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Cyber-physical networking systems (CPNSs) are made up of various physical systems that are heterogeneous in nature. Therefore, exploring universalities in CPNSs for either data or systems is desired in its fundamental theory. This paper is in the aspect of data, aiming at addressing that power laws may yet be a universality of data in CPNSs. The contributions of this paper are in triple folds. First, we provide a short tutorial about power laws. Then, we address the power laws related to some physical systems. Finally, we discuss that power-law-type data may be governed by stochastically differential equations of fractional order. As a side product, we present the point of view that the upper bound of data flow at large-time scaling and the small one also follows power laws.

Cyber-physical networking systems (CPNSs) consist of computational and physical elements integrated towards specific tasks [

In general, both data and systems in CPNS are multidimensional. For instance, data from sources to be transmitted may be from a set of sensors distributed in a certain area. Destinations receiving data may be a set of actuators, for example, a set of cars distributed in a certain area. Systems to transmit data are generally distributed.

Denote by

Note that

Without lose of generality, we rewrite (

From the point of view of applications of CPNS, we are interested in two asymptotic expressions of

The rest of paper is organized as follows. Short tutorial about power laws is explained in Section

Denote by

As usual,

The above expressions imply that the integrals in (

A typical heavy-tailed case is the Pareto distribution. Denote by

The Pareto distribution is an instance of power-law-type pdf.

A consequence of a heavy-tailed random variable in ACF is that

Denote by

We now turn to scaling descriptions. Small scaling phenomenon may be investigated by

On the one side, following Davies and Hall [

Fractal dimension is a parameter to characterize small scaling phenomenon (Mandelbrot [

On the other side, if

Statistical dependence, either SRD or LRD, is a property for large scaling phenomenon.

We address some application cases of power laws in CPNS in this section.

Let

It is worth noting that the upper bounds of teletraffic also follow power laws. In fact, the amount of teletraffic accumulated in the interval

Both the small-scale factor and the large one of teletraffic obey power law, that is,

Two scaling factors follow

In addition to teletraffic, others with respect to the Internet also follow power laws. Some are listed below.

Barabasi and Albert [

Let

The probability of web pages among sites is of power law (Huberman and Adamic [

Let

For simplicity, denote a vector by

Let

Sea-level fluctuations, river flow, and flood height follow power laws (Li et al. [

Urban growth obeys power laws (Makse et al. [

Wind engineering is an important field relating to wind power generation and disaster preventions from a view of CPNS. In this field, studying fluctuations of wind speed is essential.

The PSD introduced by von Kármán [^{−1}), and

Another famous PSD in wind engineering is the one introduced by Davenport [^{−1}) measured at height 10 m, ^{−1}) measured at height

The cases of power laws mentioned in the previous section are a few that people may be interested in from a view of CPNS. There are others that are essential in the field of CPNS, such as power laws in earthquake, see for example, the work of Pisarenko and Rodkin in [

Conventionally, a stationary random function

Let

Let

Another class of stochastically differential equations of fractional order is given by (Lim and Muniandy [

Note that (

We have discussed the elements of power laws from both a mathematical point of view and with respect to applications to a number of fields in CPNS. The purpose of this paper is to exhibit that power laws may yet serve as a universality of data in CPNS. We believe that this point of view may be useful for data modeling and analysis in CPNS.

This work was supported partly by the National Natural Science Foundation of China under the project Grant nos 60873264, 61070214, and the 973 plan under the Project no. 2011CB302801/2011CB302802.