This document summarizes a study on degeneracy in random Boolean networks. It introduces random Boolean networks and discusses experiments measuring the effects of degeneracy, redundancy, and other factors on network properties like number of attractors, states in attractors, and sensitivity to initial conditions. The discussion suggests that degeneracy and redundancy can increase robustness and evolvability by allowing variations in node functions or structures without significantly changing dynamics.

1. A Study of Degeneracy in Random Boolean
Networks
Roberto Guti´rrez1 , David A. Rosenblueth1 , James Whitacre2
e
& Carlos Gersenson1
1 Instituto de Investigaciones en Matem´ticas Aplicadas y en Sistemas
a
Universidad Nacional Aut´noma de M´xico
o e
2 Centre of Excellence for Research in Computational Intelligence and Applications
University of Birmingham
Eurepean Conference on Complex Systems, 2012

2. Introduction Experiments Discussion
Contents
1 Introduction
2 Experiments
3 Discussion

3. Introduction Experiments Discussion
Random Boolean Networks
n(t) p(t) o(t + 1)
Figure: RBN with connectivity K = 2. 0 0 1
0 1 0
1 0 0
1 1 1
Table: Lookup table for net o.
Figure: Dynamics of RBNs in a) ordered,
b) critical & c) chaotic phase, respectively.

4. Introduction Experiments Discussion
Robustness vs. Evolvability
Robustness
“A system is robust if it continues to function in the face of
perturbations” (A. Wagner, 2005).

5. Introduction Experiments Discussion
Robustness vs. Evolvability
Robustness
“A system is robust if it continues to function in the face of
perturbations” (A. Wagner, 2005).
Evolvability
The capacity to discover beneficial, heritable adaptations. (Wagner
and Altenberg, 1996).

6. Introduction Experiments Discussion
Degeneracy
definition
Describes the coexistence of structurally distinct components that
can perform similar roles or are interchangeable under certain
conditions, yet have distinct roles under other conditions (Edelman
and Gally, 2001).
Figure: Robustness and evolvability. 1
1
Image from “Degeneracy: a design principle for achieving robustness
and evolvability”, James M. Whitacre, 2010.

7. Introduction Experiments Discussion
Simple RBN (core)
Figure: RBN with N = 3, K = 2.

8. Introduction Experiments Discussion
Adding redundancy (RBN with redundancy)
Figure: RBN with N = 4, K = 2.2
2
“The Role of Redundancy in the Robustness of Random Boolean
Network” Carlos Gershenson, Stuart A. Kauffman, Ilya Shmulevich, 2006.

9. Introduction Experiments Discussion
RBN with function degeneracy
Figure: RBN with N = 4, K = 2.

10. Introduction Experiments Discussion
RBN with input degeneracy
Figure: RBN with N = 4, K = 2.

11. Introduction Experiments Discussion
RBN with output degeneracy
Figure: RBN with N = 4, K = 2.

12. Introduction Experiments Discussion
Experiments
Simple RBN (core).
Function Degeneracy.
Input Degeneracy.
Output Degegeracy.
Redundancy.

13. Introduction Experiments Discussion
Number of Attractors (A)
A Reflects how many distinct sets of states can “capture” the
dynamics of the RBN.
Figure: Dynamics of an RBN. 3
3
Image taken from http://www.sussex.ac.uk/Users/andywu/

14. Introduction Experiments Discussion
Number of Attractors (A)
Figure: RBN with NTotal = 20, K = 5, Ndeg = 5.

15. Introduction Experiments Discussion
States in Attractors (SIA)
SIA is dependent on the number and length of attractors.
Figure: RBN with NTotal = 20, K = 5, Ndeg = 5.

16. Introduction Experiments Discussion
Sensitivity to Initial Conditions I
Calculated with the normalized Hamming distance:
∆H = H(Sf , Sf ) − H(Si , Si ).
Figure: RBN with NTotal = 20, K = 5, Ndeg = 5.

17. Introduction Experiments Discussion
Sensitivity to Initial Conditions II
Figure: RBN with NTotal = 200, K = 5, Ndeg = 20.

18. Introduction Experiments Discussion
Discussion I
Figure: Propagation of mutations.

19. Introduction Experiments Discussion
Discussion II
Figure: Propagation of mutations.

20. Introduction Experiments Discussion
Discussion III
These results suggest that degeneracy, as well as redundancy,
can facilitate robustness and evolvability, allowing new
functionalities to arise from nodes with small variations of
function or structure without changing too much the
dynamics (phenotype).

21. Thanks for your attention

22. Introduction Experiments Discussion
References I
Carlos Gershenson. Introduction to Random Boolean
Networks. Centrum Leo Apostel, Vrije Universiteit Brussel.
Tononi G, Sporns O, Edelman GM Measures of degeneracy
and redundancy in biological networks 0027-8424 1999,
96:3257-3262.
James M Whitacre. Degeneracy: a design principle for
achieving robustness and evolvability School of Computer
Science, University of Birmingham, Edgbaston, UK
James M Whitacre. Degeneracy: a link between
evolvability, robustness and complexity in biological
systems School of Computer Science, University of
Birmingham, Edgbaston, UK

23. Introduction Experiments Discussion
References II
James M Whitacre. The Role of Redndancy in the
Robustness of Random Boolean Networks School of
Computer Science, University of Birmingham, Edgbaston, UK
A. Wagner Distributed robustness versus redundancy as
causes of mutational robustness BioEssays, vol. 27, pp.
176-188, 2005
Fern´ndez P., Sol´ R.(2004). The role of computation in
a e
complex regulatory networks In Koonin, E. V., Wolf, Y. I.,
and Karev, G. P., editors, Power Laws, Scale-Free Networks
and Genome Biology. Landes Bioscience.

24. Introduction Experiments Discussion
References III
Carlos Gershenson, Stuart A. Kauffman, Ilya Shmulevich
(2006). The Role of Redundancy in the Robustness of
Random Boolean Networks Artificial Life X, Proceedings of
the Tenth International Conference on the Simulation and
Synthesis of Living Systems.