Mittag-Leffler functions and their applications in network science

TL;DR

This paper develops a comprehensive framework for walk-based centrality measures in complex networks using Mittag-Leffler functions, unifying and generalizing existing indices, with extensions to dynamic networks and practical guidelines.

Contribution

Introduces a unified theory of Mittag-Leffler-based centralities that interpolate between known measures and extends to evolving networks, with computational insights.

Findings

Mittag-Leffler centralities interpolate between degree and eigenvector centralities.

The framework unifies subgraph centrality, Katz centrality, and related measures.

Numerical experiments demonstrate the effectiveness on synthetic and real networks.

Abstract

We describe a complete theory for walk-based centrality indices in complex networks defined in terms of Mittag-Leffler functions. This overarching theory includes as special cases well-known centrality measures like subgraph centrality and Katz centrality. The indices we introduce are parametrized by two numbers; by letting these vary, we show that Mittag-Leffler centralities interpolate between degree and eigenvector centrality, as well as between resolvent-based and exponential-based indices. We further discuss modeling and computational issues, and provide guidelines on parameter selection. The theory is then extended to the case of networks that evolve over time. Numerical experiments on synthetic and real-world networks are provided.