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Three hypergraph eigenvector centralities
1. 1
Three hypergraph eigenvector centralities
Austin R. Benson · Cornell University
ILAS · July 12, 2019
Slides. bit.ly/arb-ILAS-19
2. Complex systems are modeled by networks (graphs):
sets of nodes and edges capturing interacting things.
2
Collaboration
nodes are people
edges link entities
working together
Communications
nodes are people/accounts
edges show info.exchange
Physical proximity
nodes are people/animals
edges link those that interact
in close proximity
Drug compounds
nodes are substances
edge between substances that
appear in the same drug
4. Graph eigenvector centrality is a standard measurement.
4
1. A node’s centrality is proportional to the sum of its
neighbors’ centralities.
A is adjacency matrix for graphs.
2. A node’s centrality is positive: c > 0.
3. By Perron-Frobenius, .Ac = 1c<latexit 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Aij =
(
1 nodes i and j connected by an edge
0 otherwise
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sha1_base64="dJ1AtrmCTE91IjiNhbj+IaC34BM=">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</latexit>
A
<latexit 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cu /
X
(u,v)2E
cv ! c / Ac ! c = Ac, > 0
<latexit 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5. Many real-world systems are composed of“higher-order”
interactions that we can model with hypergraphs.
5
Collaboration
nodes are people
teams are made up of
small groups
Communications
nodes are people/accounts
emails often have several
recipients,not just one
Physical proximity
nodes are people/animals
people often gather in
small groups
Drug compounds
nodes are substances
drugs are made up of
several substances
6. Many real-world systems are composed of“higher-order”
interactions that we can model with hypergraphs.
6
• Adjacency matrix A for graphs.
• Adjacency tensor T for hypergraphs.
(generalizes to k-way interactions; assume symmetric)
T
<latexit 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A
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T(i, j, k) =
(
1 i, j, and k connected by a hyperedge
0 otherwise
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A(i, j) =
(
1 nodes i and j connected by an edge
0 otherwise
<latexit 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7. We would like to extend eigenvector centrality to the
case of such higher-order interactions.
7
1. A function f of a node’s centrality is proportional to the sum
of a function g of its neighbors’ centralities.
2. A node’s centrality is positive: c > 0.
f (cu) /
X
(u,v,w)2E
g(cv, cw)
<latexit 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8. • Simplest functions are .
• Linear → reduces to linear problem
(again by PF).
= # of hyperedges containing i and j.
• Matrix W called clique motif adjacency matrix
[Benson-Gleich-Leskovec 16; Yin-Benson-Leskovec-Gleich 17].
General technique called clique expansion
[Agarwal-Branson-Belongie 06].
1. Clique motif eigenvector centrality (CEC).
8
f (cu) /
X
(u,v,w)2E
g(cv, cw), c > 0.
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f (cu) = cu, g(cv, cw) = cv + cw<latexit 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c = Wc, > 0, c > 0 ! 1c = Wc<latexit 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sha1_base64="jpDv00oDu/arQOuQpO+593hsndo=">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</latexit><latexit 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Wij =
P
k T(i, j, k)<latexit 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9. How do we compute CEC?
9
• Want to compute first eigenvector of the matrix W, where
.
• Assuming hyperedges are of size at least 3 means that the
graph induced by W has a 3-clique and is not bipartite.
Thus, |𝞴n| < |𝞴1|, and the power method converges.
Wij =
P
k T(i, j, k)<latexit 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10. 10
• Nodes are tags.
• A hyperedge on k
nodes if those tags all
appear on some post.
• Look at 4-uniform
hypergraphs.