power law
topology generators", Proceeding od the 21st IEEE INFOCOM conference.
IEEE, volume 2, pages 638-647, 2002.
"""
def calc_pi(G, beta):
"""Calculate GLP Pi function for all nodes of the graph"""
# validate input parameter
if beta >= 1:
raise ValueError('beta must be < 1')
degree = dict(G.degree())
den = float(sum(degree.values()) - (G.number_of_nodes() * beta))
return {node: (degree[node] - beta) / den for node in G.nodes()}
def add_m_links(G, pi):
"""Add m links between existing nodes to the graph"""
n_nodes = G.number_of_nodes()
n_edges = G.number_of_edges()
max_n_edges = (n_nodes * (n_nodes - 1)) / 2
if n_edges + m > max_n_edges: # cannot add m links
add_node(G, pi) # add a new node instead
# return in any case because before doing another operation
# (add node or links) we need to recalculate pi
return
new_links = 0
while new_links < m:
u = random_from_pdf(pi)
v = random_from_pdf(pi)
if u != v and not G.has_edge(u, v):
G.add_edge(u, v)
new_links += 1
def add_node(G, pi):
"""Add one node to the graph and connect it to m existing nodes"""
new_node = G.number_of_nodes()
G.add_node(new_node)
new_links = 0
while new_links < m:
existing_node = random_from_pdf(pi)
if not G.has_edge(new_node, existing_node):
G.add_edge(new_node, existing_node)
new_links += 1
# validate input parameters
if n < 1 or m < 1 or m0 < 1:
raise ValueError('n, m and m0 must be a positive integers')
if beta >= 1:
raise ValueError('beta must be < 1')
if m >= m0:
raise ValueError('m must be <= m0')
if p > 1 or p < 0:
raise ValueError('p must be included between 0 and 1')
if seed is not None:
random.seed(seed)
# step 1: create a graph of m0 nodes connected by n-1 edges
G = Topology(nx.path_graph(m0))
G.graph['type'] = 'glp'
G.name = "glp_topology(%d, %d, %d, %f, %f)" % (n, m, m0, p, beta)
# Add nodes and links now
while G.number_of_nodes() < n:
pi = calc_pi(G, beta)
if random.random() < p:
# add m new links with probability p
add_m_links(G, pi)
else:
# add a new node with m new links with probability 1 - p
add_node(G, pi)
return G