"""Provides explicit constructions of expander graphs."""importitertoolsimportnetworkxasnx__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),#
[docs]defmargulis_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)ifG.is_directed()ornotG.is_multigraph():msg="`create_using` must be an undirected multigraph."raisenx.NetworkXError(msg)for(x,y)initertools.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})"returnG
[docs]defchordal_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)ifG.is_directed()ornotG.is_multigraph():msg="`create_using` must be an undirected multigraph."raisenx.NetworkXError(msg)forxinrange(p):left=(x-1)%pright=(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)ifx>0else0foryin(left,right,chord):G.add_edge(x,y)G.graph["name"]=f"chordal_cycle_graph({p})"returnG
[docs]defpaley_graph(p,create_using=None):"""Returns the Paley (p-1)/2-regular graph on p nodes. The returned graph is a graph on Z/pZ with edges between x and y if and only if x-y is a nonzero square in Z/pZ. If p = 1 mod 4, -1 is a square in Z/pZ 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 = 3 mod 4, -1 is not a square in Z/pZ and therefore either x-y or y-x is a square in Z/pZ 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 Z/pZ. 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)ifG.is_multigraph():msg="`create_using` cannot be a multigraph."raisenx.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)%pforxinrange(1,p)if(x**2)%p!=0}forxinrange(p):forx2insquare_set:G.add_edge(x,(x+x2)%p)G.graph["name"]=f"paley({p})"returnG