Introduction to Natural Computation Lecture 18 Random Boolean Networks Alberto Moraglio 1 / 25 Random Boolean Networks Extension of Cellular Automata. Introduced by Stuart Kauffman (1969) as model for genetic regulatory networks. Uncover fundamental principles of living systems. Hypothesis: living organisms can be constructed from random elements. Simplification: Boolean states for all nodes in the network. 2 / 25 Definition Random Boolean Network (RBN) A random Boolean network consists of N nodes, each with a state 0 or 1 K edges for each node to K different randomly selected nodes a lookup table for each node that determines the next state, according to states of its K neighbours. Last few lectures: fixed graph, random process This time: random graph, deterministic process Note: randomness used to create network, afterwards network remains fixed! 3 / 25 Relation to Cellular Automata http://www.metafysica.nl/boolean.html 4 / 25 Lookup Tables Recall rules from Cellular Automata: RBN: each node has its own random lookup table: inputs at time t state at time t + 1 000 1 001 0 010 0 011 1 100 1 101 0 110 1 111 1 5 / 25 Example of a Simple RBN Gershenson C. Introduction to Random Boolean Networks. ArXiv Nonlinear Sciences e-prints. 2004; 6 / 25 States State space has size 2N . After at most 2N + 1 steps a state will be repeated. 7 / 25 States State space has size 2N . After at most 2N + 1 steps a state will be repeated. Attractors As the system is deterministic, it will get stuck in an attractor. one state → point attractor / steady state two or more states → cycle attractor / state cycle The set of states that flow towards an attractor is called attractor basin. 8 / 25 Number of RBNs Q: How many possible functions exists for each node? 9 / 25 Number of RBNs Q: How many possible functions exists for each node? K A: All possible 0-1-choices for 2K inputs: 22 . 10 / 25 Number of RBNs Q: How many possible functions exists for each node? K A: All possible 0-1-choices for 2K inputs: 22 . Q: How many possible wirings to K neighbours exist for each node? 11 / 25 Number of RBNs Q: How many possible functions exists for each node? K A: All possible 0-1-choices for 2K inputs: 22 . Q: How many possible wirings to K neighbours exist for each node? A: There are N!/(N − K)! ordered combinations. 12 / 25 Number of RBNs Q: How many possible functions exists for each node? K A: All possible 0-1-choices for 2K inputs: 22 . Q: How many possible wirings to K neighbours exist for each node? A: There are N!/(N − K)! ordered combinations. Q: How large is the number of possible networks for given N, K? 13 / 25 Number of RBNs Q: How many possible functions exists for each node? K A: All possible 0-1-choices for 2K inputs: 22 . Q: How many possible wirings to K neighbours exist for each node? A: There are N!/(N − K)! ordered combinations. Q: How large is the number of possible networks for given N, K? A: Pretty big! 14 / 25 Number of RBNs Q: How many possible functions exists for each node? K A: All possible 0-1-choices for 2K inputs: 22 . Q: How many possible wirings to K neighbours exist for each node? A: There are N!/(N − K)! ordered combinations. Q: How large is the number of possible networks for given N, K? A: Pretty big! K N 22 N! − (N K)! ! 15 / 25 Number of RBNs Q: How many possible functions exists for each node? K A: All possible 0-1-choices for 2K inputs: 22 . Q: How many possible wirings to K neighbours exist for each node? A: There are N!/(N − K)! ordered combinations. Q: How large is the number of possible networks for given N, K? A: Pretty big! K N 22 N! (N − K)! ! N = 3,K = 3 : 3623878656 16 / 25 Number of RBNs Q: How many possible functions exists for each node? K A: All possible 0-1-choices for 2K inputs: 22 . Q: How many possible wirings to K neighbours exist for each node? A: There are N!/(N − K)! ordered combinations. Q: How large is the number of possible networks for given N, K? A: Pretty big! K N 22 N! (N − K)! ! N = 3,K = 3 : 3623878656 N = 4,K = 3 : 1424967069597696 17 / 25 Number of RBNs Q: How many possible functions exists for each node? K A: All possible 0-1-choices for 2K inputs: 22 . Q: How many possible wirings to K neighbours exist for each node? A: There are N!/(N − K)! ordered combinations. Q: How large is the number of possible networks for given N, K? A: Pretty big! K N 22 N! (N − K)! ! N = 3,K = 3 : 3623878656 N = 4,K = 3 : 1424967069597696 N = 8,K = 4 : 21593035501811706443110335226483118854343255930579818905600000000 18 / 25 Order vs. Chaos Random Boolean networks can be in three phases: ordered, chaotic, and critical. Gershenson C. Introduction to Random Boolean Networks. ArXiv Nonlinear Sciences e-prints. 2004; 19 / 25 Order vs. Chaos Ordered regime Occurs when K ≤ 2. System is insensitive to initial conditions and “mutations”: flip the state of a node change a connection change the lookup table of a node “Damage” typically does not spread far. 20 / 25 Order vs. Chaos Ordered regime Occurs when K ≤ 2. System is insensitive to initial conditions and “mutations”: flip the state of a node change a connection change the lookup table of a node “Damage” typically does not spread far. Chaotic regime Occurs when K ≥ 3. Small changes can have a huge impact–“butterfly effect”. Similar states tend to diverge. 21 / 25 View of an Attractor Basin N = 13,K = 3 http://www.metafysica.nl/boolean.html 22 / 25 All Attractor Basins N = 13,K = 3 http://www.metafysica.nl/boolean.html from Andrew Wuensche’s DLLab gallery 23 / 25 Other Update Schemes Attractor cycles occur because the system is deterministic and all nodes are updated synchronously. Criticism about synchronicity: genes do not march in step! Variants of RBN Asynchronous RBNs: randomly choose a node to be updated. Deterministic asynchronous RBNs: create random periods Pi,Qi for each node that remain fixed afterwards. Update node i at time t if t mod Pi ≡ Qi. Other update schemes can change the behaviour significantly. 24 / 25 Further Reading Gershenson C. Introduction to Random Boolean Networks. ArXiv Nonlinear Sciences e-prints. 2004;. 25 / 25.