Node centrality Lifting (Graph to Hypergraph)

[mbanf]

Motivation
This is a novel lifting that creates hyperedges based on central, i.e. highly influential, nodes in the network. Mapping a connection between individual nodes to specific nodes in the network architecture that have a specific and potentially competing influence on them is a very convenient scenario to be modelled via hyperedges. Using shortest path distance to identify the most influential nodes on any given node even allows for placing weights on the hyperedge connection to individual, connected nodes (i.e. the inverse shortest path distance to the corresponding most influential node that the hyperedge represents) in order to model influence decay across the network.

To define and identify influential nodes in the network, we refer to the variant of the Eigenvector Centrality that introduces additional random teleportations besides the stationary distribution of the Markov Chain, i.e. PageRank.

Background
Eigenvector Centrality is an algorithm that measures the transitive influence of nodes. Relationships originating from high-scoring nodes contribute more to the score of a node than connections from low-scoring nodes. A high eigenvector score means that a node is connected to many nodes who themselves have high scores.

The algorithm computes the eigenvector associated with the largest absolute eigenvalue. To compute that eigenvalue, the algorithm applies the power iteration approach. Within each iteration, the centrality score for each node is derived from the scores of its incoming neighbors. In the power iteration method, the eigenvector is L2-normalized after each iteration, leading to normalized results by default.

The PageRank [1] variant of Eigenvector Centrality utilises, at any step of the power iteration, an additional teleportation probability, called dampening factor $\alpha$, which decides whether to continue following the transition matrix or teleport to random positions in the process. These random teleportations have shown to be an effective way to ensure that the transition matrix and corresponding Markov chain exhibit ergodicity which makes them easier to analyze and to guarantee convergence.

Method

Our approach is applicable to both directed and undirected as well as weighted and unweighted networks. It works as follows:
(1) calculate the node centrality of all nodes in the graph.
(2) select the top $n$ most influential nodes in the graph as hyperedges based on a given quantile.
(3) Assign all nodes in the network to $m >= 1$ most influential nodes (with $m <= n$), i.e. their respective hyperedges, based on their shortest path distance $d$ to each influential node.
(4, optional) model individual connection weights per node to a hyperedge via the inverse shortest path distance (i.e. $1/d$) to the hyperedge's corresponding most influential node.

Remarks on additional influential node feature lifting proposed

Note that the algorithm can support the utilization of the ProjectionSum feature lifting to model the inverse relationship between all nodes towards their shared most influential node.
In order to model the direct influence of the influential node on all individual nodes via the hyperedge, we have, however, further implemented an (optional) straight-forward feature lifting via assignment of the hyperedge's corresponding node's features, thereby bypassing the ProjectionSum feature lifting. In combination with the individual connection weights per node to a hyperedge via the inverse shortest path distance (i.e. $1/d$) to the hyperedge's corresponding most influential node, this direct assignment of the influential node's features to the hyperedges lends itself ideally to model the decaying influence of the influential node to all nodes assigned to the hyperedge by (i.e. $1/d$) per node across the network.

Submission by Team PerelynAI
Max Schattauer (@max-perelyn), Liliya Imasheva (@liliya-imasheva) , Dominik Filipiak (@DominikFilipiak), Michael Banf (@mbanf)

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Hello @mbanf! Thank you for your submission. As we near the end of the challenge, I am collecting participant info for the purpose of selecting and announcing winners. Please email me (or have one member of your team email me) at guillermo_bernardez@ucsb.edu so I can share access to the voting form. In your email, please include:

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