Functional Multiplex PageRank

Jacopo Iacovacci,Christoph Rahmede,Ginestra Bianconi,Alex Arenas

doi:10.1209/0295-5075/116/28004

Abstract

Recently it has been recognized that many complex social, technological and biological networks have a multilayer nature and can be described by multiplex networks. Multiplex networks are formed by a set of nodes connected by links having different connotations forming the different layers of the multiplex. Characterizing the centrality of the nodes in a multiplex network is a challenging task since the centrality of the node naturally depends on the importance associated to links of a certain type. Here we propose to assign to each node of a multiplex network a centrality called Functional Multiplex PageRank that is a function of the weights given to every different pattern of connections (multilinks) existent in the multiplex network between any two nodes. Since multilinks distinguish all the possible ways in which the links in different layers can overlap, the Functional Multiplex PageRank can describe important non-linear effects when large relevance or small relevance is assigned to multilinks with overlap. Here we apply the Functional Page Rank to the multiplex airport networks, to the neuronal network of the nematode C. elegans, and to social collaboration and citation networks between scientists. This analysis reveals important differences existing between the most central nodes of these networks, and the correlations between their so-called pattern to success.

Highlights

Introduction. – Many complex interacting systems are formed by nodes related by different types of interactions forming multiplex networks [1,2,3,4]
For example scientific authors form at the same time collaboration networks and citation networks in which they cite each other [7,8], the airport network is formed by airports connected by flights operated by different airline companies [10], in the brain neurons are simultaneously connected by chemical and electrical types of connections [9, 13, 14]
The Versatility [25] emphasizes the relevance of nodes connected in many different layers and it applies to multiplex networks where corresponding nodes in different layers are connected by interlinks

Summary

Functional Multiplex PageRank

PACS 89.75.-k – Complex systems PACS 89.75.Fb – Structures and organization in complex systems PACS 89.75.Hc – Networks and genealogical trees. We propose a new centrality measure called Functional Multiplex PageRank which assigns to each node a centrality depending on the influence assigned to each type of multilink In this way it is possible to go beyond the modelling of the influence of each single layer because the influence of multilinks can capture important non-linear effects due to the overlap links. The Functional Multiplex PageRank Xi of node i depends on the tensor z with elements zm ≥ 0 defined for every type of multilink m It describes the steady state of a random walker that hops from a node j to a neighbor node i with probability αif this is possible, and otherwise performs random jumps to a random connected node of the multiplex network. We have that the Functional Multiplex PageRank Xi(z) of node i is given by

Aimj ij zmij

Xi max z

Nφ Nθ