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Three hypergraph eigenvector centralities

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Slides from ILAS talk.
July 12, 2019.

Published in: Data & Analytics
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Three hypergraph eigenvector centralities

  1. 1. 1 Three hypergraph eigenvector centralities Austin R. Benson · Cornell University ILAS · July 12, 2019 Slides. bit.ly/arb-ILAS-19
  2. 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
  3. 3. Finding central nodes is one of the most common tasks in network analysis. 3
  4. 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 <latexit 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cu / X (u,v)2E cv ! c / Ac ! c = Ac, > 0 <latexit 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  5. 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. 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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sha1_base64="EEENsY24Omo4nTUZnQskl3ljo8k=">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</latexit><latexit sha1_base64="EEENsY24Omo4nTUZnQskl3ljo8k=">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</latexit><latexit sha1_base64="EEENsY24Omo4nTUZnQskl3ljo8k=">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</latexit> A <latexit 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A(i, j) = ( 1 nodes i and j connected by an edge 0 otherwise <latexit 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  7. 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. 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. <latexit 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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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Wij = P k T(i, j, k)<latexit 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  9. 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. 10 • Nodes are tags. • A hyperedge on k nodes if those tags all appear on some post. • Look at 4-uniform hypergraphs.
  11. 11. 11 CEC ZEC HEC 1 14.04 dual-boot 14.04 2 boot boot boot 3 drivers grub2 drivers 4 12.04 partitioning 12.04 5 16.04 uefi 16.04 6 networking system-installation dual-boot 7 server 14.04 nvidia 8 dual-boot windows grub2 9 nvidia installation networking 10 grub2 12.04 partitioning <latexit 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  12. 12. 12 CEC analysis reduces to nonnegative matrix theory. There is no fun non-linearity we might expect from a tensor representation of hypergraphs.

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