Current Research Interests Complex models of innovation on rugged technology landscapes Endogenous...

Current Research Interests

• Complex models of innovation on rugged technology landscapes

• Endogenous evolutionary models of economic growth, and comparison with neoclassical rational expectations models with learning

• Applied econometrics of complex systems:– Long memory– Skewed, fat-tailed distributions (e.g. Pareto vs. Lognormal)– Spatio-temporal clustering– Graph-theoretic properties of technological trajectories

• Risk management in a Paretian universe

Percolating Complexity: Generating the Complex Patterns of the Innovation Process from a Simple

Probabilistic Lattice, with some Empirical Illustrations

Gerald Silverberg

MERIT, Maastricht University

Lyon Exystence Thematic Institute, June 2003

based on joint work with

Bart Verspagen

ECIS, Eindhoven University of Technology

What is Complexity, and What is a Complex Dynamics Model?

Negative definitions:

Dick Day (1994): Something not generated by a point attractor or a limit cycle, i.e., highly unpredictable with classical deterministic methods.

If we add noise and think about it observationally: Something that is not Gaussian and short-memory (i.e., stable ARMA process) or does not consist solely of a finite number of sharp spectral peaks above background noise.

Positive, heuristic definitions:

A complex spatio-temporal structure with significant clustering or long-range correlations

Examples: 1/f noise, power laws, fractals, long memory, intermittancy

Possible modeling approaches: exploit extended critical systems such as percolation, self-organized criticality (sandpiles, etc.)

1. Technical change is cumulative: new technologies build on each other. 2.  Technical change follows relatively ordered pathways, as can be measured ex post in technology characteristics space (see the work of Sahal, Saviotti, Foray and Gruebler, etc.). This has led to the positing of natural trajectories (Nelson), technological paradigms (Dosi), and technological guideposts (Sahal). 3.  The arrival of innovations appears to be stochastic, but more highly clustered than Poisson (overdispersion).  4.  The ‘size’ of an innovation is drawn from a highly skewed distribution (as evidenced e.g. by citation and co-citation frequencies compiled by Tratjenberg and van Raan). 5.  Technological trajectories bifurcate and also merge. 6.  There appears to be a certain arbitrariness in the path actually chosen, which could be the result of small events (path dependence or neutral theory?) and cultural and institutional biases (social construction of technology?). 7.  Incremental improvements tend to follow upon radical innovations according to rather regular laws (learning curves).

Stylized Facts About Technological Change

(a) Innovation supersample

0

1

2

3

4

5

6

7

8

9

1760

1780

1800

1820

1840

1860

1880

1900

1920

1940

1960

1980

raw data fit nb fit Poisson

Raw data and four fitted regression models for supersample data. Pure Poisson models are rejected against negative binomial

(overdispersed) models

Frequency Distribution of Patent Citations

0

50

100

150

200

250

0 20 40 60 80

Number of Citations

Num

ber o

f Pat

ents

Frequency distributions of CT scanner patent citations. Linear scale above, double log plot by rank above right, with self-citation left. Data source: Tratjenberg (1990).

Patent Citations by Rank

1

10

100

0 0.5 1 1.5 2 2.5 3

Log Rank

Cita

tions

Patent Citations +1 by Rank

1

10

100

0 0.5 1 1.5 2 2.5 3

Log Rank

Cita

tions

+ 1

Innovation Size Distributions

Scherer et al. Analysis of Innovation Returns

Source: Scherer, Harhoff and Kukies, 2000, „Uncertainty and the size

distribution of rewards from innovation“, JEE, 10: 175-200.

Scherer et al. continued

Hill Estimator: Consider n observations of a random variable Xi,

and denote by X[i] the order statistics X[1]X[2]… X[n]. Then

the Hill estimator is defined as follows:

k

iki XXknkH

1]1[][ ).ln(ln/1),(

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1 10 100

k

alph

a

0.00

0.50

1.00

1.50

2.00

2.50

3.00

1 10 100 1000

k

alph

a

Hill estimator applied to Harvard University patent portfolio data used in Scherer (1998) (left), and to Trajtenberg’s (1990) patent citation data (right).

Percolation diagram in technology-performance space. Lattice sites are filled at random. A site is viable when it

connects to the baseline.

Probability of a random site being on the infinite cluster P as a function of the percolation probability q

q

P

pc

Convergence, divergence and shortcuts, and two methods of defining a technology's competitiveness.

New innovations are generated with probability p in a region d units above and below the technological frontier.

I n n o v a t i o n s e a r c h s p a c e

Near disjoint regions represent inventions, far off discoveries science, and clusters that can never be

connected to the baseline science “fictions”.

A cluster of simultaneous invention occurs when a disjoint island of invention is suddenly joined to the frontier by a

single 'cornerstone' innovation.

Screen Shot of Run with Search Radius 8

0

1

2

3

4

5

6

7

8

9

10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

search radius

nu

mb

er o

f d

ead

lock

ed r

un

s

Number of runs, which deadlock out of batches of ten for different values of the search radius. q=0.593.

0.523

0.548

0.573

0.598

0.623

0.648

0.673

0.698

1

4

7

10

13

16

19

0

1

2

3

4

5

6

7

8

9

10

q

search radius

Number of deadlocked runs out of ten as a joint function of the search radius and the percolation probability q.

Deadlocking Statistics

1 4 7

10

13

16

19

0.523

0.553

0.583

0.613

0.643

0.673

0.703

0.733

0

50

100

150

200

250

300

350

mean height

search radius

q

The mean height of the BPF attained after 5000 periods as a function of the search radius and percolation probability q.

0

20

40

60

80

100

120

140

160

180

200

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97

innovation size

nu

mb

er o

f ca

ses

1

10

100

1 10 100 1000 10000

rank (radius=10)in

no

vati

on

siz

e

Size distribution of innovations (left) and rank-order distribution (right, double-log scale), q=0.603, m=10.

Hill estimator of Pareto for innovation distribution generated with q=0.6 and m=5 plotted on a double-log scale for values of k up to 90% of number of observations.

Hill estimator of Pareto for innovation distribution generated with q=0.645 and m=5 plotted on a double-log scale for values of k up to 90% of number of observations

Hill Estimators q=0.6, 0.645

Hill estimator of Pareto for innovation distribution generated with q=0.695 and m=5 plotted on a double-log scale for values of k up to 90% of number of observations.

LD plot for innovation distribution generated with q=0.60 and m=5 (original data and aggregated data in blocks of 10, 100 and 200 observations).

Hill Estimator q=0.695

LD Plot q=0.6

LD plot for innovation distribution generated with q=0.645 and m=5 (original data and aggregated data in blocks of 10, 100 and 200 observations).

LD plot for innovation distribution generated with q=0.695 and m=5 (original data and aggregated data in blocks of 10, 100 and 200 observations).

LD Plots q=0.645, 0.695

Temporal clustering: Innovation count time series for a threshold of 2 (left) and 10 (right) for a run with search radius 5.

0

2

4

6

8

10

12

1000 1200 1400 1600 1800 2000

0

1

2

3

4

5

1000 1200 1400 1600 1800 2000

Arrival rate and overdispersion index for search radius m and threshold theta for radical innovations

13

5202

3 4 5 67890

0.05

0.1

0.15

0.2

0.25

Arr

iva

l R

ate

m

1 3 5 20

234567890

50

100

150

Ov

erd

isp

ers

ion

m

Space-time plot of innovations: Moran clustering index highly significant

Technology space

time

Space-time clustering of cylinder sizes of Cornish steam engines (Nuvolari and Verspagen 2003)

Conclusions

• We can maximize our ignorance of technology space by percolating it with an exogenous percolation probability q

• We can impose the cumulativeness condition by requiring viable technologies to trace a path back to the baseline

• We can impose blindness and localness of the R&D search process by using m-neighborhoods of the BPF with uniform prob of testing sites

• Nevertheless we obtain, instead of a completely random innovation process, spatial and temporal clustering of innovations

• Innovation size distributions are highly skewed and possibly fat tailed. Near the critical q they appear to be Pareto, for higher q probably lognormal

Outlook

• Endogenize R&D effort and targeting using agent-based model• Endogenize q along lines of Bak-Sneppen model