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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.

No MeSH data available.


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Empirical and numerical evidences for the breakdown of the previous flux-fluctuation law Eq. (1).Nodal flux fluctuation σi versus the average flux 〈fi〉, (a) and (b) for real data from microchips1, and (c,d) and (e,f) for the numerical data from heterogeneous networks on logarithmic and linear scales, respectively. The dashed and solid curves are the predictions of the previous flux-fluctuation law Eq. (1) and our new theory Eq. (2), respectively. And the dotted curves in (a,b) are fitting of real data which can be explained by our theory. Numerical simulations are realized on scale-free networks of size N = 2000 and mean degree 〈k〉 = 2, under constant external driving M = 4000. The networks of (c,e) and (d,f) are schematically illustrated in (g) and (h), respectively.
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f1: Empirical and numerical evidences for the breakdown of the previous flux-fluctuation law Eq. (1).Nodal flux fluctuation σi versus the average flux 〈fi〉, (a) and (b) for real data from microchips1, and (c,d) and (e,f) for the numerical data from heterogeneous networks on logarithmic and linear scales, respectively. The dashed and solid curves are the predictions of the previous flux-fluctuation law Eq. (1) and our new theory Eq. (2), respectively. And the dotted curves in (a,b) are fitting of real data which can be explained by our theory. Numerical simulations are realized on scale-free networks of size N = 2000 and mean degree 〈k〉 = 2, under constant external driving M = 4000. The networks of (c,e) and (d,f) are schematically illustrated in (g) and (h), respectively.

Mentions: In spite of the success of Eq. (1) in explaining the flux-fluctuation relation observed from certain real systems, there exists a paradox. In particular, Eq. (1) implies that, as the flux increases continuously, so would the fluctuation. Consider, for example, a complex network of small size, where the total amount of traffic, or flux, is finite. In such a system, the traffic flow through various nodes will exhibit different amount of fluctuations, depending on the corresponding flux. In the special but not unlikely case where most of the traffic flow passes through a dominant node in the network, the flux is large but the fluctuation observed from it must be small, since the total amount of traffic is fixed. For other nodes in the network the opposite would occur. Thus, in a strict sense, Eq. (1) is applicable only to physical systems of infinite size with less flux heterogeneity. An example of the failure of Eq. (1) for a small system is demonstrated in Fig. 1 panels (a) and (b), where the flux-fluctuation relation obtained from a real Microchip system1 is shown. We see that, on a logarithmic scale [see panel (a)], the flux-fluctuation behavior from the real data (open diamonds) appears to mostly agree with the prediction of Eq. (1) (dashed curve). However, as indicated in panel (b), on a linear scale there is substantial deviation of the real behavior from the prediction of Eq. (1). A careful examination of the flux-fluctuation relation for model network systems, especially for small systems with weak external driving or those with a high nodal flux heterogeneity, reveals that the deviation from the theoretical prediction is systematic. Curiosity demands that we ask the following question: in small complex physical system, is there a general flux-fluctuation law?


Universal flux-fluctuation law in small systems.

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

Empirical and numerical evidences for the breakdown of the previous flux-fluctuation law Eq. (1).Nodal flux fluctuation σi versus the average flux 〈fi〉, (a) and (b) for real data from microchips1, and (c,d) and (e,f) for the numerical data from heterogeneous networks on logarithmic and linear scales, respectively. The dashed and solid curves are the predictions of the previous flux-fluctuation law Eq. (1) and our new theory Eq. (2), respectively. And the dotted curves in (a,b) are fitting of real data which can be explained by our theory. Numerical simulations are realized on scale-free networks of size N = 2000 and mean degree 〈k〉 = 2, under constant external driving M = 4000. The networks of (c,e) and (d,f) are schematically illustrated in (g) and (h), respectively.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f1: Empirical and numerical evidences for the breakdown of the previous flux-fluctuation law Eq. (1).Nodal flux fluctuation σi versus the average flux 〈fi〉, (a) and (b) for real data from microchips1, and (c,d) and (e,f) for the numerical data from heterogeneous networks on logarithmic and linear scales, respectively. The dashed and solid curves are the predictions of the previous flux-fluctuation law Eq. (1) and our new theory Eq. (2), respectively. And the dotted curves in (a,b) are fitting of real data which can be explained by our theory. Numerical simulations are realized on scale-free networks of size N = 2000 and mean degree 〈k〉 = 2, under constant external driving M = 4000. The networks of (c,e) and (d,f) are schematically illustrated in (g) and (h), respectively.
Mentions: In spite of the success of Eq. (1) in explaining the flux-fluctuation relation observed from certain real systems, there exists a paradox. In particular, Eq. (1) implies that, as the flux increases continuously, so would the fluctuation. Consider, for example, a complex network of small size, where the total amount of traffic, or flux, is finite. In such a system, the traffic flow through various nodes will exhibit different amount of fluctuations, depending on the corresponding flux. In the special but not unlikely case where most of the traffic flow passes through a dominant node in the network, the flux is large but the fluctuation observed from it must be small, since the total amount of traffic is fixed. For other nodes in the network the opposite would occur. Thus, in a strict sense, Eq. (1) is applicable only to physical systems of infinite size with less flux heterogeneity. An example of the failure of Eq. (1) for a small system is demonstrated in Fig. 1 panels (a) and (b), where the flux-fluctuation relation obtained from a real Microchip system1 is shown. We see that, on a logarithmic scale [see panel (a)], the flux-fluctuation behavior from the real data (open diamonds) appears to mostly agree with the prediction of Eq. (1) (dashed curve). However, as indicated in panel (b), on a linear scale there is substantial deviation of the real behavior from the prediction of Eq. (1). A careful examination of the flux-fluctuation relation for model network systems, especially for small systems with weak external driving or those with a high nodal flux heterogeneity, reveals that the deviation from the theoretical prediction is systematic. Curiosity demands that we ask the following question: in small complex physical system, is there a general flux-fluctuation law?

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