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prodcells.py
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prodcells.py
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# -*- coding: utf-8 -*-
"""
Created on Tue Apr 17 15:17:23 2018
@author: fajardogomez
"""
from sympy.combinatorics import Permutation
import networkx as nx
import numpy as np
import igraph as ig
import matplotlib.pyplot as plt
from scipy.sparse import dok_matrix
from scipy.sparse.linalg import aslinearoperator
from scipy.linalg.interpolative import estimate_rank
class PCELL():
LABEL_NAME = 'label'
G = None
def __init__(self, G=None):
self.set_graph(G)
# Note that cell.G can access nx methods.
def set_graph(self, G):
self.G = nx.DiGraph(G)
def draw(self, **kwargs):
fig, G = self.draw_directed_graph(self.G, **kwargs)
return fig, G
@classmethod
def draw_directed_graph(cls, G, mode='light', **kwargs):
"""
Draws the graph and saves it in .png, .svg and .eps formats
Parameters
----------
G : nx.DiGraph
Networkx generated directed graph.
mode : str, optional
Can be 'light' or 'dark' to match color themes. Light draws edges
and vertices in black, dark in white. The default is 'light'.
**kwargs :
layer_by : str, optional
Values include 'layer' and 'length.' Layer uses the reduction
process to determine position, while length uses DOW length,
regardless of how many deletions it takes to get there.
Returns
-------
None.
"""
kwargs.setdefault('layer_by','layer')
kwargs.setdefault('node_color','#0081D7' )
kwargs.setdefault('angle', 10)
kwargs.setdefault('node_size',60)
if mode=='light':
mdcolor = 'black'
face = 'white'
elif mode=='dark':
mdcolor = 'white'
face = '#0E1117'
nodes = dict(G.nodes(data=True))
if len(nodes) == 0:
layer_by = 'length'
if 'layer_by' in kwargs:
layer_by = kwargs['layer_by']
if 'filename' in kwargs:
filename = kwargs['filename']
if 'wordlen' in kwargs:
wordlen = kwargs['wordlen']
else:
wordlen = int(max([len(x.split(','))/2 for x in nodes]))
if 'node_color' in kwargs:
ncolor = kwargs['node_color']
if 'angle' in kwargs:
angle = kwargs['angle']
if 'node_size' in kwargs:
nsize = kwargs['node_size']
if layer_by == 'length':
root_len = int(max([len(n.split(','))/2 for n in nodes]))
for n in nodes:
nodes[n]['length'] = root_len - int(len(n.split(','))/2)
if layer_by == 'deletions':
layer_by = 'layer'
done = False
try:
layer_pairs = list()
layernums = [nodes[x][layer_by] for x in nodes]
i=min(layernums)
while not done:
counter=0
for n in nodes:
if int(nodes[n][layer_by]) == i:
counter +=1
if counter != 0:
layer_pairs.append((i,counter))
i+=1
if counter == 0:
done = True
num_layers = max([x for (x,y) in layer_pairs])
max_layer = max([y for (x,y) in layer_pairs])
# Estimate size of figure with length of word and number of layers
plt.figure(figsize=(2*max_layer + 0.2*wordlen, 2*num_layers))
pos = nx.multipartite_layout(G, subset_key=layer_by,
align = 'horizontal', scale=2)
except KeyError:
pos = nx.spring_layout(G)
# Shift vertex labels up
pos_higher = {}
y_off = -0.1 # offset on the y axis
if num_layers != None and num_layers < 5:
y_off -= 0.05
for k, v in pos.items():
pos_higher[k] = (v[0], v[1]+y_off)
pos_lower = {}
y_off = -0.1 # offset on the y axis
if num_layers != None and num_layers < 5:
y_off -= 0.05
for k, v in pos.items():
pos_lower[k] = (v[0], v[1]+y_off)
nx.draw_networkx_nodes(G, pos, node_color=ncolor, alpha =0.7, node_size=nsize)
nx.draw_networkx_edges(G, pos_higher, arrowstyle='->', arrowsize=10, node_size=900,
edge_cmap=plt.cm.Blues, width=2,edge_color=mdcolor, alpha = 0.7)
node_labels=dict()
# Labels the vertices
for node in G.nodes():
nodelbl = str(node)
if nodelbl == '':
nodelbl = '$\epsilon$'
node_labels[node]=nodelbl
text = nx.draw_networkx_labels(G, pos_higher, node_labels, font_size=10, font_color =mdcolor)
max_node_len = max([len(str(x)) for x in G.nodes()])
# Rotate node labels so they don't overlap
for _,t in text.items():
t.set_rotation(angle)
ax = plt.gca()
ax.invert_yaxis()
ax.set_axis_off()
ax.patch.set_facecolor(face)
plt.show()
fig = plt.gcf()
fig.patch.set_facecolor(face)
return fig, G
def change_layer(self, node, new_layer):
"""Change the layer node is in to new_layer."""
nodes = dict(self.G.nodes(data=True))
if node not in nodes:
print(str(node) + ' is not a node in the graph.')
else:
nodes[node]['layer'] = new_layer
@classmethod
def generate_simplex(cls, n):
ret = nx.DiGraph()
if n <= 0:
return ret
for i in range(0, n + 1):
for j in range(0, i):
ret.add_edge(str(j), str(i)) # order-orientation
return ret
@classmethod
def generate_w(cls, p):
"""
Returns the Cartesian product of simplices of dimensions as
determined by the partition p.
"""
simplices = list()
W = None
# Creates simplices of the needed dimensions
for i in p:
simplex = cls.generate_simplex(i)
simplices.append(simplex)
# Computes the Cartesian product W, a graph to search for in G
for j in simplices:
if W is None:
W = j
continue
W = nx.cartesian_product(W, j)
return W
def n_cells(self, n):
"""Returns a list of n-dimensional prodsimplicial cells."""
ret = list()
tmp = list()
used_vertices = dict()
# igraph version of G
iG = ig.Graph.from_networkx(self.G)
if n == 0:
nodes = list(self.G.nodes())
for node in nodes:
g = nx.DiGraph()
g.add_node(node)
tmp.append({
'graph':g,
'isom':{'0':str(node)},
'part':(0,),
'orientation':1,
'vertices': '_'.join(sorted([str(node)]))})
ret = sorted(tmp, key=lambda d: d['vertices'])
return ret
else:
# Creates a partition to compute all products of dimension n
for p in partition(n):
if p not in used_vertices:
used_vertices[p] = list()
W = self.generate_w(p)
# igraph version of W
iW = ig.Graph.from_networkx(W)
isomorphisms = iG.get_subisomorphisms_lad(iW, induced=True)
# Check for duplicate graphs with different isomorphisms
for ism in isomorphisms:
subg_nodes = [iG.vs[x]['_nx_name'] for x in ism]
W_nodes = [iW.vs[x]['_nx_name'] for x in list(range(len(ism)))]
isom_map = {W_nodes[i]: subg_nodes[i] for i in range(len(subg_nodes))}
S = self.G.subgraph(subg_nodes)
# Standard source node
st_sc_node = min(list(isom_map.keys()))
# Neighbors of source in standard cell, in descending order.
st_sc_nbrs = sorted(list(W.neighbors(st_sc_node)))
st_sc_nbrs.reverse()
# Cell source node
sc_node = isom_map[st_sc_node]
# If the standard cell translates to an even permutation of
# this, the cell is poitively oriented
sc_nbrs = sorted(list(S.neighbors(sc_node)))
# Translated neighbors of the source in the standard cell W
tr_sc_nbrs = [isom_map[nbr] for nbr in st_sc_nbrs]
# The permutation that maps sc_nbrs to tr_sc_nbrs
tr_perm = Permutation([sc_nbrs.index(x) for x in tr_sc_nbrs])
if set(subg_nodes)not in used_vertices[p]:
used_vertices[p].append(set(subg_nodes))
if tr_perm.parity() == 0:
orientation = 1
else:
orientation = -1
tmp.append(dict({"graph": S,
"isom": isom_map,
"part": p,
"orientation": orientation,
"vertices" : '_'.join(sorted(subg_nodes))
}))
ret = sorted(tmp, key=lambda d: d['vertices'])
return ret
@classmethod
def boundary_els(cls, p):
"""
Computes the boundary of the cell determined by partiton p.
Parameters
----------
p : tuple
A partition of a positive integer.
Returns
-------
ret : dict
List of dictionaries containing the cell and information about its
orientation (the exponent for (-1) in the formula).
"""
# Initializes the partition as a list
plist = list(p)
ret = list()
for m in range(0,len(plist)):
n = plist[m]
# Sn is a factor of W
Sn = cls.generate_simplex(n)
# Removes one node at a time from Sn and computes the induced
# subgraph
for i in range(0,n+1):
nbunch = list(Sn.nodes())
if len(nbunch) > 0:
nbunch.remove(str(i))
bd = Sn.subgraph(nbunch)
Wbd = nx.DiGraph()
# Sets the "prefix" factor
if m==0:
Wbd=bd
elif m>0:
Wpre = nx.DiGraph()
part = plist[:m]
Wpre = cls.generate_w(part)
Wbd = nx.cartesian_product(Wpre, bd)
a = sum(plist[:m]) + i
# Adds the "suffix" factors
part2 = plist[m+1:len(p)]
for j in part2:
Wbd = nx.cartesian_product(Wbd, cls.generate_simplex(j))
ret.append(dict({"bdry_el":Wbd, "multi": a}))
return ret
def facets(self, p, isom):
"""Returns a dictionary containing the facets of the cell defined by p"""
bd_els = self.boundary_els(p)
ret = list()
# For each face in the standard boundary set, map to the graph
for elem in bd_els:
new_isom = dict()
nodes = elem["bdry_el"].nodes()
# face_nodes = set()
for node in nodes:
# face_nodes.add(isom[node])
new_isom[node] = isom[node]
a = elem["multi"]
ret.append(dict({"multi": a, "isom": new_isom, "bdry_el": elem["bdry_el"]})) #"face":rface,
return ret
def boundary_op(self, n):
"""Outputs a sparse matrix corresponding to the boundary operator."""
# Source cells are the domain - n dimensional cells
source_cells = list()
Ncells = list(self.n_cells(n))
# Target cells are the range - (n-1) dimensional cells
target_cells = list()
# Keep track of used node sets and partitions
used_nodes = list()
used_partitions = list()
source_cell_or = dict()
target_cell_or = dict()
if n == 0:
d = len(Ncells)
S = dok_matrix((1,d), dtype=np.float64)
else:
nm1cells = list(self.n_cells(n-1))
for i in range(len(nm1cells)):
elem = nm1cells[i]
p = elem["part"]
vset = set(elem["isom"].values())
if p not in used_partitions:
used_partitions.append(p)
if vset not in used_nodes:
used_nodes.append(vset)
target_cell_or[(used_partitions.index(p), used_nodes.index(vset))] = elem["orientation"]
# Each target cell is uniquely identified by a partition and node set
target_cells.append((used_partitions.index(p), used_nodes.index(vset)))
for i in range(len(Ncells)):
elem = Ncells[i]
p = elem["part"]
vset = set(elem["isom"].values())
if p not in used_partitions:
used_partitions.append(p)
if vset not in used_nodes:
used_nodes.append(vset)
source_cell_or[(used_partitions.index(p), used_nodes.index(vset))] = elem["orientation"]
# Each source cell is uniquely identified by a partition and node set
source_cells.append((used_partitions.index(p), used_nodes.index(vset)))
target_cells.sort(key=lambda x:x[1])
source_cells.sort(key=lambda x:x[1])
# if range is empty (domain is vertices), make a row of of ones of length source_cells
if len(target_cells)==0:
d = len(source_cells)########
S = dok_matrix((1, d), dtype=np.float64)
S[0, 0:d] = 1
else:
source_cells_dict = {
source_cells[j]: j for j in range(len(source_cells))}
target_cells_dict = {
target_cells[i]: i for i in range(len(target_cells))}
d = len(source_cells)
f = len(target_cells)
S = dok_matrix((f, d), dtype=np.float64)
# For each cell, compute the standard boundary and translate it
for k in range(len(source_cells)):
s_cell = Ncells[k]
vset = set(s_cell["isom"].values())
part = s_cell["part"]
s_pair = (used_partitions.index(part), used_nodes.index(vset))
# Boundary elements of the given source cell
t_cell_aux = PCELL(s_cell["graph"]).facets(part, s_cell["isom"])
# For each face
for elem in t_cell_aux:
# Orientation of the face in the standard cell
# induced by the boundary operator
# a = sum of dimensions up to bdry factor
a = elem["multi"]
subg_nodes = list(elem["isom"].values())
subG = self.G.subgraph(subg_nodes)
# Standard cell
W = elem["bdry_el"]
isom_map = elem["isom"]
st_sc_node = min(list(isom_map.keys()))
st_sc_nbrs = sorted(list(W.neighbors(st_sc_node)))
# Standard cell vertices are ordered in decreasing
# lexicographic order
st_sc_nbrs.reverse()
sc_node = isom_map[st_sc_node]
sc_nbrs = sorted(list(subG.neighbors(sc_node)))
# Actual cell vertices are ordered in increasing
# lexicographic order
tr_sc_nbrs = [isom_map[nbr] for nbr in st_sc_nbrs]
# Permutation between st_sc_nbrs and tr_sc_nbrs
tr_perm = Permutation([sc_nbrs.index(x) for x in tr_sc_nbrs])
bd_vset = set(elem["isom"].values())
# Orientation of the target cell as a standard face
inherit_or = 1 if tr_perm.parity() == 0 else -1
for t_pair in target_cells:
# Find the target cell corresponding to the boundary element
if bd_vset == used_nodes[t_pair[1]]:
entry = -1 if a % 2==1 else 1
entry *= inherit_or
entry *= source_cell_or[s_pair]
i = target_cells_dict[t_pair]
j = source_cells_dict[s_pair]
S[i, j] = entry# S[i, j] = (-1)^a
return S
def betti_number(self, i, eps=None):
"""Computes the nth Betti number."""
boundop_ip1 = self.boundary_op(i+1)
boundop_i = self.boundary_op(i)
if i == 0:
boundop_i_rank = 0
else:
if eps is None:
try:
boundop_i_rank = np.linalg.matrix_rank(boundop_i.toarray())
except (np.linalg.LinAlgError, ValueError):
boundop_i_rank = boundop_i.shape[1]
else:
boundop_i_rank = estimate_rank(aslinearoperator(boundop_i), eps)
if eps is None:
try:
boundop_ip1_rank = np.linalg.matrix_rank(boundop_ip1.toarray())
except (np.linalg.LinAlgError, ValueError):
boundop_ip1_rank = boundop_ip1.shape[1]
else:
boundop_ip1_rank = estimate_rank(aslinearoperator(boundop_ip1), eps)
betti = (boundop_i.shape[1] - boundop_i_rank) - boundop_ip1_rank
return betti
def partition(n):
"""Returns a set of partitions of n, each as a tuple"""
ret = set()
ret.add((n, ))
for x in range(1, n):
for y in partition(n - x):
ret.add(tuple(sorted((x, ) + y)))
return ret