Sandpiles¶
Functions and classes for mathematical sandpiles.
Version: 2.4
AUTHOR:
David Perkinson (June 4, 2015) Upgraded from version 2.3 to 2.4.
MAJOR CHANGES
Eliminated dependence on 4ti2, substituting the use of Polyhedron methods. Thus, no optional packages are necessary.
Fixed bug in
Sandpile.__init__
so that now multigraphs are handled correctly.Created
sandpiles
to handle examples of Sandpiles in analogy withgraphs
,simplicial_complexes
, andpolytopes
. In the process, we implemented a much faster way of producing the sandpile grid graph.Added support for open and closed sandpile Markov chains.
Added support for Weierstrass points.
Implemented the Cori-Le Borgne algorithm for computing ranks of divisors on complete graphs.
NEW METHODS
Sandpile: avalanche_polynomial, genus, group_gens, help, jacobian_representatives, markov_chain, picard_representatives, smith_form, stable_configs, stationary_density, tutte_polynomial.
SandpileConfig: burst_size, help.
SandpileDivisor: help, is_linearly_equivalent, is_q_reduced, is_weierstrass_pt, polytope, polytope_integer_pts, q_reduced, rank, simulate_threshold, stabilize, weierstrass_div, weierstrass_gap_seq, weierstrass_pts, weierstrass_rank_seq.
MINOR CHANGES
The
sink
argument toSandpile.__init__
now defaults to the first vertex.A SandpileConfig or SandpileDivisor may now be multiplied by an integer.
Sped up
__add__
method for SandpileConfig and SandpileDivisor.Enhanced string representation of a Sandpile (via
_repr_
and thename
methods).Recurrents for complete graphs and cycle graphs are computed more quickly.
The stabilization code for SandpileConfig has been made more efficient.
Added optional probability distribution arguments to
add_random
methods.
Marshall Hampton (2010-1-10) modified for inclusion as a module within Sage library.
David Perkinson (2010-12-14) added show3d(), fixed bug in resolution(), replaced elementary_divisors() with invariant_factors(), added show() for SandpileConfig and SandpileDivisor.
David Perkinson (2010-9-18): removed is_undirected, added show(), added verbose arguments to several functions to display SandpileConfigs and divisors as lists of integers
David Perkinson (2010-12-19): created separate SandpileConfig, SandpileDivisor, and Sandpile classes
David Perkinson (2009-07-15): switched to using config_to_list instead of .values(), thus fixing a few bugs when not using integer labels for vertices.
David Perkinson (2009): many undocumented improvements
David Perkinson (2008-12-27): initial version
EXAMPLES:
For general help, enter Sandpile.help()
, SandpileConfig.help()
, and
SandpileDivisor.help()
. Miscellaneous examples appear below.
A weighted directed graph given as a Python dictionary:
sage: from sage.sandpiles import *
sage: g = {0: {},
....: 1: {0: 1, 2: 1, 3: 1},
....: 2: {1: 1, 3: 1, 4: 1},
....: 3: {1: 1, 2: 1, 4: 1},
....: 4: {2: 1, 3: 1}}
The associated sandpile with 0 chosen as the sink:
sage: S = Sandpile(g,0)
Or just:
sage: S = Sandpile(g)
A picture of the graph:
sage: S.show() # long time
The relevant Laplacian matrices:
sage: S.laplacian()
[ 0 0 0 0 0]
[-1 3 -1 -1 0]
[ 0 -1 3 -1 -1]
[ 0 -1 -1 3 -1]
[ 0 0 -1 -1 2]
sage: S.reduced_laplacian()
[ 3 -1 -1 0]
[-1 3 -1 -1]
[-1 -1 3 -1]
[ 0 -1 -1 2]
The number of elements of the sandpile group for S:
sage: S.group_order()
8
The structure of the sandpile group:
sage: S.invariant_factors()
[1, 1, 1, 8]
The elements of the sandpile group for S:
sage: S.recurrents()
[{1: 2, 2: 2, 3: 2, 4: 1},
{1: 2, 2: 2, 3: 2, 4: 0},
{1: 2, 2: 1, 3: 2, 4: 0},
{1: 2, 2: 2, 3: 0, 4: 1},
{1: 2, 2: 0, 3: 2, 4: 1},
{1: 2, 2: 2, 3: 1, 4: 0},
{1: 2, 2: 1, 3: 2, 4: 1},
{1: 2, 2: 2, 3: 1, 4: 1}]
The maximal stable element (2 grains of sand on vertices 1, 2, and 3, and 1 grain of sand on vertex 4:
sage: S.max_stable()
{1: 2, 2: 2, 3: 2, 4: 1}
sage: S.max_stable().values()
[2, 2, 2, 1]
The identity of the sandpile group for S:
sage: S.identity()
{1: 2, 2: 2, 3: 2, 4: 0}
An arbitrary sandpile configuration:
sage: c = SandpileConfig(S,[1,0,4,-3])
sage: c.equivalent_recurrent()
{1: 2, 2: 2, 3: 2, 4: 0}
Some group operations:
sage: m = S.max_stable()
sage: i = S.identity()
sage: m.values()
[2, 2, 2, 1]
sage: i.values()
[2, 2, 2, 0]
sage: m + i # coordinate-wise sum
{1: 4, 2: 4, 3: 4, 4: 1}
sage: m - i
{1: 0, 2: 0, 3: 0, 4: 1}
sage: m & i # add, then stabilize
{1: 2, 2: 2, 3: 2, 4: 1}
sage: e = m + m
sage: e
{1: 4, 2: 4, 3: 4, 4: 2}
sage: ~e # stabilize
{1: 2, 2: 2, 3: 2, 4: 0}
sage: a = -m
sage: a & m
{1: 0, 2: 0, 3: 0, 4: 0}
sage: a * m # add, then find the equivalent recurrent
{1: 2, 2: 2, 3: 2, 4: 0}
sage: a^3 # a*a*a
{1: 2, 2: 2, 3: 2, 4: 1}
sage: a^(-1) == m
True
sage: a < m # every coordinate of a is < that of m
True
Firing an unstable vertex returns resulting configuration:
sage: c = S.max_stable() + S.identity()
sage: c.fire_vertex(1)
{1: 1, 2: 5, 3: 5, 4: 1}
sage: c
{1: 4, 2: 4, 3: 4, 4: 1}
Fire all unstable vertices:
sage: c.unstable()
[1, 2, 3]
sage: c.fire_unstable()
{1: 3, 2: 3, 3: 3, 4: 3}
Stabilize c, returning the resulting configuration and the firing vector:
sage: c.stabilize(True)
[{1: 2, 2: 2, 3: 2, 4: 1}, {1: 6, 2: 8, 3: 8, 4: 8}]
sage: c
{1: 4, 2: 4, 3: 4, 4: 1}
sage: S.max_stable() & S.identity() == c.stabilize()
True
The number of superstable configurations of each degree:
sage: S.h_vector()
[1, 3, 4]
sage: S.postulation()
2
the saturated homogeneous toppling ideal:
sage: S.ideal()
Ideal (x1 - x0, x3*x2 - x0^2, x4^2 - x0^2, x2^3 - x4*x3*x0, x4*x2^2 - x3^2*x0, x3^3 - x4*x2*x0, x4*x3^2 - x2^2*x0) of Multivariate Polynomial Ring in x4, x3, x2, x1, x0 over Rational Field
its minimal free resolution:
sage: S.resolution()
'R^1 <-- R^7 <-- R^15 <-- R^13 <-- R^4'
and its Betti numbers:
sage: S.betti()
0 1 2 3 4
------------------------------------
0: 1 1 - - -
1: - 2 2 - -
2: - 4 13 13 4
------------------------------------
total: 1 7 15 13 4
Some various ways of creating Sandpiles:
sage: S = sandpiles.Complete(4) # for more options enter ``sandpile.TAB``
sage: S = sandpiles.Wheel(6)
A multidigraph with loops (vertices 0, 1, 2; for example, there is a directed edge from vertex 2 to vertex 1 of weight 3, which can be thought of as three directed edges of the form (2,3). There is also a single loop at vertex 2 and an edge (2,0) of weight 2):
sage: S = Sandpile({0:[1,2], 1:[0,0,2], 2:[0,0,1,1,1,2], 3:[2]})
Using the graph library (vertex 1 is specified as the sink; omitting this would make the sink vertex 0 by default):
sage: S = Sandpile(graphs.PetersenGraph(),1)
Distribution of avalanche sizes:
sage: S = sandpiles.Grid(10,10)
sage: m = S.max_stable()
sage: a = []
sage: for i in range(1000):
....: m = m.add_random()
....: m, f = m.stabilize(True)
....: a.append(sum(f.values()))
sage: p = list_plot([[log(i+1),log(a.count(i))] for i in [0..max(a)] if a.count(i)])
sage: p.axes_labels(['log(N)','log(D(N))'])
sage: t = text("Distribution of avalanche sizes", (2,2), rgbcolor=(1,0,0))
sage: show(p+t,axes_labels=['log(N)','log(D(N))']) # long time
Working with sandpile divisors:
sage: S = sandpiles.Complete(4)
sage: D = SandpileDivisor(S, [0,0,0,5])
sage: E = D.stabilize(); E
{0: 1, 1: 1, 2: 1, 3: 2}
sage: D.is_linearly_equivalent(E)
True
sage: D.q_reduced()
{0: 4, 1: 0, 2: 0, 3: 1}
sage: S = sandpiles.Complete(4)
sage: D = SandpileDivisor(S, [0,0,0,5])
sage: E = D.stabilize(); E
{0: 1, 1: 1, 2: 1, 3: 2}
sage: D.is_linearly_equivalent(E)
True
sage: D.q_reduced()
{0: 4, 1: 0, 2: 0, 3: 1}
sage: D.rank()
2
sage: sorted(D.effective_div(), key=str)
[{0: 0, 1: 0, 2: 0, 3: 5},
{0: 0, 1: 0, 2: 4, 3: 1},
{0: 0, 1: 4, 2: 0, 3: 1},
{0: 1, 1: 1, 2: 1, 3: 2},
{0: 4, 1: 0, 2: 0, 3: 1}]
sage: sorted(D.effective_div(False))
[[0, 0, 0, 5], [0, 0, 4, 1], [0, 4, 0, 1], [1, 1, 1, 2], [4, 0, 0, 1]]
sage: D.rank()
2
sage: D.rank(True)
(2, {0: 2, 1: 1, 2: 0, 3: 0})
sage: E = D.rank(True)[1] # E proves the rank is not 3
sage: E.values()
[2, 1, 0, 0]
sage: E.deg()
3
sage: rank(D - E)
-1
sage: (D - E).effective_div()
[]
sage: D.weierstrass_pts()
(0, 1, 2, 3)
sage: D.weierstrass_rank_seq(0)
(2, 1, 0, 0, 0, -1)
sage: D.weierstrass_pts()
(0, 1, 2, 3)
sage: D.weierstrass_rank_seq(0)
(2, 1, 0, 0, 0, -1)
- class sage.sandpiles.sandpile.Sandpile(g, sink=None)¶
Bases:
sage.graphs.digraph.DiGraph
Class for Dhar’s abelian sandpile model.
- all_k_config(k)¶
The constant configuration with all values set to \(k\).
INPUT:
k
– integerOUTPUT:
SandpileConfig
EXAMPLES:
sage: s = sandpiles.Diamond() sage: s.all_k_config(7) {1: 7, 2: 7, 3: 7}
- all_k_div(k)¶
The divisor with all values set to \(k\).
INPUT:
k
– integerOUTPUT:
SandpileDivisor
EXAMPLES:
sage: S = sandpiles.House() sage: S.all_k_div(7) {0: 7, 1: 7, 2: 7, 3: 7, 4: 7}
- avalanche_polynomial(multivariable=True)¶
The avalanche polynomial. See NOTE for details.
INPUT:
multivariable
– (default:True
) booleanOUTPUT:
polynomial
EXAMPLES:
sage: s = sandpiles.Complete(4) sage: s.avalanche_polynomial() 9*x0*x1*x2 + 2*x0*x1 + 2*x0*x2 + 2*x1*x2 + 3*x0 + 3*x1 + 3*x2 + 24 sage: s.avalanche_polynomial(False) 9*x0^3 + 6*x0^2 + 9*x0 + 24
Note
For each nonsink vertex \(v\), let \(x_v\) be an indeterminate. If \((r,v)\) is a pair consisting of a recurrent \(r\) and nonsink vertex \(v\), then for each nonsink vertex \(w\), let \(n_w\) be the number of times vertex \(w\) fires in the stabilization of \(r + v\). Let \(M(r,v)\) be the monomial \(\prod_w x_w^{n_w}\), i.e., the exponent records the vector of \(n_w\) as \(w\) ranges over the nonsink vertices. The avalanche polynomial is then the sum of \(M(r,v)\) as \(r\) ranges over the recurrents and \(v\) ranges over the nonsink vertices. If
multivariable
isFalse
, then set all the indeterminates equal to each other (and, thus, only count the number of vertex firings in the stabilizations, forgetting which particular vertices fired).
- betti(verbose=True)¶
The Betti table for the homogeneous toppling ideal. If
verbose
isTrue
, it prints the standard Betti table, otherwise, it returns a less formatted table.INPUT:
verbose
– (default:True
) booleanOUTPUT:
Betti numbers for the sandpile
EXAMPLES:
sage: S = sandpiles.Diamond() sage: S.betti() 0 1 2 3 ------------------------------ 0: 1 - - - 1: - 2 - - 2: - 4 9 4 ------------------------------ total: 1 6 9 4 sage: S.betti(False) [1, 6, 9, 4]
- betti_complexes()¶
The support-complexes with non-trivial homology. (See NOTE.)
OUTPUT:
list (of pairs [divisors, corresponding simplicial complex])
EXAMPLES:
sage: S = Sandpile({0:{},1:{0: 1, 2: 1, 3: 4},2:{3: 5},3:{1: 1, 2: 1}},0) sage: p = S.betti_complexes() sage: p[0] [{0: -8, 1: 5, 2: 4, 3: 1}, Simplicial complex with vertex set (1, 2, 3) and facets {(3,), (1, 2)}] sage: S.resolution() 'R^1 <-- R^5 <-- R^5 <-- R^1' sage: S.betti() 0 1 2 3 ------------------------------ 0: 1 - - - 1: - 5 5 - 2: - - - 1 ------------------------------ total: 1 5 5 1 sage: len(p) 11 sage: p[0][1].homology() {0: Z, 1: 0} sage: p[-1][1].homology() {0: 0, 1: 0, 2: Z}
Note
A
support-complex
is the simplicial complex formed from the supports of the divisors in a linear system.
- burning_config()¶
The minimal burning configuration.
OUTPUT:
dict (configuration)
EXAMPLES:
sage: g = {0:{},1:{0:1,3:1,4:1},2:{0:1,3:1,5:1}, ....: 3:{2:1,5:1},4:{1:1,3:1},5:{2:1,3:1}} sage: S = Sandpile(g,0) sage: S.burning_config() {1: 2, 2: 0, 3: 1, 4: 1, 5: 0} sage: S.burning_config().values() [2, 0, 1, 1, 0] sage: S.burning_script() {1: 1, 2: 3, 3: 5, 4: 1, 5: 4} sage: script = S.burning_script().values() sage: script [1, 3, 5, 1, 4] sage: matrix(script)*S.reduced_laplacian() [2 0 1 1 0]
Note
The burning configuration and script are computed using a modified version of Speer’s script algorithm. This is a generalization to directed multigraphs of Dhar’s burning algorithm.
A burning configuration is a nonnegative integer-linear combination of the rows of the reduced Laplacian matrix having nonnegative entries and such that every vertex has a path from some vertex in its support. The corresponding burning script gives the integer-linear combination needed to obtain the burning configuration. So if \(b\) is the burning configuration, \(\sigma\) is its script, and \(\tilde{L}\) is the reduced Laplacian, then \(\sigma\cdot \tilde{L} = b\). The minimal burning configuration is the one with the minimal script (its components are no larger than the components of any other script for a burning configuration).
The following are equivalent for a configuration \(c\) with burning configuration \(b\) having script \(\sigma\):
\(c\) is recurrent;
\(c+b\) stabilizes to \(c\);
the firing vector for the stabilization of \(c+b\) is \(\sigma\).
- burning_script()¶
A script for the minimal burning configuration.
OUTPUT:
dict
EXAMPLES:
sage: g = {0:{},1:{0:1,3:1,4:1},2:{0:1,3:1,5:1}, ....: 3:{2:1,5:1},4:{1:1,3:1},5:{2:1,3:1}} sage: S = Sandpile(g,0) sage: S.burning_config() {1: 2, 2: 0, 3: 1, 4: 1, 5: 0} sage: S.burning_config().values() [2, 0, 1, 1, 0] sage: S.burning_script() {1: 1, 2: 3, 3: 5, 4: 1, 5: 4} sage: script = S.burning_script().values() sage: script [1, 3, 5, 1, 4] sage: matrix(script)*S.reduced_laplacian() [2 0 1 1 0]
Note
The burning configuration and script are computed using a modified version of Speer’s script algorithm. This is a generalization to directed multigraphs of Dhar’s burning algorithm.
A burning configuration is a nonnegative integer-linear combination of the rows of the reduced Laplacian matrix having nonnegative entries and such that every vertex has a path from some vertex in its support. The corresponding burning script gives the integer-linear combination needed to obtain the burning configuration. So if \(b\) is the burning configuration, \(s\) is its script, and \(L_{\mathrm{red}}\) is the reduced Laplacian, then \(s\cdot L_{\mathrm{red}}= b\). The minimal burning configuration is the one with the minimal script (its components are no larger than the components of any other script for a burning configuration).
The following are equivalent for a configuration \(c\) with burning configuration \(b\) having script \(s\):
\(c\) is recurrent;
\(c+b\) stabilizes to \(c\);
the firing vector for the stabilization of \(c+b\) is \(s\).
- canonical_divisor()¶
The canonical divisor. This is the divisor with \(\deg(v)-2\) grains of sand on each vertex (not counting loops). Only for undirected graphs.
OUTPUT:
SandpileDivisor
EXAMPLES:
sage: S = sandpiles.Complete(4) sage: S.canonical_divisor() {0: 1, 1: 1, 2: 1, 3: 1} sage: s = Sandpile({0:[1,1],1:[0,0,1,1,1]},0) sage: s.canonical_divisor() # loops are disregarded {0: 0, 1: 0}
Warning
The underlying graph must be undirected.
- dict()¶
A dictionary of dictionaries representing a directed graph.
OUTPUT:
dict
EXAMPLES:
sage: S = sandpiles.Diamond() sage: S.dict() {0: {1: 1, 2: 1}, 1: {0: 1, 2: 1, 3: 1}, 2: {0: 1, 1: 1, 3: 1}, 3: {1: 1, 2: 1}} sage: S.sink() 0
- genus()¶
The genus: (# non-loop edges) - (# vertices) + 1. Only defined for undirected graphs.
OUTPUT:
integer
EXAMPLES:
sage: sandpiles.Complete(4).genus() 3 sage: sandpiles.Cycle(5).genus() 1
- groebner()¶
A Groebner basis for the homogeneous toppling ideal. It is computed with respect to the standard sandpile ordering (see
ring
).OUTPUT:
Groebner basis
EXAMPLES:
sage: S = sandpiles.Diamond() sage: S.groebner() [x3*x2^2 - x1^2*x0, x2^3 - x3*x1*x0, x3*x1^2 - x2^2*x0, x1^3 - x3*x2*x0, x3^2 - x0^2, x2*x1 - x0^2]
- group_gens(verbose=True)¶
A minimal list of generators for the sandpile group. If
verbose
isFalse
then the generators are represented as lists of integers.INPUT:
verbose
– (default:True
) booleanOUTPUT:
list of SandpileConfig (or of lists of integers if
verbose
isFalse
)EXAMPLES:
sage: s = sandpiles.Cycle(5) sage: s.group_gens() [{1: 0, 2: 1, 3: 1, 4: 1}] sage: s.group_gens()[0].order() 5 sage: s = sandpiles.Complete(5) sage: s.group_gens(False) [[2, 3, 2, 2], [2, 2, 3, 2], [2, 2, 2, 3]] sage: [i.order() for i in s.group_gens()] [5, 5, 5] sage: s.invariant_factors() [1, 5, 5, 5]
- group_order()¶
The size of the sandpile group.
OUTPUT:
integer
EXAMPLES:
sage: S = sandpiles.House() sage: S.group_order() 11
- h_vector()¶
The number of superstable configurations in each degree. Equivalently, this is the list of first differences of the Hilbert function of the (homogeneous) toppling ideal.
OUTPUT:
list of nonnegative integers
EXAMPLES:
sage: s = sandpiles.Grid(2,2) sage: s.hilbert_function() [1, 5, 15, 35, 66, 106, 146, 178, 192] sage: s.h_vector() [1, 4, 10, 20, 31, 40, 40, 32, 14]
- static help(verbose=True)¶
List of Sandpile-specific methods (not inherited from
Graph
). Ifverbose
, include short descriptions.INPUT:
verbose
– (default:True
) booleanOUTPUT:
printed string
EXAMPLES:
sage: Sandpile.help() # long time For detailed help with any method FOO listed below, enter "Sandpile.FOO?" or enter "S.FOO?" for any Sandpile S. all_k_config -- The constant configuration with all values set to k. all_k_div -- The divisor with all values set to k. avalanche_polynomial -- The avalanche polynomial. betti -- The Betti table for the homogeneous toppling ideal. betti_complexes -- The support-complexes with non-trivial homology. burning_config -- The minimal burning configuration. burning_script -- A script for the minimal burning configuration. canonical_divisor -- The canonical divisor. dict -- A dictionary of dictionaries representing a directed graph. genus -- The genus: (# non-loop edges) - (# vertices) + 1. groebner -- A Groebner basis for the homogeneous toppling ideal. group_gens -- A minimal list of generators for the sandpile group. group_order -- The size of the sandpile group. h_vector -- The number of superstable configurations in each degree. help -- List of Sandpile-specific methods (not inherited from "Graph"). hilbert_function -- The Hilbert function of the homogeneous toppling ideal. ideal -- The saturated homogeneous toppling ideal. identity -- The identity configuration. in_degree -- The in-degree of a vertex or a list of all in-degrees. invariant_factors -- The invariant factors of the sandpile group. is_undirected -- Is the underlying graph undirected? jacobian_representatives -- Representatives for the elements of the Jacobian group. laplacian -- The Laplacian matrix of the graph. markov_chain -- The sandpile Markov chain for configurations or divisors. max_stable -- The maximal stable configuration. max_stable_div -- The maximal stable divisor. max_superstables -- The maximal superstable configurations. min_recurrents -- The minimal recurrent elements. nonsink_vertices -- The nonsink vertices. nonspecial_divisors -- The nonspecial divisors. out_degree -- The out-degree of a vertex or a list of all out-degrees. picard_representatives -- Representatives of the divisor classes of degree d in the Picard group. points -- Generators for the multiplicative group of zeros of the sandpile ideal. postulation -- The postulation number of the toppling ideal. recurrents -- The recurrent configurations. reduced_laplacian -- The reduced Laplacian matrix of the graph. reorder_vertices -- A copy of the sandpile with vertex names permuted. resolution -- A minimal free resolution of the homogeneous toppling ideal. ring -- The ring containing the homogeneous toppling ideal. show -- Draw the underlying graph. show3d -- Draw the underlying graph. sink -- The sink vertex. smith_form -- The Smith normal form for the Laplacian. solve -- Approximations of the complex affine zeros of the sandpile ideal. stable_configs -- Generator for all stable configurations. stationary_density -- The stationary density of the sandpile. superstables -- The superstable configurations. symmetric_recurrents -- The symmetric recurrent configurations. tutte_polynomial -- The Tutte polynomial of the underlying graph. unsaturated_ideal -- The unsaturated, homogeneous toppling ideal. version -- The version number of Sage Sandpiles. zero_config -- The all-zero configuration. zero_div -- The all-zero divisor.
- hilbert_function()¶
The Hilbert function of the homogeneous toppling ideal.
OUTPUT:
list of nonnegative integers
EXAMPLES:
sage: s = sandpiles.Wheel(5) sage: s.hilbert_function() [1, 5, 15, 31, 45] sage: s.h_vector() [1, 4, 10, 16, 14]
- ideal(gens=False)¶
The saturated homogeneous toppling ideal. If
gens
isTrue
, the generators for the ideal are returned instead.INPUT:
gens
– (default:False
) booleanOUTPUT:
ideal or, optionally, the generators of an ideal
EXAMPLES:
sage: S = sandpiles.Diamond() sage: S.ideal() Ideal (x2*x1 - x0^2, x3^2 - x0^2, x1^3 - x3*x2*x0, x3*x1^2 - x2^2*x0, x2^3 - x3*x1*x0, x3*x2^2 - x1^2*x0) of Multivariate Polynomial Ring in x3, x2, x1, x0 over Rational Field sage: S.ideal(True) [x2*x1 - x0^2, x3^2 - x0^2, x1^3 - x3*x2*x0, x3*x1^2 - x2^2*x0, x2^3 - x3*x1*x0, x3*x2^2 - x1^2*x0] sage: S.ideal().gens() # another way to get the generators [x2*x1 - x0^2, x3^2 - x0^2, x1^3 - x3*x2*x0, x3*x1^2 - x2^2*x0, x2^3 - x3*x1*x0, x3*x2^2 - x1^2*x0]
- identity(verbose=True)¶
The identity configuration. If
verbose
isFalse
, the configuration are converted to a list of integers.INPUT:
verbose
– (default:True
) booleanOUTPUT:
SandpileConfig or a list of integers If
verbose
isFalse
, the configuration are converted to a list of integers.EXAMPLES:
sage: s = sandpiles.Diamond() sage: s.identity() {1: 2, 2: 2, 3: 0} sage: s.identity(False) [2, 2, 0] sage: s.identity() & s.max_stable() == s.max_stable() True
- in_degree(v=None)¶
The in-degree of a vertex or a list of all in-degrees.
INPUT:
v
– (optional) vertex nameOUTPUT:
integer or dict
EXAMPLES:
sage: s = sandpiles.House() sage: s.in_degree() {0: 2, 1: 2, 2: 3, 3: 3, 4: 2} sage: s.in_degree(2) 3
- invariant_factors()¶
The invariant factors of the sandpile group.
OUTPUT:
list of integers
EXAMPLES:
sage: s = sandpiles.Grid(2,2) sage: s.invariant_factors() [1, 1, 8, 24]
- is_undirected()¶
Is the underlying graph undirected?
True
if \((u,v)\) is and edge if and only if \((v,u)\) is an edge, each edge with the same weight.OUTPUT:
boolean
EXAMPLES:
sage: sandpiles.Complete(4).is_undirected() True sage: s = Sandpile({0:[1,2], 1:[0,2], 2:[0]}, 0) sage: s.is_undirected() False
- jacobian_representatives(verbose=True)¶
Representatives for the elements of the Jacobian group. If
verbose
isFalse
, then lists representing the divisors are returned.INPUT:
verbose
– (default:True
) booleanOUTPUT:
list of SandpileDivisor (or of lists representing divisors)
EXAMPLES:
For an undirected graph, divisors of the form
s - deg(s)*sink
ass
varies over the superstables forms a distinct set of representatives for the Jacobian group.:sage: s = sandpiles.Complete(3) sage: s.superstables(False) [[0, 0], [0, 1], [1, 0]] sage: s.jacobian_representatives(False) [[0, 0, 0], [-1, 0, 1], [-1, 1, 0]]
If the graph is directed, the representatives described above may by equivalent modulo the rowspan of the Laplacian matrix:
sage: s = Sandpile({0: {1: 1, 2: 2}, 1: {0: 2, 2: 4}, 2: {0: 4, 1: 2}},0) sage: s.group_order() 28 sage: s.jacobian_representatives() [{0: -5, 1: 3, 2: 2}, {0: -4, 1: 3, 2: 1}]
Let \(\tau\) be the nonnegative generator of the kernel of the transpose of the Laplacian, and let \(tau_s\) be it sink component, then the sandpile group is isomorphic to the direct sum of the cyclic group of order \(\tau_s\) and the Jacobian group. In the example above, we have:
sage: s.laplacian().left_kernel() Free module of degree 3 and rank 1 over Integer Ring Echelon basis matrix: [14 5 8]
Note
The Jacobian group is the set of all divisors of degree zero modulo the integer rowspan of the Laplacian matrix.
- laplacian()¶
The Laplacian matrix of the graph. Its rows encode the vertex firing rules.
OUTPUT:
matrix
EXAMPLES:
sage: G = sandpiles.Diamond() sage: G.laplacian() [ 2 -1 -1 0] [-1 3 -1 -1] [-1 -1 3 -1] [ 0 -1 -1 2]
Warning
The function
laplacian_matrix
should be avoided. It returns the indegree version of the Laplacian.
- markov_chain(state, distrib=None)¶
The sandpile Markov chain for configurations or divisors. The chain starts at
state
. See NOTE for details.INPUT:
state
– SandpileConfig, SandpileDivisor, or list representing one of thesedistrib
– (optional) list of nonnegative numbers summing to 1 (representing a prob. dist.)
OUTPUT:
generator for Markov chain (see NOTE)
EXAMPLES:
sage: s = sandpiles.Complete(4) sage: m = s.markov_chain([0,0,0]) sage: next(m) # random {1: 0, 2: 0, 3: 0} sage: next(m).values() # random [0, 0, 0] sage: next(m).values() # random [0, 0, 0] sage: next(m).values() # random [0, 0, 0] sage: next(m).values() # random [0, 1, 0] sage: next(m).values() # random [0, 2, 0] sage: next(m).values() # random [0, 2, 1] sage: next(m).values() # random [1, 2, 1] sage: next(m).values() # random [2, 2, 1] sage: m = s.markov_chain(s.zero_div(), [0.1,0.1,0.1,0.7]) sage: next(m).values() # random [0, 0, 0, 1] sage: next(m).values() # random [0, 0, 1, 1] sage: next(m).values() # random [0, 0, 1, 2] sage: next(m).values() # random [1, 1, 2, 0] sage: next(m).values() # random [1, 1, 2, 1] sage: next(m).values() # random [1, 1, 2, 2] sage: next(m).values() # random [1, 1, 2, 3] sage: next(m).values() # random [1, 1, 2, 4] sage: next(m).values() # random [1, 1, 3, 4]
Note
The
closed sandpile Markov chain
has state space consisting of the configurations on a sandpile. It transitions from a state by choosing a vertex at random (according to the probability distributiondistrib
), dropping a grain of sand at that vertex, and stabilizing. If the chosen vertex is the sink, the chain stays at the current state.The
open sandpile Markov chain
has state space consisting of the recurrent elements, i.e., the state space is the sandpile group. It transitions from the configuration \(c\) by choosing a vertex \(v\) at random according todistrib
. The next state is the stabilization of \(c+v\). If \(v\) is the sink vertex, then the stabilization of \(c+v\) is defined to be \(c\).Note that in either case, if
distrib
is specified, its length is equal to the total number of vertices (including the sink).REFERENCES:
- max_stable()¶
The maximal stable configuration.
OUTPUT:
SandpileConfig (the maximal stable configuration)
EXAMPLES:
sage: S = sandpiles.House() sage: S.max_stable() {1: 1, 2: 2, 3: 2, 4: 1}
- max_stable_div()¶
The maximal stable divisor.
OUTPUT:
SandpileDivisor (the maximal stable divisor)
EXAMPLES:
sage: s = sandpiles.Diamond() sage: s.max_stable_div() {0: 1, 1: 2, 2: 2, 3: 1} sage: s.out_degree() {0: 2, 1: 3, 2: 3, 3: 2}
- max_superstables(verbose=True)¶
The maximal superstable configurations. If the underlying graph is undirected, these are the superstables of highest degree. If
verbose
isFalse
, the configurations are converted to lists of integers.INPUT:
verbose
– (default:True
) booleanOUTPUT:
tuple of SandpileConfig
EXAMPLES:
sage: s = sandpiles.Diamond() sage: s.superstables(False) [[0, 0, 0], [0, 0, 1], [1, 0, 1], [0, 2, 0], [2, 0, 0], [0, 1, 1], [1, 0, 0], [0, 1, 0]] sage: s.max_superstables(False) [[1, 0, 1], [0, 2, 0], [2, 0, 0], [0, 1, 1]] sage: s.h_vector() [1, 3, 4]
- min_recurrents(verbose=True)¶
The minimal recurrent elements. If the underlying graph is undirected, these are the recurrent elements of least degree. If
verbose
isFalse
, the configurations are converted to lists of integers.INPUT:
verbose
– (default:True
) booleanOUTPUT:
list of SandpileConfig
EXAMPLES:
sage: s = sandpiles.Diamond() sage: s.recurrents(False) [[2, 2, 1], [2, 2, 0], [1, 2, 0], [2, 0, 1], [0, 2, 1], [2, 1, 0], [1, 2, 1], [2, 1, 1]] sage: s.min_recurrents(False) [[1, 2, 0], [2, 0, 1], [0, 2, 1], [2, 1, 0]] sage: [i.deg() for i in s.recurrents()] [5, 4, 3, 3, 3, 3, 4, 4]
- nonsink_vertices()¶
The nonsink vertices.
OUTPUT:
list of vertices
EXAMPLES:
sage: s = sandpiles.Grid(2,3) sage: s.nonsink_vertices() [(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)]
- nonspecial_divisors(verbose=True)¶
The nonspecial divisors. Only for undirected graphs. (See NOTE.)
INPUT:
verbose
– (default:True
) booleanOUTPUT:
list (of divisors)
EXAMPLES:
sage: S = sandpiles.Complete(4) sage: ns = S.nonspecial_divisors() sage: D = ns[0] sage: D.values() [-1, 0, 1, 2] sage: D.deg() 2 sage: [i.effective_div() for i in ns] [[], [], [], [], [], []]
Note
The “nonspecial divisors” are those divisors of degree \(g-1\) with empty linear system. The term is only defined for undirected graphs. Here, \(g = |E| - |V| + 1\) is the genus of the graph (not counting loops as part of \(|E|\)). If
verbose
isFalse
, the divisors are converted to lists of integers.Warning
The underlying graph must be undirected.
- out_degree(v=None)¶
The out-degree of a vertex or a list of all out-degrees.
INPUT:
v
- (optional) vertex nameOUTPUT:
integer or dict
EXAMPLES:
sage: s = sandpiles.House() sage: s.out_degree() {0: 2, 1: 2, 2: 3, 3: 3, 4: 2} sage: s.out_degree(2) 3
- picard_representatives(d, verbose=True)¶
Representatives of the divisor classes of degree \(d\) in the Picard group. (Also see the documentation for
jacobian_representatives
.)INPUT:
d
– integerverbose
– (default:True
) boolean
OUTPUT:
list of SandpileDivisors (or lists representing divisors)
EXAMPLES:
sage: s = sandpiles.Complete(3) sage: s.superstables(False) [[0, 0], [0, 1], [1, 0]] sage: s.jacobian_representatives(False) [[0, 0, 0], [-1, 0, 1], [-1, 1, 0]] sage: s.picard_representatives(3,False) [[3, 0, 0], [2, 0, 1], [2, 1, 0]]
- points()¶
Generators for the multiplicative group of zeros of the sandpile ideal.
OUTPUT:
list of complex numbers
EXAMPLES:
The sandpile group in this example is cyclic, and hence there is a single generator for the group of solutions.
sage: S = sandpiles.Complete(4) sage: S.points() [[-I, I, 1], [-I, 1, I]]
- postulation()¶
The postulation number of the toppling ideal. This is the largest weight of a superstable configuration of the graph.
OUTPUT:
nonnegative integer
EXAMPLES:
sage: s = sandpiles.Complete(4) sage: s.postulation() 3
- recurrents(verbose=True)¶
The recurrent configurations. If
verbose
isFalse
, the configurations are converted to lists of integers.INPUT:
verbose
– (default:True
) booleanOUTPUT:
list of recurrent configurations
EXAMPLES:
sage: r = Sandpile(graphs.HouseXGraph(),0).recurrents() sage: r[:3] [{1: 2, 2: 3, 3: 3, 4: 1}, {1: 1, 2: 3, 3: 3, 4: 0}, {1: 1, 2: 3, 3: 3, 4: 1}] sage: sandpiles.Complete(4).recurrents(False) [[2, 2, 2], [2, 2, 1], [2, 1, 2], [1, 2, 2], [2, 2, 0], [2, 0, 2], [0, 2, 2], [2, 1, 1], [1, 2, 1], [1, 1, 2], [2, 1, 0], [2, 0, 1], [1, 2, 0], [1, 0, 2], [0, 2, 1], [0, 1, 2]] sage: sandpiles.Cycle(4).recurrents(False) [[1, 1, 1], [0, 1, 1], [1, 0, 1], [1, 1, 0]]
- reduced_laplacian()¶
The reduced Laplacian matrix of the graph.
OUTPUT:
matrix
EXAMPLES:
sage: S = sandpiles.Diamond() sage: S.laplacian() [ 2 -1 -1 0] [-1 3 -1 -1] [-1 -1 3 -1] [ 0 -1 -1 2] sage: S.reduced_laplacian() [ 3 -1 -1] [-1 3 -1] [-1 -1 2]
Note
This is the Laplacian matrix with the row and column indexed by the sink vertex removed.
- reorder_vertices()¶
A copy of the sandpile with vertex names permuted.
After reordering, vertex \(u\) comes before vertex \(v\) in the list of vertices if \(u\) is closer to the sink.
OUTPUT:
Sandpile
EXAMPLES:
sage: S = Sandpile({0:[1], 2:[0,1], 1:[2]}) sage: S.dict() {0: {1: 1}, 1: {2: 1}, 2: {0: 1, 1: 1}} sage: T = S.reorder_vertices()
The vertices 1 and 2 have been swapped:
sage: T.dict() {0: {1: 1}, 1: {0: 1, 2: 1}, 2: {0: 1}}
- resolution(verbose=False)¶
A minimal free resolution of the homogeneous toppling ideal. If
verbose
isTrue
, then all of the mappings are returned. Otherwise, the resolution is summarized.INPUT:
verbose
– (default:False
) booleanOUTPUT:
free resolution of the toppling ideal
EXAMPLES:
sage: S = Sandpile({0: {}, 1: {0: 1, 2: 1, 3: 4}, 2: {3: 5}, 3: {1: 1, 2: 1}},0) sage: S.resolution() # a Gorenstein sandpile graph 'R^1 <-- R^5 <-- R^5 <-- R^1' sage: S.resolution(True) [ [ x1^2 - x3*x0 x3*x1 - x2*x0 x3^2 - x2*x1 x2*x3 - x0^2 x2^2 - x1*x0], [ x3 x2 0 x0 0] [ x2^2 - x1*x0] [-x1 -x3 x2 0 -x0] [-x2*x3 + x0^2] [ x0 x1 0 x2 0] [-x3^2 + x2*x1] [ 0 0 -x1 -x3 x2] [x3*x1 - x2*x0] [ 0 0 x0 x1 -x3], [ x1^2 - x3*x0] ] sage: r = S.resolution(True) sage: r[0]*r[1] [0 0 0 0 0] sage: r[1]*r[2] [0] [0] [0] [0] [0]
- ring()¶
The ring containing the homogeneous toppling ideal.
OUTPUT:
ring
EXAMPLES:
sage: S = sandpiles.Diamond() sage: S.ring() Multivariate Polynomial Ring in x3, x2, x1, x0 over Rational Field sage: S.ring().gens() (x3, x2, x1, x0)
Note
The indeterminate
xi
corresponds to the \(i\)-th vertex as listed my the methodvertices
. The term-ordering is degrevlex with indeterminates ordered according to their distance from the sink (larger indeterminates are further from the sink).
- show(**kwds)¶
Draw the underlying graph.
INPUT:
kwds
– (optional) arguments passed to the show method for Graph or DiGraphEXAMPLES:
sage: S = Sandpile({0:[], 1:[0,3,4], 2:[0,3,5], 3:[2,5], 4:[1,1], 5:[2,4]}) sage: S.show() sage: S.show(graph_border=True, edge_labels=True)
- show3d(**kwds)¶
Draw the underlying graph.
INPUT:
kwds
– (optional) arguments passed to the show method for Graph or DiGraphEXAMPLES:
sage: S = sandpiles.House() sage: S.show3d() # long time
- sink()¶
The sink vertex.
OUTPUT:
sink vertex
EXAMPLES:
sage: G = sandpiles.House() sage: G.sink() 0 sage: H = sandpiles.Grid(2,2) sage: H.sink() (0, 0) sage: type(H.sink()) <... 'tuple'>
- smith_form()¶
The Smith normal form for the Laplacian. In detail: a list of integer matrices \(D, U, V\) such that \(ULV = D\) where \(L\) is the transpose of the Laplacian, \(D\) is diagonal, and \(U\) and \(V\) are invertible over the integers.
OUTPUT:
list of integer matrices
EXAMPLES:
sage: s = sandpiles.Complete(4) sage: D,U,V = s.smith_form() sage: D [1 0 0 0] [0 4 0 0] [0 0 4 0] [0 0 0 0] sage: U*s.laplacian()*V == D # Laplacian symmetric => transpose not necessary True
- solve()¶
Approximations of the complex affine zeros of the sandpile ideal.
OUTPUT:
list of complex numbers
EXAMPLES:
sage: S = Sandpile({0: {}, 1: {2: 2}, 2: {0: 4, 1: 1}}, 0) sage: S.solve() [[-0.707107000000000 + 0.707107000000000*I, 0.707107000000000 - 0.707107000000000*I], [-0.707107000000000 - 0.707107000000000*I, 0.707107000000000 + 0.707107000000000*I], [-I, -I], [I, I], [0.707107000000000 + 0.707107000000000*I, -0.707107000000000 - 0.707107000000000*I], [0.707107000000000 - 0.707107000000000*I, -0.707107000000000 + 0.707107000000000*I], [1, 1], [-1, -1]] sage: len(_) 8 sage: S.group_order() 8
Note
The solutions form a multiplicative group isomorphic to the sandpile group. Generators for this group are given exactly by
points()
.
- stable_configs(smax=None)¶
Generator for all stable configurations. If
smax
is provided, then the generator gives all stable configurations less than or equal tosmax
. Ifsmax
does not represent a stable configuration, then each component ofsmax
is replaced by the corresponding component of the maximal stable configuration.INPUT:
smax
– (optional) SandpileConfig or list representing a SandpileConfigOUTPUT:
generator for all stable configurations
EXAMPLES:
sage: s = sandpiles.Complete(3) sage: a = s.stable_configs() sage: next(a) {1: 0, 2: 0} sage: [i.values() for i in a] [[0, 1], [1, 0], [1, 1]] sage: b = s.stable_configs([1,0]) sage: list(b) [{1: 0, 2: 0}, {1: 1, 2: 0}]
- stationary_density()¶
The stationary density of the sandpile.
OUTPUT:
rational number
EXAMPLES:
sage: s = sandpiles.Complete(3) sage: s.stationary_density() 10/9 sage: s = Sandpile(digraphs.DeBruijn(2,2),'00') sage: s.stationary_density() 9/8
Note
The stationary density of a sandpile is the sum \(\sum_c (\deg(c) + \deg(s))\) where \(\deg(s)\) is the degree of the sink and the sum is over all recurrent configurations.
REFERENCES:
- superstables(verbose=True)¶
The superstable configurations. If
verbose
isFalse
, the configurations are converted to lists of integers. Superstables for undirected graphs are also known asG-parking functions
.INPUT:
verbose
– (default:True
) booleanOUTPUT:
list of SandpileConfig
EXAMPLES:
sage: sp = Sandpile(graphs.HouseXGraph(),0).superstables() sage: sp[:3] [{1: 0, 2: 0, 3: 0, 4: 0}, {1: 1, 2: 0, 3: 0, 4: 1}, {1: 1, 2: 0, 3: 0, 4: 0}] sage: sandpiles.Complete(4).superstables(False) [[0, 0, 0], [0, 0, 1], [0, 1, 0], [1, 0, 0], [0, 0, 2], [0, 2, 0], [2, 0, 0], [0, 1, 1], [1, 0, 1], [1, 1, 0], [0, 1, 2], [0, 2, 1], [1, 0, 2], [1, 2, 0], [2, 0, 1], [2, 1, 0]] sage: sandpiles.Cycle(4).superstables(False) [[0, 0, 0], [1, 0, 0], [0, 1, 0], [0, 0, 1]]
- symmetric_recurrents(orbits)¶
The symmetric recurrent configurations.
INPUT:
orbits
- list of lists partitioning the verticesOUTPUT:
list of recurrent configurations
EXAMPLES:
sage: S = Sandpile({0: {}, ....: 1: {0: 1, 2: 1, 3: 1}, ....: 2: {1: 1, 3: 1, 4: 1}, ....: 3: {1: 1, 2: 1, 4: 1}, ....: 4: {2: 1, 3: 1}}) sage: S.symmetric_recurrents([[1],[2,3],[4]]) [{1: 2, 2: 2, 3: 2, 4: 1}, {1: 2, 2: 2, 3: 2, 4: 0}] sage: S.recurrents() [{1: 2, 2: 2, 3: 2, 4: 1}, {1: 2, 2: 2, 3: 2, 4: 0}, {1: 2, 2: 1, 3: 2, 4: 0}, {1: 2, 2: 2, 3: 0, 4: 1}, {1: 2, 2: 0, 3: 2, 4: 1}, {1: 2, 2: 2, 3: 1, 4: 0}, {1: 2, 2: 1, 3: 2, 4: 1}, {1: 2, 2: 2, 3: 1, 4: 1}]
Note
The user is responsible for ensuring that the list of orbits comes from a group of symmetries of the underlying graph.
- tutte_polynomial()¶
The Tutte polynomial of the underlying graph. Only defined for undirected sandpile graphs.
OUTPUT:
polynomial
EXAMPLES:
sage: s = sandpiles.Complete(4) sage: s.tutte_polynomial() x^3 + y^3 + 3*x^2 + 4*x*y + 3*y^2 + 2*x + 2*y sage: s.tutte_polynomial().subs(x=1) y^3 + 3*y^2 + 6*y + 6 sage: s.tutte_polynomial().subs(x=1).coefficients() == s.h_vector() True
- unsaturated_ideal()¶
The unsaturated, homogeneous toppling ideal.
OUTPUT:
ideal
EXAMPLES:
sage: S = sandpiles.Diamond() sage: S.unsaturated_ideal().gens() [x1^3 - x3*x2*x0, x2^3 - x3*x1*x0, x3^2 - x2*x1] sage: S.ideal().gens() [x2*x1 - x0^2, x3^2 - x0^2, x1^3 - x3*x2*x0, x3*x1^2 - x2^2*x0, x2^3 - x3*x1*x0, x3*x2^2 - x1^2*x0]
- static version()¶
The version number of Sage Sandpiles.
OUTPUT:
string
EXAMPLES:
sage: Sandpile.version() Sage Sandpiles Version 2.4 sage: S = sandpiles.Complete(3) sage: S.version() Sage Sandpiles Version 2.4
- zero_config()¶
The all-zero configuration.
OUTPUT:
SandpileConfig
EXAMPLES:
sage: s = sandpiles.Diamond() sage: s.zero_config() {1: 0, 2: 0, 3: 0}
- zero_div()¶
The all-zero divisor.
OUTPUT:
SandpileDivisor
EXAMPLES:
sage: S = sandpiles.House() sage: S.zero_div() {0: 0, 1: 0, 2: 0, 3: 0, 4: 0}
- class sage.sandpiles.sandpile.SandpileConfig(S, c)¶
Bases:
dict
Class for configurations on a sandpile.
- add_random(distrib=None)¶
Add one grain of sand to a random vertex. Optionally, a probability distribution,
distrib
, may be placed on the vertices or the nonsink vertices. See NOTE for details.INPUT:
distrib
– (optional) list of nonnegative numbers summing to 1 (representing a prob. dist.)OUTPUT:
SandpileConfig
EXAMPLES:
sage: s = sandpiles.Complete(4) sage: c = s.zero_config() sage: c.add_random() # random {1: 0, 2: 1, 3: 0} sage: c {1: 0, 2: 0, 3: 0} sage: c.add_random([0.1,0.1,0.8]) # random {1: 0, 2: 0, 3: 1} sage: c.add_random([0.7,0.1,0.1,0.1]) # random {1: 0, 2: 0, 3: 0}
We compute the “sizes” of the avalanches caused by adding random grains of sand to the maximal stable configuration on a grid graph. The function
stabilize()
returns the firing vector of the stabilization, a dictionary whose values say how many times each vertex fires in the stabilization.:sage: S = sandpiles.Grid(10,10) sage: m = S.max_stable() sage: a = [] sage: for i in range(1000): ....: m = m.add_random() ....: m, f = m.stabilize(True) ....: a.append(sum(f.values())) sage: p = list_plot([[log(i+1),log(a.count(i))] for i in [0..max(a)] if a.count(i)]) sage: p.axes_labels(['log(N)','log(D(N))']) sage: t = text("Distribution of avalanche sizes", (2,2), rgbcolor=(1,0,0)) sage: show(p+t,axes_labels=['log(N)','log(D(N))']) # long time
Note
If
distrib
isNone
, then the probability is the uniform probability on the nonsink vertices. Otherwise, there are two possibilities:(i) the length of
distrib
is equal to the number of vertices, anddistrib
represents a probability distribution on all of the vertices. In that case, the sink may be chosen at random, in which case, the configuration is unchanged.(ii) Otherwise, the length of
distrib
must be equal to the number of nonsink vertices, anddistrib
represents a probability distribution on the nonsink vertices.Warning
If
distrib != None
, the user is responsible for assuring the sum of its entries is 1 and that its length is equal to the number of sink vertices or the number of nonsink vertices.
- burst_size(v)¶
The burst size of the configuration with respect to the given vertex.
INPUT:
v
– vertexOUTPUT:
integer
EXAMPLES:
sage: s = sandpiles.Diamond() sage: [i.burst_size(0) for i in s.recurrents()] [1, 1, 1, 1, 1, 1, 1, 1] sage: [i.burst_size(1) for i in s.recurrents()] [0, 0, 1, 2, 1, 2, 0, 2]
Note
To define
c.burst(v)
, if \(v\) is not the sink, let \(c'\) be the unique recurrent for which the stabilization of \(c' + v\) is \(c\). The burst size is then the amount of sand that goes into the sink during this stabilization. If \(v\) is the sink, the burst size is defined to be 1.REFERENCES:
- deg()¶
The degree of the configuration.
OUTPUT:
integer
EXAMPLES:
sage: S = sandpiles.Complete(3) sage: c = SandpileConfig(S, [1,2]) sage: c.deg() 3
- dualize()¶
The difference with the maximal stable configuration.
OUTPUT:
SandpileConfig
EXAMPLES:
sage: S = sandpiles.Cycle(3) sage: c = SandpileConfig(S, [1,2]) sage: S.max_stable() {1: 1, 2: 1} sage: c.dualize() {1: 0, 2: -1} sage: S.max_stable() - c == c.dualize() True
- equivalent_recurrent(with_firing_vector=False)¶
The recurrent configuration equivalent to the given configuration. Optionally, return the corresponding firing vector.
INPUT:
with_firing_vector
– (default:False
) booleanOUTPUT:
SandpileConfig or [SandpileConfig, firing_vector]
EXAMPLES:
sage: S = sandpiles.Diamond() sage: c = SandpileConfig(S, [0,0,0]) sage: c.equivalent_recurrent() == S.identity() True sage: x = c.equivalent_recurrent(True) sage: r = vector([x[0][v] for v in S.nonsink_vertices()]) sage: f = vector([x[1][v] for v in S.nonsink_vertices()]) sage: cv = vector(c.values()) sage: r == cv - f*S.reduced_laplacian() True
Note
Let \(L\) be the reduced Laplacian, \(c\) the initial configuration, \(r\) the returned configuration, and \(f\) the firing vector. Then \(r = c - f\cdot L\).
- equivalent_superstable(with_firing_vector=False)¶
The equivalent superstable configuration. Optionally, return the corresponding firing vector.
INPUT:
with_firing_vector
– (default:False
) booleanOUTPUT:
SandpileConfig or [SandpileConfig, firing_vector]
EXAMPLES:
sage: S = sandpiles.Diamond() sage: m = S.max_stable() sage: m.equivalent_superstable().is_superstable() True sage: x = m.equivalent_superstable(True) sage: s = vector(x[0].values()) sage: f = vector(x[1].values()) sage: mv = vector(m.values()) sage: s == mv - f*S.reduced_laplacian() True
Note
Let \(L\) be the reduced Laplacian, \(c\) the initial configuration, \(s\) the returned configuration, and \(f\) the firing vector. Then \(s = c - f\cdot L\).
- fire_script(sigma)¶
Fire the given script. In other words, fire each vertex the number of times indicated by
sigma
.INPUT:
sigma
– SandpileConfig or (list or dict representing a SandpileConfig)OUTPUT:
SandpileConfig
EXAMPLES:
sage: S = sandpiles.Cycle(4) sage: c = SandpileConfig(S, [1,2,3]) sage: c.unstable() [2, 3] sage: c.fire_script(SandpileConfig(S,[0,1,1])) {1: 2, 2: 1, 3: 2} sage: c.fire_script(SandpileConfig(S,[2,0,0])) == c.fire_vertex(1).fire_vertex(1) True
- fire_unstable()¶
Fire all unstable vertices.
OUTPUT:
SandpileConfig
EXAMPLES:
sage: S = sandpiles.Cycle(4) sage: c = SandpileConfig(S, [1,2,3]) sage: c.fire_unstable() {1: 2, 2: 1, 3: 2}
- fire_vertex(v)¶
Fire the given vertex.
INPUT:
v
– vertexOUTPUT:
SandpileConfig
EXAMPLES:
sage: S = sandpiles.Cycle(3) sage: c = SandpileConfig(S, [1,2]) sage: c.fire_vertex(2) {1: 2, 2: 0}
- static help(verbose=True)¶
List of SandpileConfig methods. If
verbose
, include short descriptions.INPUT:
verbose
– (default:True
) booleanOUTPUT:
printed string
EXAMPLES:
sage: SandpileConfig.help() Shortcuts for SandpileConfig operations: ~c -- stabilize c & d -- add and stabilize c * c -- add and find equivalent recurrent c^k -- add k times and find equivalent recurrent (taking inverse if k is negative) For detailed help with any method FOO listed below, enter "SandpileConfig.FOO?" or enter "c.FOO?" for any SandpileConfig c. add_random -- Add one grain of sand to a random vertex. burst_size -- The burst size of the configuration with respect to the given vertex. deg -- The degree of the configuration. dualize -- The difference with the maximal stable configuration. equivalent_recurrent -- The recurrent configuration equivalent to the given configuration. equivalent_superstable -- The equivalent superstable configuration. fire_script -- Fire the given script. fire_unstable -- Fire all unstable vertices. fire_vertex -- Fire the given vertex. help -- List of SandpileConfig methods. is_recurrent -- Is the configuration recurrent? is_stable -- Is the configuration stable? is_superstable -- Is the configuration superstable? is_symmetric -- Is the configuration symmetric? order -- The order of the equivalent recurrent element. sandpile -- The configuration's underlying sandpile. show -- Show the configuration. stabilize -- The stabilized configuration. support -- The vertices containing sand. unstable -- The unstable vertices. values -- The values of the configuration as a list.
- is_recurrent()¶
Is the configuration recurrent?
OUTPUT:
boolean
EXAMPLES:
sage: S = sandpiles.Diamond() sage: S.identity().is_recurrent() True sage: S.zero_config().is_recurrent() False
- is_stable()¶
Is the configuration stable?
OUTPUT:
boolean
EXAMPLES:
sage: S = sandpiles.Diamond() sage: S.max_stable().is_stable() True sage: (2*S.max_stable()).is_stable() False sage: (S.max_stable() & S.max_stable()).is_stable() True
- is_superstable()¶
Is the configuration superstable?
OUTPUT:
boolean
EXAMPLES:
sage: S = sandpiles.Diamond() sage: S.zero_config().is_superstable() True
- is_symmetric(orbits)¶
Is the configuration symmetric? Return
True
if the values of the configuration are constant over the vertices in each sublist oforbits
.INPUT:
orbits
– list of lists of verticesOUTPUT:
boolean
EXAMPLES:
sage: S = Sandpile({0: {}, ....: 1: {0: 1, 2: 1, 3: 1}, ....: 2: {1: 1, 3: 1, 4: 1}, ....: 3: {1: 1, 2: 1, 4: 1}, ....: 4: {2: 1, 3: 1}}) sage: c = SandpileConfig(S, [1, 2, 2, 3]) sage: c.is_symmetric([[2,3]]) True
- order()¶
The order of the equivalent recurrent element.
OUTPUT:
integer
EXAMPLES:
sage: S = sandpiles.Diamond() sage: c = SandpileConfig(S,[2,0,1]) sage: c.order() 4 sage: ~(c + c + c + c) == S.identity() True sage: c = SandpileConfig(S,[1,1,0]) sage: c.order() 1 sage: c.is_recurrent() False sage: c.equivalent_recurrent() == S.identity() True
- sandpile()¶
The configuration’s underlying sandpile.
OUTPUT:
Sandpile
EXAMPLES:
sage: S = sandpiles.Diamond() sage: c = S.identity() sage: c.sandpile() Diamond sandpile graph: 4 vertices, sink = 0 sage: c.sandpile() == S True
- show(sink=True, colors=True, heights=False, directed=None, **kwds)¶
Show the configuration.
INPUT:
sink
– (default:True
) whether to show the sinkcolors
– (default:True
) whether to color-code the amount of sand on each vertexheights
– (default:False
) whether to label each vertex with the amount of sanddirected
– (optional) whether to draw directed edgeskwds
– (optional) arguments passed to the show method for Graph
EXAMPLES:
sage: S = sandpiles.Diamond() sage: c = S.identity() sage: c.show() sage: c.show(directed=False) sage: c.show(sink=False,colors=False,heights=True)
- stabilize(with_firing_vector=False)¶
The stabilized configuration. Optionally returns the corresponding firing vector.
INPUT:
with_firing_vector
– (default:False
) booleanOUTPUT:
SandpileConfig
or[SandpileConfig, firing_vector]
EXAMPLES:
sage: S = sandpiles.House() sage: c = 2*S.max_stable() sage: c._set_stabilize() sage: '_stabilize' in c.__dict__ True sage: S = sandpiles.House() sage: c = S.max_stable() + S.identity() sage: c.stabilize(True) [{1: 1, 2: 2, 3: 2, 4: 1}, {1: 2, 2: 2, 3: 3, 4: 3}] sage: S.max_stable() & S.identity() == c.stabilize() True sage: ~c == c.stabilize() True
- support()¶
The vertices containing sand.
OUTPUT:
list - support of the configuration
EXAMPLES:
sage: S = sandpiles.Diamond() sage: c = S.identity() sage: c {1: 2, 2: 2, 3: 0} sage: c.support() [1, 2]
- unstable()¶
The unstable vertices.
OUTPUT:
list of vertices
EXAMPLES:
sage: S = sandpiles.Cycle(4) sage: c = SandpileConfig(S, [1,2,3]) sage: c.unstable() [2, 3]
- values()¶
The values of the configuration as a list.
The list is sorted in the order of the vertices.
OUTPUT:
list of integers
boolean
EXAMPLES:
sage: S = Sandpile({'a':['c','b'], 'b':['c','a'], 'c':['a']},'a') sage: c = SandpileConfig(S, {'b':1, 'c':2}) sage: c {'b': 1, 'c': 2} sage: c.values() [1, 2] sage: S.nonsink_vertices() ['b', 'c']
- class sage.sandpiles.sandpile.SandpileDivisor(S, D)¶
Bases:
dict
Class for divisors on a sandpile.
- Dcomplex()¶
The support-complex. (See NOTE.)
OUTPUT:
simplicial complex
EXAMPLES:
sage: S = sandpiles.House() sage: p = SandpileDivisor(S, [1,2,1,0,0]).Dcomplex() sage: p.homology() {0: 0, 1: Z x Z, 2: 0} sage: p.f_vector() [1, 5, 10, 4] sage: p.betti() {0: 1, 1: 2, 2: 0}
Note
The “support-complex” is the simplicial complex determined by the supports of the linearly equivalent effective divisors.
- add_random(distrib=None)¶
Add one grain of sand to a random vertex.
INPUT:
distrib
– (optional) list of nonnegative numbers representing a probability distribution on the verticesOUTPUT:
SandpileDivisor
EXAMPLES:
sage: s = sandpiles.Complete(4) sage: D = s.zero_div() sage: D.add_random() # random {0: 0, 1: 0, 2: 1, 3: 0} sage: D.add_random([0.1,0.1,0.1,0.7]) # random {0: 0, 1: 0, 2: 0, 3: 1}
Warning
If
distrib
is notNone
, the user is responsible for assuring the sum of its entries is 1.
- betti()¶
The Betti numbers for the support-complex. (See NOTE.)
OUTPUT:
dictionary of integers
EXAMPLES:
sage: S = sandpiles.Cycle(3) sage: D = SandpileDivisor(S, [2,0,1]) sage: D.betti() {0: 1, 1: 1}
Note
The “support-complex” is the simplicial complex determined by the supports of the linearly equivalent effective divisors.
- deg()¶
The degree of the divisor.
OUTPUT:
integer
EXAMPLES:
sage: S = sandpiles.Cycle(3) sage: D = SandpileDivisor(S, [1,2,3]) sage: D.deg() 6
- dualize()¶
The difference with the maximal stable divisor.
OUTPUT:
SandpileDivisor
EXAMPLES:
sage: S = sandpiles.Cycle(3) sage: D = SandpileDivisor(S, [1,2,3]) sage: D.dualize() {0: 0, 1: -1, 2: -2} sage: S.max_stable_div() - D == D.dualize() True
- effective_div(verbose=True, with_firing_vectors=False)¶
All linearly equivalent effective divisors. If
verbose
isFalse
, the divisors are converted to lists of integers. Ifwith_firing_vectors
isTrue
then a list of firing vectors is also given, each of which prescribes the vertices to be fired in order to obtain an effective divisor.INPUT:
verbose
– (default:True
) booleanwith_firing_vectors
– (default:False
) boolean
OUTPUT:
list (of divisors)
EXAMPLES:
sage: s = sandpiles.Complete(4) sage: D = SandpileDivisor(s,[4,2,0,0]) sage: sorted(D.effective_div(), key=str) [{0: 0, 1: 2, 2: 0, 3: 4}, {0: 0, 1: 2, 2: 4, 3: 0}, {0: 0, 1: 6, 2: 0, 3: 0}, {0: 1, 1: 3, 2: 1, 3: 1}, {0: 2, 1: 0, 2: 2, 3: 2}, {0: 4, 1: 2, 2: 0, 3: 0}] sage: sorted(D.effective_div(False)) [[0, 2, 0, 4], [0, 2, 4, 0], [0, 6, 0, 0], [1, 3, 1, 1], [2, 0, 2, 2], [4, 2, 0, 0]] sage: sorted(D.effective_div(with_firing_vectors=True), key=str) [({0: 0, 1: 2, 2: 0, 3: 4}, (0, -1, -1, -2)), ({0: 0, 1: 2, 2: 4, 3: 0}, (0, -1, -2, -1)), ({0: 0, 1: 6, 2: 0, 3: 0}, (0, -2, -1, -1)), ({0: 1, 1: 3, 2: 1, 3: 1}, (0, -1, -1, -1)), ({0: 2, 1: 0, 2: 2, 3: 2}, (0, 0, -1, -1)), ({0: 4, 1: 2, 2: 0, 3: 0}, (0, 0, 0, 0))] sage: a = _[2] sage: a[0].values() [0, 6, 0, 0] sage: vector(D.values()) - s.laplacian()*a[1] (0, 6, 0, 0) sage: sorted(D.effective_div(False, True)) [([0, 2, 0, 4], (0, -1, -1, -2)), ([0, 2, 4, 0], (0, -1, -2, -1)), ([0, 6, 0, 0], (0, -2, -1, -1)), ([1, 3, 1, 1], (0, -1, -1, -1)), ([2, 0, 2, 2], (0, 0, -1, -1)), ([4, 2, 0, 0], (0, 0, 0, 0))] sage: D = SandpileDivisor(s,[-1,0,0,0]) sage: D.effective_div(False,True) []
- fire_script(sigma)¶
Fire the given script. In other words, fire each vertex the number of times indicated by
sigma
.INPUT:
sigma
– SandpileDivisor or (list or dict representing a SandpileDivisor)OUTPUT:
SandpileDivisor
EXAMPLES:
sage: S = sandpiles.Cycle(3) sage: D = SandpileDivisor(S, [1,2,3]) sage: D.unstable() [1, 2] sage: D.fire_script([0,1,1]) {0: 3, 1: 1, 2: 2} sage: D.fire_script(SandpileDivisor(S,[2,0,0])) == D.fire_vertex(0).fire_vertex(0) True
- fire_unstable()¶
Fire all unstable vertices.
OUTPUT:
SandpileDivisor
EXAMPLES:
sage: S = sandpiles.Cycle(3) sage: D = SandpileDivisor(S, [1,2,3]) sage: D.fire_unstable() {0: 3, 1: 1, 2: 2}
- fire_vertex(v)¶
Fire the given vertex.
INPUT:
v
– vertexOUTPUT:
SandpileDivisor
EXAMPLES:
sage: S = sandpiles.Cycle(3) sage: D = SandpileDivisor(S, [1,2,3]) sage: D.fire_vertex(1) {0: 2, 1: 0, 2: 4}
- static help(verbose=True)¶
List of SandpileDivisor methods. If
verbose
, include short descriptions.INPUT:
verbose
– (default:True
) booleanOUTPUT:
printed string
EXAMPLES:
sage: SandpileDivisor.help() For detailed help with any method FOO listed below, enter "SandpileDivisor.FOO?" or enter "D.FOO?" for any SandpileDivisor D. Dcomplex -- The support-complex. add_random -- Add one grain of sand to a random vertex. betti -- The Betti numbers for the support-complex. deg -- The degree of the divisor. dualize -- The difference with the maximal stable divisor. effective_div -- All linearly equivalent effective divisors. fire_script -- Fire the given script. fire_unstable -- Fire all unstable vertices. fire_vertex -- Fire the given vertex. help -- List of SandpileDivisor methods. is_alive -- Is the divisor stabilizable? is_linearly_equivalent -- Is the given divisor linearly equivalent? is_q_reduced -- Is the divisor q-reduced? is_symmetric -- Is the divisor symmetric? is_weierstrass_pt -- Is the given vertex a Weierstrass point? polytope -- The polytope determining the complete linear system. polytope_integer_pts -- The integer points inside divisor's polytope. q_reduced -- The linearly equivalent q-reduced divisor. rank -- The rank of the divisor. sandpile -- The divisor's underlying sandpile. show -- Show the divisor. simulate_threshold -- The first unstabilizable divisor in the closed Markov chain. stabilize -- The stabilization of the divisor. support -- List of vertices at which the divisor is nonzero. unstable -- The unstable vertices. values -- The values of the divisor as a list. weierstrass_div -- The Weierstrass divisor. weierstrass_gap_seq -- The Weierstrass gap sequence at the given vertex. weierstrass_pts -- The Weierstrass points (vertices). weierstrass_rank_seq -- The Weierstrass rank sequence at the given vertex.
- is_alive(cycle=False)¶
Is the divisor stabilizable? In other words, will the divisor stabilize under repeated firings of all unstable vertices? Optionally returns the resulting cycle.
INPUT:
cycle
– (default:False
) booleanOUTPUT:
boolean or optionally, a list of SandpileDivisors
EXAMPLES:
sage: S = sandpiles.Complete(4) sage: D = SandpileDivisor(S, {0: 4, 1: 3, 2: 3, 3: 2}) sage: D.is_alive() True sage: D.is_alive(True) [{0: 4, 1: 3, 2: 3, 3: 2}, {0: 3, 1: 2, 2: 2, 3: 5}, {0: 1, 1: 4, 2: 4, 3: 3}]
- is_linearly_equivalent(D, with_firing_vector=False)¶
Is the given divisor linearly equivalent? Optionally, returns the firing vector. (See NOTE.)
INPUT:
D
– SandpileDivisor or list, tuple, etc. representing a divisorwith_firing_vector
– (default:False
) boolean
OUTPUT:
boolean or integer vector
EXAMPLES:
sage: s = sandpiles.Complete(3) sage: D = SandpileDivisor(s,[2,0,0]) sage: D.is_linearly_equivalent([0,1,1]) True sage: D.is_linearly_equivalent([0,1,1],True) (0, -1, -1) sage: v = vector(D.is_linearly_equivalent([0,1,1],True)) sage: vector(D.values()) - s.laplacian()*v (0, 1, 1) sage: D.is_linearly_equivalent([0,0,0]) False sage: D.is_linearly_equivalent([0,0,0],True) ()
Note
If
with_firing_vector
isFalse
, returns eitherTrue
orFalse
.If
with_firing_vector
isTrue
then: (i) ifself
is linearly equivalent to \(D\), returns a vector \(v\) such thatself - v*self.laplacian().transpose() = D
. Otherwise, (ii) ifself
is not linearly equivalent to \(D\), the output is the empty vector,()
.
- is_q_reduced()¶
Is the divisor \(q\)-reduced? This would mean that \(self = c + kq\) where \(c\) is superstable, \(k\) is an integer, and \(q\) is the sink vertex.
OUTPUT:
boolean
EXAMPLES:
sage: s = sandpiles.Complete(4) sage: D = SandpileDivisor(s,[2,-3,2,0]) sage: D.is_q_reduced() False sage: SandpileDivisor(s,[10,0,1,2]).is_q_reduced() True
For undirected or, more generally, Eulerian graphs, \(q\)-reduced divisors are linearly equivalent if and only if they are equal. The same does not hold for general directed graphs:
sage: s = Sandpile({0:[1],1:[1,1]}) sage: D = SandpileDivisor(s,[-1,1]) sage: Z = s.zero_div() sage: D.is_q_reduced() True sage: Z.is_q_reduced() True sage: D == Z False sage: D.is_linearly_equivalent(Z) True
- is_symmetric(orbits)¶
Is the divisor symmetric? Return
True
if the values of the configuration are constant over the vertices in each sublist oforbits
.INPUT:
orbits
– list of lists of verticesOUTPUT:
boolean
EXAMPLES:
sage: S = sandpiles.House() sage: S.dict() {0: {1: 1, 2: 1}, 1: {0: 1, 3: 1}, 2: {0: 1, 3: 1, 4: 1}, 3: {1: 1, 2: 1, 4: 1}, 4: {2: 1, 3: 1}} sage: D = SandpileDivisor(S, [0,0,1,1,3]) sage: D.is_symmetric([[2,3], [4]]) True
- is_weierstrass_pt(v='sink')¶
Is the given vertex a Weierstrass point?
INPUT:
v
– (default:sink
) vertexOUTPUT:
boolean
EXAMPLES:
sage: s = sandpiles.House() sage: K = s.canonical_divisor() sage: K.weierstrass_rank_seq() # sequence at the sink vertex, 0 (1, 0, -1) sage: K.is_weierstrass_pt() False sage: K.weierstrass_rank_seq(4) (1, 0, 0, -1) sage: K.is_weierstrass_pt(4) True
Note
The vertex \(v\) is a (generalized) Weierstrass point for divisor \(D\) if the sequence of ranks \(r(D - nv)\) for \(n = 0, 1, 2, \dots\) is not \(r(D), r(D)-1, \dots, 0, -1, -1, \dots\)
- polytope()¶
The polytope determining the complete linear system.
OUTPUT:
polytope
EXAMPLES:
sage: s = sandpiles.Complete(4) sage: D = SandpileDivisor(s,[4,2,0,0]) sage: p = D.polytope() sage: p.inequalities() (An inequality (-3, 1, 1) x + 2 >= 0, An inequality (1, 1, 1) x + 4 >= 0, An inequality (1, -3, 1) x + 0 >= 0, An inequality (1, 1, -3) x + 0 >= 0) sage: D = SandpileDivisor(s,[-1,0,0,0]) sage: D.polytope() The empty polyhedron in QQ^3
Note
For a divisor \(D\), this is the intersection of (i) the polyhedron determined by the system of inequalities \(L^t x \leq D\) where \(L^t\) is the transpose of the Laplacian with (ii) the hyperplane \(x_{\mathrm{sink\_vertex}} = 0\). The polytope is thought of as sitting in \((n-1)\)-dimensional Euclidean space where \(n\) is the number of vertices.
- polytope_integer_pts()¶
The integer points inside divisor’s polytope. The polytope referred to here is the one determining the divisor’s complete linear system (see the documentation for
polytope
).OUTPUT:
tuple of integer vectors
EXAMPLES:
sage: s = sandpiles.Complete(4) sage: D = SandpileDivisor(s,[4,2,0,0]) sage: sorted(D.polytope_integer_pts()) [(-2, -1, -1), (-1, -2, -1), (-1, -1, -2), (-1, -1, -1), (0, -1, -1), (0, 0, 0)] sage: D = SandpileDivisor(s,[-1,0,0,0]) sage: D.polytope_integer_pts() ()
- q_reduced(verbose=True)¶
The linearly equivalent \(q\)-reduced divisor.
INPUT:
verbose
– (default:True
) booleanOUTPUT:
SandpileDivisor or list representing SandpileDivisor
EXAMPLES:
sage: s = sandpiles.Complete(4) sage: D = SandpileDivisor(s,[2,-3,2,0]) sage: D.q_reduced() {0: -2, 1: 1, 2: 2, 3: 0} sage: D.q_reduced(False) [-2, 1, 2, 0]
Note
The divisor \(D\) is \(qreduced if \) where \(c\) is superstable, \(k\) is an integer, and \(q\) is the sink.
- rank(with_witness=False)¶
The rank of the divisor. Optionally returns an effective divisor \(E\) such that \(D - E\) is not winnable (has an empty complete linear system).
INPUT:
with_witness
– (default:False
) booleanOUTPUT:
integer or (integer, SandpileDivisor)
EXAMPLES:
sage: S = sandpiles.Complete(4) sage: D = SandpileDivisor(S,[4,2,0,0]) sage: D.rank() 3 sage: D.rank(True) (3, {0: 3, 1: 0, 2: 1, 3: 0}) sage: E = _[1] sage: (D - E).rank() -1 Riemann-Roch theorem:: sage: D.rank() - (S.canonical_divisor()-D).rank() == D.deg() + 1 - S.genus() True Riemann-Roch theorem:: sage: D.rank() - (S.canonical_divisor()-D).rank() == D.deg() + 1 - S.genus() True sage: S = Sandpile({0:[1,1,1,2],1:[0,0,0,1,1,1,2,2],2:[2,2,1,1,0]},0) # multigraph with loops sage: D = SandpileDivisor(S,[4,2,0]) sage: D.rank(True) (2, {0: 1, 1: 1, 2: 1}) sage: S = Sandpile({0:[1,2], 1:[0,2,2], 2: [0,1]},0) # directed graph sage: S.is_undirected() False sage: D = SandpileDivisor(S,[0,2,0]) sage: D.effective_div() [{0: 0, 1: 2, 2: 0}, {0: 2, 1: 0, 2: 0}] sage: D.rank(True) (0, {0: 0, 1: 0, 2: 1}) sage: E = D.rank(True)[1] sage: (D - E).effective_div() []
Note
The rank of a divisor \(D\) is -1 if \(D\) is not linearly equivalent to an effective divisor (i.e., the dollar game represented by \(D\) is unwinnable). Otherwise, the rank of \(D\) is the largest integer \(r\) such that \(D - E\) is linearly equivalent to an effective divisor for all effective divisors \(E\) with \(\deg(E) = r\).
- sandpile()¶
The divisor’s underlying sandpile.
OUTPUT:
Sandpile
EXAMPLES:
sage: S = sandpiles.Diamond() sage: D = SandpileDivisor(S,[1,-2,0,3]) sage: D.sandpile() Diamond sandpile graph: 4 vertices, sink = 0 sage: D.sandpile() == S True
- show(heights=True, directed=None, **kwds)¶
Show the divisor.
INPUT:
heights
– (default:True
) whether to label each vertex with the amount of sanddirected
– (optional) whether to draw directed edgeskwds
– (optional) arguments passed to the show method for Graph
EXAMPLES:
sage: S = sandpiles.Diamond() sage: D = SandpileDivisor(S,[1,-2,0,2]) sage: D.show(graph_border=True,vertex_size=700,directed=False)
- simulate_threshold(distrib=None)¶
The first unstabilizable divisor in the closed Markov chain. (See NOTE.)
INPUT:
distrib
– (optional) list of nonnegative numbers representing a probability distribution on the verticesOUTPUT:
SandpileDivisor
EXAMPLES:
sage: s = sandpiles.Complete(4) sage: D = s.zero_div() sage: D.simulate_threshold() # random {0: 2, 1: 3, 2: 1, 3: 2} sage: n(mean([D.simulate_threshold().deg() for _ in range(10)])) # random 7.10000000000000 sage: n(s.stationary_density()*s.num_verts()) 6.93750000000000
Note
Starting at
self
, repeatedly choose a vertex and add a grain of sand to it. Return the first unstabilizable divisor that is reached. Also see themarkov_chain
method for the underlying sandpile.
- stabilize(with_firing_vector=False)¶
The stabilization of the divisor. If not stabilizable, return an error.
INPUT:
with_firing_vector
– (default:False
) booleanEXAMPLES:
sage: s = sandpiles.Complete(4) sage: D = SandpileDivisor(s,[0,3,0,0]) sage: D.stabilize() {0: 1, 1: 0, 2: 1, 3: 1} sage: D.stabilize(with_firing_vector=True) [{0: 1, 1: 0, 2: 1, 3: 1}, {0: 0, 1: 1, 2: 0, 3: 0}]
- support()¶
List of vertices at which the divisor is nonzero.
OUTPUT:
list representing the support of the divisor
EXAMPLES:
sage: S = sandpiles.Cycle(4) sage: D = SandpileDivisor(S, [0,0,1,1]) sage: D.support() [2, 3] sage: S.vertices() [0, 1, 2, 3]
- unstable()¶
The unstable vertices.
OUTPUT:
list of vertices
EXAMPLES:
sage: S = sandpiles.Cycle(3) sage: D = SandpileDivisor(S, [1,2,3]) sage: D.unstable() [1, 2]
- values()¶
The values of the divisor as a list.
The list is sorted in the order of the vertices.
OUTPUT:
list of integers
boolean
EXAMPLES:
sage: S = Sandpile({'a':['c','b'], 'b':['c','a'], 'c':['a']},'a') sage: D = SandpileDivisor(S, {'a':0, 'b':1, 'c':2}) sage: D {'a': 0, 'b': 1, 'c': 2} sage: D.values() [0, 1, 2] sage: S.vertices() ['a', 'b', 'c']
- weierstrass_div(verbose=True)¶
The Weierstrass divisor. Its value at a vertex is the weight of that vertex as a Weierstrass point. (See
SandpileDivisor.weierstrass_gap_seq
.)INPUT:
verbose
– (default:True
) booleanOUTPUT:
SandpileDivisor
EXAMPLES:
sage: s = sandpiles.Diamond() sage: D = SandpileDivisor(s,[4,2,1,0]) sage: [D.weierstrass_rank_seq(v) for v in s] [(5, 4, 3, 2, 1, 0, 0, -1), (5, 4, 3, 2, 1, 0, -1), (5, 4, 3, 2, 1, 0, 0, 0, -1), (5, 4, 3, 2, 1, 0, 0, -1)] sage: D.weierstrass_div() {0: 1, 1: 0, 2: 2, 3: 1} sage: k5 = sandpiles.Complete(5) sage: K = k5.canonical_divisor() sage: K.weierstrass_div() {0: 9, 1: 9, 2: 9, 3: 9, 4: 9}
- weierstrass_gap_seq(v='sink', weight=True)¶
The Weierstrass gap sequence at the given vertex. If
weight
isTrue
, then also compute the weight of each gap value.INPUT:
v
– (default:sink
) vertexweight
– (default:True
) boolean
OUTPUT:
list or (list of list) of integers
EXAMPLES:
sage: s = sandpiles.Cycle(4) sage: D = SandpileDivisor(s,[2,0,0,0]) sage: [D.weierstrass_gap_seq(v,False) for v in s.vertices()] [(1, 3), (1, 2), (1, 3), (1, 2)] sage: [D.weierstrass_gap_seq(v) for v in s.vertices()] [((1, 3), 1), ((1, 2), 0), ((1, 3), 1), ((1, 2), 0)] sage: D.weierstrass_gap_seq() # gap sequence at sink vertex, 0 ((1, 3), 1) sage: D.weierstrass_rank_seq() # rank sequence at the sink vertex (1, 0, 0, -1)
Note
The integer \(k\) is a Weierstrass gap for the divisor \(D\) at vertex \(v\) if the rank of \(D - (k-1)v\) does not equal the rank of \(D - kv\). Let \(r\) be the rank of \(D\) and let \(k_i\) be the \(i\)-th gap at \(v\). The Weierstrass weight of \(v\) for \(D\) is the sum of \((k_i - i)\) as \(i\) ranges from \(1\) to \(r + 1\). It measure the difference between the sequence \(r, r - 1, ..., 0, -1, -1, ...\) and the rank sequence \(\mathrm{rank}(D), \mathrm{rank}(D - v), \mathrm{rank}(D - 2v), \dots\)
- weierstrass_pts(with_rank_seq=False)¶
The Weierstrass points (vertices). Optionally, return the corresponding rank sequences.
INPUT:
with_rank_seq
– (default:False
) booleanOUTPUT:
tuple of vertices or list of (vertex, rank sequence)
EXAMPLES:
sage: s = sandpiles.House() sage: K = s.canonical_divisor() sage: K.weierstrass_pts() (4,) sage: K.weierstrass_pts(True) [(4, (1, 0, 0, -1))]
Note
The vertex \(v\) is a (generalized) Weierstrass point for divisor \(D\) if the sequence of ranks \(r(D - nv)\) for \(n = 0, 1, 2, \dots\) is not \(r(D), r(D)-1, \dots, 0, -1, -1, \dots\)
- weierstrass_rank_seq(v='sink')¶
The Weierstrass rank sequence at the given vertex. Computes the rank of the divisor \(D - nv\) starting with \(n=0\) and ending when the rank is \(-1\).
INPUT:
v
– (default:sink
) vertexOUTPUT:
tuple of int
EXAMPLES:
sage: s = sandpiles.House() sage: K = s.canonical_divisor() sage: [K.weierstrass_rank_seq(v) for v in s.vertices()] [(1, 0, -1), (1, 0, -1), (1, 0, -1), (1, 0, -1), (1, 0, 0, -1)]
- sage.sandpiles.sandpile.admissible_partitions(S, k)¶
The partitions of the vertices of \(S\) into \(k\) parts, each of which is connected.
INPUT:
S
– Sandpilek
– integerOUTPUT:
list of partitions
EXAMPLES:
sage: from sage.sandpiles.sandpile import admissible_partitions sage: from sage.sandpiles.sandpile import partition_sandpile sage: S = sandpiles.Cycle(4) sage: P = [admissible_partitions(S, i) for i in [2,3,4]] sage: P [[{{0, 2, 3}, {1}}, {{0, 3}, {1, 2}}, {{0, 1, 3}, {2}}, {{0}, {1, 2, 3}}, {{0, 1}, {2, 3}}, {{0, 1, 2}, {3}}], [{{0, 3}, {1}, {2}}, {{0}, {1}, {2, 3}}, {{0}, {1, 2}, {3}}, {{0, 1}, {2}, {3}}], [{{0}, {1}, {2}, {3}}]] sage: for p in P: ....: sum([partition_sandpile(S, i).betti(verbose=False)[-1] for i in p]) 6 8 3 sage: S.betti() 0 1 2 3 ------------------------------ 0: 1 - - - 1: - 6 8 3 ------------------------------ total: 1 6 8 3
- sage.sandpiles.sandpile.aztec_sandpile(n)¶
The aztec diamond graph.
INPUT:
n
– integerOUTPUT:
dictionary for the aztec diamond graph
EXAMPLES:
sage: from sage.sandpiles.sandpile import aztec_sandpile sage: T = aztec_sandpile(2) sage: sorted(len(v) for u, v in T.items()) [3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 8] sage: Sandpile(T,(0, 0)).group_order() 4542720
Note
This is the aztec diamond graph with a sink vertex added. Boundary vertices have edges to the sink so that each vertex has degree 4.
- sage.sandpiles.sandpile.firing_graph(S, eff)¶
Creates a digraph with divisors as vertices and edges between two divisors \(D\) and \(E\) if firing a single vertex in \(D\) gives \(E\).
INPUT:
S
– Sandpileeff
– list of divisorsOUTPUT:
DiGraph
EXAMPLES:
sage: S = sandpiles.Cycle(6) sage: D = SandpileDivisor(S, [1,1,1,1,2,0]) sage: eff = D.effective_div() sage: firing_graph(S,eff).show3d(edge_size=.005,vertex_size=0.01) # long time
- sage.sandpiles.sandpile.glue_graphs(g, h, glue_g, glue_h)¶
Glue two graphs together.
INPUT:
g
,h
– dictionaries for directed multigraphsglue_h
,glue_g
– dictionaries for a vertex
OUTPUT:
dictionary for a directed multigraph
EXAMPLES:
sage: from sage.sandpiles.sandpile import glue_graphs sage: x = {0: {}, 1: {0: 1}, 2: {0: 1, 1: 1}, 3: {0: 1, 1: 1, 2: 1}} sage: y = {0: {}, 1: {0: 2}, 2: {1: 2}, 3: {0: 1, 2: 1}} sage: glue_x = {1: 1, 3: 2} sage: glue_y = {0: 1, 1: 2, 3: 1} sage: z = glue_graphs(x,y,glue_x,glue_y); z {'sink': {}, 'x0': {'sink': 1, 'x1': 1, 'x3': 2, 'y1': 2, 'y3': 1}, 'x1': {'x0': 1}, 'x2': {'x0': 1, 'x1': 1}, 'x3': {'x0': 1, 'x1': 1, 'x2': 1}, 'y1': {'sink': 2}, 'y2': {'y1': 2}, 'y3': {'sink': 1, 'y2': 1}} sage: S = Sandpile(z,'sink') sage: S.h_vector() [1, 6, 17, 31, 41, 41, 31, 17, 6, 1] sage: S.resolution() 'R^1 <-- R^7 <-- R^21 <-- R^35 <-- R^35 <-- R^21 <-- R^7 <-- R^1'
Note
This method makes a dictionary for a graph by combining those for \(g\) and \(h\). The sink of \(g\) is replaced by a vertex that is connected to the vertices of \(g\) as specified by
glue_g
the vertices of \(h\) as specified inglue_h
. The sink of the glued graph is'sink'
.Both
glue_g
andglue_h
are dictionaries with entries of the formv:w
wherev
is the vertex to be connected to andw
is the weight of the connecting edge.
- sage.sandpiles.sandpile.min_cycles(G, v)¶
Minimal length cycles in the digraph \(G\) starting at vertex \(v\).
INPUT:
G
– DiGraphv
– vertex ofG
OUTPUT:
list of lists of vertices
EXAMPLES:
sage: from sage.sandpiles.sandpile import min_cycles, sandlib sage: T = sandlib('gor') sage: [min_cycles(T, i) for i in T.vertices()] [[], [[1, 3]], [[2, 3, 1], [2, 3]], [[3, 1], [3, 2]]]
- sage.sandpiles.sandpile.parallel_firing_graph(S, eff)¶
Creates a digraph with divisors as vertices and edges between two divisors \(D\) and \(E\) if firing all unstable vertices in \(D\) gives \(E\).
INPUT:
S
– Sandpileeff
– list of divisorsOUTPUT:
DiGraph
EXAMPLES:
sage: S = sandpiles.Cycle(6) sage: D = SandpileDivisor(S, [1,1,1,1,2,0]) sage: eff = D.effective_div() sage: parallel_firing_graph(S,eff).show3d(edge_size=.005,vertex_size=0.01) # long time
- sage.sandpiles.sandpile.partition_sandpile(S, p)¶
Each set of vertices in \(p\) is regarded as a single vertex, with and edge between \(A\) and \(B\) if some element of \(A\) is connected by an edge to some element of \(B\) in \(S\).
INPUT:
S
– Sandpilep
– partition of the vertices ofS
OUTPUT:
Sandpile
EXAMPLES:
sage: from sage.sandpiles.sandpile import admissible_partitions, partition_sandpile sage: S = sandpiles.Cycle(4) sage: P = [admissible_partitions(S, i) for i in [2,3,4]] sage: for p in P: ....: sum([partition_sandpile(S, i).betti(verbose=False)[-1] for i in p]) 6 8 3 sage: S.betti() 0 1 2 3 ------------------------------ 0: 1 - - - 1: - 6 8 3 ------------------------------ total: 1 6 8 3
- sage.sandpiles.sandpile.random_DAG(num_verts, p=0.5, weight_max=1)¶
A random directed acyclic graph with
num_verts
vertices. The method starts with the sink vertex and adds vertices one at a time. Each vertex is connected only to only previously defined vertices, and the probability of each possible connection is given by the argumentp
. The weight of an edge is a random integer between1
andweight_max
.INPUT:
num_verts
– positive integerp
– (default: 0,5) real number such that \(0 < p \leq 1\)weight_max
– (default: 1) positive integer
OUTPUT:
a dictionary, encoding the edges of a directed acyclic graph with sink \(0\)
EXAMPLES:
sage: from sage.sandpiles.sandpile import random_DAG sage: d = DiGraph(random_DAG(5, .5)); d doctest:...: DeprecationWarning: method random_DAG is deprecated. Please use digraphs.RandomDirectedAcyclicGraph instead. See https://trac.sagemath.org/30479 for details. Digraph on 5 vertices
- sage.sandpiles.sandpile.sandlib(selector=None)¶
Returns the sandpile identified by
selector
. If no argument is given, a description of the sandpiles in the sandlib is printed.INPUT:
selector
– (optional) identifier or NoneOUTPUT:
sandpile or description
EXAMPLES:
sage: from sage.sandpiles.sandpile import sandlib sage: sandlib() Sandpiles in the sandlib: ci1 : complete intersection, non-DAG but equivalent to a DAG generic : generic digraph with 6 vertices genus2 : Undirected graph of genus 2 gor : Gorenstein but not a complete intersection kite : generic undirected graphs with 5 vertices riemann-roch1 : directed graph with postulation 9 and 3 maximal weight superstables riemann-roch2 : directed graph with a superstable not majorized by a maximal superstable sage: S = sandlib('gor') sage: S.resolution() 'R^1 <-- R^5 <-- R^5 <-- R^1'
- sage.sandpiles.sandpile.triangle_sandpile(n)¶
A triangular sandpile. Each nonsink vertex has out-degree six. The vertices on the boundary of the triangle are connected to the sink.
INPUT:
n
– integerOUTPUT:
Sandpile
EXAMPLES:
sage: from sage.sandpiles.sandpile import triangle_sandpile sage: T = triangle_sandpile(5) sage: T.group_order() 135418115000
- sage.sandpiles.sandpile.wilmes_algorithm(M)¶
Computes an integer matrix \(L\) with the same integer row span as \(M\) and such that \(L\) is the reduced Laplacian of a directed multigraph.
INPUT:
M
– square integer matrix of full rankOUTPUT:
integer matrix (
L
)EXAMPLES:
sage: P = matrix([[2,3,-7,-3],[5,2,-5,5],[8,2,5,4],[-5,-9,6,6]]) sage: wilmes_algorithm(P) [ 3279 -79 -1599 -1600] [ -1 1539 -136 -1402] [ 0 -1 1650 -1649] [ 0 0 -1658 1658]
REFERENCES: