计算机代写|复杂网络代写complex network代考|Kauffman’s Hypothesis

Doug I. Jones

Doug I. Jones

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计算机代写|复杂网络代写complex network代考|Kauffman’s Hypothesis

计算机代写|复杂网络代写complex network代考|Kauffman’s Hypothesis

The main characteristic of the chaotic behaviour of physical dynamical systems is their high sensitivity to small perturbation of the initial conditions that leads to very different final effects. In other words, for a very similar initial conditions, the system evolution is unpredictable and can result in a widely different outcome. Kauffman [17-19] uses the term chaos to describe the behaviour of finite, discrete, deterministic networks. Note, that contemporary theories use the definition of chaos based on the Lyapunov coefficients for functions in infinite, continuous spaces. The term (deterministic) chaos is not reserved for just one of the continuous/infinite versus discrete/finite systems, but can, and has been applied to both. Specifically, chaos theory can and has been used for finite discrete networks $[2,3]$. For different experiments different assumptions about the nature of the underlying (finite and discrete) process can be made. Those assumptions sometimes involve using network with infinite number of nodes, as well as relying on certain specific ways of calculating the nodes value with the use of probability distributions, e.g. [9], or asynchronous function updates [5]. Some assumptions, e.g. in relation to asynchronous update function activation allow researchers to use Lyapunov exponents and Jacobians in their analysis, e.g. [4]. Note however, that by making some of those assumptions the researchers have missed some properties that are inherent in finite discrete networks, but absent in the context of networks with infinite and continuous functions. For example, in the traditional chaos theory for networks with continuous functions it does not make sense to define the probability of an event where a given node in a network obtains a specific input values (node inputs) that has already happened in the past. In contrast, such a probability can be defined and measured for finite and discrete networks, see for example [1]. Analogously, the concept of the attractor length, or path length to an attractor can be defined in discrete finite systems, but cannot be defined in the context of continuous space. In discrete step-wise synchronous systems it can be simply defined as the number of transformation steps in a process. This is an important difference and it allows investigating real world phenomena that map to finite and discrete networks in a more detailed way in comparison to traditional chaos theory based on continuous spaces. Current modelling methodologies assume, that the finite and discrete networks can be either stable (i.e. ordered) or unstable (i.e. chaotic). By slightly changing the network parameters, a fast jump between chaos and order called a phase transition appears. Only for network parameters in the immediate vicinity of this phase transition, the changes are not too small and not too large, to have properties suitable for describing stability of adaptive evolution of modelled realworld objects. This is commonly known as Kauffman’s hypothesis and captured by a succinct phrase: life evolves on the edge of chaos. In this article we describe our experimental finding of half-chaotic networks. A half-chaotic network will exhibit both, very large and also very small changes for similar disturbances. This range of network behaviours is not taken into account by the current modelling methodologies. The discovery of half-chaotic regimes is not a refutation of the existing theories. It provides an evidence that in some areas the current theory is not adequate for describing complexity of the discrete systems that are being modelled. Similarly, Newtonian mechanics is not false, but provides false expectations when used to predict behaviour for objects moving near the speed of light.

计算机代写|复杂网络代写complex network代考|Experiments and Network Parameters

The experimental basis used to classify a particular network as chaotic or ordered is the distribution of damage size, i.e. size distribution of avalanches [27] caused by small disturbances. Figure 1 depicts the levels of damage equilibrium (maximum damage, $d m x$ ) calculated for chaotic networks from Derrida’s annealed approximation model $[8,9]$. Chaotic networks are not suitable for modelling most of the interesting real-world complex systems. Half-chaotic networks with parameters $s$ and $K$ can be used for modelling real-world complex systems given the existence of the order they exhibit (see the middle of Fig.2). In the case of the random network the chaos area is determined by the parameters $s$ and $K$, but half-chaotic networks with such parameters already show enough order (see Fig. 2) to be used for modelling. In Fig. 2, there are two peaks in the damage distribution. The right is near a Derrida equilibrium level [11]; it contains the results of chaotic responses to small disturbances. The left peak contains damage so minimal that they are taken as ordered. The shares of ordered and chaotic responses are shown in the middle of Fig. 2. For typical ordered or chaotic systems, there is only one of the peaks respectively (very similar to one of shown in Fig. 2). Thus, systems exhibiting distributions depicted in Fig. 2 are neither ordered nor chaotic. We call them half-chaotic. Those systems are distinct from ordered or chaotic systems, that were the only two states previously assumed under the Kauffman’s models. The existence of this third state for discrete systems was unknown before our experimental work.

In this work and in our experiments we have used non-Boolean Kauffman networks, i.e. we continue to use the term Kauffman network even for networks with $s>2$. The experiments are run on networks with $N=400$ nodes; each node has $K$ inputs and $k$ outputs. $K$ is fixed to 3 subject to a specific network configuration. Note that $k$ may differ for different nodes. Signals on inputs and outputs have $s=4$ equally probable variants. Kauffman used only $s=2$ and his networks were Boolean. In random Boolean network the phase transition between order and chaos appears at $K=2$. Other researchers have attempted to introduce more, yet not equally probable, signal states, see for example $[20$, 24]. This was done for the ordered phase of fully random networks. We use connectivity $K>2$, as has been suggested earlier by other researchers in [3,25], and $s>2$, which for random networks practically always results in chaos. Note, that we use networks that are not fully random, in half-chaos, which also offers the ordered peak on the left side of the figure, and are much more adequate for describing real-world phenomena $[11,12]$. A node, using its function and input signals, calculates its state – output signal, and then it sends the output signal to all $k$ of its outputs. The calculation takes a single time step: if the input signals are from time $t$, then the new state is sent to the receivers at time $t+1$. This type of calculation of network function is called synchronous.

计算机代写|复杂网络代写complex network代考|Kauffman’s Hypothesis


计算机代写|复杂网络代写complex network代考|Kauffman’s Hypothesis


计算机代写|复杂网络代写complex network代考|Experiments and Network Parameters

将特定网络划分为混沌或有序的实验依据是损伤大小的分布,即由小扰动引起的雪崩的大小分布[27]。图1描述了根据Derrida的退火近似模型$[8,9]$计算的混沌网络的损害平衡水平(最大损害,$d m x$)。混沌网络不适合建模大多数有趣的现实世界的复杂系统。具有参数$s$和$K$的半混沌网络可以用于模拟现实世界的复杂系统,只要它们表现出一定的秩序(见图2中间)。在随机网络的情况下,混沌区域由参数$s$和$K$决定,但具有这些参数的半混沌网络已经显示出足够的有序(见图2),可以用于建模。在图2中,损伤分布有两个峰。右侧接近德里达均衡水平[11];它包含了对小扰动的混沌响应的结果。左边的峰值包含的伤害非常小,所以它们被认为是有序的。有序响应和混沌响应的份额如图2中间所示。对于典型的有序或混沌系统,分别只有一个峰值(与图2中所示的一个非常相似)。因此,呈现图2中所示分布的系统既不是有序的,也不是混沌的。我们称之为半混沌。这些系统不同于有序系统或混沌系统,这是考夫曼模型之前假设的仅有的两种状态。在我们的实验工作之前,离散系统的第三种状态的存在是未知的。


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术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。



有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。





随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。


多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


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