Multiplex PageRank (1306.3576v3)

Abstract: 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.

• The paper introduces Multiplex PageRank to rank nodes in multiplex networks by incorporating inter-layer dependencies.
• The method applies biased random walks and presents four variants—Additive, Multiplicative, Combined, and Neutral—to model different influence dynamics.
• Empirical analysis on online social networks demonstrates that Multiplex PageRank uncovers synergistic effects that traditional single-layer methods overlook.

Multiplex PageRank: A Centrality Measure for Complex Networks

The paper "Multiplex PageRank" presents an innovative approach to addressing the challenges inherent in ranking nodes within multiplex networks. Multiplex networks, characterized by nodes interacting across multiple layers or types of connections, capture the complex dynamics of real-world systems more accurately than traditional single-layer networks. The authors propose an extension of the PageRank algorithm, originally developed for ranking web pages, to encompass the multifaceted nature of these networks.

Centrality in Multiplex Networks

The centrality of nodes—a measure of their importance or influence within a network—presents particular challenges in multiplex networks due to the interdependence of the layers. Existing measures like PageRank, which are traditionally applied to single-layer networks, do not capture how the importance of a node in one layer may influence its standing in another layer. The paper introduces Multiplex PageRank as a solution, drawing on the concept of biased random walks to accommodate inter-layer dependencies.

Multiplex PageRank Variants

The authors delineate four versions of the Multiplex PageRank centrality measure: Additive, Multiplicative, Combined, and Neutral. Each version reflects different dynamics of how the importance of nodes in one layer affects their ranking in another:

1. Additive Multiplex PageRank: This version models a scenario where the centrality gained in one layer simply adds to the node's centrality in another layer.
2. Multiplicative Multiplex PageRank: Here, a node's importance in one layer multiplies its ability to benefit from its interactions in another layer, capturing a synergetic effect.
3. Combined Multiplex PageRank: This approach integrates both additive and multiplicative effects, providing a comprehensive view of inter-layer influence.
4. Neutral Multiplex PageRank: This serves as a baseline where no inter-layer effects are considered, essentially reducing the problem to a traditional PageRank in a single network.

Empirical Application and Key Findings

To illustrate these concepts, the authors apply the Multiplex PageRank to an online social network comprising two interaction types: instant messaging and forum postings. Their analysis reveals that considering the multiplex nature of the network uncovers unique node rankings that single-layer analysis would miss. Particularly, the multiplicative version highlights nodes that exhibit significant synergistic interactions across layers.

Implications and Future Directions

The formulation of the Multiplex PageRank provides significant insights into the structural properties and dynamics of complex systems. It presents empirical researchers and practitioners with a powerful tool for uncovering hidden patterns of influence and connectivity that are prevalent in numerous real-world networks, such as social media, transport systems, and biological networks.

The paper's exploration of multiplicative effects suggests avenues for future research, particularly in exploring how these dynamics play out in networks with varying degrees of correlation between layers. Additionally, the methodological framework set forth opens up possibilities for integrating other centrality measures beyond PageRank, such as eigenvector centrality, to multiplex settings.

As researchers continue to unravel the intricacies of multiplex networks, the contributions of this paper provide a foundational step towards a deeper understanding of complex interconnected systems. The Multiplex PageRank not only enriches the toolkit available for network analysis but sets the stage for further theoretical and practical advancements in the field of complex networks.