  Many complex systems can be described as multiplex networks in which the same
nodes can interact with one another in different layers, thus forming a set of
interacting and co-evolving networks. Examples of such multiplex systems are
social networks where people are involved in different types of relationships
and interact through various forms of communication media. The ranking of nodes
in multiplex networks is one of the most pressing and challenging tasks that
research on complex networks is currently facing. When pairs of nodes can be
connected through multiple links and in multiple layers, the ranking of nodes
should necessarily reflect the importance of nodes in one layer as well as
their importance in other interdependent layers. In this paper, we draw on the
idea of biased random walks to define the Multiplex PageRank centrality measure
in which the effects of the interplay between networks on the centrality of
nodes are directly taken into account. In particular, depending on the
intensity of the interaction between layers, we define the Additive,
Multiplicative, Combined, and Neutral versions of Multiplex PageRank, and show
how each version reflects the extent to which the importance of a node in one
layer affects the importance the node can gain in another layer. We discuss
these measures and apply them to an online multiplex social network. Findings
indicate that taking the multiplex nature of the network into account helps
uncover the emergence of rankings of nodes that differ from the rankings
obtained from one single layer. Results provide support in favor of the
salience of multiplex centrality measures, like Multiplex PageRank, for
assessing the prominence of nodes embedded in multiple interacting networks,
and for shedding a new light on structural properties that would otherwise
remain undetected if each of the interacting networks were analyzed in
isolation.
