"""Generates graphs resembling the Internet Autonomous System network"""
import networkx as nx
from networkx.utils import py_random_state
__all__ = ["random_internet_as_graph"]
def uniform_int_from_avg(a, m, seed):
"""Pick a random integer with uniform probability.
Returns a random integer uniformly taken from a distribution with
minimum value 'a' and average value 'm', X~U(a,b), E[X]=m, X in N where
b = 2*m - a.
Notes
-----
p = (b-floor(b))/2
X = X1 + X2; X1~U(a,floor(b)), X2~B(p)
E[X] = E[X1] + E[X2] = (floor(b)+a)/2 + (b-floor(b))/2 = (b+a)/2 = m
"""
from math import floor
assert m >= a
b = 2 * m - a
p = (b - floor(b)) / 2
X1 = round(seed.random() * (floor(b) - a) + a)
if seed.random() < p:
X2 = 1
else:
X2 = 0
return X1 + X2
def choose_pref_attach(degs, seed):
"""Pick a random value, with a probability given by its weight.
Returns a random choice among degs keys, each of which has a
probability proportional to the corresponding dictionary value.
Parameters
----------
degs: dictionary
It contains the possible values (keys) and the corresponding
probabilities (values)
seed: random state
Returns
-------
v: object
A key of degs or None if degs is empty
"""
if len(degs) == 0:
return None
s = sum(degs.values())
if s == 0:
return seed.choice(list(degs.keys()))
v = seed.random() * s
nodes = list(degs.keys())
i = 0
acc = degs[nodes[i]]
while v > acc:
i += 1
acc += degs[nodes[i]]
return nodes[i]
class AS_graph_generator:
"""Generates random internet AS graphs."""
def __init__(self, n, seed):
"""Initializes variables. Immediate numbers are taken from [1].
Parameters
----------
n: integer
Number of graph nodes
seed: random state
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
GG: AS_graph_generator object
References
----------
[1] A. Elmokashfi, A. Kvalbein and C. Dovrolis, "On the Scalability of
BGP: The Role of Topology Growth," in IEEE Journal on Selected Areas
in Communications, vol. 28, no. 8, pp. 1250-1261, October 2010.
"""
self.seed = seed
self.n_t = min(n, round(self.seed.random() * 2 + 4)) # num of T nodes
self.n_m = round(0.15 * n) # number of M nodes
self.n_cp = round(0.05 * n) # number of CP nodes
self.n_c = max(0, n - self.n_t - self.n_m - self.n_cp) # number of C nodes
self.d_m = 2 + (2.5 * n) / 10000 # average multihoming degree for M nodes
self.d_cp = 2 + (1.5 * n) / 10000 # avg multihoming degree for CP nodes
self.d_c = 1 + (5 * n) / 100000 # average multihoming degree for C nodes
self.p_m_m = 1 + (2 * n) / 10000 # avg num of peer edges between M and M
self.p_cp_m = 0.2 + (2 * n) / 10000 # avg num of peer edges between CP, M
self.p_cp_cp = 0.05 + (2 * n) / 100000 # avg num of peer edges btwn CP, CP
self.t_m = 0.375 # probability M's provider is T
self.t_cp = 0.375 # probability CP's provider is T
self.t_c = 0.125 # probability C's provider is T
def t_graph(self):
"""Generates the core mesh network of tier one nodes of a AS graph.
Returns
-------
G: Networkx Graph
Core network
"""
self.G = nx.Graph()
for i in range(self.n_t):
self.G.add_node(i, type="T")
for r in self.regions:
self.regions[r].add(i)
for j in self.G.nodes():
if i != j:
self.add_edge(i, j, "peer")
self.customers[i] = set()
self.providers[i] = set()
return self.G
def add_edge(self, i, j, kind):
if kind == "transit":
customer = str(i)
else:
customer = "none"
self.G.add_edge(i, j, type=kind, customer=customer)
def choose_peer_pref_attach(self, node_list):
"""Pick a node with a probability weighted by its peer degree.
Pick a node from node_list with preferential attachment
computed only on their peer degree
"""
d = {}
for n in node_list:
d[n] = self.G.nodes[n]["peers"]
return choose_pref_attach(d, self.seed)
def choose_node_pref_attach(self, node_list):
"""Pick a node with a probability weighted by its degree.
Pick a node from node_list with preferential attachment
computed on their degree
"""
degs = dict(self.G.degree(node_list))
return choose_pref_attach(degs, self.seed)
def add_customer(self, i, j):
"""Keep the dictionaries 'customers' and 'providers' consistent."""
self.customers[j].add(i)
self.providers[i].add(j)
for z in self.providers[j]:
self.customers[z].add(i)
self.providers[i].add(z)
def add_node(self, i, kind, reg2prob, avg_deg, t_edge_prob):
"""Add a node and its customer transit edges to the graph.
Parameters
----------
i: object
Identifier of the new node
kind: string
Type of the new node. Options are: 'M' for middle node, 'CP' for
content provider and 'C' for customer.
reg2prob: float
Probability the new node can be in two different regions.
avg_deg: float
Average number of transit nodes of which node i is customer.
t_edge_prob: float
Probability node i establish a customer transit edge with a tier
one (T) node
Returns
-------
i: object
Identifier of the new node
"""
regs = 1 # regions in which node resides
if self.seed.random() < reg2prob: # node is in two regions
regs = 2
node_options = set()
self.G.add_node(i, type=kind, peers=0)
self.customers[i] = set()
self.providers[i] = set()
self.nodes[kind].add(i)
for r in self.seed.sample(list(self.regions), regs):
node_options = node_options.union(self.regions[r])
self.regions[r].add(i)
edge_num = uniform_int_from_avg(1, avg_deg, self.seed)
t_options = node_options.intersection(self.nodes["T"])
m_options = node_options.intersection(self.nodes["M"])
if i in m_options:
m_options.remove(i)
d = 0
while d < edge_num and (len(t_options) > 0 or len(m_options) > 0):
if len(m_options) == 0 or (
len(t_options) > 0 and self.seed.random() < t_edge_prob
): # add edge to a T node
j = self.choose_node_pref_attach(t_options)
t_options.remove(j)
else:
j = self.choose_node_pref_attach(m_options)
m_options.remove(j)
self.add_edge(i, j, "transit")
self.add_customer(i, j)
d += 1
return i
def add_m_peering_link(self, m, to_kind):
"""Add a peering link between two middle tier (M) nodes.
Target node j is drawn considering a preferential attachment based on
other M node peering degree.
Parameters
----------
m: object
Node identifier
to_kind: string
type for target node j (must be always M)
Returns
-------
success: boolean
"""
# candidates are of type 'M' and are not customers of m
node_options = self.nodes["M"].difference(self.customers[m])
# candidates are not providers of m
node_options = node_options.difference(self.providers[m])
# remove self
if m in node_options:
node_options.remove(m)
# remove candidates we are already connected to
for j in self.G.neighbors(m):
if j in node_options:
node_options.remove(j)
if len(node_options) > 0:
j = self.choose_peer_pref_attach(node_options)
self.add_edge(m, j, "peer")
self.G.nodes[m]["peers"] += 1
self.G.nodes[j]["peers"] += 1
return True
else:
return False
def add_cp_peering_link(self, cp, to_kind):
"""Add a peering link to a content provider (CP) node.
Target node j can be CP or M and it is drawn uniformly among the nodes
belonging to the same region as cp.
Parameters
----------
cp: object
Node identifier
to_kind: string
type for target node j (must be M or CP)
Returns
-------
success: boolean
"""
node_options = set()
for r in self.regions: # options include nodes in the same region(s)
if cp in self.regions[r]:
node_options = node_options.union(self.regions[r])
# options are restricted to the indicated kind ('M' or 'CP')
node_options = self.nodes[to_kind].intersection(node_options)
# remove self
if cp in node_options:
node_options.remove(cp)
# remove nodes that are cp's providers
node_options = node_options.difference(self.providers[cp])
# remove nodes we are already connected to
for j in self.G.neighbors(cp):
if j in node_options:
node_options.remove(j)
if len(node_options) > 0:
j = self.seed.sample(list(node_options), 1)[0]
self.add_edge(cp, j, "peer")
self.G.nodes[cp]["peers"] += 1
self.G.nodes[j]["peers"] += 1
return True
else:
return False
def graph_regions(self, rn):
"""Initializes AS network regions.
Parameters
----------
rn: integer
Number of regions
"""
self.regions = {}
for i in range(rn):
self.regions["REG" + str(i)] = set()
def add_peering_links(self, from_kind, to_kind):
"""Utility function to add peering links among node groups."""
peer_link_method = None
if from_kind == "M":
peer_link_method = self.add_m_peering_link
m = self.p_m_m
if from_kind == "CP":
peer_link_method = self.add_cp_peering_link
if to_kind == "M":
m = self.p_cp_m
else:
m = self.p_cp_cp
for i in self.nodes[from_kind]:
num = uniform_int_from_avg(0, m, self.seed)
for _ in range(num):
peer_link_method(i, to_kind)
def generate(self):
"""Generates a random AS network graph as described in [1].
Returns
-------
G: Graph object
Notes
-----
The process steps are the following: first we create the core network
of tier one nodes, then we add the middle tier (M), the content
provider (CP) and the customer (C) nodes along with their transit edges
(link i,j means i is customer of j). Finally we add peering links
between M nodes, between M and CP nodes and between CP node couples.
For a detailed description of the algorithm, please refer to [1].
References
----------
[1] A. Elmokashfi, A. Kvalbein and C. Dovrolis, "On the Scalability of
BGP: The Role of Topology Growth," in IEEE Journal on Selected Areas
in Communications, vol. 28, no. 8, pp. 1250-1261, October 2010.
"""
self.graph_regions(5)
self.customers = {}
self.providers = {}
self.nodes = {"T": set(), "M": set(), "CP": set(), "C": set()}
self.t_graph()
self.nodes["T"] = set(self.G.nodes())
i = len(self.nodes["T"])
for _ in range(self.n_m):
self.nodes["M"].add(self.add_node(i, "M", 0.2, self.d_m, self.t_m))
i += 1
for _ in range(self.n_cp):
self.nodes["CP"].add(self.add_node(i, "CP", 0.05, self.d_cp, self.t_cp))
i += 1
for _ in range(self.n_c):
self.nodes["C"].add(self.add_node(i, "C", 0, self.d_c, self.t_c))
i += 1
self.add_peering_links("M", "M")
self.add_peering_links("CP", "M")
self.add_peering_links("CP", "CP")
return self.G
[docs]
@py_random_state(1)
@nx._dispatchable(graphs=None, returns_graph=True)
def random_internet_as_graph(n, seed=None):
"""Generates a random undirected graph resembling the Internet AS network
Parameters
----------
n: integer in [1000, 10000]
Number of graph nodes
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
G: Networkx Graph object
A randomly generated undirected graph
Notes
-----
This algorithm returns an undirected graph resembling the Internet
Autonomous System (AS) network, it uses the approach by Elmokashfi et al.
[1]_ and it grants the properties described in the related paper [1]_.
Each node models an autonomous system, with an attribute 'type' specifying
its kind; tier-1 (T), mid-level (M), customer (C) or content-provider (CP).
Each edge models an ADV communication link (hence, bidirectional) with
attributes:
- type: transit|peer, the kind of commercial agreement between nodes;
- customer: <node id>, the identifier of the node acting as customer
('none' if type is peer).
References
----------
.. [1] A. Elmokashfi, A. Kvalbein and C. Dovrolis, "On the Scalability of
BGP: The Role of Topology Growth," in IEEE Journal on Selected Areas
in Communications, vol. 28, no. 8, pp. 1250-1261, October 2010.
"""
GG = AS_graph_generator(n, seed)
G = GG.generate()
return G
