"""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