"""Provides explicit constructions of expander graphs.
"""
import itertools
import networkx as nx
__all__ = [
"margulis_gabber_galil_graph",
"chordal_cycle_graph",
"paley_graph",
"maybe_regular_expander",
"is_regular_expander",
"random_regular_expander_graph",
]
# Other discrete torus expanders can be constructed by using the following edge
# sets. For more information, see Chapter 4, "Expander Graphs", in
# "Pseudorandomness", by Salil Vadhan.
#
# For a directed expander, add edges from (x, y) to:
#
# (x, y),
# ((x + 1) % n, y),
# (x, (y + 1) % n),
# (x, (x + y) % n),
# (-y % n, x)
#
# For an undirected expander, add the reverse edges.
#
# Also appearing in the paper of Gabber and Galil:
#
# (x, y),
# (x, (x + y) % n),
# (x, (x + y + 1) % n),
# ((x + y) % n, y),
# ((x + y + 1) % n, y)
#
# and:
#
# (x, y),
# ((x + 2*y) % n, y),
# ((x + (2*y + 1)) % n, y),
# ((x + (2*y + 2)) % n, y),
# (x, (y + 2*x) % n),
# (x, (y + (2*x + 1)) % n),
# (x, (y + (2*x + 2)) % n),
#
[docs]
@nx._dispatchable(graphs=None, returns_graph=True)
def margulis_gabber_galil_graph(n, create_using=None):
r"""Returns the Margulis-Gabber-Galil undirected MultiGraph on `n^2` nodes.
The undirected MultiGraph is regular with degree `8`. Nodes are integer
pairs. The second-largest eigenvalue of the adjacency matrix of the graph
is at most `5 \sqrt{2}`, regardless of `n`.
Parameters
----------
n : int
Determines the number of nodes in the graph: `n^2`.
create_using : NetworkX graph constructor, optional (default MultiGraph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : graph
The constructed undirected multigraph.
Raises
------
NetworkXError
If the graph is directed or not a multigraph.
"""
G = nx.empty_graph(0, create_using, default=nx.MultiGraph)
if G.is_directed() or not G.is_multigraph():
msg = "`create_using` must be an undirected multigraph."
raise nx.NetworkXError(msg)
for x, y in itertools.product(range(n), repeat=2):
for u, v in (
((x + 2 * y) % n, y),
((x + (2 * y + 1)) % n, y),
(x, (y + 2 * x) % n),
(x, (y + (2 * x + 1)) % n),
):
G.add_edge((x, y), (u, v))
G.graph["name"] = f"margulis_gabber_galil_graph({n})"
return G
[docs]
@nx._dispatchable(graphs=None, returns_graph=True)
def chordal_cycle_graph(p, create_using=None):
"""Returns the chordal cycle graph on `p` nodes.
The returned graph is a cycle graph on `p` nodes with chords joining each
vertex `x` to its inverse modulo `p`. This graph is a (mildly explicit)
3-regular expander [1]_.
`p` *must* be a prime number.
Parameters
----------
p : a prime number
The number of vertices in the graph. This also indicates where the
chordal edges in the cycle will be created.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : graph
The constructed undirected multigraph.
Raises
------
NetworkXError
If `create_using` indicates directed or not a multigraph.
References
----------
.. [1] Theorem 4.4.2 in A. Lubotzky. "Discrete groups, expanding graphs and
invariant measures", volume 125 of Progress in Mathematics.
Birkhäuser Verlag, Basel, 1994.
"""
G = nx.empty_graph(0, create_using, default=nx.MultiGraph)
if G.is_directed() or not G.is_multigraph():
msg = "`create_using` must be an undirected multigraph."
raise nx.NetworkXError(msg)
for x in range(p):
left = (x - 1) % p
right = (x + 1) % p
# Here we apply Fermat's Little Theorem to compute the multiplicative
# inverse of x in Z/pZ. By Fermat's Little Theorem,
#
# x^p = x (mod p)
#
# Therefore,
#
# x * x^(p - 2) = 1 (mod p)
#
# The number 0 is a special case: we just let its inverse be itself.
chord = pow(x, p - 2, p) if x > 0 else 0
for y in (left, right, chord):
G.add_edge(x, y)
G.graph["name"] = f"chordal_cycle_graph({p})"
return G
[docs]
@nx._dispatchable(graphs=None, returns_graph=True)
def paley_graph(p, create_using=None):
r"""Returns the Paley $\frac{(p-1)}{2}$ -regular graph on $p$ nodes.
The returned graph is a graph on $\mathbb{Z}/p\mathbb{Z}$ with edges between $x$ and $y$
if and only if $x-y$ is a nonzero square in $\mathbb{Z}/p\mathbb{Z}$.
If $p \equiv 1 \pmod 4$, $-1$ is a square in $\mathbb{Z}/p\mathbb{Z}$ and therefore $x-y$ is a square if and
only if $y-x$ is also a square, i.e the edges in the Paley graph are symmetric.
If $p \equiv 3 \pmod 4$, $-1$ is not a square in $\mathbb{Z}/p\mathbb{Z}$ and therefore either $x-y$ or $y-x$
is a square in $\mathbb{Z}/p\mathbb{Z}$ but not both.
Note that a more general definition of Paley graphs extends this construction
to graphs over $q=p^n$ vertices, by using the finite field $F_q$ instead of $\mathbb{Z}/p\mathbb{Z}$.
This construction requires to compute squares in general finite fields and is
not what is implemented here (i.e `paley_graph(25)` does not return the true
Paley graph associated with $5^2$).
Parameters
----------
p : int, an odd prime number.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : graph
The constructed directed graph.
Raises
------
NetworkXError
If the graph is a multigraph.
References
----------
Chapter 13 in B. Bollobas, Random Graphs. Second edition.
Cambridge Studies in Advanced Mathematics, 73.
Cambridge University Press, Cambridge (2001).
"""
G = nx.empty_graph(0, create_using, default=nx.DiGraph)
if G.is_multigraph():
msg = "`create_using` cannot be a multigraph."
raise nx.NetworkXError(msg)
# Compute the squares in Z/pZ.
# Make it a set to uniquify (there are exactly (p-1)/2 squares in Z/pZ
# when is prime).
square_set = {(x**2) % p for x in range(1, p) if (x**2) % p != 0}
for x in range(p):
for x2 in square_set:
G.add_edge(x, (x + x2) % p)
G.graph["name"] = f"paley({p})"
return G
[docs]
@nx.utils.decorators.np_random_state("seed")
@nx._dispatchable(graphs=None, returns_graph=True)
def maybe_regular_expander(n, d, *, create_using=None, max_tries=100, seed=None):
r"""Utility for creating a random regular expander.
Returns a random $d$-regular graph on $n$ nodes which is an expander
graph with very good probability.
Parameters
----------
n : int
The number of nodes.
d : int
The degree of each node.
create_using : Graph Instance or Constructor
Indicator of type of graph to return.
If a Graph-type instance, then clear and use it.
If a constructor, call it to create an empty graph.
Use the Graph constructor by default.
max_tries : int. (default: 100)
The number of allowed loops when generating each independent cycle
seed : (default: None)
Seed used to set random number generation state. See :ref`Randomness<randomness>`.
Notes
-----
The nodes are numbered from $0$ to $n - 1$.
The graph is generated by taking $d / 2$ random independent cycles.
Joel Friedman proved that in this model the resulting
graph is an expander with probability
$1 - O(n^{-\tau})$ where $\tau = \lceil (\sqrt{d - 1}) / 2 \rceil - 1$. [1]_
Examples
--------
>>> G = nx.maybe_regular_expander(n=200, d=6, seed=8020)
Returns
-------
G : graph
The constructed undirected graph.
Raises
------
NetworkXError
If $d % 2 != 0$ as the degree must be even.
If $n - 1$ is less than $ 2d $ as the graph is complete at most.
If max_tries is reached
See Also
--------
is_regular_expander
random_regular_expander_graph
References
----------
.. [1] Joel Friedman,
A Proof of Alon’s Second Eigenvalue Conjecture and Related Problems, 2004
https://arxiv.org/abs/cs/0405020
"""
import numpy as np
if n < 1:
raise nx.NetworkXError("n must be a positive integer")
if not (d >= 2):
raise nx.NetworkXError("d must be greater than or equal to 2")
if not (d % 2 == 0):
raise nx.NetworkXError("d must be even")
if not (n - 1 >= d):
raise nx.NetworkXError(
f"Need n-1>= d to have room for {d//2} independent cycles with {n} nodes"
)
G = nx.empty_graph(n, create_using)
if n < 2:
return G
cycles = []
edges = set()
# Create d / 2 cycles
for i in range(d // 2):
iterations = max_tries
# Make sure the cycles are independent to have a regular graph
while len(edges) != (i + 1) * n:
iterations -= 1
# Faster than random.permutation(n) since there are only
# (n-1)! distinct cycles against n! permutations of size n
cycle = seed.permutation(n - 1).tolist()
cycle.append(n - 1)
new_edges = {
(u, v)
for u, v in nx.utils.pairwise(cycle, cyclic=True)
if (u, v) not in edges and (v, u) not in edges
}
# If the new cycle has no edges in common with previous cycles
# then add it to the list otherwise try again
if len(new_edges) == n:
cycles.append(cycle)
edges.update(new_edges)
if iterations == 0:
raise nx.NetworkXError("Too many iterations in maybe_regular_expander")
G.add_edges_from(edges)
return G
[docs]
@nx.utils.not_implemented_for("directed")
@nx.utils.not_implemented_for("multigraph")
@nx._dispatchable(preserve_edge_attrs={"G": {"weight": 1}})
def is_regular_expander(G, *, epsilon=0):
r"""Determines whether the graph G is a regular expander. [1]_
An expander graph is a sparse graph with strong connectivity properties.
More precisely, this helper checks whether the graph is a
regular $(n, d, \lambda)$-expander with $\lambda$ close to
the Alon-Boppana bound and given by
$\lambda = 2 \sqrt{d - 1} + \epsilon$. [2]_
In the case where $\epsilon = 0$ then if the graph successfully passes the test
it is a Ramanujan graph. [3]_
A Ramanujan graph has spectral gap almost as large as possible, which makes them
excellent expanders.
Parameters
----------
G : NetworkX graph
epsilon : int, float, default=0
Returns
-------
bool
Whether the given graph is a regular $(n, d, \lambda)$-expander
where $\lambda = 2 \sqrt{d - 1} + \epsilon$.
Examples
--------
>>> G = nx.random_regular_expander_graph(20, 4)
>>> nx.is_regular_expander(G)
True
See Also
--------
maybe_regular_expander
random_regular_expander_graph
References
----------
.. [1] Expander graph, https://en.wikipedia.org/wiki/Expander_graph
.. [2] Alon-Boppana bound, https://en.wikipedia.org/wiki/Alon%E2%80%93Boppana_bound
.. [3] Ramanujan graphs, https://en.wikipedia.org/wiki/Ramanujan_graph
"""
import numpy as np
from scipy.sparse.linalg import eigsh
if epsilon < 0:
raise nx.NetworkXError("epsilon must be non negative")
if not nx.is_regular(G):
return False
_, d = nx.utils.arbitrary_element(G.degree)
A = nx.adjacency_matrix(G, dtype=float)
lams = eigsh(A, which="LM", k=2, return_eigenvectors=False)
# lambda2 is the second biggest eigenvalue
lambda2 = min(lams)
# Use bool() to convert numpy scalar to Python Boolean
return bool(abs(lambda2) < 2 ** np.sqrt(d - 1) + epsilon)
[docs]
@nx.utils.decorators.np_random_state("seed")
@nx._dispatchable(graphs=None, returns_graph=True)
def random_regular_expander_graph(
n, d, *, epsilon=0, create_using=None, max_tries=100, seed=None
):
r"""Returns a random regular expander graph on $n$ nodes with degree $d$.
An expander graph is a sparse graph with strong connectivity properties. [1]_
More precisely the returned graph is a $(n, d, \lambda)$-expander with
$\lambda = 2 \sqrt{d - 1} + \epsilon$, close to the Alon-Boppana bound. [2]_
In the case where $\epsilon = 0$ it returns a Ramanujan graph.
A Ramanujan graph has spectral gap almost as large as possible,
which makes them excellent expanders. [3]_
Parameters
----------
n : int
The number of nodes.
d : int
The degree of each node.
epsilon : int, float, default=0
max_tries : int, (default: 100)
The number of allowed loops, also used in the maybe_regular_expander utility
seed : (default: None)
Seed used to set random number generation state. See :ref`Randomness<randomness>`.
Raises
------
NetworkXError
If max_tries is reached
Examples
--------
>>> G = nx.random_regular_expander_graph(20, 4)
>>> nx.is_regular_expander(G)
True
Notes
-----
This loops over `maybe_regular_expander` and can be slow when
$n$ is too big or $\epsilon$ too small.
See Also
--------
maybe_regular_expander
is_regular_expander
References
----------
.. [1] Expander graph, https://en.wikipedia.org/wiki/Expander_graph
.. [2] Alon-Boppana bound, https://en.wikipedia.org/wiki/Alon%E2%80%93Boppana_bound
.. [3] Ramanujan graphs, https://en.wikipedia.org/wiki/Ramanujan_graph
"""
G = maybe_regular_expander(
n, d, create_using=create_using, max_tries=max_tries, seed=seed
)
iterations = max_tries
while not is_regular_expander(G, epsilon=epsilon):
iterations -= 1
G = maybe_regular_expander(
n=n, d=d, create_using=create_using, max_tries=max_tries, seed=seed
)
if iterations == 0:
raise nx.NetworkXError(
"Too many iterations in random_regular_expander_graph"
)
return G
