Reduce & Relabel

Homology calculations are only available in OAT for one type of hypergraph: a list of sorted-lists of integers. However we provide tools to translate back and forth between this and other data fromats, e.g. a dictionary of lists of strings.

Below are two examples.

import oat_python as oat

See also

• The HypernetX library provides many useful tools for working with hypergraphs, including tools to remove duplicate nodes and edges.

• Restricted Barycentric Subdivision (Vietoris-Rips) shows how to relabel nodes/edges, and use proper labels in plotting.

Relabel a hypergraph

Here we assign integer labels to the nodes of a hypergraph whose nodes are labeled with strings.

Define a hypergraph

E = { "A": ["x"], "B": ["y"], "C": ["x","y","z","zz"], "D": ["w","ww","x","y"], "DD": ["w","ww","x","y"] }

Relabel with integers

relabeled_hg, label_translator = oat.hypergraph.relabel(E)

The relabeled hypergraph:

relabeled_hg

[[2], [3], [2, 3, 4, 5], [0, 1, 2, 3], [0, 1, 2, 3]]

The label translator maps the new integer labels back to original labels and vice versa

for key, val in label_translator.items():
    print(f"{key}:  {val}")

new_node_for_old_node:  {'w': 0, 'ww': 1, 'x': 2, 'y': 3, 'z': 4, 'zz': 5}
new_edge_for_old_edge:  {'A': 0, 'B': 1, 'C': 2, 'D': 3, 'DD': 4}
old_node_for_new_node:  ['w', 'ww', 'x', 'y', 'z', 'zz']
old_edge_for_new_edge:  ['A', 'B', 'C', 'D', 'DD']

Relabel and reduce a hypergraph

Here we perform two actions in one step:

• reduce the hypergraph by removing duplicate hyperedges (meaning hyperedges with the same vertex set) and duplicate nodes (meaning nodes that belong to the same set of hyperedges)

• reformat the reduced hypergraph as a list of sorted-lists of integers

Define a hypergraph

E = { "A": ["x"], "B": ["y"], "C": ["x","y","z","zz"], "D": ["w","ww","x","y"], "DD": ["w","ww","x","y"] }

Collapse out redundant information, returning a list of lists

reduced_hg, label_translator = oat.hypergraph.reduce_hypergraph_with_labels(E)

The reduced hypergraph edges (duplicate nodes have been removed; the remaining nodes have been relabeled as integers)

reduced_hg

[[0], [2], [0, 1, 2], [0, 2, 3]]

The label translator maps

• new integer labels back to sets of original labels

• single original labels back to new integer labels

for key, val in label_translator.items():
    print(f"{key}:  {val}")

new_node_for_old_node:  {'x': 0, 'z': 1, 'y': 2, 'zz': 1, 'w': 3, 'ww': 3}
new_edge_for_old_edge:  {'A': 0, 'B': 1, 'C': 2, 'D': 3, 'DD': 3}
old_nodes_for_new_node:  [['x'], ['z', 'zz'], ['y'], ['w', 'ww']]
old_edges_for_new_edge:  [['A'], ['B'], ['C'], ['D', 'DD']]