崎岖景观模型

Rugged-Landscape Models 崎岖景观模型

（属性的集合）Like spatial and hedonic models, the rugged-landscape model defines an entity as a collection of attributes. Each set of attributes maps to a value. The goal is to modify attributes to construct an entity of highest value.

（属性的相互依赖）We apply the model to reveal how interdependence in the effects of attributes makes innovation difficult and leads to path dependence in the solutions found and also leads to a greater variety of solutions. We also see how more difficult problems benefit from a greater diversity of problem solving approaches.

The Fitness Landscape

（特征参数-适合度）The fitness landscape model assumes species have features or traits that contribute to their fitness, loosely defined as their reproductive potential, and that individual members of a population differ in how much they have of a particular trait. If we plot the amount of the trait on the horizontal axis and the fitness of species on the vertical axis, we produce a graph known as a fitness landscape in which points of high elevation correspond to high fitness.

Rugged Landscapes

（多属性，相互作用）When we allow for multiple attributes and for the contribution of one attribute to interact with those of others, we produce a rugged landscape—that is, a landscape with multiple peaks.
当我们允许多个属性存在，且一个属性的贡献与其他属性发生交互时，就会产生崎岖景观——即拥有多个峰值的景观。

Interdependent effects between variables create ruggedness on the landscape.Ruggedness has several implications. First, different approaches to finding the highest point on a rugged landscape may locate different peaks. So too may different starting points. Thus, ruggedness creates sensitivity to initial conditions and the possibility of path dependence. Each of these implies that landscape ruggedness contributes to outcome diversity. Ruggedness also implies the possibility of suboptimal outcomes. These are represented as local peaks on the landscape.

例如，在我对于主观幸福感的研究中，绝对收入、相对收入、社会公平、婚姻状况等，都具有相关性。而且有些属性是相互作用的，比如绝对收入和社会公平等。如果我们的出发点不同，会得到不同的结论。

（多样性）To find an optimal or near-optimal peak on a rugged landscape requires either diversity or sophistication. The value of diversity should be self-evident. If distinct heuristics locate different peaks, then applying multiple, diverse heuristics to a problem will produce multiple, diverse local peaks, and one can choose the best from among these.1 The same result will occur if one applies the same heuristic from different starting points: distinct local optima will be found and the best among them can be chosen.

The NK Model

The NK model allows us to formalize the connection between interactions and ruggedness.
NK模型使我们能够将“交互作用”与“崎岖性”之间的联系形式化。

An object consists of N bits, s ∈ {0, 1}^N.
一个长度为N的二进制字符串

The value of an object is V (s) = V_k1(s₁, {s_1k}) + V_k2(s₂, {s_2k}) + ··· + V_k2(s₂, {s_2k}) where {s_ik} equals a randomly selected set of k bits other than i, and V_k1(s₁, {s_1k}) is a random number drawn from the interval [0, 1].
K，指的是与该字符串的每一位交互以确定这一位的值的其他位的数量。

K=0: Results in a linear function of the bits.
如果K等于0，则代表没有交互发生，说明这是一个线性的关系。也可以说，每个属性对总价值的贡献独立于其他属性。

K = N − 1: Any bit change produces a new random contribution from each bit.
如果K等于N-1，则所有的位都相互交互，每个字符串的值都是随机的。

（单翻转算法-演化）Single-flip algorithm chooses each attribute in sequence and switches the attribute’s state. If changing this attribute results in a higher value, the switch is adopted. Otherwise the attribute is returned to its original state. The choice of this algorithm can be motivated in two ways. It can be interpreted as a crude model of genetic mutation, where good variants take over in the population and bad ones die. It is also the most natural way to represent a hill-climbing algorithm in this space.

Ruggedness and Dancing Landscapes

（组合学&保持）The NK model implies that we want a moderate degree of interdependence as that creates higher peaks. We find that the logic consists of two parts. The first rests on combinatorics: the number of pairs of combinations increases with the square of the number of pairs and the cube of the number of triples. Thus, interdependent effects create more possibilities of beneficial interactions. The second part rests on the fact that we need only keep the better combinations.

（可能性-变化-发现&随机）When we have an abundance of possibilities, we prefer variation. The same logic applies here:combining pairs (and triples) produces abundant possibilities. And we would have preferred that these many possibilities had high variation in value. We then have a greater likelihood that one of them has a very high value. Given that interdependent effects increase variation, on the whole they are advantageous,but only up to a point. As we have just seen, too many make the landscape random. Ideally, then, we have a moderate number of interactions.

（相互作用的演化-复杂系统）if the number and size of interactions can evolve or adapt, then systems should naturally evolve to rugged landscapes with high peaks. This would suggest that systems tend toward complexity and not equilibrium or randomness.

（适应是全局动态的）We have taken the landscape as fixed. In ecological and social systems, the landscape that a species or firm confronts depends on the actions and attributes of others. An adaptation by a competing species, or a change in strategy by another firm, will shift and rearrange the fitness landscapes of competitors. One reason that we see so much complexity may well be that much of our world consists of adaptive and purposive actors maneuvering on dancing landscapes. To make sense of that complexity, we need many models.