Universal Behavior of Load Distribution in Scale-free Networks

Universal Behavior of Load Distribution in Scale-free Networks

October 22, 2018 | K.-I. Goh, B. Kahng, and D. Kim
This study investigates the load distribution in scale-free networks, where the degree distribution follows a power-law with exponent γ. Load, defined as the accumulated number of data packets passing through a vertex, is found to follow a power-law distribution with exponent δ ≈ 2.2, independent of γ in the range 2 < γ ≤ 3 and for both undirected and directed networks. This suggests that δ is a universal quantity for scale-free networks. Scale-free networks are characterized by a power-law degree distribution, common in real-world systems like the internet, social networks, and biological systems. Unlike random networks, scale-free networks have a small-world property, where the average distance between nodes is small, due to the presence of highly connected hubs. These hubs play a critical role in reducing network diameter and are heavily loaded with data packets. The study defines load as the total number of data packets passing through a vertex, and finds that it follows a power-law distribution with exponent δ ≈ 2.2, regardless of the network structure as long as γ is in the range 2 < γ ≤ 3. This universality is confirmed through simulations of both static and evolving scale-free networks, as well as directed networks. The load distribution is also found to be independent of the mean degree of the network. The results show that for γ > 3, the load distribution decays exponentially, indicating a fundamental difference in transport properties between scale-free networks with γ > 3 and those with 2 < γ ≤ 3. The universality of δ is further supported by analysis of real-world networks, such as the co-authorship network, where the load distribution follows a power-law with δ ≈ 2.2. The study also examines the effect of time delay on load distribution, finding that it does not alter the power-law behavior. Additionally, the load distribution in small-world networks, which are not scale-free, is found to follow a different pattern, showing a combination of two Poisson-type decays. In conclusion, the study demonstrates that the load exponent δ ≈ 2.2 is a universal quantity for scale-free networks, independent of the degree exponent γ in the range 2 < γ ≤ 3. This finding has implications for understanding the transport dynamics and robustness of scale-free networks.This study investigates the load distribution in scale-free networks, where the degree distribution follows a power-law with exponent γ. Load, defined as the accumulated number of data packets passing through a vertex, is found to follow a power-law distribution with exponent δ ≈ 2.2, independent of γ in the range 2 < γ ≤ 3 and for both undirected and directed networks. This suggests that δ is a universal quantity for scale-free networks. Scale-free networks are characterized by a power-law degree distribution, common in real-world systems like the internet, social networks, and biological systems. Unlike random networks, scale-free networks have a small-world property, where the average distance between nodes is small, due to the presence of highly connected hubs. These hubs play a critical role in reducing network diameter and are heavily loaded with data packets. The study defines load as the total number of data packets passing through a vertex, and finds that it follows a power-law distribution with exponent δ ≈ 2.2, regardless of the network structure as long as γ is in the range 2 < γ ≤ 3. This universality is confirmed through simulations of both static and evolving scale-free networks, as well as directed networks. The load distribution is also found to be independent of the mean degree of the network. The results show that for γ > 3, the load distribution decays exponentially, indicating a fundamental difference in transport properties between scale-free networks with γ > 3 and those with 2 < γ ≤ 3. The universality of δ is further supported by analysis of real-world networks, such as the co-authorship network, where the load distribution follows a power-law with δ ≈ 2.2. The study also examines the effect of time delay on load distribution, finding that it does not alter the power-law behavior. Additionally, the load distribution in small-world networks, which are not scale-free, is found to follow a different pattern, showing a combination of two Poisson-type decays. In conclusion, the study demonstrates that the load exponent δ ≈ 2.2 is a universal quantity for scale-free networks, independent of the degree exponent γ in the range 2 < γ ≤ 3. This finding has implications for understanding the transport dynamics and robustness of scale-free networks.
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