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 = 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
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 = G.degree()
den = float(sum(degree.values()) + G.number_of_nodes())
return {node: (degree[node] + 1) / den for node in G.nodes_iter()}
# 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(G.nodes()) # pick up node randomly (uniform)
if len(G.edge[i]) is 0: # if i has no edges, I cannot rewire
break
j = random.choice(list(
G.edge[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
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_iter():
G.node[v]['latitude'] = (ymin + (ymax - ymin)) * random.random()
G.node[v]['longitude'] = (xmin + (xmax - xmin)) * random.random()
l = {}
nodes = 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
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
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
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
