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

#### 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

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.

# 复杂网络代写

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