Abstract
The level and nature of complexity is widely regarded as an important determinant of a number of economic, technological and organizational phenomena. A popular modeling tool for the representation of complexity in economics and organizational sciences is the NK model that represents the complexity stemming from the interactions among the elements of a system. This paper proposes an enhanced model for complexity that, though maintaining the core design (and properties) of the NK model, provides a more intuitive and richer representation of complexity, extending its applications and deepening the understanding of its effects on economic systems. The proposed pseudo-NK (pNK) model is defined on real-valued variables, as opposed to the binary variables required by NK, so as to allow for richer and more intuitive definitions of distance and search strategies. It also admits as a source of complexity not only the number of interactions, as in NK, but also their intensity, opening a novel way to express and measure the level of complexity. Finally, instead of relying on statistical properties of a large dataset of random values, pNK is defined as a deterministic function, far simpler to implement, to interpret and to calibrate for specific requirements. The paper replicates known results and presents original ones; in both cases, the proposed model proves a powerful tool for the investigation of the role of complexity, particularly in agent-based models.
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Notes
Altenberg (1997) is credited as the first to propose a solution to this problem.
Actually, it is possible to extend the model to admit multiple global maxima, though we will ignore this possibility.
Here we adopt a specific functional form providing the properties required to pNK; however, there is a whole class of functions that may be used, depending on specific requirements.
The proposed model can easily be modified changing the functional form for Eqs. 2 and 3. The same properties of the model discussed here are maintained provided that \(\mu _{i}(\vec {x})\) is a positive function of \(a_{i,j} x_{j}\) and that \(\phi _{i}(\vec {x})\) is an inverse function of \(|x_{i} - \mu _{i}(\vec {x})|\). Using different functional forms for these elements will change the density of local peaks and the overall shape of the landscape, though maintaining the same properties discussed here.
Different functional forms can be generated by altering the expression of the fitness contributions \(\phi _{i}\). The only requirement is that the \(\phi _{i}\) is inverted related to the difference \(|\mu _{i}-x_{i}|\).
All the experiments reported in the paper are produced with simulation programs implemented with Laboratory for Simulation Development-LSD (Valente 2008). The LSD platform is available for download at www.labsimdev.org. The code for the models and the specific exercises, together for the instructions on their use, is available upon request.
Valleys, trivially, lead always to the global optimum, so we need not consider these areas.
Tests with different values of Δ showed the irrelevance of the value chosen for this parameter, but for the level of detail of the graphs. The value used for the simulations is 0.05, so that the portion of the graphs shown (from 98 to 102 on both dimensions) is effectively considered as a square lattice made of 80 units on each side and composed of 6,400 points. Therefore, the 30,000 runs generate, on average, about five random searches started from each point of the landscape.
In NK, the fitness is computed out of a sample of random numbers the dimension of which is proportional to 2K. Hence, rougher landscapes use a larger sample than smoother ones, increasing the probability of finding a higher value.
We do not consider the speed of research of the two research strategies, that is, the number of steps required on average to reach the global optimum. In fact, we may expect that the greedy strategy is faster than the random one, requiring a smaller number of steps to reach the eventual peak. In the following section, we will discuss the issue of speed of research showing how pNK is able, contrary to NK, to provide an intuitively appropriate representation of the length of research patterns.
Such a random structure actually leads quickly to the production of fully uncorrelated structures, even for relatively low K values (Frenken et al. 1999).
For simplicity, we consider only C values that are integer divisors of N, though the results would not change otherwise. It would only mean that the last class would contain a smaller number of variables than the other classes.
For obvious reasons of comparability, we assume that each interdependency in pNK is maximum, i.e. \(|a_{i,j}|=1\) for each i and j dimensions within the same block.
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This paper was presented to the International Schumpeter Society Conference, 2010, Aalborg (DK). I wish to thank Tommaso Ciarli for valuable comments and suggestions during the development of the model. Luigi Marengo, Lee Altenberg and an anonymous referee provided encouraging comments and corrections to earlier versions. Any remaining error or imprecision is, obviously, my sole responsibility.
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Valente, M. An NK-like model for complexity. J Evol Econ 24, 107–134 (2014). https://doi.org/10.1007/s00191-013-0334-4
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DOI: https://doi.org/10.1007/s00191-013-0334-4