1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
|
"""Provides explicit constructions of expander graphs.
"""
import itertools
import networkx as nx
__all__ = ["margulis_gabber_galil_graph", "chordal_cycle_graph", "paley_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),
#
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
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
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
|
