Abstract
The basis of Google’s acclaimed PageRank is an artificial mixing of the Markov chain representing the connectivity structure of the network under study with a maximally connected network where every node is connected to every other node. The rate with which the original network is mixed with the strongly connected one is called damping factor. The choice of the damping factor can influence the ranking of the nodes. As we show in this paper, the ranks of transient nodes, i.e., nodes not belonging to a strongly connected component without outgoing links in the original network, tend to zero as the damping factor increases. In this paper we develop a new methodology for obtaining a meaningful ranking of nodes without having to resort to mixing the network with an artificial one. Our new ranking relies on an adjusted definition of the ergodic projector of the Markov chain representing the original network. We will show how the new ergodic projector leads to a more structural way of ranking (transient) nodes. Numerical examples are provided to illustrate the impact of this new ranking approach.
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Notes
Positivity of w is needed for the row normalization later on. Possibly a scaling is required to make the weights positive in practical applications.
The term random walk also applies, but since the ranking is inspired by Google’s PageRank we will use their concept of random surfer.
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This article is part of the Topical Collection: Special Issue on Performance Analysis and Optimization of Discrete Event Systems
Guest Editors: Christos G. Cassandras and Alessandro Giua
Part of this work was done while Joost Berkhout was with Vrije Universiteit Amsterdam
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Berkhout, J., Heidergott, B.F. Ranking nodes in general networks: a Markov multi-chain approach. Discrete Event Dyn Syst 28, 3–33 (2018). https://doi.org/10.1007/s10626-017-0248-7
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DOI: https://doi.org/10.1007/s10626-017-0248-7