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Percolation Centrality (Centrality Measure)
• Last Updated : 04 Jan, 2018

Mathematical Definition

The Percolation Centrality is defined for a given node, at a given time, as the proportion of ‘percolated paths’ that go through that node. A ‘percolated path’ is a shortest path between a pair of nodes, where the source node is percolated (e.g., infected). The target node can be percolated or non-percolated, or in a partially percolated state.

where is total number of shortest paths from node to node and is the number of those paths that pass through . The percolation state of the node at time is denoted by and two special cases are when which indicates a non-percolated state at time whereas when which indicates a fully percolated state at time . The values in between indicate partially percolated states ( e.g., in a network of townships, this would be the percentage of people infected in that town).

The attached weights to the percolation paths depend on the percolation levels assigned to the source nodes, based on the premise that the higher the percolation level of a source node is, the more important are the paths that originate from that node. Nodes which lie on shortest paths originating from highly percolated nodes are therefore potentially more important to the percolation. The definition of PC may also be extended to include target node weights as well. Percolation centrality calculations run in time with an efficient implementation adopted from Brandes’ fast algorithm and if the calculation needs to consider target nodes weights, the worst case time is .

References
https://en.wikipedia.org/wiki/Centrality

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