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Universal flux-fluctuation law in small systems.

Huang ZG, Dong JQ, Huang L, Lai YC - Sci Rep (2014)

Bottom Line: For realistic complex systems of small sizes, this law breaks down when both the average flux and fluctuation become large.Here we demonstrate the failure of this law in small systems using real data and model complex networked systems, derive analytically a modified flux-fluctuation law, and validate it through computations of a large number of complex networked systems.Our law is more general in that its predictions agree with numerics and it reduces naturally to the previous law in the limit of large system size, leading to new insights into the flow dynamics in small-size complex systems with significant implications for the statistical and scaling behaviors of small systems, a topic of great recent interest.

View Article: PubMed Central - PubMed

Affiliation: 1] Institute of Computational Physics and Complex Systems and Key Laboratory for Magnetism and Magnetic Materials of MOE, Lanzhou University, Lanzhou 730000, China [2] School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ 85287, USA.

ABSTRACT
The relation between flux and fluctuation is fundamental to complex physical systems that support and transport flows. A recently obtained law predicts monotonous enhancement of fluctuation as the average flux is increased, which in principle is valid but only for large systems. For realistic complex systems of small sizes, this law breaks down when both the average flux and fluctuation become large. Here we demonstrate the failure of this law in small systems using real data and model complex networked systems, derive analytically a modified flux-fluctuation law, and validate it through computations of a large number of complex networked systems. Our law is more general in that its predictions agree with numerics and it reduces naturally to the previous law in the limit of large system size, leading to new insights into the flow dynamics in small-size complex systems with significant implications for the statistical and scaling behaviors of small systems, a topic of great recent interest.

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Related in: MedlinePlus

Flux-fluctuation relationship in small systems.Nodal flux fluctuation σi versus the average flux 〈fi〉 for small networks of size N = 50. The mean degree is 〈k〉 = 4 for (a,b) and 〈k〉 = 2 for (c,d). Panels (e,f) are results from a modified network of mean degree 〈k〉 = 2 but with a super-hub node. In panels (a,c,e) the flux-fluctuation relations are plotted on a logarithmic scale while in panels (b,d,f), a linear scale is used. The dashed curves are obtained from Eq. (1) and the solid curves are predictions from our modified flux-fluctuation law Eq. (2). All systems are under a constant external drive defined by M = 4000.
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f2: Flux-fluctuation relationship in small systems.Nodal flux fluctuation σi versus the average flux 〈fi〉 for small networks of size N = 50. The mean degree is 〈k〉 = 4 for (a,b) and 〈k〉 = 2 for (c,d). Panels (e,f) are results from a modified network of mean degree 〈k〉 = 2 but with a super-hub node. In panels (a,c,e) the flux-fluctuation relations are plotted on a logarithmic scale while in panels (b,d,f), a linear scale is used. The dashed curves are obtained from Eq. (1) and the solid curves are predictions from our modified flux-fluctuation law Eq. (2). All systems are under a constant external drive defined by M = 4000.

Mentions: We next test networked systems of much smaller size, which are ubiquitous in the physical world such as functional biological networks composed of a few proteins, quantum communication networks of a limited number of quantum repeaters, and local ad hoc computer/communication networks supporting a small group of users. For a variety of combinations of (small) network size and mean degree, we obtain essentially the same results as in Fig. 1: large deviations from the previous flux-fluctuation law [Eq. (1)] and excellent agreement with our theory. Representative examples are shown in Fig. 2 for a number of networked systems, all of N = 50 nodes but with different structural parameters. A constant external drive with parameter M = 4000 and the shortest path-length routing protocol are used. Plotted in each panel in Fig. 2 are predictions from Eq. (1) (dashed curves) and from our theory Eq. (2) (solid curves), as well as simulation results (open circles). We see that, for small systems, the disagreement between numerics and Eq. (1) becomes more drastic, especially for smaller mean-degree value. However, in all cases, our new law [Eq. (2)] agrees excellently with the corresponding numerical results, indicating the importance and necessity of including our correctional term [−〈fi〉2/〈M〉] in the flux-fluctuation law for small sparse networked systems.


Universal flux-fluctuation law in small systems.

Huang ZG, Dong JQ, Huang L, Lai YC - Sci Rep (2014)

Flux-fluctuation relationship in small systems.Nodal flux fluctuation σi versus the average flux 〈fi〉 for small networks of size N = 50. The mean degree is 〈k〉 = 4 for (a,b) and 〈k〉 = 2 for (c,d). Panels (e,f) are results from a modified network of mean degree 〈k〉 = 2 but with a super-hub node. In panels (a,c,e) the flux-fluctuation relations are plotted on a logarithmic scale while in panels (b,d,f), a linear scale is used. The dashed curves are obtained from Eq. (1) and the solid curves are predictions from our modified flux-fluctuation law Eq. (2). All systems are under a constant external drive defined by M = 4000.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4209461&req=5

f2: Flux-fluctuation relationship in small systems.Nodal flux fluctuation σi versus the average flux 〈fi〉 for small networks of size N = 50. The mean degree is 〈k〉 = 4 for (a,b) and 〈k〉 = 2 for (c,d). Panels (e,f) are results from a modified network of mean degree 〈k〉 = 2 but with a super-hub node. In panels (a,c,e) the flux-fluctuation relations are plotted on a logarithmic scale while in panels (b,d,f), a linear scale is used. The dashed curves are obtained from Eq. (1) and the solid curves are predictions from our modified flux-fluctuation law Eq. (2). All systems are under a constant external drive defined by M = 4000.
Mentions: We next test networked systems of much smaller size, which are ubiquitous in the physical world such as functional biological networks composed of a few proteins, quantum communication networks of a limited number of quantum repeaters, and local ad hoc computer/communication networks supporting a small group of users. For a variety of combinations of (small) network size and mean degree, we obtain essentially the same results as in Fig. 1: large deviations from the previous flux-fluctuation law [Eq. (1)] and excellent agreement with our theory. Representative examples are shown in Fig. 2 for a number of networked systems, all of N = 50 nodes but with different structural parameters. A constant external drive with parameter M = 4000 and the shortest path-length routing protocol are used. Plotted in each panel in Fig. 2 are predictions from Eq. (1) (dashed curves) and from our theory Eq. (2) (solid curves), as well as simulation results (open circles). We see that, for small systems, the disagreement between numerics and Eq. (1) becomes more drastic, especially for smaller mean-degree value. However, in all cases, our new law [Eq. (2)] agrees excellently with the corresponding numerical results, indicating the importance and necessity of including our correctional term [−〈fi〉2/〈M〉] in the flux-fluctuation law for small sparse networked systems.

Bottom Line: For realistic complex systems of small sizes, this law breaks down when both the average flux and fluctuation become large.Here we demonstrate the failure of this law in small systems using real data and model complex networked systems, derive analytically a modified flux-fluctuation law, and validate it through computations of a large number of complex networked systems.Our law is more general in that its predictions agree with numerics and it reduces naturally to the previous law in the limit of large system size, leading to new insights into the flow dynamics in small-size complex systems with significant implications for the statistical and scaling behaviors of small systems, a topic of great recent interest.

View Article: PubMed Central - PubMed

Affiliation: 1] Institute of Computational Physics and Complex Systems and Key Laboratory for Magnetism and Magnetic Materials of MOE, Lanzhou University, Lanzhou 730000, China [2] School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ 85287, USA.

ABSTRACT
The relation between flux and fluctuation is fundamental to complex physical systems that support and transport flows. A recently obtained law predicts monotonous enhancement of fluctuation as the average flux is increased, which in principle is valid but only for large systems. For realistic complex systems of small sizes, this law breaks down when both the average flux and fluctuation become large. Here we demonstrate the failure of this law in small systems using real data and model complex networked systems, derive analytically a modified flux-fluctuation law, and validate it through computations of a large number of complex networked systems. Our law is more general in that its predictions agree with numerics and it reduces naturally to the previous law in the limit of large system size, leading to new insights into the flow dynamics in small-size complex systems with significant implications for the statistical and scaling behaviors of small systems, a topic of great recent interest.

No MeSH data available.


Related in: MedlinePlus