Cooperative Game Theoretic Solution Concepts for top-$k$ Problems

TL;DR

This paper explores how cooperative game theory can identify the most critical nodes in networks, proposing post-processing methods and a combined algorithm to improve top-$k$ node selection.

Contribution

It introduces novel post-processing techniques and a hybrid algorithm integrating cooperative game solutions with greedy hill-climbing for better top-$k$ node identification.

Findings

Proposed several post-processing methods for cooperative game values.

Developed a standalone algorithm combining game theory and greedy hill-climbing.

Provided reasoning and analysis for each proposed method.

Abstract

The problem of finding the $k$ most critical nodes, referred to as the $top\text{-}k$ problem, is a very important one in several contexts such as information diffusion and preference aggregation in social networks, clustering of data points, etc. It has been observed in the literature that the value allotted to a node by most of the popular cooperative game theoretic solution concepts, acts as a good measure of appropriateness of that node (or a data point) to be included in the $top\text{-}k$ set, by itself. However, in general, nodes having the highest $k$ values are not the desirable $top\text{-}k$ nodes, because the appropriateness of a node to be a part of the $top\text{-}k$ set depends on other nodes in the set. As this is not explicitly captured by cooperative game theoretic solution concepts, it is necessary to post-process the obtained values in order to output the suitable $top\text{-}k$ nodes. In this paper, we propose several such post-processing methods and give reasoning behind each of them, and also propose a standalone algorithm that combines cooperative game theoretic solution concepts with the popular greedy hill-climbing algorithm.