# 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

1. Eliminated dependence on 4ti2, substituting the use of Polyhedron methods. Thus, no optional packages are necessary.

2. Fixed bug in Sandpile.__init__ so that now multigraphs are handled correctly.

3. Created sandpiles to handle examples of Sandpiles in analogy with graphs, simplicial_complexes, and polytopes. In the process, we implemented a much faster way of producing the sandpile grid graph.

4. Added support for open and closed sandpile Markov chains.

5. Added support for Weierstrass points.

6. 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 to Sandpile.__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 the name 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, 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)

Class for Dhar’s abelian sandpile model.

all_k_config(k)

The constant configuration with all values set to $$k$$.

INPUT:

k – integer

OUTPUT:

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 – integer

OUTPUT:

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) boolean

OUTPUT:

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 is False, 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 is True, it prints the standard Betti table, otherwise, it returns a less formatted table.

INPUT:

verbose – (default: True) boolean

OUTPUT:

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 is False then the generators are represented as lists of integers.

INPUT:

verbose – (default: True) boolean

OUTPUT:

list of SandpileConfig (or of lists of integers if verbose is False)

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). If verbose, include short descriptions.

INPUT:

verbose – (default: True) boolean

OUTPUT:

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 is True, the generators for the ideal are returned instead.

INPUT:

gens – (default: False) boolean

OUTPUT:

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 is False, the configuration are converted to a list of integers.

INPUT:

verbose – (default: True) boolean

OUTPUT:

SandpileConfig or a list of integers If verbose is False, 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 name

OUTPUT:

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 is False, then lists representing the divisors are returned.

INPUT:

verbose – (default: True) boolean

OUTPUT:

list of SandpileDivisor (or of lists representing divisors)

EXAMPLES:

For an undirected graph, divisors of the form s - deg(s)*sink as s 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 these

• distrib – (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 distribution distrib), 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 to distrib. 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 is False, the configurations are converted to lists of integers.

INPUT:

verbose – (default: True) boolean

OUTPUT:

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 is False, the configurations are converted to lists of integers.

INPUT:

verbose – (default: True) boolean

OUTPUT:

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) boolean

OUTPUT:

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 is False, 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 name

OUTPUT:

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 – integer

• verbose – (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 is False, the configurations are converted to lists of integers.

INPUT:

verbose – (default: True) boolean

OUTPUT:

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 is True, then all of the mappings are returned. Otherwise, the resolution is summarized.

INPUT:

verbose – (default: False) boolean

OUTPUT:

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 method vertices. 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 DiGraph

EXAMPLES:

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 DiGraph

EXAMPLES:

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 to smax. If smax does not represent a stable configuration, then each component of smax is replaced by the corresponding component of the maximal stable configuration.

INPUT:

smax – (optional) SandpileConfig or list representing a SandpileConfig

OUTPUT:

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 is False, the configurations are converted to lists of integers. Superstables for undirected graphs are also known as G-parking functions.

INPUT:

verbose – (default: True) boolean

OUTPUT:

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 vertices

OUTPUT:

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 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()
{1: 0, 2: 1, 3: 0}
sage: c
{1: 0, 2: 0, 3: 0}
{1: 0, 2: 0, 3: 1}
{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, 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 is None, 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, and distrib 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, and distrib 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 – vertex

OUTPUT:

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) boolean

OUTPUT:

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) boolean

OUTPUT:

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 – vertex

OUTPUT:

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) boolean

OUTPUT:

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.

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 of orbits.

INPUT:

orbits – list of lists of vertices

OUTPUT:

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 sink

• colors – (default: True) whether to color-code the amount of sand on each vertex

• heights – (default: False) whether to label each vertex with the amount of sand

• directed – (optional) whether to draw directed edges

• kwds – (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) boolean

OUTPUT:

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 one grain of sand to a random vertex.

INPUT:

distrib – (optional) list of nonnegative numbers representing a probability distribution on the vertices

OUTPUT:

SandpileDivisor

EXAMPLES:

sage: s = sandpiles.Complete(4)
sage: D = s.zero_div()
{0: 0, 1: 0, 2: 1, 3: 0}
{0: 0, 1: 0, 2: 0, 3: 1}


Warning

If distrib is not None, 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 is False, the divisors are converted to lists of integers. If with_firing_vectors is True 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) boolean

• with_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 – vertex

OUTPUT:

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) boolean

OUTPUT:

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.
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) boolean

OUTPUT:

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 divisor

• with_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 is False, returns either True or False.

• If with_firing_vector is True then: (i) if self is linearly equivalent to $$D$$, returns a vector $$v$$ such that self - v*self.laplacian().transpose() = D. Otherwise, (ii) if self 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 of orbits.

INPUT:

orbits – list of lists of vertices

OUTPUT:

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) vertex

OUTPUT:

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) boolean

OUTPUT:

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) boolean

OUTPUT:

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 sand

• directed – (optional) whether to draw directed edges

• kwds – (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 vertices

OUTPUT:

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 the markov_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) boolean

EXAMPLES:

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) boolean

OUTPUT:

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 is True, then also compute the weight of each gap value.

INPUT:

• v – (default: sink) vertex

• weight – (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) boolean

OUTPUT:

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) vertex

OUTPUT:

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)]


The partitions of the vertices of $$S$$ into $$k$$ parts, each of which is connected.

INPUT:

S – Sandpile

k – integer

OUTPUT:

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 – integer

OUTPUT:

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 – Sandpile

eff – list of divisors

OUTPUT:

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 multigraphs

• glue_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 in glue_h. The sink of the glued graph is 'sink'.

Both glue_g and glue_h are dictionaries with entries of the form v:w where v is the vertex to be connected to and w 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 – DiGraph

• v – vertex of G

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 – Sandpile

eff – list of divisors

OUTPUT:

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 – Sandpile

p – partition of the vertices of S

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 argument p. The weight of an edge is a random integer between 1 and weight_max.

INPUT:

• num_verts – positive integer

• p – (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
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 None

OUTPUT:

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 – integer

OUTPUT:

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 rank

OUTPUT:

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: