pNK: A model for Complexity - UFPR pNK...Introduction pNK Fitness Landscape 2-D pNK model generic pNK model Literature applications pNK: A model for Complexity Marco VALENTE1 1LEM,

Introduction pNK Fitness Landscape 2-D pNK model generic pNK model Literature applications

pNK : A model for Complexity

Marco VALENTE1

1LEM, S. Anna School of Advanced Studies, Pisa.Science and Policy Research Unit, University of Sussex.

University of L’Aquila.


Introduction

Models for complexity are used to investigate how differentsolving strategies perform on different classes of problems.

A model for complexity is made of two modules:

Problem space: a set of potential solutions eachassociated to a performance value. Typically representedby a multimensional space is associated to a fitness value.Search strategy: a routine representing the strategyadopted by an individual exploring the problem spacesearching for ever higher fitness values, i.e. bettersolutions.


Introduction

Models for complexity are used to investigate how differentsolving strategies perform on different classes of problems.

A model for complexity is made of two modules:

Problem space: a set of potential solutions eachassociated to a performance value. Typically representedby a multimensional space is associated to a fitness value.Search strategy: a routine representing the strategyadopted by an individual exploring the problem spacesearching for ever higher fitness values, i.e. bettersolutions.


Introduction

NK is the first model to represent explicitly the complexity asstemming from the interaction among elements of a system,and considers bounded rational and myopic strategies.

The agents facing the problem do not know the performance ofall the possibilities, but only those of the currently occupiedlocation and of its immediate neighbours.


Outline

Present a definition of complexity.Present a model for complexity on 1-, 2-, andN-dimensions.Discuss general results.Summarize results of applications to economics ofinnovation and organizational studies.


NK Fitness landscapes

Kauffman proposed NK as a metaphor for biological evolution.Strings of binary values (the DNA space) of length N areassigned random values representing the fitness of the speciesassociated to each DNA.

Each generation brings the mutation of one bit of the DNA. Theold species is replaced by the new one only if the mutationbrings a fitness increment.

Evolution leads to global optimum if the dimensions (genes) ofthe DNA space do not interact. Increasing the number ofinteractions (K ) among genes it is more and more likely toencounter local optima, sub-optimal species neighbouring onlyinferior ones.













Formally, we can represent the fitness landscape as follows.The problem space is represented by N variables:~x = x1, x2, ..., xN.

The fitness function associates a real-valued performancemeasure for each point of the landscape: f (~x) ∈ [0,1].

Taking any two points x1, x2 we then have either f (~x1) > f (~x2) orf (~x1) < f (~x2).

There is a global optimum which is a point ~x∗ producing thehighest fitness of all f (~x∗) = 1.





















A search strategy is rule stating how an artificial agent decidesto move assuming it does not have a global knowledge of theproblem space, but can only experiment the fitness value ofone point at a time.

The goal of the agent is to reach the highest possible fitnessvalue.



Agents are assumed to have a myopic and bounded rationalapproach.

This means they can only observe points in the immediateneighborhood of the point they are currently on.

Moreover, they use a simply rule of thumb: switch to a newpoint only if the new point has higher fitness then the currentone.













Assuming the space is one-dimension the agents apply thefollowing strategy:

1 Choose an initial point of the space randomly x0 for thestep t = 0.

2 Compute the fitness of the current point ft = f (xt ).3 Choose randomly one of two points: xt + ∆ or xt −∆. Call

this point the experimental solution xt .4 Compute the fitness of the fitness of the experimental

solution with that of the currently held one. If f (xt ) > ft thanset xt+1 = xt . Otherwise, set xt+1 = xt .

5 Set t = t + 1. Return to 2.





2 Compute the fitness of the current point ft = f (xt ).

3 Choose randomly one of two points: xt + ∆ or xt −∆. Callthis point the experimental solution xt .

4 Compute the fitness of the fitness of the experimentalsolution with that of the currently held one. If f (xt ) > ft thanset xt+1 = xt . Otherwise, set xt+1 = xt .







this point the experimental solution xt .

4 Compute the fitness of the fitness of the experimentalsolution with that of the currently held one. If f (xt ) > ft thanset xt+1 = xt . Otherwise, set xt+1 = xt .




















The search strategy generates a pattern starting from arandomly chosen point and concluding to a peak, a point whereno more fitness-improving steps are possible:

x0, x1, ..., xt , ..., xΩ

The performance of a strategy is measured by the averagevalue over many repetitions of the final points reached.

The performance varies depending on the shape of the fitnesslandscape.



The search strategy generates a pattern starting from arandomly chosen point and concluding to a peak, a point whereno more fitness-improving steps are possible:

x0, x1, ..., xt , ..., xΩ

The performance of a strategy is measured by the averagevalue over many repetitions of the final points reached.

The performance varies depending on the shape of the fitnesslandscape.


1-D pNK landscape

Introduzione Complessita’ Complessita’ in 2-D Casualita’ ed ingordigia Complessita’ in N-D

1D landscape - single peak

-5 -4 -3 -2 -1 0 1 2 3 4 5

1


1-D pNK landscape

Introduzione Complessita’ Complessita’ in 2-D Casualita’ ed ingordigia Complessita’ in N-D

1D landscape - multiple peaks

-5 -4 -3 -2 -1 0 1 2 3 4 5

1



Clearly, the performance depends on the starting point.

We call local optima peaks different from the globally highestfitness points. We call basin of attraction the set of pointsfrom which you can reach a peak.



We can expand the same issue on a multi-dimensional space.The strategy becomes:

1 Choose an initial point of the space randomly ~x0 for thestep t = 0.

2 Compute the fitness of the current point ft = f (~xt ).3 Choose randomly one dimension, i .4 Choose randomly one of the two points where all

dimensions are kept constant, and the i th is set to: xi,t + ∆

or xi,t −∆. Call this point the experimental solution ~xt .5 Compute the fitness of the fitness of the experimental

solution with that of the currently held one. If f (~xt ) > ft thanset ~xt+1 = ~xt . Otherwise, set ~xt+1 = ~xt .

6 set t = t + 1. Return to 2.





2 Compute the fitness of the current point ft = f (~xt ).

3 Choose randomly one dimension, i .4 Choose randomly one of the two points where all











or xi,t −∆. Call this point the experimental solution ~xt .

5 Compute the fitness of the fitness of the experimentalsolution with that of the currently held one. If f (~xt ) > ft thanset ~xt+1 = ~xt . Otherwise, set ~xt+1 = ~xt .






















We can build a generic multi-dimensional fitness functionallowing to determine the complexity of the problem space asexplored by a one-bit mutation search strategy.

The complexity is introduced by tuning interaction betweendimensions. Interaction between i and j implies that whether itis better to increase or decrease a variable depends on thevalue of the other variable.

The intensity of interaction expresses the range of valuesfor which one variable is affected by the value of the othervariables.












pNK structure

Formally, a complex fitness landscape can be defined asfollows.The fitness function f (~x) : ~x ∈ <N → [0,M] is defined as theaverage of N fitness contributions φi (~x), one for eachdimension i of the problem space:

f (~x) =

∑Ni=1 φi (~x)

N

φi(~x) =M

(1 + |xi − µi(~x)|)

µi(~x) = ci +N∑

j=1

ai,jxj


pNK structure


f (~x) =

∑Ni=1 φi (~x)

N

φi(~x) =M

(1 + |xi − µi(~x)|)

µi(~x) = ci +N∑

j=1

ai,jxj


pNK structure


f (~x) =

∑Ni=1 φi (~x)

N

φi(~x) =M

(1 + |xi − µi(~x)|)

µi(~x) = ci +N∑

j=1

ai,jxj


pNK structure

The parameter M determines the maximum fitness value.Parameters ci determines the optimum ~x∗ : f (~x∗) = M.The coefficients ai,j ∈ [0,1] determine the influence ofdimension j on the contribution of dimension i .

f (~x) =

∑Ni=1 φi (~x)

N

φi(~x) =M

(1 + |xi − µi(~x)|)

µi(~x) = ci +N∑

j=1

ai,jxj


2-D pNK landscape

Example of fitness landscape

98 98.5

99 99.5

100 100.5

101 101.5

102X1 98

98.5

99

99.5

100

100.5

101

101.5

102

X2

0.3

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0.5

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0.9

1


Landscape ai ,j = 0

0.3

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1

Fitness |a|=0

98 98.5 99 99.5 100 100.5 101 101.5 102 98

98.5

99

99.5

100

100.5

101

101.5

102


Landscape ai ,j = 0.25

0.3

0.4

0.5

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0.9

1

Fitness |a|=0.25

98 98.5 99 99.5 100 100.5 101 101.5 102 98

98.5

99

99.5

100

100.5

101

101.5

102



0.3

0.4

0.5

0.6

0.7

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1

Fitness |a|=0.50

98 98.5 99 99.5 100 100.5 101 101.5 102 98

98.5

99

99.5

100

100.5

101

101.5

102



0.3

0.4

0.5

0.6

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1

Fitness |a|=0.75

98 98.5 99 99.5 100 100.5 101 101.5 102 98

98.5

99

99.5

100

100.5

101

101.5

102


Landscape ai ,j = 1

0.3

0.4

0.5

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0.9

1

Fitness |a|=1.00

98 98.5 99 99.5 100 100.5 101 101.5 102 98

98.5

99

99.5

100

100.5

101

101.5

102


One-dimensional search strategy

The simplest search strategy generates a sequence of stepsproducing a pattern of points starting from an initial one andending on a peak, a point from which no other step can beperformed.

Each step operates the following procedure:

1 Choose randomly one dimension.2 Make a step ∆ in one direction on the chosen dimension.3 If the fitness increases, move to the new point.4 If the fitness decreases, stay in the same point.


One-dimensional search strategy

The simplest search strategy generates a sequence of stepsproducing a pattern of points starting from an initial one andending on a peak, a point from which no other step can beperformed.

Each step operates the following procedure:

1 Choose randomly one dimension.2 Make a step ∆ in one direction on the chosen dimension.3 If the fitness increases, move to the new point.4 If the fitness decreases, stay in the same point.


Searching landscapes

We can investigate the performance of the search strategybased on a single dimension (one-bit mutation) on landscapeswith different levels of interaction between the dimensions.

We compute the (average) final fitness value for searchesstarting from each point of the landscape.


Final fitness without interactions: ai ,j = 0.0

Fitness landscapes with no interactions between thedimensions allow always to reach the maximum fitness point,independently from the starting point.

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fitness |a|=0

98 98.5 99 99.5 100 100.5 101 101.5 102 98

98.5

99

99.5

100

100.5

101

101.5

102


Final fitness with complexity

When the dimensions are subject to even a slight degree ofinteraction only a search starting from certain areas (orextremely lucky ones) manage to reach the top values.

In other cases the pattern may get trapped on a “ridge” whereboth horizontal or vertical moves generate a fitness fall.Therefore, they constitute local optima.

The stronger the interaction the larger the basins of attraction oflocal optima.


Final fitness with complexity: ai ,j = 0.25

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fitness |a|=0.25

98 98.5 99 99.5 100 100.5 101 101.5 102 98

98.5

99

99.5

100

100.5

101

101.5

102

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Random search |a|=0.25

98 98.5 99 99.5 100 100.5 101 101.5 102 98

98.5

99

99.5

100

100.5

101

101.5

102



0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fitness |a|=0.50

98 98.5 99 99.5 100 100.5 101 101.5 102 98

98.5

99

99.5

100

100.5

101

101.5

102

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1


98 98.5 99 99.5 100 100.5 101 101.5 102 98

98.5

99

99.5

100

100.5

101

101.5

102



0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fitness |a|=0.75

98 98.5 99 99.5 100 100.5 101 101.5 102 98

98.5

99

99.5

100

100.5

101

101.5

102

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1


98 98.5 99 99.5 100 100.5 101 101.5 102 98

98.5

99

99.5

100

100.5

101

101.5

102



0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fitness |a|=1.00

98 98.5 99 99.5 100 100.5 101 101.5 102 98

98.5

99

99.5

100

100.5

101

101.5

102

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1


98 98.5 99 99.5 100 100.5 101 101.5 102 98

98.5

99

99.5

100

100.5

101

101.5

102


Conclusions

The outcome depends on the shape of the landscape and thestrategy adopted.

Smarter strategies may avoid some local peaks. For exampleallowing diagonal steps or accepting fitness decreasing to getout of local peaks.

In both cases you increase the costs of exploration and thepossibility of un-expected consequences.


Conclusions





Conclusions





Random vs. Greedy strategy

We can compare the results produced by the one-bit mutationsearch strategy with an alternative one.

Instead of choosing randomly which dimension to test, agreedy strategy opts systematically for the dimension providingthe highest fitness improvements.

Which strategy score best? We can test the two strategies fordifferent levels of complexity.



0.7

0.75

0.8

0.85

0.9

0.95

1

0 0.2 0.4 0.6 0.8 1

Av. fitn

ess

|a|

RandomGreedy



The greedy strategy is better in highly complex problems,where almost certainly the strategy gets immediately stuck in alocal optima. It ensures you get the best local optima available.

The random strategy is better for moderately complexproblems, because it avoids early convergence, the rushing tothe best local peak preventing the access to slowly increasingpatterns leading closer to the global optimum.



The greedy strategy is better in highly complex problems,where almost certainly the strategy gets immediately stuck in alocal optima. It ensures you get the best local optima available.

The random strategy is better for moderately complexproblems, because it avoids early convergence, the rushing tothe best local peak preventing the access to slowly increasingpatterns leading closer to the global optimum.


Structure and intensity of complexity

Moving to N > 2 we have two possible sources of complexity.Besides the intensity of interaction, also the number ofinteractions, called K , generates complexity, i.e. local peaks.


Structure and intensity of complexity

0.3

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1

0 5 10 15 20 25

K

Average Fitness

|a|=0.0|a|=0.1|a|=0.2|a|=0.3|a|=0.4|a|=0.5|a|=0.6|a|=0.7|a|=0.8|a|=0.9|a|=1.0


Exploring search strategies

We can drop the assumption of one-bit mutation strategy andconsider the results that may be produced under theassumption C-bits mutations.

With C-bits mutation agents can change the value of up to Cdimensions at the same time, essentially has the possibilitymoving “diagonally” within a sub-set of the problem dimensions.



We can drop the assumption of one-bit mutation strategy andconsider the results that may be produced under theassumption C-bits mutations.

With C-bits mutation agents can change the value of up to Cdimensions at the same time, essentially has the possibilitymoving “diagonally” within a sub-set of the problem dimensions.



N = 24, K = 4, C = 1,2,3,4,6,12,24

1 7500 15000 22500 30000

0.50476

0.628531

0.752302

0.876073

0.999844

Cl. 1 Cl. 2 Cl. 3 Cl. 4

Cl. 6 Cl. 8 Cl. 12



If variables interacts in groups of K then search strategieswith C = K can reach the global optimum with certainty.

if C < K then the search strategies will get stuck to localpeaks, but will increase very quickly the fitness from thestarting, low fitness points.if C is large agents have a large number of potentialmutations. Therefore, they are slow to increase fitness.



If variables interacts in groups of K then search strategieswith C = K can reach the global optimum with certainty.if C < K then the search strategies will get stuck to localpeaks, but will increase very quickly the fitness from thestarting, low fitness points.

if C is large agents have a large number of potentialmutations. Therefore, they are slow to increase fitness.



If variables interacts in groups of K then search strategieswith C = K can reach the global optimum with certainty.if C < K then the search strategies will get stuck to localpeaks, but will increase very quickly the fitness from thestarting, low fitness points.if C is large agents have a large number of potentialmutations. Therefore, they are slow to increase fitness.


NK in Social sciences

Real world problems are obviously far more complex that theNK fitness landscape. However, also search strategies aresmarter and more sophisticated.

As economist or scholar of organizational studies it can beuseful to analyse the performance of search strategies subjectsto bounded rationality and limited/biased information applied toproblems with different degrees of complexity.


NK in Social sciences

Real world problems are obviously far more complex that theNK fitness landscape. However, also search strategies aresmarter and more sophisticated.

As economist or scholar of organizational studies it can beuseful to analyse the performance of search strategies subjectsto bounded rationality and limited/biased information applied toproblems with different degrees of complexity.


Long Jumps

D.Levinthal introduced NK to Organization Sciences, using themetaphor to represent the differences between localadjustments and long jumps, risky and radical changesnecessary to get out from local optima.


Decomposition and modularization

When the interactions are limited to a subset of dimensions weface a modular landscape. The search strategies candecompose the space exploring in parallel subsets ofdimensions, called blocks.

The comparison between the size of the landscape’s modulesand the search strategies blocks is called the mirroringhypothesis.



1 Only blocks equal or larger than modules can reach theglobal optimum.

2 The search time is exponentially proportional to the size ofthe blocks.

3 Complex landscapes (large modules) are better exploredwith smaller blocks when time is relevant.












R&D strategies for technological innovation

Firms’ willing to find better technologies face the tradeoffbetween ambition of the research and the expected timing ofresults.

1 Firms aiming at high value innovations need to use largeblocks producing no result for a long time: short-termcompetition favors modular innovations producing fasterinnovations.

2 Firms aiming at fast innovations apply small blocks andthus will never access high value technologies: long-termcompetition favor integral innovations producing bettertechnologies.


Market dynamics and division of labor

Division of labor in a market concerns the segments of aproductive process (e.g the components of a products)produced and improved within a company.

1 Disintegrated markets are composed by many firms eachdealing with a component of the final product.

2 Integrated markets are composed by firms includingmany or all the components of final product.



Division of labor in a market concerns the segments of aproductive process (e.g the components of a products)produced and improved within a company.

1 Disintegrated markets are composed by many firms eachdealing with a component of the final product.

2 Integrated markets are composed by firms includingmany or all the components of final product.



Innovation differs radically in the two markets1 Disintegrated firms provide quick innovations in parallel

each in a component (block), but need to respect theconstraints of the dominant design.

2 Integrated firms are slow to make small improvements,but have the capabilities to generate new designs.



Through time markets undergo waves ofintegration-disintegration:

1 New designs are generated by integrated firms2 Once a dominant design is imposed disintegrated firms

improves aggressively their components3 When each component exploits fully its potential the

design becomes the bottleneck4 New integrated firms explore new designs5 ....


Organizational structures: information and power

Organizations structures define sub-structures (departments)dealing with a specific tasks. These are technically proficient intheir work, able to find smart solutions to specific problemsunder their mandate but ignore the interactions with otherdepartments.

Higher layers (top management) have indirect knowledge ofthe departments’ potential, but are aware of the interactionsand have the power to decide which proposed change toimplement.

It is possible to analyse the results provided by differentorganizational structures and decisional strategies applied todifferent complexity landscapes .












Patents, innovation and welfare

Patents incentivate innovations, but generate monopoly powerreducing welfare.

Many firms complain of patent litigation by patent trolls.

Question: is it better to have strong patents or weak IPRprotection?

Answer: it depends on the complexity of the technologicalspace.





















It is possible to show that we have two answers, broadlyoverlapping with the type of firms.

1 In case of “simple” landscapes (no interactions) patentsare necessary to protect innovations that can easilytranslate in market advantage.

2 In case of “complex” landscapes (many interactions)patents are useless because copying one module isuseless if you don’t have global knowledge.

3 Patents can be negative for social welfare becauseprovides the power to block innovation on a whole systemdue to a patent on a single component.

4 Network externalities protects sufficiently without the needof patents.























Mental models and landscapes translation

It can be proven that any landscape, even with maximumcomplexity, can be translated into another, possibly with alarger number of dimensions, of arbitrary complexity, even null.

The process of discovery or radical innovation can be framed intwo alternative ways:

1 A massive search with a sophisticated strategy applied ona complex landscape made of few variables.

2 A quick and trivial search applied on a new landscape,translated from the original with the purpose of removingthe interactions: a new representation of the problemmapping the original one onto a new one with moredimensions and fewer interactions.














References

Valente, M. , 2014 , “AN NK-LIKE MODEL FOR COMPLEXITY”,Journal of Evolutionary Economics, 24(1), pp. 107-134.

Marengo, L., Pasquali, C., Valente, M. and Dosi, G. , 2012 ,“APPROPRIABILITY, PATENTS, AND RATES OF INNOVATION INCOMPLEX PRODUCTS INDUSTRIES”, Economics of Innovation andNew Technology, 21(8), pp. 753-773.

Ciarli, T., Leoncini, R., Montresor, S. and Valente, M. , 2008 ,“TECHNOLOGICAL CHANGE AND THE VERTICAL ORGANIZATIONOF INDUSTRIES”, Journal of Evolutionary Economics 18(3-4), pp.367-87.


References

Brusoni, S., Marengo, L., Prencipe, A. and Valente, M. , 2007 , “THEVALUE AND COSTS OF MODULARITY: A PROBLEM-SOLVINGPERSPECTIVE”, European Management Review, 4, pp. 121-132.

Frenken, K., Marengo, L. and Valente, M. , 1999,“INTERDEPENDENCIES, NEAR-DECOMPOSABILITY ANDADAPTATION”, in T. Brenner, (ed.), “Computational Techniques forModelling Learning in Economics”, Kluwer Academics.

Levinthal DA (1997) “Adaptation on rugged landscapes”. ManagementScience 43(7):934?950.

Documents

pNK: A model for Complexity - UFPR pNK...Introduction pNK Fitness Landscape 2-D pNK model generic pNK model Literature applications pNK: A model for Complexity Marco VALENTE1 1LEM,