Homology and Betti Numbers

We have been talking about the elements of topological data analysis. In my previous post, I introduced simplicial complexes, concerning the ways to connect points together. In topology, it is the shape and geometry, not distances, which matter ( although while constructing the distance does play a role).

With the simplicial complexes, we can go ahead to describe its topology. We will use the techniques in algebraic topology without going into too much details. The techniques involves homology, but a full explanation of it requires the concepts of normal subgroup, kernel, image, quotient group in group theory. I will not talk about them, although I admit that there is no easy ways to talk about computational topology without touching them. I highly recommend the readers can refer to Zomorodian’s textbook for more details. [Zomorodian 2009]

I will continue with the Python class

SimplicialComplex

that I wrote in the previous blog post. Suppose we have an k-simplex, then the n-th face is any combinations with n+1 vertices. A simplicial complex is such that a face contained in a face is also a face of the complex. In this, we can define the boundary operator by

\partial_k \sigma = \sum_i (-1)^i [v_0 v_1 \ldots \hat{v}_i \ldots v_k],

where \hat{v}_i indicates the i-th vertex be removed. This operator gives all the boundary faces of a face \sigma. The faces being operated are k-faces, and this operator will be mapped to a (k-1)-faces. Then the boundary operator can be seen as a (n_k \times n_{k-1})-matrix, where n_k is the number of k-faces. This can be easily calculated with the following method:

class SimplicialComplex:
  ...
  def boundary_operator(self, i):
    source_simplices = self.n_faces(i)
    target_simplices = self.n_faces(i-1)

    if len(target_simplices)==0:
      S = dok_matrix((1, len(source_simplices)), dtype=np.float32)
      S[0, 0:len(source_simplices)] = 1
    else:
      source_simplices_dict = {}
      for j in range(len(source_simplices)):
        source_simplices_dict[source_simplices[j]] = j
      target_simplices_dict = {}
      for i in range(len(target_simplices)):
        target_simplices_dict[target_simplices[i]] = i

      S = dok_matrix((len(target_simplices), len(source_simplices)), dtype=np.float32)
      for source_simplex in source_simplices:
        for a in range(len(source_simplex)):
          target_simplex = source_simplex[:a]+source_simplex[(a+1):]
          i = target_simplices_dict[target_simplex]
          j = source_simplices_dict[source_simplex]
          S[i, j] = -1 if a % 2==1 else 1 # S[i, j] = (-1)**a
   return S

With the boundary operator, we can calculate the Betti numbers that characterize uniquely the topology of the shapes. Actually it involves the concept of homology groups that we are going to omit. To calculate the k-th Betti numbers, we calculate:

\beta_k = \text{rank} (\text{ker} \partial_k) - \text{rank} (\text{Im} \partial_{k+1}).

By rank-nullity theorem, [Jackson]

\text{rank} (\text{ker} \partial_k) +\text{rank} (\text{Im} \partial_k) = \text{dim} (\partial_k)

the Betti number is then

\beta_k = \text{dim} (\partial_k) - \text{rank}(\text{Im} \partial_k)) - \text{rank} (\text{Im} \partial_{k+1})

where the rank of the image of an operator can be easily computed using the rank method available in numpy. Then the method of calculating the Betti number is

class SimplicialComplex:
  ...
  def betti_number(self, i):
    boundop_i = self.boundary_operator(i)
    boundop_ip1 = self.boundary_operator(i+1)

    if i==0:
      boundop_i_rank = 0
    else:
      try:
        boundop_i_rank = np.linalg.matrix_rank(boundop_i.toarray())
      except np.linalg.LinAlgError:
        boundop_i_rank = boundop_i.shape[1]
    try:
      boundop_ip1_rank = np.linalg.matrix_rank(boundop_ip1.toarray())
    except np.linalg.LinAlgError:
      boundop_ip1_rank = boundop_ip1.shape[1]

    return ((boundop_i.shape[1]-boundop_i_rank)-boundop_ip1_rank)

If we draw a simplicial complex on a 2-dimensional plane, we almost have \beta_0, \beta_1 and \beta_2. $\beta_0$ indicates the number of components, \beta_1 the number of bases for a tunnel, and \beta_2 the number of voids.

Let’s have some examples. Suppose we have a triangle, not filled.

e1 = [(0, 1), (1, 2), (2, 0)]
sc1 = SimplicialComplex(e1)

Then the Betti numbers are:


In [5]: sc1.betti_number(0)
Out[5]: 1

In [6]: sc1.betti_number(1)
Out[6]: 1

In [7]: sc1.betti_number(2)
Out[7]: 0

Let’s try another example with multiple components.

e2 = [(1,2), (2,3), (3,1), (4,5,6), (6,7), (7,4)]
sc2 = SimplicialComplex(e2)

We can graphically represent it using networkx:

import networkx as nx
import matplotlib.pyplot as plt
n2 = nx.Graph()
n2.add_edges_from(sc2.n_faces(1))
nx.draw(n2)
plt.show()
Simplicial Complex of e2
Simplicial Complex of e2

And its Betti numbers are as follow:


In [13]: sc2.betti_number(0)
Out[13]: 2

In [14]: sc2.betti_number(1)
Out[14]: 2

In [15]: sc2.betti_number(2)
Out[15]: 0

A better illustration is the Wolfram Demonstration, titled “Simplicial Homology of the Alpha Complex”.

On top of the techniques in this current post, we can describe the homology of discrete points using persistent homology, which I will describe in my future posts. I will probably spend a post on homotopy in comparison to other types of quantitative problems.

Continue reading “Homology and Betti Numbers”

Constructing Connectivities

In my previous blog post, I introduced the newly emerged topological data analysis (TDA). Unlike most of the other data analytic algorithms, TDA, concerning the topology as its name tells, cares for the connectivity of points, instead of the distance (according to a metric, whether it is Euclidean, Manhattan, Minkowski or any other). What is the best tools to describe topology?

Physicists use a lot homotopy. But for the sake of computation, it is better to use a scheme that are suited for discrete computation. It turns out that there are useful tools in algebraic topology: homology. But to understand homology, we need to understand what a simplicial complex is.

Gunnar Carlsson [Carlsson 2009] and Afra Zomorodian [Zomorodian 2011] wrote good reviews about them, although from a different path in introducing the concept. I first followed Zomorodian’s review [Zomorodian 2011], then his book [Zomorodian 2009] that filled in a lot of missing links in his review, to a certain point. I recently started reading Carlsson’s review.

One must first understand what a simplicial complex is. Without giving too much technical details, simplicial complex is basically a shape connecting points together. A line is a 1-simplex, connecting two points. A triangle is a 2-simplex. A tetrahedron is a 3-complex. There are other more complicated and unnamed complexes. Any subsets of a simplicial complex are faces. For example, the sides of the triangle are faces. The faces and the sides are the faces of the tetrahedron. (Refer to Wolfram MathWorld for more details. There are a lot of good tutorials online.)

Implementing Simplicial Complex

We can easily encoded this into a python code. I wrote a class SimplicialComplex in Python to implement this. We first import necessary libraries:

import numpy as np
from itertools import combinations
from scipy.sparse import dok_matrix
from operator import add

The first line imports the numpy library, the second the iteration tools necessary for extracting the faces for simplicial complex, the third the sparse matrix implementation in the scipy library (applied on something that I will not go over in this blog entry), and the fourth for some reduce operation.

We want to describe the simplicial complexes in the order of some labels (which can be anything, such as integers or strings). If it is a point, then it can be represented as tuples, as below:

 (1,) 

Or if it is a line (a 1-simplex), then

 (1, 2) 

Or a 2-simplex as a triangle, then

 (1, 2, 3) 

I think you get the gist. The integers 1, 2, or 3 here are simply labels. We can easily store this in the class:

class SimplicialComplex:
  def __init__(self, simplices=[]):
    self.import_simplices(simplices=simplices)

  def import_simplices(self, simplices=[]):
    self.simplices = map(lambda simplex: tuple(sorted(simplex)), simplices)
    self.face_set = self.faces()

You might observe the last line of the codes above. And it is for calculating all the faces of this complex, and it is implemented in this way:

  def faces(self):
    faceset = set()
    for simplex in self.simplices:
      numnodes = len(simplex)
      for r in range(numnodes, 0, -1):
        for face in combinations(simplex, r):
          faceset.add(face)
    return faceset

The faces are intuitively sides of a 2D shape (2-simplex), or planes of a 3D shape (3-simplex). But the faces of a 3-simplex includes the faces of all its faces. All the faces are saved in a field called faceset. If the user wants to retrieve the faces in a particular dimension, they can call this method:

  def n_faces(self, n):
    return filter(lambda face: len(face)==n+1, self.face_set)

There are other methods that I am not going over in this blog entry. Now let us demonstrate how to use the class by implementing a tetrahedron.

sc = SimplicialComplex([('a', 'b', 'c', 'd')])

If we want to extract the faces, then enter:

sc.faces()

which outputs:

{('a',),
 ('a', 'b'),
 ('a', 'b', 'c'),
 ('a', 'b', 'c', 'd'),
 ('a', 'b', 'd'),
 ('a', 'c'),
 ('a', 'c', 'd'),
 ('a', 'd'),
 ('b',),
 ('b', 'c'),
 ('b', 'c', 'd'),
 ('b', 'd'),
 ('c',),
 ('c', 'd'),
 ('d',)}

We have gone over the basis of simplicial complex, which is the foundation of TDA. We appreciate that the simplicial complex deals only with the connectivity of points instead of the distances between the points. And the homology groups will be calculated based on this. However, how do we obtain the simplicial complex from the discrete data we have? Zomorodian’s review [Zomorodian 2011] gave a number of examples, but I will only go through two of them only. And from this, you can see that to establish the connectivity between points, we still need to apply some sort of distance metrics.

Alpha Complex

An alpha complex is the nerve of the cover of the restricted Voronoi regions. (Refer the details to Zomorodian’s review [Zomorodian 2011], this Wolfram MathWorld entry, or this Wolfram Demonstration.) We can extend the class SimplicialComplex to get a class AlphaComplex:

from scipy.spatial import Delaunay, distance
from operator import or_
from functools import partial

def facesiter(simplex):
  for i in range(len(simplex)):
    yield simplex[:i]+simplex[(i+1):]

def flattening_simplex(simplices):
  for simplex in simplices:
    for point in simplex:
      yield point

def get_allpoints(simplices):
  return set(flattening_simplex(simplices))

def contain_detachededges(simplex, distdict, epsilon):
  if len(simplex)==2:
    return (distdict[simplex[0], simplex[1]] > 2*epsilon)
  else:
    return reduce(or_, map(partial(contain_detachededges, distdict=distdict, epsilon=epsilon), facesiter(simplex)))

class AlphaComplex(SimplicialComplex):
  def __init__(self, points, epsilon, labels=None, distfcn=distance.euclidean):
    self.pts = points
    self.labels = range(len(self.pts)) if labels==None or len(labels)!=len(self.pts) else labels
    self.epsilon = epsilon
    self.distfcn = distfcn
    self.import_simplices(self.construct_simplices(self.pts, self.labels, self.epsilon, self.distfcn))

  def calculate_distmatrix(self, points, labels, distfcn):
    distdict = {}
    for i in range(len(labels)):
      for j in range(len(labels)):
        distdict[(labels[i], labels[j])] = distfcn(points[i], points[j])
    return distdict

  def construct_simplices(self, points, labels, epsilon, distfcn):
    delaunay = Delaunay(points)
    delaunay_simplices = map(tuple, delaunay.simplices)
    distdict = self.calculate_distmatrix(points, labels, distfcn)

    simplices = []
    for simplex in delaunay_simplices:
      faces = list(facesiter(simplex))
      detached = map(partial(contain_detachededges, distdict=distdict, epsilon=epsilon), faces)
      if reduce(or_, detached):
        if len(simplex)>2:
          for face, notkeep in zip(faces, detached):
            if not notkeep:
              simplices.append(face)
      else:
        simplices.append(simplex)
    simplices = map(lambda simplex: tuple(sorted(simplex)), simplices)
    simplices = list(set(simplices))

    allpts = get_allpoints(simplices)
    for point in (set(labels)-allpts):
      simplices += [(point,)]

    return simplices

The scipy package already has a package to calculate Delaunay region. The function contain_detachededges is for constructing the restricted Voronoi region from the calculated Delaunay region.

This class demonstrates how an Alpha Complex is constructed, but this runs slowly once the number of points gets big!

Vietoris-Rips (VR) Complex

Another commonly used complex is called the Vietoris-Rips (VR) Complex, which connects points as the edge of a graph if they are close enough. (Refer to Zomorodian’s review [Zomorodian 2011] or this Wikipedia page for details.) To implement this, import the famous networkx originally designed for network analysis.

import networkx as nx
from scipy.spatial import distance
from itertools import product

class VietorisRipsComplex(SimplicialComplex):
  def __init__(self, points, epsilon, labels=None, distfcn=distance.euclidean):
    self.pts = points
    self.labels = range(len(self.pts)) if labels==None or len(labels)!=len(self.pts) else labels
    self.epsilon = epsilon
    self.distfcn = distfcn
    self.network = self.construct_network(self.pts, self.labels, self.epsilon, self.distfcn)
    self.import_simplices(map(tuple, list(nx.find_cliques(self.network))))

  def construct_network(self, points, labels, epsilon, distfcn):
    g = nx.Graph()
    g.add_nodes_from(labels)
    zips = zip(points, labels)
    for pair in product(zips, zips):
      if pair[0][1]!=pair[1][1]:
        dist = distfcn(pair[0][0], pair[1][0])
        if dist<epsilon:
          g.add_edge(pair[0][1], pair[1][1])
    return g

The intuitiveness and efficiencies are the reasons that VR complexes are widely used.

For more details about the Alpha Complexes, VR Complexes and the related Čech Complexes, refer to this page.

More…

There are other commonly used complexes used, including Witness Complex, Cubical Complex etc., which I leave no introductions. Upon building the complexes, we can analyze the topology by calculating their homology groups, Betti numbers, the persistent homology etc. I wish to write more about it soon.

Taken from Wolfram Mathworld
Taken from Wolfram Mathworld

Continue reading “Constructing Connectivities”

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