203 lines
6.0 KiB
Python
203 lines
6.0 KiB
Python
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"""Provides explicit constructions of expander graphs.
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"""
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import itertools
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import networkx as nx
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__all__ = ["margulis_gabber_galil_graph", "chordal_cycle_graph", "paley_graph"]
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# Other discrete torus expanders can be constructed by using the following edge
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# sets. For more information, see Chapter 4, "Expander Graphs", in
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# "Pseudorandomness", by Salil Vadhan.
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#
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# For a directed expander, add edges from (x, y) to:
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#
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# (x, y),
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# ((x + 1) % n, y),
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# (x, (y + 1) % n),
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# (x, (x + y) % n),
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# (-y % n, x)
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#
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# For an undirected expander, add the reverse edges.
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#
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# Also appearing in the paper of Gabber and Galil:
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#
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# (x, y),
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# (x, (x + y) % n),
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# (x, (x + y + 1) % n),
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# ((x + y) % n, y),
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# ((x + y + 1) % n, y)
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#
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# and:
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#
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# (x, y),
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# ((x + 2*y) % n, y),
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# ((x + (2*y + 1)) % n, y),
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# ((x + (2*y + 2)) % n, y),
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# (x, (y + 2*x) % n),
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# (x, (y + (2*x + 1)) % n),
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# (x, (y + (2*x + 2)) % n),
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#
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def margulis_gabber_galil_graph(n, create_using=None):
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r"""Returns the Margulis-Gabber-Galil undirected MultiGraph on `n^2` nodes.
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The undirected MultiGraph is regular with degree `8`. Nodes are integer
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pairs. The second-largest eigenvalue of the adjacency matrix of the graph
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is at most `5 \sqrt{2}`, regardless of `n`.
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Parameters
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----------
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n : int
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Determines the number of nodes in the graph: `n^2`.
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create_using : NetworkX graph constructor, optional (default MultiGraph)
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Graph type to create. If graph instance, then cleared before populated.
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Returns
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-------
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G : graph
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The constructed undirected multigraph.
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Raises
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------
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NetworkXError
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If the graph is directed or not a multigraph.
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"""
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G = nx.empty_graph(0, create_using, default=nx.MultiGraph)
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if G.is_directed() or not G.is_multigraph():
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msg = "`create_using` must be an undirected multigraph."
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raise nx.NetworkXError(msg)
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for (x, y) in itertools.product(range(n), repeat=2):
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for (u, v) in (
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((x + 2 * y) % n, y),
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((x + (2 * y + 1)) % n, y),
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(x, (y + 2 * x) % n),
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(x, (y + (2 * x + 1)) % n),
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):
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G.add_edge((x, y), (u, v))
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G.graph["name"] = f"margulis_gabber_galil_graph({n})"
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return G
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def chordal_cycle_graph(p, create_using=None):
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"""Returns the chordal cycle graph on `p` nodes.
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The returned graph is a cycle graph on `p` nodes with chords joining each
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vertex `x` to its inverse modulo `p`. This graph is a (mildly explicit)
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3-regular expander [1]_.
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`p` *must* be a prime number.
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Parameters
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----------
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p : a prime number
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The number of vertices in the graph. This also indicates where the
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chordal edges in the cycle will be created.
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create_using : NetworkX graph constructor, optional (default=nx.Graph)
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Graph type to create. If graph instance, then cleared before populated.
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Returns
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-------
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G : graph
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The constructed undirected multigraph.
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Raises
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------
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NetworkXError
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If `create_using` indicates directed or not a multigraph.
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References
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----------
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.. [1] Theorem 4.4.2 in A. Lubotzky. "Discrete groups, expanding graphs and
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invariant measures", volume 125 of Progress in Mathematics.
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Birkhäuser Verlag, Basel, 1994.
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"""
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G = nx.empty_graph(0, create_using, default=nx.MultiGraph)
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if G.is_directed() or not G.is_multigraph():
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msg = "`create_using` must be an undirected multigraph."
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raise nx.NetworkXError(msg)
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for x in range(p):
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left = (x - 1) % p
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right = (x + 1) % p
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# Here we apply Fermat's Little Theorem to compute the multiplicative
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# inverse of x in Z/pZ. By Fermat's Little Theorem,
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#
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# x^p = x (mod p)
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#
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# Therefore,
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#
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# x * x^(p - 2) = 1 (mod p)
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#
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# The number 0 is a special case: we just let its inverse be itself.
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chord = pow(x, p - 2, p) if x > 0 else 0
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for y in (left, right, chord):
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G.add_edge(x, y)
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G.graph["name"] = f"chordal_cycle_graph({p})"
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return G
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def paley_graph(p, create_using=None):
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"""Returns the Paley (p-1)/2-regular graph on p nodes.
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The returned graph is a graph on Z/pZ with edges between x and y
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if and only if x-y is a nonzero square in Z/pZ.
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If p = 1 mod 4, -1 is a square in Z/pZ and therefore x-y is a square if and
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only if y-x is also a square, i.e the edges in the Paley graph are symmetric.
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If p = 3 mod 4, -1 is not a square in Z/pZ and therefore either x-y or y-x
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is a square in Z/pZ but not both.
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Note that a more general definition of Paley graphs extends this construction
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to graphs over q=p^n vertices, by using the finite field F_q instead of Z/pZ.
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This construction requires to compute squares in general finite fields and is
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not what is implemented here (i.e paley_graph(25) does not return the true
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Paley graph associated with 5^2).
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Parameters
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----------
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p : int, an odd prime number.
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create_using : NetworkX graph constructor, optional (default=nx.Graph)
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Graph type to create. If graph instance, then cleared before populated.
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Returns
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-------
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G : graph
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The constructed directed graph.
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Raises
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------
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NetworkXError
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If the graph is a multigraph.
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References
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----------
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Chapter 13 in B. Bollobas, Random Graphs. Second edition.
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Cambridge Studies in Advanced Mathematics, 73.
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Cambridge University Press, Cambridge (2001).
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"""
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G = nx.empty_graph(0, create_using, default=nx.DiGraph)
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if G.is_multigraph():
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msg = "`create_using` cannot be a multigraph."
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raise nx.NetworkXError(msg)
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# Compute the squares in Z/pZ.
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# Make it a set to uniquify (there are exactly (p-1)/2 squares in Z/pZ
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# when is prime).
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square_set = {(x ** 2) % p for x in range(1, p) if (x ** 2) % p != 0}
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for x in range(p):
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for x2 in square_set:
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G.add_edge(x, (x + x2) % p)
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G.graph["name"] = f"paley({p})"
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return G
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