11. Getting started with XGI#
XGI is a Python library to make working with and analyzing complex systems with higher-order interactions easy. We have collected a lot of useful functions, algorithms, and tools for working with hypergraphs and simplicial complexes to make life easier.
We will
Create and load hypergraphs
Visualize hypergraphs and simplicial complexes
Show how to use the stats interface
Give an example of how to compare an empirical dataset to a null model.
We start off by loading the XGI library.
install with pip install xgi.
https://xgi.readthedocs.io/
import xgi
import numpy as np
import pandas as pd
from IPython.display import display
from matplotlib.pyplot import cm
import matplotlib.pyplot as plt
import networkx as nx
You can easily check the version of XGI that you have installed by using the __version__ property:
xgi.__version__
'0.8.2'
11.1. Creating a hypergraph#
We want to start with a hypergraph, and this can be done in several ways:
Build a hypergraph node-by-node and edge-by-edge (less common, but can be helpful in writing your own generative models)
Load an existing dataset
Sample from a random generative model
Let’s start with the first method.
11.1.1. Building a hypergraph#
H = xgi.Hypergraph()
H.add_nodes_from([1, 2, 3, 4])
edges = [(1, 2), (1, 2, 4), (3, 4)]
H.add_edges_from(edges)
H.nodes
NodeView((1, 2, 3, 4))
H.edges.members(dtype=dict)
{0: {1, 2}, 1: {1, 2, 4}, 2: {3, 4}}
xgi.draw(H)
(<Axes: >,
(<matplotlib.collections.PathCollection at 0x293463fd0>,
<matplotlib.collections.LineCollection at 0x293463760>,
<matplotlib.collections.PatchCollection at 0x293463b80>))
S = xgi.SimplicialComplex()
S.add_nodes_from(range(1, 5))
S.add_simplices_from(edges)
S.nodes
NodeView((1, 2, 3, 4))
S.edges.members()
[frozenset({1, 2}),
frozenset({1, 2, 4}),
frozenset({3, 4}),
frozenset({2, 4}),
frozenset({1, 4})]
H.edges.members()
[{1, 2}, {1, 2, 4}, {3, 4}]
H.nodes.memberships()
{1: {0, 1}, 2: {0, 1}, 3: {2}, 4: {1, 2}}
xgi.draw(S)
(<Axes: >,
(<matplotlib.collections.PathCollection at 0x293a4f6a0>,
<matplotlib.collections.LineCollection at 0x293a4f970>,
<matplotlib.collections.PatchCollection at 0x1081aa460>))
H_build = xgi.Hypergraph()
H_build.add_edge([1, 2], id="a")
H_build.add_node(0)
H_build.add_edges_from([[3, 4], [0, 2, 3]])
H_build.add_nodes_from([9, 10])
XGI automatically assigns unique edge IDs (if a user doesn’t specify the ID)
H_build.edges
EdgeView(('a', 0, 1))
Why NodeViews and EdgeViews? These allow users to access many different properties and data structures from nodes and edges in a much simpler way. We will cover this more in depth later. For now, we can get the edges of which each node is a part and the nodes in each edge as follows:
print(H_build.nodes.memberships())
print(H_build.edges.members())
print(H_build.nodes.memberships(2))
print(H_build.edges.members("a"))
{1: {'a'}, 2: {1, 'a'}, 0: {1}, 3: {0, 1}, 4: {0}, 9: set(), 10: set()}
[{1, 2}, {3, 4}, {0, 2, 3}]
{1, 'a'}
{1, 2}
11.1.2. Loading datasets#
Moving on to method 2, one can load datasets in several different ways. First, we provide a companion data repository, xgi-data, where users can easily load several datasets in standard format:
H_enron = xgi.load_xgi_data("email-enron")
print(H_enron)
Hypergraph named email-Enron with 148 nodes and 10885 hyperedges
H_enron.edges
EdgeView(('0', '1', '2', '3', '4', '5', '6', '7', '8', '9', '10', '11', '12', '13', '14', '15', '16', '17', '18', '19', '20', '21', '22', '23', '24', '25', '26', '27', '28', '29', '30', '31', '32', '33', '34', '35', '36', '37', '38', '39', '40', '41', '42', '43', '44', '45', '46', '47', '48', '49', '50', '51', '52', '53', '54', '55', '56', '57', '58', '59', '60', '61', '62', '63', '64', '65', '66', '67', '68', '69', '70', '71', '72', '73', '74', '75', '76', '77', '78', '79', '80', '81', '82', '83', '84', '85', '86', '87', '88', '89', '90', '91', '92', '93', '94', '95', '96', '97', '98', '99', '100', '101', '102', '103', '104', '105', '106', '107', '108', '109', '110', '111', '112', '113', '114', '115', '116', '117', '118', '119', '120', '121', '122', '123', '124', '125', '126', '127', '128', '129', '130', '131', '132', '133', '134', '135', '136', '137', '138', '139', '140', '141', '142', '143', '144', '145', '146', '147', '148', '149', '150', '151', '152', '153', '154', '155', '156', 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{'18', '4', '44'},
{'108', '125', '44'},
{'125', '18', '44'},
{'108', '125', '44'},
{'125', '20', '44'},
{'108', '125', '44'},
{'108', '125', '44'},
{'108', '125', '44'},
{'108', '125', '44'},
{'33', '46', '76'},
{'36', '46', '53'},
{'33', '46', '52'},
{'17', '33', '46'},
{'17', '33', '46'},
{'17', '36', '46'},
{'17', '36', '46'},
{'47', '5', '87'},
{'47', '5', '55'},
{'114', '47', '87'},
{'114', '47', '87'},
{'114', '47', '86'},
{'45', '46', '52'},
{'17', '46', '52'},
{'110', '27', '53'},
{'138', '45', '53'},
{'45', '53', '69'},
{'132', '54', '55'},
{'25', '54', '90'},
{'137', '54', '62'},
{'136', '137', '54'},
{'137', '143', '54'},
{'25', '54', '90'},
{'104', '132', '55'},
{'47', '5', '55'},
{'47', '5', '55'},
{'114', '132', '55'},
{'47', '5', '55'},
{'47', '5', '55'},
{'114', '25', '55'},
{'114', '25', '55'},
{'47', '5', '55'},
{'132', '25', '55'},
{'132', '25', '55'},
{'47', '5', '55'},
{'132', '25', '55'},
{'132', '25', '55'},
{'132', '25', '55'},
{'47', '5', '55'},
{'130', '148', '55'},
{'114', '132', '55'},
{'130', '148', '55'},
{'104', '25', '55'},
{'104', '25', '55'},
{'104', '25', '55'},
{'104', '25', '55'},
{'104', '25', '55'},
{'104', '25', '55'},
{'104', '25', '55'},
{'104', '25', '55'},
{'104', '25', '55'},
{'104', '25', '55'},
{'104', '25', '55'},
{'104', '25', '55'},
{'104', '25', '55'},
{'104', '25', '55'},
{'104', '25', '55'},
{'132', '55', '61'},
{'104', '25', '55'},
{'132', '55', '61'},
{'104', '25', '55'},
{'104', '25', '55'},
{'104', '25', '55'},
{'104', '25', '55'},
{'104', '25', '55'},
{'104', '25', '55'},
{'104', '25', '55'},
{'104', '25', '55'},
{'104', '25', '55'},
{'104', '25', '55'},
{'104', '25', '55'},
{'104', '25', '55'},
{'104', '25', '55'},
{'104', '25', '55'},
{'114', '132', '55'},
{'114', '132', '55'},
{'132', '43', '55'},
{'147', '55', '61'},
{'147', '55', '61'},
{'114', '132', '55'},
{'147', '55', '61'},
{'147', '55', '61'},
{'5', '55', '87'},
{'5', '55', '87'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'114', '132', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'114', '132', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'114', '132', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'5', '55', '87'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'114', '132', '55'},
{'132', '25', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'114', '132', '55'},
{'114', '132', '55'},
{'25', '47', '55'},
{'5', '55', '87'},
{'114', '5', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'147', '55', '61'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'147', '55', '61'},
{'25', '47', '55'},
{'147', '55', '61'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'5', '55', '87'},
{'147', '55', '61'},
{'25', '47', '55'},
{'25', '47', '55'},
{'147', '55', '61'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'147', '55', '61'},
{'25', '47', '55'},
{'25', '47', '55'},
{'132', '47', '55'},
{'25', '47', '55'},
{'147', '55', '61'},
{'147', '55', '61'},
{'147', '55', '61'},
{'25', '47', '55'},
{'132', '47', '55'},
{'5', '55', '87'},
{'132', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'132', '47', '55'},
{'147', '55', '61'},
{'25', '47', '55'},
{'25', '47', '55'},
{'147', '55', '61'},
{'147', '55', '61'},
{'114', '132', '55'},
{'25', '47', '55'},
{'114', '132', '55'},
{'5', '55', '87'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'25', '47', '55'},
{'104', '132', '55'},
{'147', '55', '61'},
{'25', '47', '55'},
{'25', '47', '55'},
{'147', '55', '61'},
{'147', '55', '61'},
{'147', '55', '61'},
{'5', '55', '87'},
{'147', '55', '61'},
{'5', '55', '87'},
{'147', '55', '61'},
{'5', '55', '87'},
{'2', '55', '95'},
{'132', '25', '55'},
{'132', '25', '55'},
{'132', '25', '55'},
{'5', '55', '87'},
{'132', '25', '55'},
{'114', '132', '55'},
{'5', '55', '87'},
{'2', '55', '95'},
{'2', '55', '95'},
{'2', '55', '61'},
{'2', '55', '95'},
{'2', '55', '95'},
{'2', '55', '95'},
{'2', '55', '95'},
{'2', '55', '95'},
{'2', '55', '95'},
{'114', '132', '55'},
{'132', '47', '55'},
{'2', '55', '95'},
{'2', '55', '95'},
{'2', '55', '95'},
{'2', '55', '95'},
{'2', '55', '95'},
{'2', '55', '95'},
{'47', '55', '90'},
{'2', '55', '95'},
{'2', '55', '95'},
{'2', '55', '95'},
{'2', '55', '95'},
{'5', '55', '87'},
{'114', '47', '55'},
{'2', '55', '95'},
{'43', '56', '57'},
{'115', '56', '63'},
{'111', '117', '56'},
{'140', '56', '61'},
{'115', '12', '56'},
{'115', '140', '56'},
{'52', '56', '76'},
{'52', '56', '76'},
{'140', '56', '61'},
{'140', '56', '61'},
{'118', '57', '64'},
{'116', '125', '57'},
{'116', '125', '57'},
{'116', '125', '57'},
{'116', '125', '57'},
{'116', '125', '57'},
{'116', '125', '57'},
{'20', '44', '57'},
{'125', '44', '57'},
{'116', '125', '57'},
{'20', '44', '57'},
{'23', '57', '63'},
{'116', '125', '57'},
{'125', '17', '57'},
{'125', '20', '57'},
{'116', '125', '57'},
{'116', '24', '57'},
{'118', '57', '64'},
{'118', '57', '63'},
{'116', '125', '57'},
{'116', '125', '57'},
{'41', '58', '62'},
{'14', '28', '60'},
{'140', '61', '63'},
{'140', '61', '63'},
{'15', '61', '63'},
{'12', '147', '61'},
{'140', '61', '63'},
{'140', '61', '63'},
{'140', '61', '63'},
{'140', '61', '63'},
{'140', '61', '63'},
{'140', '61', '63'},
{'140', '61', '63'},
{'140', '61', '63'},
{'140', '61', '63'},
{'62', '65', '97'},
{'41', '62', '65'},
{'120', '128', '62'},
{'23', '63', '93'},
{'23', '63', '93'},
{'12', '23', '63'},
{'115', '23', '63'},
{'23', '63', '93'},
{'140', '63', '93'},
{'140', '23', '63'},
{'23', '43', '63'},
{'41', '63', '85'},
{'116', '64', '8'},
{'4', '65', '98'},
{'6', '65', '70'},
{'65', '71', '97'},
{'62', '65', '71'},
{'6', '65', '70'},
{'112', '6', '65'},
{'6', '65', '71'},
{'1', '41', '65'},
{'112', '6', '65'},
{'112', '6', '65'},
{'112', '6', '65'},
{'112', '6', '65'},
{'112', '6', '65'},
{'112', '6', '65'},
{'122', '65', '97'},
{'6', '65', '97'},
{'45', '53', '68'},
{'17', '45', '68'},
{'17', '45', '68'},
{'138', '27', '68'},
{'68', '69', '96'},
{'138', '146', '68'},
{'13', '138', '69'},
{'45', '69', '96'},
{'68', '69', '96'},
{'138', '36', '69'},
{'6', '65', '70'},
{'6', '65', '70'},
{'6', '65', '70'},
{'39', '59', '70'},
{'145', '39', '70'},
{'145', '39', '70'},
{'39', '59', '70'},
{'6', '65', '70'},
{'39', '59', '70'},
{'39', '59', '70'},
{'145', '39', '70'},
{'145', '39', '70'},
{'6', '65', '70'},
{'6', '65', '70'},
{'104', '25', '73'},
{'45', '46', '76'},
{'33', '45', '76'},
{'33', '45', '76'},
{'33', '45', '76'},
{'33', '52', '76'},
{'3', '61', '77'},
{'3', '61', '77'},
{'115', '23', '77'},
{'23', '63', '77'},
{'12', '57', '77'},
{'23', '52', '77'},
{'2', '72', '80'},
{'40', '80', '95'},
{'117', '129', '85'},
{'117', '66', '85'},
{'135', '37', '85'},
{'85', '89', '99'},
{'11', '58', '85'},
{'23', '63', '85'},
{'1', '41', '85'},
{'117', '66', '85'},
{'85', '89', '99'},
{'117', '66', '85'},
{'117', '66', '85'},
{'79', '85', '99'},
{'112', '33', '86'},
{'123', '71', '86'},
{'142', '71', '86'},
{'142', '71', '86'},
{'142', '71', '86'},
{'123', '142', '86'},
{'142', '71', '86'},
{'136', '142', '86'},
{'142', '71', '86'},
{'123', '142', '86'},
{'123', '142', '86'},
{'123', '142', '86'},
{'142', '71', '86'},
{'104', '132', '86'},
{'142', '71', '86'},
{'136', '137', '86'},
{'142', '71', '86'},
{'142', '71', '86'},
{'142', '71', '86'},
{'142', '71', '86'},
{'136', '137', '86'},
{'136', '137', '86'},
{'136', '137', '86'},
{'136', '137', '86'},
{'136', '142', '86'},
{'25', '54', '90'},
{'25', '54', '90'},
{'25', '54', '90'},
{'25', '5', '90'},
{'25', '54', '90'},
{'25', '54', '90'},
{'25', '54', '90'},
{'25', '54', '90'},
{'25', '54', '90'},
{'25', '54', '90'},
{'142', '86', '90'},
{'25', '54', '90'},
{'25', '54', '90'},
{'25', '54', '90'},
{'25', '54', '90'},
{'25', '54', '90'},
{'143', '25', '90'},
{'25', '54', '90'},
{'25', '54', '90'},
...]
xgi.load_xgi_data("contact-high-school")
<xgi.core.hypergraph.Hypergraph at 0x2933d7850>
This dataset, for example, has a corresponding datasheet explaining its characteristics. The nodes (individuals) in this dataset contain associated email addresses and the edges (emails) contain associated timestamps. These attributes can be accessed by simply typing H.nodes[id] or H.edges[id] respectively.
print(f"The hypergraph has {H_enron.num_nodes} nodes and {H_enron.num_edges} edges")
# We can also print a summary of the hypergraph
print(H_enron)
The hypergraph has 148 nodes and 10885 edges
Hypergraph named email-Enron with 148 nodes and 10885 hyperedges
print("The first 10 node IDs are:")
print(list(H_enron.nodes)[:10])
print("\nThe first 10 edge IDs are:")
print(list(H_enron.edges)[:10])
print("\nThe attributes of node '4' are")
print(H_enron.nodes["4"])
print("\nThe attributes of edge '6' are")
print(H_enron.edges["6"])
The first 10 node IDs are:
['4', '1', '117', '129', '51', '41', '65', '107', '122', '29']
The first 10 edge IDs are:
['0', '1', '2', '3', '4', '5', '6', '7', '8', '9']
The attributes of node '4' are
{'name': 'robert.badeer@enron.com'}
The attributes of edge '6' are
{'timestamp': '2000-02-22T08:07:00'}
We can clean up this dataset to remove isolates, multi-edges, singletons, and to replace all IDs with integer IDs.
H_enron_cleaned = H_enron.cleanup(
multiedges=False, singletons=False, isolates=False, relabel=True, in_place=False
)
print(H_enron_cleaned)
print("The first 10 node IDs are:")
print(list(H_enron_cleaned.nodes)[:10])
Hypergraph named email-Enron with 143 nodes and 1459 hyperedges
The first 10 node IDs are:
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
11.1.3. Random null models#
Lastly, we can create synthetic hypergraphs using random generative models. For example, we can extract the degree and edge size sequence from this dataset and wire them together at random (according to the Chung-Lu model) to create a random null model:
N = 10
ps = [0.5, 0.02]
H = xgi.random_hypergraph(N, ps, seed=1)
pos = xgi.barycenter_spring_layout(H, seed=1)
# pos = xgi.random_layout(H, seed=1)
xgi.draw(H, node_labels=True, hyperedge_labels=False, pos=pos)
(<Axes: >,
(<matplotlib.collections.PathCollection at 0x2a45e3fd0>,
<matplotlib.collections.LineCollection at 0x2a47a1fd0>,
<matplotlib.collections.PatchCollection at 0x2a477f850>))
We check whether this new hypergraph is connected and if not, the sizes of the connected components:
H_random = H
connected = xgi.is_connected(H_random)
if not connected:
size, num = np.unique(
[len(cc) for cc in xgi.connected_components(H_random)], return_counts=True
)
print(size)
print(num)
print("The connected components:")
# display(pd.DataFrame([size, num], columns=["Component size", "Number"]))
11.2. Converting between different representations#
11.2.1. Incidence matrix#
Any hypergraph can be expressed as an \(N \times M\) incidence matrix, \(I\), where \(N\) is the number of nodes and \(M\) is the number of edges. Rows indicate the node ID and the columns indicate the edge ID. \(I_{i,j}=1\) if node \(i\) is a member of edge \(j\) and zero otherwise.
I = xgi.incidence_matrix(H_enron_cleaned, sparse=False)
plt.spy(I, aspect="auto")
plt.xlabel("Hyperedges", fontsize=16)
plt.ylabel("Nodes", fontsize=16)
plt.show()
11.2.2. Adjacency matrix#
We can represent a hypergraph with an \(N\times N\) adjacency matrix, \(A\), where \(N\) is the number of nodes. The adjacency matrix is a lossy format; different hypergraphs can create the same adjacency matrix. \(A_{i,j} = 1\) if there is at least one hyperedge containing both nodes \(i\) and \(j\).
A = xgi.adjacency_matrix(H_enron_cleaned, sparse=False, order=4)
plt.spy(A)
<matplotlib.image.AxesImage at 0x2a5f68400>
orders = np.array(xgi.unique_edge_sizes(H_enron_cleaned)) - 1
weights = np.ones_like(orders)
xgi.multiorder_laplacian(H_enron_cleaned, weights, orders)
array([[172.34239802, -17.2342398 , 0. , ..., 0. ,
0. , 0. ],
[-17.2342398 , 517.02719407, -17.2342398 , ..., 0. ,
0. , 0. ],
[ 0. , -17.2342398 , 413.62175525, ..., 0. ,
0. , 0. ],
...,
[ 0. , 0. , 0. , ..., 34.4684796 ,
0. , 0. ],
[ 0. , 0. , 0. , ..., 0. ,
68.93695921, 0. ],
[ 0. , 0. , 0. , ..., 0. ,
0. , 51.70271941]])
We can do much more such as computing different properties, reading/writing, converting to/from different data structures, hypergraph null models and much more!
See the Read The Docs for more information: https://xgi.readthedocs.io
11.3. Visualization#
The first step for drawing a hypergraph is to choose a layout for the nodes. At the moment the available layouts are:
random_layout: positions nodes uniformly at random in the unit square.pairwise_spring_layout: positions the nodes using the Fruchterman-Reingold force-directed algorithm on the projected graph.barycenter_spring_layoutandweighted_barycenter_spring_layout: slight modification ofpairwise_spring_layout
Each layout returns a dictionary that maps nodes ID into (x, y) coordinates.
seed = 0
H_viz = xgi.random_hypergraph(20, [0.08, 0.01, 0.001], seed=seed)
pos1 = xgi.random_layout(H_viz)
pos2 = xgi.pairwise_spring_layout(H_viz)
pos3 = xgi.barycenter_spring_layout(H_viz)
pos4 = xgi.weighted_barycenter_spring_layout(H_viz)
xgi.draw(H_viz, pos1)
(<Axes: >,
(<matplotlib.collections.PathCollection at 0x2a5fed460>,
<matplotlib.collections.LineCollection at 0x29339a340>,
<matplotlib.collections.PatchCollection at 0x2a5f3f880>))
We can easily add labels to the nodes to more easily compare the visualization to the hypergraph:
xgi.draw(H_build, node_labels=True, hyperedge_labels=True)
(<Axes: >,
(<matplotlib.collections.PathCollection at 0x2a6040d60>,
<matplotlib.collections.LineCollection at 0x2a6040970>,
<matplotlib.collections.PatchCollection at 0x2a6040d90>))
Colors of the hyperedges are designed to match the hyperedge size, but the colormap can be tweaked:
# Sequential colormap
cmap = cm.Paired
xgi.draw(H_viz, pos4, edge_fc_cmap=cmap)
(<Axes: >,
(<matplotlib.collections.PathCollection at 0x2a60a17c0>,
<matplotlib.collections.LineCollection at 0x2a6093fa0>,
<matplotlib.collections.PatchCollection at 0x2a60a16d0>))
Other parameters can be changed as well:
cmap = cm.Reds
dyad_lc = "gray"
dyad_lw = 4
node_fc = "black"
node_ec = "white"
node_lw = 2
node_size = 20
xgi.draw(
H_viz,
pos4,
edge_fc_cmap=cmap,
dyad_lc=dyad_lc,
dyad_lw=dyad_lw,
node_fc=node_fc,
node_ec=node_ec,
node_lw=node_lw,
node_size=node_size,
)
(<Axes: >,
(<matplotlib.collections.PathCollection at 0x2a60c7070>,
<matplotlib.collections.LineCollection at 0x2a60ea2b0>,
<matplotlib.collections.PatchCollection at 0x2a60dd1c0>))
Simplicial complexes can be visualized using the same functions for node layout and drawing.
11.3.1. Technical note#
By definition, a simplicial complex object contains all sub-simplices. This would make the visualization heavy since all sub-simplices contained in a maximal simplex would overlap. The automatic solution for this, implemented by default in all the layouts, is to convert the simplicial complex into a hypergraph composed by solely by its maximal simplices.
11.3.2. Visual note#
To visually distinguish simplicial complexes from hypergraphs, the draw function will also show all links contained in each maximal simplices (while omitting simplices of intermediate orders).
SC = xgi.SimplicialComplex()
SC.add_simplices_from([[3, 4, 5], [3, 6], [6, 7, 8, 9], [1, 4, 10, 11, 12], [1, 4]])
pos = xgi.pairwise_spring_layout(SC)
xgi.draw(SC, pos)
(<Axes: >,
(<matplotlib.collections.PathCollection at 0x2a5f80070>,
<matplotlib.collections.LineCollection at 0x2a5f686a0>,
<matplotlib.collections.PatchCollection at 0x2a5f74bb0>))
We can even color/draw the nodes and edges based on statistics!
n = 100
is_connected = False
while not is_connected:
H = xgi.random_hypergraph(n, [0.03, 0.0002, 0.00001])
is_connected = xgi.is_connected(H)
pos = xgi.barycenter_spring_layout(H)
H.nodes.degree.asdict()
{0: 2,
1: 11,
2: 3,
3: 4,
4: 4,
5: 6,
6: 2,
7: 2,
8: 4,
9: 6,
10: 9,
11: 6,
12: 5,
13: 12,
14: 8,
15: 11,
16: 6,
17: 8,
18: 6,
19: 6,
20: 11,
21: 4,
22: 6,
23: 10,
24: 11,
25: 12,
26: 2,
27: 4,
28: 3,
29: 4,
30: 2,
31: 12,
32: 9,
33: 2,
34: 5,
35: 7,
36: 2,
37: 7,
38: 3,
39: 3,
40: 9,
41: 7,
42: 7,
43: 5,
44: 5,
45: 6,
46: 7,
47: 5,
48: 6,
49: 7,
50: 5,
51: 4,
52: 2,
53: 3,
54: 5,
55: 3,
56: 9,
57: 6,
58: 8,
59: 5,
60: 7,
61: 2,
62: 5,
63: 7,
64: 7,
65: 5,
66: 3,
67: 8,
68: 4,
69: 9,
70: 9,
71: 8,
72: 6,
73: 9,
74: 3,
75: 4,
76: 3,
77: 9,
78: 7,
79: 9,
80: 6,
81: 4,
82: 6,
83: 4,
84: 10,
85: 8,
86: 2,
87: 7,
88: 7,
89: 7,
90: 8,
91: 4,
92: 5,
93: 5,
94: 1,
95: 8,
96: 7,
97: 3,
98: 4,
99: 3}
plt.figure(figsize=(10, 10))
ax = plt.subplot(111)
xgi.draw(
H,
pos,
node_size=H.nodes.degree,
node_lw=H.nodes.average_neighbor_degree,
node_fc=H.nodes.degree,
ax=ax,
)
(<Axes: >,
(<matplotlib.collections.PathCollection at 0x2a5f0be50>,
<matplotlib.collections.LineCollection at 0x2a5deeee0>,
<matplotlib.collections.PatchCollection at 0x2a5faad90>))
11.4. The stats package#
The stats package is one of the features that sets xgi apart from other libraries. It
provides a common interface to all statistics that can be computed from a network, its
nodes, or edges.
Consider the degree of the nodes of a hypergraph H. After computing the values of the
degrees, one may wish to store them in a dict, a list, an array, a dataframe, etc.
Through the stats package, xgi provides a simple interface that seamlessly allows for
this type conversion. This is done via the NodeStat class.
import xgi
import numpy as np
import pandas as pd
H_stats = xgi.Hypergraph([[1, 2, 3], [2, 3, 4, 5], [3, 4, 5]])
H_stats.degree()
{1: 1, 2: 2, 3: 3, 4: 2, 5: 2}
H_stats.nodes.degree
NodeStat('degree')
This NodeStat object allows us to choose the datatype, specify keywords, and get basic statistics from these properties
print("As a dictionary:")
print(H_stats.nodes.degree.asdict())
print("\nAs a list:")
print(H_stats.nodes.degree.aslist())
print("\nAs a numpy array:")
print(H_stats.nodes.degree.asnumpy())
print("\nAs a pandas dataframe:")
print(H_stats.nodes.degree.aspandas())
As a dictionary:
{1: 1, 2: 2, 3: 3, 4: 2, 5: 2}
As a list:
[1, 2, 3, 2, 2]
As a numpy array:
[1 2 3 2 2]
As a pandas dataframe:
1 1
2 2
3 3
4 2
5 2
Name: degree, dtype: int64
H_stats.nodes.degree.aspandas().plot()
<Axes: >
print(H_stats.nodes.degree(order=2).asdict())
{1: 1, 2: 1, 3: 2, 4: 1, 5: 1}
st = H_stats.nodes.degree
np.round([st.max(), st.min(), st.mean(), st.median(), st.var(), st.std()], 3)
array([3. , 1. , 2. , 2. , 0.4 , 0.632])
Likewise, for edges,
st = H_stats.edges.size
st
EdgeStat('size')
np.round([st.max(), st.min(), st.mean(), st.median(), st.var(), st.std()], 3)
array([4. , 3. , 3.333, 3. , 0.222, 0.471])
The interface for attributes is very similar. If we add nodal attributes, for example
H_stats.add_nodes_from(
[
(1, {"color": "red", "name": "horse"}),
(2, {"color": "blue", "name": "pony"}),
(3, {"color": "yellow", "name": "zebra"}),
(4, {"color": "red", "name": "orangutan", "age": 20}),
(5, {"color": "blue", "name": "fish", "age": 2}),
]
)
print(H_stats.nodes.attrs("color").asdict())
{1: 'red', 2: 'blue', 3: 'yellow', 4: 'red', 5: 'blue'}
print(H_stats.nodes.attrs[1])
{'color': 'red', 'name': 'horse'}
One can also filter nodes and edges by their attributes as well as any associated statistic. For example,
print(H_stats.nodes.filterby("degree", 2))
print(H_stats.nodes.filterby_attr("color", "blue"))
[2, 4, 5]
[2, 5]
H_stats.edges.filterby("size", 3).members()
[{1, 2, 3}, {3, 4, 5}]
H_stats.nodes.multi(["degree", "clustering_coefficient"]).aspandas().groupby(
"degree"
).agg("mean")
| clustering_coefficient | |
|---|---|
| degree | |
| 1 | 1.000000 |
| 2 | 0.888889 |
| 3 | 0.666667 |
(
H_stats.nodes.multi(["degree", "clustering_coefficient", "average_neighbor_degree"])
.aspandas()
.groupby("degree")
.agg("mean")
.plot(marker="o")
)
<Axes: xlabel='degree'>
For even more functionality, see the Read The Docs: https://xgi.readthedocs.io
From this plot we can see that the original dataset is more assortative than the random null model.
11.5. Dynamics#
We can also simulate the order parameter of the Kuramoto model for hypergraphs.
n = 100
H = xgi.random_hypergraph(n, [0.05, 0.001], seed=None)
omega = np.random.normal(0, 0.05, n)
theta = np.linspace(0, 2 * np.pi, n)
timesteps = 1000
dt = 0.01
theta, t = xgi.simulate_kuramoto(
H, k2=2, k3=3, omega=omega, theta=theta, timesteps=timesteps, dt=dt
)
R = xgi.compute_kuramoto_order_parameter(theta)
plt.plot(t, np.sin(theta), c="grey", alpha=0.2)
plt.show()
plt.figure()
plt.plot(t, R)
plt.xlabel("time")
plt.ylabel("Kuramoto order parameter")
plt.show()
For more details, visit https://xgi.readthedocs.io