Hierarchy measure for complex networks (1202.0191v2)

Abstract: Nature, technology and society are full of complexity arising from the intricate web of the interactions among the units of the related systems (e.g., proteins, computers, people). Consequently, one of the most successful recent approaches to capturing the fundamental features of the structure and dynamics of complex systems has been the investigation of the networks associated with the above units (nodes) together with their relations (edges). Most complex systems have an inherently hierarchical organization and, correspondingly, the networks behind them also exhibit hierarchical features. Indeed, several papers have been devoted to describing this essential aspect of networks, however, without resulting in a widely accepted, converging concept concerning the quantitative characterization of the level of their hierarchy. Here we develop an approach and propose a quantity (measure) which is simple enough to be widely applicable, reveals a number of universal features of the organization of real-world networks and, as we demonstrate, is capable of capturing the essential features of the structure and the degree of hierarchy in a complex network. The measure we introduce is based on a generalization of the m-reach centrality, which we first extend to directed/partially directed graphs. Then, we define the global reaching centrality (GRC), which is the difference between the maximum and the average value of the generalized reach centralities over the network. We investigate the behavior of the GRC considering both a synthetic model with an adjustable level of hierarchy and real networks. Results for real networks show that our hierarchy measure is related to the controllability of the given system. We also propose a visualization procedure for large complex networks that can be used to obtain an overall qualitative picture about the nature of their hierarchical structure.

Analysis of a Hierarchical Measure for Complex Networks

The paper "Hierarchy Measure for Complex Networks" by Mones, Vicsek, and Vicsek addresses the challenge of quantitatively characterizing hierarchies within complex networks, a fundamental aspect lacking a universally accepted metric. This paper introduces the global reaching centrality (GRC), a robust measure designed to overcome limitations of previous hierarchy measures by being applicable across various network types, including those with both directed and undirected interactions, and addressing parameters like loops and edge weights.

Quantitative Approach to Hierarchical Structures

The authors propose the GRC as an extension of m-reach centrality, a node's capacity to influence other nodes within a network. The GRC is defined as the difference between the maximum local reaching centrality and the average local reaching centrality across the entire network. This approach captures the heterogeneity in node influence, serving as a proxy for hierarchy levels within network structures.

By analyzing synthetic networks with tunable hierarchy levels, such as the adjustable hierarchical network, and classical networks like Erdos-Renyi (ER) and Scale-Free (SF) models, the paper examines how GRC varies with changes in network structure. The findings suggest that the GRC effectively differentiates between hierarchical structures and more egalitarian networks, with higher GRC values correlating with higher hierarchical levels.

Real-world Network Analysis and Implications

This metric was applied to various real-world networks ranging from food webs and neuronal networks to organizational structures. The results demonstrate significant variability in the degree of hierarchy, as reflected by the GRC, across different types of networks. For instance, food webs typically exhibited high GRC values indicating a pronounced hierarchical structure, while organizational trust networks showed much lower GRC values, confirming their less hierarchical nature.

Furthermore, the paper evaluates the relationship between the GRC and network controllability, a critical aspect of understanding how network architecture influences the ease of directing system behavior. The paper notes that networks with higher GRC tend to be more challenging to control, a surprising contradiction to the notion that hierarchical structures are optimal for control.

Visualizing Network Hierarchies

The proposed method also advances hierarchical visualization techniques. By facilitating layout generation through the GRC-based approach, the authors provide a way to visually distinguish the hierarchical nuances in large networks effectively. This visualization method enhances understanding beyond numerical results, offering an intuitive grasp of a network's hierarchical complexity.

Future Directions

While the GRC presents a significant advance in measuring hierarchy, future research could explore its applications in diverse domains, particularly those involving dynamic networks where edges are continuously evolving. Moreover, the interplay between hierarchy and other network characteristics like modularity and resilience deserves further investigation. The potential integration of GRC with structural controllability analysis could yield novel insights into the design and regulation of complex systems in engineering, biology, and social sciences.

In conclusion, the paper provides a comprehensive framework for hierarchy analysis in networks, emphasizing the utility of the GRC as a universal metric. Its implications span theoretical and practical realms, offering a foundation for deeper exploration into the organization and control of complex networks.