Fixed-Size Dynamic Scale-Free Networks: Modeling, Stationarity, and Resilience

2026-01-05

IEEE Transactions on Systems, Man, and Cybernetics: Systems

AI Summary

This study addresses the limitation of traditional scale-free network models, which rely on node growth and preferential attachment and thus fail to capture real-world systems with fixed node counts but dynamically evolving links. To overcome this, the authors propose a dynamic network model based on state-dependent random walks, where node degree evolution is formulated as a stochastic process featuring stagnation and a variable diffusion coefficient. Theoretical analysis and numerical simulations demonstrate that, even on a fixed-size network, the model spontaneously generates a stable power-law degree distribution and successfully reproduces structural characteristics observed in three empirical networks. Furthermore, resilience assessments reveal that the model accurately captures network robustness under targeted attacks, thereby elucidating its intrinsic mechanisms of resilience.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Multiagent Systems: Agent-Based Simulation and Emergent Behavior
• Planning, Routing, and Scheduling: Model-Based Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Computational social science

Abstract

Many real-world scale-free networks, such as neural networks and online communication networks, consist of a fixed number of nodes but exhibit dynamic edge fluctuations. However, traditional models frequently overlook scenarios where the node count remains constant, instead prioritizing node growth. In this work, we depart from the assumptions of node number variation and preferential attachment to present an innovative model that conceptualizes node degree fluctuations as a state-dependent random walk process with stasis and variable diffusion coefficient. We show that this model yields stochastic dynamic networks with stable scale-free properties. Through comprehensive theoretical and numerical analyses, we demonstrate that the degree distribution converges to a power-law distribution, provided that the lowest degree state within the network is not an absorbing state. Furthermore, we investigate the resilience of the fraction of the largest component and the average shortest path length following deliberate attacks on the network. By using three real-world networks, we confirm that the proposed model accurately replicates actual data. The proposed model thus elucidates mechanisms by which networks, devoid of growth and preferential attachment features, can still exhibit power-law distributions and be used to simulate and study the resilience of attacked fixed-size scale-free networks.

Problem

• scale-free networks
• fixed-size networks
• dynamic networks
• degree distribution
• network resilience

Innovation

• fixed-size networks
• dynamic scale-free networks
• state-dependent random walk
• power-law degree distribution
• network resilience