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 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 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 = G.degree()
den = float(sum(degree.values()))
return {node: degree[node] / 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 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
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
