With the rapid development of complex network science and the complexity of communication network, it is difficult to research its technical application. In order to solve these problems, we firstly design the cellular mobile communication network as a fractal six-star network <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$G_{t,s}$</tex-math></inline-formula> from the view of complex network science. Secondly, we investigate the analytical expression of the mean first-passage time <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M_{G_{t,s}}$</tex-math></inline-formula> of the random walk and the average path length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\left\langle W \right\rangle$</tex-math></inline-formula> . Both of them are proved to be proportional to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$s$</tex-math></inline-formula> . Thirdly, taking into account network transmission efficiency and time delay, the recursive expression of diameter <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$D_{t}$</tex-math></inline-formula> is derived, which increases exponentially with respect to the size of the network. Fourthly, it is verified that the experimental simulations are in perfect consistent with the theoretical analysis, which shows that the designed network is reasonable. Fifthly, to further proving the stability of the six-star network, the robustness of the network is analyzed, which indicates the robustness and density are positively correlated. Finally, the average value of experiments is used to analyze the decisive effect of the number of nodes removed after network failure on robustness, which is of great significance to the research of communication network in reality. 

Mean First-Passage Time and Robustness of Complex Cellular Mobile Communication Network

This study aims to analyze the coherence and properties in generalized Bethe tree networks within a uniform noise distribution. We handle two distinct categories of consensus algorithms within the balanced <inline-formula><tex-math notation="LaTeX">$2^{p}$</tex-math></inline-formula>-ary trees, denoted as <inline-formula><tex-math notation="LaTeX">$B_{k}^{p}$</tex-math></inline-formula>, designed for both leaderless and leader-follower networks. First, we derive the precise expressions for two cases of consensus algorithms based on the regular network topology. Second, we investigate an in-depth analysis of network properties from three perspectives: (1) Kirchhoff index of balanced <inline-formula><tex-math notation="LaTeX">$2^{p}$</tex-math></inline-formula>-ary tree networks. (2) Global average first-passage time of balanced <inline-formula><tex-math notation="LaTeX">$2^{p}$</tex-math></inline-formula>-ary tree networks. (3) The robustness of network coherence. Finally, we further obtain exact solutions for the first two perspectives, gaining insights into the impacts of network parameters on consensus. We reveal: (1) Single-leader networks show better coherence than leaderless networks. (2) Different network parameters result in different consensus distribution patterns, indicating that the topology structures of complex networks significantly influence the robustness of network coherence.

The Coherence and Properties Analysis of Balanced 2
 p
 -Ary Tree Networks

The complexity–entropy curve (CEC) is a valuable tool for characterizing the structure of time series and finds broad application across various research fields. Despite its widespread usage, the original permutation complexity–entropy curve (PCEC), which is founded on permutation entropy (PE), exhibits a notable limitation: its inability to take the means and amplitudes of time series into considerations. This oversight can lead to inaccuracies in differentiating time series. In this paper, drawing inspiration from dispersion entropy (DE), we propose the dispersion complexity–entropy curve (DCEC) to enhance the capability of CEC in uncovering the concealed structures within nonlinear time series. Our approach initiates with simulated data including the logistic map, color noises, and various chaotic systems. The outcomes of our simulated experiments consistently showcase the effectiveness of DCEC in distinguishing nonlinear time series with diverse characteristics. Furthermore, we extend the application of DCEC to real-world data, thereby asserting its practical utility. A novel approach is proposed, wherein DCEC-based feature extraction is combined with multivariate support vector machine for the diagnosis of various types of bearing faults. This combination achieved a high accuracy rate in our experiments. Additionally, we employ DCEC to assess stock indices from different countries and periods, thereby facilitating an analysis of the complexity inherent in financial markets. Our findings reveal significant insights into the dynamic regularities and distinct structures of these indices, offering a novel perspective for analyzing financial time series. Collectively, these applications underscore the potential of DCEC as an effective tool for the nonlinear time series analysis. 

Dispersion complexity–entropy curves: An effective method to characterize the structures of nonlinear time series

Abstract Complexity-entropy curves(CEC) is a useful tool to detect the structure of time series. It’s widely applied in many research areas since it can distinguish the chaotic system and the stochastic process well. However, the original permutation complexity-entropy curves(PCEC) based on permutation entropy(PE) has a defect for it can not take means and amplitudes of time series into consideration, which may lead to some errors when distinguishing the systems. In this paper, we propose dispersion complexity-entropy curves(DCEC) to overcome the defects of PCEC. In addition, we expand the curves from two-dimension to threedimension. We first compare DCEC with PCEC by simulated data. The result shows that DCEC are not only more distinguishable but also more robust with the change of parameters when detecting periodic series. Then, we apply our method to real-world data to illustrate its practicability. We propose a creative feature extraction method based on DCEC and combine it with MSVM to diagnose the different types of bearing fault, which obtains perfect results for the accuracy achieves 100%. We also apply DCEC to stock indices in different countries and different periods to analyze the complexity degree of financial markets. The results successfully detect American financial crisis in 2008 and the rapid development of the economy in China during 2014-2018. These demonstrate that DCEC can serve as an effective method to analyze complex systems. 

Dispersion complexity-entropy curves: an effective method to detect the structures of complex systems

Maximal planar networks with exponential degree distribution

Local differential privacy (LDP for short), as an emerging standard privacy-preserving technique that is suitable for privacy preserving data analysis, has been widely deployed into various real-world scenarios to analyze massive data. In this study, we are mainly concerned with piecewise transformation technique (PTT for short) for analyzing numerical data under LDP. We first provide the principled framework for PTT in the context of LDP. Then, we show that (1) many PTTs are asymptotically optimal when used to obtain an unbiased estimator for mean of numerical data, and (2) there is PTT that reaches the theoretical lower bound on variance. Next, we prove that (1) there does not exist strictly better PTT than the well-used Duchi's scheme in terms of the consistency-noisy-variance, also, the latter is not always better than arbitrary PTT, (2) however, one has the ability to find a great number of PTTs that are consistently better than the latter according to the worst-case-noisy-variance. When we are restricted to the high privacy level <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon \in (0,1]$</tex-math></inline-formula> , enough PTTs turn out to have smaller variance than the well-known Laplace mechanism. Lastly, we demonstrate that for a family of PTTs, the correspondingly theoretical lower bound of noisy variance follows <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(\epsilon ^{-2})$</tex-math></inline-formula> when considering <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon \in (0,1]$</tex-math></inline-formula> . 

PTT: Piecewise Transformation Technique for Analyzing Numerical Data under Local Differential Privacy

The family of planar graphs is a particularly important family and models many real-world networks. In this paper, we propose a principled framework based on the widely-known Apollonian packing process to generate new planar network, i.e., Type-II Apollonian network $\mathcal{A}_{t}$. The manipulation is different from that of the typical Apollonian network, and is proceeded in terms of the iterative addition of triangle instead of vertex. As a consequence, network $\mathcal{A}_{t}$ turns out to be hamiltonian and eulerian, however, the typical Apollonian network is not. Then, we in-depth study some fundamental structural properties on network $\mathcal{A}_{t}$, and verify that network $\mathcal{A}_{t}$ is sparse like most real-world networks, has scale-free feature and small-world property, and exhibits disassortative mixing structure. Next, we design an effective algorithm for solving the problem of how to enumerate spanning trees on network $\mathcal{A}_{t}$, and derive the asymptotic solution of the spanning tree entropy, which suggests that Type-II Apollonian network is more reliable to a random removal of edges than the typical Apollonian network. Additionally, we study trapping problem on network $\mathcal{A}_{t}$, and use average trapping time as metric to show that Type-II Apollonian network $\mathcal{A}_{t}$ has better structure for fast information diffusion than the typical Apollonian network.

Type-II Apollonian Model

In this article, we propose a principled framework using Vertex-based and Edge-based uniform generation mechanisms to build stochastic uniform growth tree networks that have a wide range of applications in various fields including physics, engineering, chemistry, ect., and then uncover the associated structural features analytically. When considering vertex-degree distribution, there exist three different classes of forms in the thermodynamic limit, i.e., exponential distribution, power-law distribution along with multiple-point distribution. At meantime, three distinct structural shapes are observed in the study of fractal phenomena, that is, fractal feature, critical phenomenon and non-fractal property. In addition, we obtain the analytical solution to fractal dimension for fractal structure from the probability point of view. More importantly, some well-known models, for instance, Vicsek fractal and T-graph, fall into our framework. Next, we precisely consider two families of stochastic uniform growth tree networks generated through the proposed framework. Specifically, we derive the analytic solution to mean hitting time <inline-formula><tex-math notation="LaTeX">$\langle \mathcal {H}\rangle$</tex-math><alternatives><mml:math><mml:mrow><mml:mo>〈</mml:mo><mml:mi mathvariant="script">H</mml:mi><mml:mo>〉</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="ma-ieq1-3206210.gif"/></alternatives></inline-formula> for measuring efficiency of delivering information on networks in a random-walk-based manner, and find that the introduction of randomness certainly enriches the scaling exponent of quantity <inline-formula><tex-math notation="LaTeX">$\langle \mathcal {H}\rangle$</tex-math><alternatives><mml:math><mml:mrow><mml:mo>〈</mml:mo><mml:mi mathvariant="script">H</mml:mi><mml:mo>〉</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="ma-ieq2-3206210.gif"/></alternatives></inline-formula>. Finally, we conduct extensive experiments, which suggests that computer simulations are in good agreement with theoretical analysis.

Structure Diversity and Mean Hitting Time for Random Walks on Stochastic Uniform Growth Tree Networks

As known, the commonly-utilized ways to determine mean first-passage time <inline-formula><tex-math notation="LaTeX">$\overline{\mathcal{F}}$</tex-math><alternatives><mml:math><mml:mover><mml:mi mathvariant="script">F</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:math><inline-graphic xlink:href="ma-ieq1-3392786.gif"/></alternatives></inline-formula> for random walk on networks are mainly based on Laplacian spectra. However, methods of this type can become prohibitively complicated and even fail to work when the Laplacian matrix of network under consideration is difficult to describe in the first place. In this paper, we propose an effective approach to determining quantity <inline-formula><tex-math notation="LaTeX">$\overline{\mathcal{F}}$</tex-math><alternatives><mml:math><mml:mover><mml:mi mathvariant="script">F</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:math><inline-graphic xlink:href="ma-ieq2-3392786.gif"/></alternatives></inline-formula> on some widely-studied tree networks. To this end, we first build up a general formula between Wiener index <inline-formula><tex-math notation="LaTeX">$\mathcal{W}$</tex-math><alternatives><mml:math><mml:mi mathvariant="script">W</mml:mi></mml:math><inline-graphic xlink:href="ma-ieq3-3392786.gif"/></alternatives></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\overline{\mathcal{F}}$</tex-math><alternatives><mml:math><mml:mover><mml:mi mathvariant="script">F</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:math><inline-graphic xlink:href="ma-ieq4-3392786.gif"/></alternatives></inline-formula> on a tree. This enables us to convert issues to answer into calculation of <inline-formula><tex-math notation="LaTeX">$\mathcal{W}$</tex-math><alternatives><mml:math><mml:mi mathvariant="script">W</mml:mi></mml:math><inline-graphic xlink:href="ma-ieq5-3392786.gif"/></alternatives></inline-formula> on networks in question. As opposed to most of previous work focusing on deterministic growth trees, our goal is to consider stochastic case. Towards this end, we establish a principled framework where randomness is introduced into the process of growing trees. As an immediate consequence, the previously published results upon deterministic cases are thoroughly covered by formulas established in this paper. Additionally, it is also straightforward to obtain Kirchhoff index on our tree networks using the proposed approach. Most importantly, our approach is more manageable than some other methods including spectral technique in situations considered herein.

Determining mean first-passage time for random walks on stochastic uniform growth tree networks

There is little attention paid to stochastic tree networks in comparison with the corresponding deterministic analogs in the current study of fractal trees. In this paper, we propose a principled framework for producing a family of stochastic growth tree networks T possessing fractal characteristic, where t represents the time step and parameter m is the number of vertices newly created for each existing vertex at generation. To this end, we introduce two types of generative ways, i.e., Edge-Operation and Edge-Vertex-Operation. More interestingly, the resulting stochastic trees turn out to have an identical fractal dimension d = ln ⁡ 2 ( m + 1 ) / ln ⁡ 2 regardless of the introduction of randomness in the growth process. At the same time, we also study many other structural parameters including diameter and degree distribution. In both extreme cases, our tree networks are deterministic and follow multiple-point degree distribution and power-law degree distribution, respectively. Additionally, we consider random walks on stochastic growth tree networks T and derive an expectation estimation for mean hitting time ⟨ H ⟩ in an effective combinatorial manner instead of commonly used spectral methods. The result shows that on average, the scaling of mean hitting time ⟨ H ⟩ obeys ⟨ H ⟩ = | T |, where | T | represents vertex number and exponent λ is equivalent to 1 + ln ⁡ 2 / ln ⁡ 2 ( m + 1 ). In the meantime, we conduct extensive experimental simulations and observe that empirical analysis is in strong agreement with theoretical results.

Stochastic growth tree networks with an identical fractal dimension: Construction and mean hitting time for random walks.

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