Controlling edge dynamics in complex networks (1112.5945v1)

Abstract: The interaction of distinct units in physical, social, biological and technological systems naturally gives rise to complex network structures. Networks have constantly been in the focus of research for the last decade, with considerable advances in the description of their structural and dynamical properties. However, much less effort has been devoted to studying the controllability of the dynamics taking place on them. Here we introduce and evaluate a dynamical process defined on the edges of a network, and demonstrate that the controllability properties of this process significantly differ from simple nodal dynamics. Evaluation of real-world networks indicates that most of them are more controllable than their randomized counterparts. We also find that transcriptional regulatory networks are particularly easy to control. Analytic calculations show that networks with scale-free degree distributions have better controllability properties than uncorrelated networks, and positively correlated in- and out-degrees enhance the controllability of the proposed dynamics.

• The paper introduces switchboard dynamics (SBD) on network edges, offering a novel control method that contrasts with traditional nodal approaches.
• It presents an algorithmic framework to identify the minimal set of driver nodes, demonstrating that out-hubs ease control in balanced or scale-free networks.
• Empirical analysis on 38 diverse networks confirms that SBD enhances controllability, especially in regulatory and social systems.

Controlling Edge Dynamics in Complex Networks

The paper of complex networks has focused largely on the structural and dynamical properties of networks which naturally arise across various domains like social, biological, and technological systems. While network science has made significant strides in modeling and understanding these properties, less attention has been directed towards the controllability of the dynamics that occur on these networks. Traditional approaches to network controllability have centered on nodal dynamics. However, this paper introduces a novel dynamical process, termed switchboard dynamics (SBD), that occurs on the edges of networks. This paradigm shift emphasizes that the controllability properties of edge-centric dynamics can markedly differ from those of nodal dynamics.

Dynamics on Network Edges

The proposed switchboard dynamics is characterized by its focus on the edges of directed networks. In this framework, each node functions as a switchboard, distributing input signals arriving via incoming edges to outgoing ones through a linear operator. The dynamics are influenced by external controls applied via offset vectors at the nodes, thereby affecting the state of outbound edges. The equations governing this setup relate the states of outbound edges to the inbound ones using a mixing matrix and damping terms. This setup allows for a comparison between driven and driver nodes, with driven nodes representing edges directly controlled by offsets.

Controllability Analysis

The analysis of controllability is conducted through the lens of structural controllability, a concept extended from nodal to edge dynamics. While prior work by Liu et al. highlighted that nodal dynamics require a higher number of driver nodes as hub connectivity increases, the introduction of SBD presents a more favorable controllability perspective. In SBD, out-hubs can facilitate control over many state variables with fewer driver nodes, contrasting with how hubs complicate control in nodal dynamics.

The paper provides an algorithmic approach to determine optimal control configurations for SBD by considering the joint degree distribution of the network. The minimum set of driver nodes is needed to control SBD, which tends to be minimal for networks that are either balanced or have a scale-free degree distribution, aligning with the analytical models provided.

Evaluation on Real Networks

The methodology was tested on 38 networks across various domains, including regulatory, trust, metabolic, and social systems. These evaluations reveal that most real-world networks are more controllable under SBD than random Erdős–Rényi networks of equivalent size. Specifically, regulatory networks such as those found in transcriptional regulation are particularly susceptible to control via SBD, owing to their inherent structure, which favors the concentration of control at process hubs.

Future Implications

The findings suggest a reevaluation of control strategies for complex systems wherein the network's edge dynamics might present new avenues for efficient control. This edge-focused approach aligns with observed network characteristics in natural systems where robust hierarchies and out-hubs are prevalent. Given the distinct behavior of edge and nodal dynamics, this research urges careful selection of dynamic models when studying real-world networks.

Looking forward, this area of research opens potential avenues in the practical implementation of network control strategies in technology, biology, and beyond. Questions regarding noise resilience, non-linear dynamics, and scenarios where only partial system control is necessary remain to be explored, indicating fruitful areas for future research. Overall, the paper's results contribute critically to the understanding of how underlying structural properties influence complex network controllability.