""" .. _dowker_rbs_homology_gallery: Restricted Barycentric Subdivision (Vietoris-Rips) =============================================================== The Restricted Barycentric Subdivision Homology of a hypergraph is the homology of the clique complex of the simple graph :math:`G` such that - the vertices of :math:`G` are the hyperedges of the hypergraph - the edges of :math:`G` are the pairs :math:`\{e,h\}` such that :math:`e \subseteq h` or :math:`h \subseteq e` OAT doesn't offer a specialized function to compute this homology, but it does allow one to compute the *persistent homology* of any weighted graph. So, as a work around, we assign weight 0 to every edge of `G` and compute its persistent homology. This notebook shows how to implement this strategy. """ # %% import oat_python as oat import copy import plotly.graph_objects as go import numpy as np import networkx as nx import hypernetx as hnx import sklearn # %% [markdown] # Define a hypergraph # ----------------------------------------- # %% # The hypergraph has edges labeled "A", "B", "C", "D", with the following vertices: E = { "A": ["x"], "B": ["y"], "C": ["x","y","z",], "D": ["x","y","w",], } # %% [markdown] # Plot the hypergraph using HyperNetX # ----------------------------------------- # %% hnxgraph = hnx.Hypergraph(E) hnx.drawing.draw(hnxgraph) # %% [markdown] # Compute homology # ----------------------------------------- # %% # convert to a list of edges G = list( E.values() ) # graph representing the edge containment poset (forgetting direction) containment = oat.hypergraph.edge_containment_graph_symmetrized( G ) # graph whose edges form the set complement of the containment graph anticontainment = nx.complement( containment ) # adjacency matrix of the anticontainment graph anti_adjacency = nx.adjacency_matrix( anticontainment ).todense() dissimilarity_matrix = oat.dissimilarity.sparse_matrix_for_dense( dissimilarity_matrix = anti_adjacency, max_dissimilarity = 0.5 ) # decomposition boundary matrix decomposition = oat.core.vietoris_rips.BoundaryMatrixDecomposition( dissimilarity_matrix = dissimilarity_matrix, max_homology_dimension = 1, ) # %% # The persistent homology dataframe contains the persistent homology of the Vietoris-Rips complex. # Because every edge in the underlying graph has weight 0, this is the same as the RBS homology of the hypergraph. # The dataframe contains a cycle basis for the RBS homology, in which each cycle technically # represets a homology class that is born at filtration parameter 0 and never dies. persistent_homology = decomposition.persistent_homology_dataframe( return_cycle_representatives = True, return_bounding_chains = True, ) persistent_homology # %% [markdown] # Inspect a cycle representative # ----------------------------------------- # %% cycle = persistent_homology["cycle_representative"][1] cycle # %% [markdown] # Relabel each vertex # ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ # # Recall that a vertex in the Vietoris-Rips complex corresponds to a hyperedge in the initial hypergraph. # These hyperedges have labels "A", "B", "C", "D", so lets relabel the vertices in the cycle representative accordingly. # %% [markdown] # Recall that each vertex in RBS homology represents an edge in the reduced hypergraph (which corresponds to a set of hyperedges in the initial hypergraph). Here we relabel each vertex with **one** of the edges that maps to it. # %% original_edge_labels= { p: k for p,k in enumerate(E.keys()) } cycle = copy.deepcopy(persistent_homology["cycle_representative"][1]) cycle["simplex"] = [ [original_edge_labels[x] for x in simplex] for simplex in cycle["simplex"] ] cycle # %% [markdown] # Plot a cycle representative # ----------------------------------------- # # %% edges = persistent_homology["cycle_representative"][1]["simplex"] # the edges in the cycle points = oat.plot.vertex_embedding_for_simplices( simplices = edges.tolist(), dimension = 3, ) fig = oat.plot.fig_3d_for_simplices( edges, points=points, kwargs_points = dict( mode="markers+text", text= [ val for val in original_edge_labels.values() ], marker_size=8, textfont_size=18, ), ) fig.update_layout(template="plotly_dark") # %% [markdown] # Suspend and repeat # --------------------- # %% F = { "A": ["x"], "B": ["y"], "C": ["x","y","z",], "D": ["x","y","w",], "E":["x","y","z","w","a"], "F":["x","y","z","w","b"] } hnxgraph = hnx.Hypergraph(F) hnx.drawing.draw(hnxgraph) # %% # convert to a list of edges G = list( F.values() ) # graph representing the edge containment poset (forgetting direction) containment = oat.hypergraph.edge_containment_graph_symmetrized( G ) # graph whose edges form the set complement of the containment graph anticontainment = nx.complement( containment ) # adjacency matrix of the anticontainment graph anti_adjacency = nx.adjacency_matrix( anticontainment ).todense() dissimilarity_matrix = oat.dissimilarity.sparse_matrix_for_dense( dissimilarity_matrix = anti_adjacency, max_dissimilarity = 0.5 ) # decomposition boundary matrix decomposition = oat.core.vietoris_rips.BoundaryMatrixDecomposition( dissimilarity_matrix = dissimilarity_matrix, max_homology_dimension = 2, ) # %% # The persistent homology dataframe: persistent_homology = decomposition.persistent_homology_dataframe( return_cycle_representatives = True, return_bounding_chains = True, ) persistent_homology # %% # A cycle representative: cycle = persistent_homology["cycle_representative"][1] cycle # %% # A cycle with reverted vertex labels original_edge_labels = { p: k for p,k in enumerate(F.keys()) } cycle = copy.deepcopy(persistent_homology["cycle_representative"][1]) cycle["simplex"] = [ [original_edge_labels[x] for x in simplex] for simplex in cycle["simplex"] ] cycle # %% # Plot the cycle representative triangles = persistent_homology["cycle_representative"][1]["simplex"] # the edges in the cycle points = oat.plot.vertex_embedding_for_simplices( simplices = triangles.tolist(), dimension = 3, ) fig = oat.plot.fig_3d_for_simplices( triangles, points=points, kwargs_points = dict( mode="markers+text", text= [ val for val in original_edge_labels.values() ], marker_size=8, textfont_size=18, ), ) fig.update_layout(template="plotly_dark") # %% [markdown] # Make triangles opaque and grey, for a different style. # %% # The trace for triangles is stored in fig.data[2] fig.data[2].update( color = 'grey', opacity = 1.0) fig