def make_small_graph(graph_description, create_using=None):
if graph_description[0] not in ("adjacencylist", "edgelist"):
raise NetworkXError("ltype must be either adjacencylist or edgelist")
ltype = graph_description[0]
name = graph_description[1]
n = graph_description[2]
G = empty_graph(n, create_using)
nodes = G.nodes()
if ltype == "adjacencylist":
adjlist = graph_description[3]
if len(adjlist) != n:
raise NetworkXError("invalid graph_description")
G.add_edges_from([(u - 1, v) for v in nodes for u in adjlist[v]])
elif ltype == "edgelist":
edgelist = graph_description[3]
for e in edgelist:
v1 = e[0] - 1
v2 = e[1] - 1
if v1 < 0 or v1 > n - 1 or v2 < 0 or v2 > n - 1:
raise NetworkXError("invalid graph_description")
G.add_edge(v1, v2)
G.name = name
return G
def gn_graph(n, kernel=None, create_using=None, seed=None):
G = empty_graph(1, create_using, default=nx.DiGraph)
if not G.is_directed():
raise nx.NetworkXError("create_using must indicate a Directed Graph")
if kernel is None:
def kernel(x):
return x
if n == 1:
return G
G.add_edge(1, 0) # get started
ds = [1, 1] # degree sequence
for source in range(2, n):
# compute distribution from kernel and degree
dist = [kernel(d) for d in ds]
# choose target from discrete distribution
target = discrete_sequence(1, distribution=dist, seed=seed)[0]
G.add_edge(source, target)
ds.append(1) # the source has only one link (degree one)
ds[target] += 1 # add one to the target link degree
return G
def random_shell_graph(constructor, seed=None):
G = empty_graph(0)
glist = []
intra_edges = []
nnodes = 0
# create gnm graphs for each shell
for (n, m, d) in constructor:
inter_edges = int(m * d)
intra_edges.append(m - inter_edges)
g = nx.convert_node_labels_to_integers(
gnm_random_graph(n, inter_edges, seed=seed), first_label=nnodes
)
glist.append(g)
nnodes += n
G = nx.operators.union(G, g)
# connect the shells randomly
for gi in range(len(glist) - 1):
nlist1 = list(glist[gi])
nlist2 = list(glist[gi + 1])
total_edges = intra_edges[gi]
edge_count = 0
while edge_count < total_edges:
u = seed.choice(nlist1)
v = seed.choice(nlist2)
if u == v or G.has_edge(u, v):
continue
else:
G.add_edge(u, v)
edge_count = edge_count + 1
return G
def newman_watts_strogatz_graph(n, k, p, seed=None):
if k > n:
raise nx.NetworkXError("k>=n, choose smaller k or larger n")
# If k == n the graph return is a complete graph
if k == n:
return nx.complete_graph(n)
G = empty_graph(n)
nlist = list(G.nodes())
fromv = nlist
# connect the k/2 neighbors
for j in range(1, k // 2 + 1):
tov = fromv[j:] + fromv[0:j] # the first j are now last
for i, value in enumerate(fromv):
G.add_edge(value, tov[i])
# for each edge u-v, with probability p, randomly select existing
# node w and add new edge u-w
e = list(G.edges())
for (u, v) in e:
if seed.random() < p:
w = seed.choice(nlist)
# no self-loops and reject if edge u-w exists
# is that the correct NWS model?
while w == u or G.has_edge(u, w):
w = seed.choice(nlist)
if G.degree(u) >= n - 1:
break # skip this rewiring
else:
G.add_edge(u, w)
return G
def dense_gnm_random_graph(n, m, seed=None):
mmax = n * (n - 1) / 2
if m >= mmax:
G = complete_graph(n)
else:
G = empty_graph(n)
if n == 1 or m >= mmax:
return G
u = 0
v = 1
t = 0
k = 0
while True:
if seed.randrange(mmax - t) < m - k:
G.add_edge(u, v)
k += 1
if k == m:
return G
t += 1
v += 1
if v == n: # go to next row of adjacency matrix
u += 1
v = u + 1
def sedgewick_maze_graph(create_using=None):
G = empty_graph(0, create_using)
G.add_nodes_from(range(8))
G.add_edges_from([[0, 2], [0, 7], [0, 5]])
G.add_edges_from([[1, 7], [2, 6]])
G.add_edges_from([[3, 4], [3, 5]])
G.add_edges_from([[4, 5], [4, 7], [4, 6]])
G.name = "Sedgewick Maze"
return G
def make_small_undirected_graph(graph_description, create_using=None):
"""
Return a small undirected graph described by graph_description.
See make_small_graph.
"""
G = empty_graph(0, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
return make_small_graph(graph_description, G)
def grid_graph(dim, periodic=False):
"""Returns the *n*-dimensional grid graph.
The dimension *n* is the length of the list `dim` and the size in
each dimension is the value of the corresponding list element.
Parameters
----------
dim : list or tuple of numbers or iterables of nodes
'dim' is a tuple or list with, for each dimension, either a number
that is the size of that dimension or an iterable of nodes for
that dimension. The dimension of the grid_graph is the length
of `dim`.
periodic : bool or iterable
If `periodic` is True, all dimensions are periodic. If False all
dimensions are not periodic. If `periodic` is iterable, it should
yield `dim` bool values each of which indicates whether the
corresponding axis is periodic.
Returns
-------
NetworkX graph
The (possibly periodic) grid graph of the specified dimensions.
Examples
--------
To produce a 2 by 3 by 4 grid graph, a graph on 24 nodes:
>>> from networkx import grid_graph
>>> G = grid_graph(dim=(2, 3, 4))
>>> len(G)
24
>>> G = grid_graph(dim=(range(7, 9), range(3, 6)))
>>> len(G)
6
"""
if not dim:
return empty_graph(0)
if len(dim) == 2:
return grid_2d_graph(dim[0], dim[1])
try:
func = (cycle_graph if p else path_graph for p in periodic)
except TypeError:
func = repeat(cycle_graph if periodic else path_graph)
G = next(func)(dim[0])
for current_dim in dim[1:]:
Gnew = next(func)(current_dim)
G = _cartesian_product(Gnew, G)
# graph G is done but has labels of the form (1, (2, (3, 1))) so relabel
H = relabel_nodes(G, flatten)
return H
def grid_2d_graph(m, n, periodic=False, create_using=None):
"""Returns the two-dimensional grid graph.
The grid graph has each node connected to its four nearest neighbors.
Parameters
----------
m, n : int or iterable container of nodes
If an integer, nodes are from `range(n)`.
If a container, elements become the coordinate of the nodes.
periodic : bool or iterable
If `periodic` is True, both dimensions are periodic. If False, none
are periodic. If `periodic` is iterable, it should yield 2 bool
values indicating whether the 1st and 2nd axes, respectively, are
periodic.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
NetworkX graph
The (possibly periodic) grid graph of the specified dimensions.
"""
G = empty_graph(0, create_using)
row_name, rows = m
col_name, cols = n
G.add_nodes_from((i, j) for i in rows for j in cols)
G.add_edges_from(
((i, j), (pi, j)) for pi, i in pairwise(rows) for j in cols)
G.add_edges_from(
((i, j), (i, pj)) for i in rows for pj, j in pairwise(cols))
try:
periodic_r, periodic_c = periodic
except TypeError:
periodic_r = periodic_c = periodic
if periodic_r and len(rows) > 2:
first = rows[0]
last = rows[-1]
G.add_edges_from(((first, j), (last, j)) for j in cols)
if periodic_c and len(cols) > 2:
first = cols[0]
last = cols[-1]
G.add_edges_from(((i, first), (i, last)) for i in rows)
# both directions for directed
if G.is_directed():
G.add_edges_from((v, u) for u, v in G.edges())
return G
def gnc_graph(n, create_using=None, seed=None):
G = empty_graph(1, create_using, default=nx.DiGraph)
if not G.is_directed():
raise nx.NetworkXError("create_using must indicate a Directed Graph")
if n == 1:
return G
for source in range(1, n):
target = seed.randrange(0, source)
for succ in G.successors(target):
G.add_edge(source, succ)
G.add_edge(source, target)
return G
def powerlaw_cluster_graph(n, m, p, seed=None):
if m < 1 or n < m:
raise nx.NetworkXError(
"NetworkXError must have m>1 and m<n, m=%d,n=%d" % (m, n)
)
if p > 1 or p < 0:
raise nx.NetworkXError("NetworkXError p must be in [0,1], p=%f" % (p))
G = empty_graph(m) # add m initial nodes (m0 in barabasi-speak)
repeated_nodes = list(G.nodes()) # list of existing nodes to sample from
# with nodes repeated once for each adjacent edge
source = m # next node is m
while source < n: # Now add the other n-1 nodes
possible_targets = _random_subset(repeated_nodes, m, seed)
# do one preferential attachment for new node
target = possible_targets.pop()
G.add_edge(source, target)
repeated_nodes.append(target) # add one node to list for each new link
count = 1
while count < m: # add m-1 more new links
if seed.random() < p: # clustering step: add triangle
neighborhood = [
nbr
for nbr in G.neighbors(target)
if not G.has_edge(source, nbr) and not nbr == source
]
if neighborhood: # if there is a neighbor without a link
nbr = seed.choice(neighborhood)
G.add_edge(source, nbr) # add triangle
repeated_nodes.append(nbr)
count = count + 1
continue # go to top of while loop
# else do preferential attachment step if above fails
target = possible_targets.pop()
G.add_edge(source, target)
repeated_nodes.append(target)
count = count + 1
repeated_nodes.extend([source] * m) # add source node to list m times
source += 1
return G
def LCF_graph(n, shift_list, repeats, create_using=None):
if n <= 0:
return empty_graph(0, create_using)
# start with the n-cycle
G = cycle_graph(n, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
G.name = "LCF_graph"
nodes = sorted(list(G))
n_extra_edges = repeats * len(shift_list)
# edges are added n_extra_edges times
# (not all of these need be new)
if n_extra_edges < 1:
return G
for i in range(n_extra_edges):
shift = shift_list[i % len(shift_list)] # cycle through shift_list
v1 = nodes[i % n] # cycle repeatedly through nodes
v2 = nodes[(i + shift) % n]
G.add_edge(v1, v2)
return G
def fast_gnp_random_graph(n, p, seed=None, directed=False):
G = empty_graph(n)
if p <= 0 or p >= 1:
return nx.gnp_random_graph(n, p, seed=seed, directed=directed)
w = -1
lp = math.log(1.0 - p)
if directed:
G = nx.DiGraph(G)
# Nodes in graph are from 0,n-1 (start with v as the first node index).
v = 0
while v < n:
lr = math.log(1.0 - seed.random())
w = w + 1 + int(lr / lp)
if v == w: # avoid self loops
w = w + 1
while v < n <= w:
w = w - n
v = v + 1
if v == w: # avoid self loops
w = w + 1
if v < n:
G.add_edge(v, w)
else:
# Nodes in graph are from 0,n-1 (start with v as the second node index).
v = 1
while v < n:
lr = math.log(1.0 - seed.random())
w = w + 1 + int(lr / lp)
while w >= v and v < n:
w = w - v
v = v + 1
if v < n:
G.add_edge(v, w)
return G
def extended_barabasi_albert_graph(n, m, p, q, seed=None):
if m < 1 or m >= n:
msg = "Extended Barabasi-Albert network needs m>=1 and m<n, m=%d, n=%d"
raise nx.NetworkXError(msg % (m, n))
if p + q >= 1:
msg = "Extended Barabasi-Albert network needs p + q <= 1, p=%d, q=%d"
raise nx.NetworkXError(msg % (p, q))
# Add m initial nodes (m0 in barabasi-speak)
G = empty_graph(m)
# List of nodes to represent the preferential attachment random selection.
# At the creation of the graph, all nodes are added to the list
# so that even nodes that are not connected have a chance to get selected,
# for rewiring and adding of edges.
# With each new edge, nodes at the ends of the edge are added to the list.
attachment_preference = []
attachment_preference.extend(range(m))
# Start adding the other n-m nodes. The first node is m.
new_node = m
while new_node < n:
a_probability = seed.random()
# Total number of edges of a Clique of all the nodes
clique_degree = len(G) - 1
clique_size = (len(G) * clique_degree) / 2
# Adding m new edges, if there is room to add them
if a_probability < p and G.size() <= clique_size - m:
# Select the nodes where an edge can be added
elligible_nodes = [nd for nd, deg in G.degree() if deg < clique_degree]
for i in range(m):
# Choosing a random source node from elligible_nodes
src_node = seed.choice(elligible_nodes)
# Picking a possible node that is not 'src_node' or
# neighbor with 'src_node', with preferential attachment
prohibited_nodes = list(G[src_node])
prohibited_nodes.append(src_node)
# This will raise an exception if the sequence is empty
dest_node = seed.choice(
[nd for nd in attachment_preference if nd not in prohibited_nodes]
)
# Adding the new edge
G.add_edge(src_node, dest_node)
# Appending both nodes to add to their preferential attachment
attachment_preference.append(src_node)
attachment_preference.append(dest_node)
# Adjusting the elligible nodes. Degree may be saturated.
if G.degree(src_node) == clique_degree:
elligible_nodes.remove(src_node)
if (
G.degree(dest_node) == clique_degree
and dest_node in elligible_nodes
):
elligible_nodes.remove(dest_node)
# Rewiring m edges, if there are enough edges
elif p <= a_probability < (p + q) and m <= G.size() < clique_size:
# Selecting nodes that have at least 1 edge but that are not
# fully connected to ALL other nodes (center of star).
# These nodes are the pivot nodes of the edges to rewire
elligible_nodes = [nd for nd, deg in G.degree() if 0 < deg < clique_degree]
for i in range(m):
# Choosing a random source node
node = seed.choice(elligible_nodes)
# The available nodes do have a neighbor at least.
neighbor_nodes = list(G[node])
# Choosing the other end that will get dettached
src_node = seed.choice(neighbor_nodes)
# Picking a target node that is not 'node' or
# neighbor with 'node', with preferential attachment
neighbor_nodes.append(node)
dest_node = seed.choice(
[nd for nd in attachment_preference if nd not in neighbor_nodes]
)
# Rewire
G.remove_edge(node, src_node)
G.add_edge(node, dest_node)
# Adjusting the preferential attachment list
attachment_preference.remove(src_node)
attachment_preference.append(dest_node)
# Adjusting the elligible nodes.
# nodes may be saturated or isolated.
if G.degree(src_node) == 0 and src_node in elligible_nodes:
elligible_nodes.remove(src_node)
if dest_node in elligible_nodes:
if G.degree(dest_node) == clique_degree:
elligible_nodes.remove(dest_node)
else:
if G.degree(dest_node) == 1:
elligible_nodes.append(dest_node)
# Adding new node with m edges
else:
# Select the edges' nodes by preferential attachment
targets = _random_subset(attachment_preference, m, seed)
G.add_edges_from(zip([new_node] * m, targets))
# Add one node to the list for each new edge just created.
attachment_preference.extend(targets)
# The new node has m edges to it, plus itself: m + 1
attachment_preference.extend([new_node] * (m + 1))
new_node += 1
return G
def random_regular_graph(d, n, seed=None):
if (n * d) % 2 != 0:
raise nx.NetworkXError("n * d must be even")
if not 0 <= d < n:
raise nx.NetworkXError("the 0 <= d < n inequality must be satisfied")
if d == 0:
return empty_graph(n)
def _suitable(edges, potential_edges):
# Helper subroutine to check if there are suitable edges remaining
# If False, the generation of the graph has failed
if not potential_edges:
return True
for s1 in potential_edges:
for s2 in potential_edges:
# Two iterators on the same dictionary are guaranteed
# to visit it in the same order if there are no
# intervening modifications.
if s1 == s2:
# Only need to consider s1-s2 pair one time
break
if s1 > s2:
s1, s2 = s2, s1
if (s1, s2) not in edges:
return True
return False
def _try_creation():
# Attempt to create an edge set
edges = set()
stubs = list(range(n)) * d
while stubs:
potential_edges = defaultdict(lambda: 0)
seed.shuffle(stubs)
stubiter = iter(stubs)
for s1, s2 in zip(stubiter, stubiter):
if s1 > s2:
s1, s2 = s2, s1
if s1 != s2 and ((s1, s2) not in edges):
edges.add((s1, s2))
else:
potential_edges[s1] += 1
potential_edges[s2] += 1
if not _suitable(edges, potential_edges):
return None # failed to find suitable edge set
stubs = [
node
for node, potential in potential_edges.items()
for _ in range(potential)
]
return edges
# Even though a suitable edge set exists,
# the generation of such a set is not guaranteed.
# Try repeatedly to find one.
edges = _try_creation()
while edges is None:
edges = _try_creation()
G = nx.Graph()
G.add_edges_from(edges)
return G