Triangle example¶

Simple example of a filtered triangle is given in examples/triangle/triangle.cpp. Its equivalent in Python appears in examples/triangle/triangle.py, and we describe it next.

from dionysus import Simplex, Filtration, StaticPersistence, \
vertex_cmp, data_cmp, data_dim_cmp \

complex = [Simplex((0,),        0),                 # A
Simplex((1,),        1),                 # B
Simplex((2,),        2),                 # C
Simplex((0,1),       2.5),               # AB
Simplex((1,2),       2.9),               # BC
Simplex((0,2),       3.5),               # CA
Simplex((0,1,2),     5)]                 # ABC

print "Complex:", complex
print "Vertex: ", sorted(complex, vertex_cmp)
print "Data:   ", sorted(complex, data_cmp)
print "DataDim:", sorted(complex, data_dim_cmp)

f = Filtration(complex, data_cmp)
print "Complex in the filtration order:", ', '.join((str(s) for s in f))

p = StaticPersistence(f)
print "Persistence initialized"
p.pair_simplices(True)
print "Simplices paired"

smap = p.make_simplex_map(f)
for i in p:
print i.sign(), i.pair().sign()
print "%s (%d) - %s (%d)" % (smap[i], i.sign(), smap[i.pair()], i.pair().sign())
print "Cycle (%d):" % len(i.cycle), " + ".join((str(smap[ii]) for ii in i.cycle))

print "Number of unpaired simplices:", len([i for i in p if i.unpaired()])

After the necessary imports, the complex is setup explicitly as a list of simplices. Each Simplex constructor takes an iterable sequence of vertices, and optionally a data value.

A filtration f is initialized using the Filtration class, which takes a list of simplices (or anything iterable) and a comparison that defines in what order the simplices should come in the filtration. In this case we use data_cmp(), which simply compares simplices’ data attributes.

StaticPersistence is initialized with the filtration, and its method pair_simplices() pairs the simplices of the filtration:

p = StaticPersistence(f)
p.pair_simplices()

Subsequently, we iterate over p to access a representation of each simplex in the filtration order. We output each simplex, its sign, and its pair. The auxilliary smap = p.make_simplex_map(f) remaps the indices of StaticPersistence into the simplices in the filtration. Naturally, one could use this to access the data attribute of the simplices to output the actual persistence diagram, as is done in the Alpha shape example and the Rips complex example.