An Introduction to Complex Systems Science

Transcription

1 DEIS, Campus of Cesena Alma Mater Studiorum Università di Bologna

2 Disclaimer The field of Complex systems science is wide and it involves numerous themes and disciplines. This talk just provides an informal introduction to some relevant topics in this area.

3 Outline 1 Complex systems Main concepts 2 Basics Random Boolean Networks Applications of Boolean Networks 3

4 Outline Complex systems Main concepts 1 Complex systems Main concepts 2 Basics Random Boolean Networks Applications of Boolean Networks 3

5 Complex systems science Complex systems Main concepts CSS A new field of science studying how parts of a system give rise to the collective behaviours of the system, and how the system interacts with its environment. It focuses on certain questions about parts, relationships. wholes and

6 Complex systems Complex systems Main concepts Examples of complex systems are: The brain The society The ecosystem The cell The ant colonies The stock market...

7 Complex systems science Complex systems Main concepts CSS is interdisciplinary and it involves: Mathematics Physics Computer science Biology Economy Philosophy...just to mention some.

8 Complex systems science Complex systems Main concepts Three main interrelated approaches to the modern study of complex systems: 1 How interactions give rise to patterns of behaviour 2 Understanding the ways of describing complex systems 3 Understanding the process of formation of complex systems through pattern formation and evolution

9 Complex systems science Complex systems Main concepts Some prominent research topics in CSS: Evolution & emergence Systems biology Information & computation Physics of Complexity

10 Reductionism vs. Holism Complex systems Main concepts Reductionism: an approach to understanding the nature of complex things by reducing them to the interactions of their parts. Holism: idea that all the properties of a system cannot be determined or explained by its component parts alone. Summarised with the sentence The whole is more than the sum of its parts.

11 Complex vs. Complicated Complex systems Main concepts Complex: from Latin (cum + plexere); it means intertwined. Complicated: from Latin (cum + plicare); it means folded together.

12 Properties of complex systems Complex systems Main concepts Complex systems enjoy (some of) these properties: Composed of many elements Nonlinear interactions Network topology Positive and negative feedbacks Adaptive and evolvable Robust Levels of organisation

13 Outline Complex systems Main concepts 1 Complex systems Main concepts 2 Basics Random Boolean Networks Applications of Boolean Networks 3

14 Emergence Complex systems Main concepts Emergence refers to understanding how collective properties arise from the properties of parts. A common case of emergence is self-organisation

15 Self-organisation Complex systems Main concepts Dynamical mechanisms whereby structures appear at the global level from interactions among lower-level components. Creation of spatio-temporal structures Possible coexistence of several stable states (multistability) Existence of bifurcations when some parameters are varied

16 Example: Bénard cells Complex systems Main concepts

17 Further examples Complex systems Main concepts

18 Model of a system Complex systems Main concepts Model A model is an abstract and schematic representation of a system. It is also usually a formal representation of the system. It makes it possible to: investigate some properties of the system make predictions on the future It is usually in the form of a set of objects and the relations among them

19 Properties of a model Complex systems Main concepts It represents only a portion of the system It only captures some of the system s features The abstraction process involves simplification, aggregation and omission of details

20 Example: the logistic map Complex systems Main concepts x t+1 = rx t (1 x t ) x i [0, 1] r [0, + [ Simple model of population growth Different kinds of behaviour depending on the values of r

21 Logistic map: steady states Complex systems Main concepts r 3 single value 3 < r < 3.57 repeated sequence of values r 3.57 sequence of values without apparent structure

22 Attractors Complex systems Main concepts Attractor Portion of the state space towards which a dynamical system evolves over time. Fixed point (Limit) Cycle Strange attractor

23 Logistic map: attractors Complex systems Main concepts r 3 fixed point 3 < r < 3.57 cycle r 3.57 strange attractor

24 Deterministic chaos Complex systems Main concepts Deterministic model Sensitivity to initial conditions In practice, it is impossible to make long term predictions The attractor is a strange attractor

25 Strange attractor Complex systems Main concepts A strange attractor is a fractal Non-integer dimension Self-similarity

26 Complexity Complex systems Main concepts Complexity lies at the edge of order and chaos

27 Complexity Complex systems Main concepts Statistical complexity of a system Complexity = Entropy Disequilibrium

28 Outline Basics Random Boolean Networks Applications of Boolean Networks 1 Complex systems Main concepts 2 Basics Random Boolean Networks Applications of Boolean Networks 3

29 Basics Random Boolean Networks Applications of Boolean Networks

30 Basics Random Boolean Networks Applications of Boolean Networks Introduced by Stuart Kauffman in 1969 as a genetic regulatory network model Discrete-time / discrete-state dynamical system Non trivial (complex) dynamics

31 Structure Basics Random Boolean Networks Applications of Boolean Networks Oriented graph of N nodes

32 Structure Basics Random Boolean Networks Applications of Boolean Networks Oriented graph of N nodes Node i:

33 Structure Basics Random Boolean Networks Applications of Boolean Networks Oriented graph of N nodes Node i: - Boolean value x i X 1 X 3 X 2

34 Structure Basics Random Boolean Networks Applications of Boolean Networks Oriented graph of N nodes Node i: - Boolean value x i - Boolean function f i X 1 AND X 3 OR X 2 OR

35 Structure Basics Random Boolean Networks Applications of Boolean Networks Oriented graph of N nodes Node i: - Boolean value x i - Boolean function f i Boolean function arguments are variables associated to input nodes of i Node state (i.e., Boolean variable) updated as a function of f i X 1 AND X 2 OR X 3 OR

36 Dynamics Basics Random Boolean Networks Applications of Boolean Networks System state at time t: s(t) = (x 1 (t),..., x N (t)) Dynamics controls node update Synchronous vs. asynchronous dynamics

37 Dynamics Basics Random Boolean Networks Applications of Boolean Networks System state at time t: s(t) = (x 1 (t),..., x N (t)) Dynamics controls node update Synchronous vs. asynchronous dynamics Synchronous dynamics (and deterministic update rules): One successor per state Cardinality of state space 2 N

38 Dynamics Transition function Basics Random Boolean Networks Applications of Boolean Networks t t + 1 x 1 x 2 x 3 x 1 x 2 x X 1 AND X 2 OR X 3 OR

39 Dynamics Trajectory in state space Basics Random Boolean Networks Applications of Boolean Networks

40 Dynamics Trajectory in state space Basics Random Boolean Networks Applications of Boolean Networks 100 Trajectory composed of two parts: Transient Attractor 101 Attractors: Fixed points Cycles

41 Dynamics Trajectory in state space Basics Random Boolean Networks Applications of Boolean Networks 100 Basin of attraction of A: set of states belonging to the trajectory ending at attractor A

42 Dynamics Basics Random Boolean Networks Applications of Boolean Networks

43 Why are BNs interesting? Basics Random Boolean Networks Applications of Boolean Networks Minimal complex system Several important phenomena in genetics can be reproduced Tight connections with the satisfiability problem

44 Outline Basics Random Boolean Networks Applications of Boolean Networks 1 Complex systems Main concepts 2 Basics Random Boolean Networks Applications of Boolean Networks 3

45 Random Model Basics Random Boolean Networks Applications of Boolean Networks K inputs per node Inputs chosen at random, no self-arcs Random Boolean functions: each entry of truth table has probability p = 0.5 of being set to 1

46 Random Properties Basics Random Boolean Networks Applications of Boolean Networks K = 1: ORDER Frozen dynamics Extremely robust: small perturbations die out quickly

47 Random Properties Basics Random Boolean Networks Applications of Boolean Networks K 3: (pseudo) CHAOS Very long cycles ( 2 N ) Sensitivity to initial conditions Not robust: small perturbations spread quickly throughout the system

48 Random Properties Basics Random Boolean Networks Applications of Boolean Networks K = 2: CRITICALITY Short cycles ( low degree polynomial of N) Robust: small perturbations die out (in the long term) or keep small Second order phase transition

49 Critical parameters Basics Random Boolean Networks Applications of Boolean Networks From the theory: K c = [2p c (1 p c )] 1

50 Extensions and variants Basics Random Boolean Networks Applications of Boolean Networks Asynchronous Probabilistic Multivalued logics Continuous variables ruled by differential equations (e.g., Glass networks) Multiple interacting BNs

51 Outline Basics Random Boolean Networks Applications of Boolean Networks 1 Complex systems Main concepts 2 Basics Random Boolean Networks Applications of Boolean Networks 3

52 BNs in biology Basics Random Boolean Networks Applications of Boolean Networks Cellular dynamics models Models of specific genetic regulatory networks Cancer and stem cell models

53 Cell dynamics Basics Random Boolean Networks Applications of Boolean Networks Main result Attractor (or set-of) cell type

54 Cell dynamics Basics Random Boolean Networks Applications of Boolean Networks Main result Attractor (or set-of) cell type Cancer and stem cell models

55 Robustness & adaptiveness Basics Random Boolean Networks Applications of Boolean Networks Main result Critical BNs make it possible to achieve the best balance between robustness and adaptiveness.

56 Robustness & adaptiveness Basics Random Boolean Networks Applications of Boolean Networks Main result Critical BNs make it possible to achieve the best balance between robustness and adaptiveness. Real cells are critical

57 KO avalanches Basics Random Boolean Networks Applications of Boolean Networks Main result BNs can reproduce the same avalanche distribution as real genetic networks.

58 KO avalanches Basics Random Boolean Networks Applications of Boolean Networks Main result BNs can reproduce the same avalanche distribution as real genetic networks. Same results with several kinds of BNs (e.g., Glass nets)

59 Basics Random Boolean Networks Applications of Boolean Networks BNs in engineering and computer science Satisfiability problem Learning systems Boolean network robotics

60 Boolean network robotics Basics Random Boolean Networks Applications of Boolean Networks Dynamical system theory and complexity science are rich sources for: analysing artificial agents and robots design principles and guidelines

61 Boolean network robotics Basics Random Boolean Networks Applications of Boolean Networks Dynamical system theory and complexity science are rich sources for: analysing artificial agents and robots design principles and guidelines Boolean network robotics Boolean network robotics concerns the use of, and other models from complex systems science, as robot programs.

62

63 System s behaviour depends on the structure of relation among the components Useful models from graph theory Recent research stream in CSS

64 Graph as a structure model Key ideas: Represent the entities of the system as graph vertices (nodes) Represent the relations between entities as edges (arcs) A vertex can be a single element or a sub-system

65 Examples of networks Technological nets: Internet, telephone, power grids, transportation, etc. Social nets: friendship, collaboration, etc. Nets of information: WWW, citations, tec. Biological nets: biochemical, neural, ecological, etc.

66 Random graphs First model of the topology of a complex system Interesting theoretical results Baseline for comparison with other topologies

67 Random graphs First model of the topology of a complex system Interesting theoretical results Baseline for comparison with other topologies Strictly speaking, a random graph model is defined in terms of an ensemble of graphs generated through a given procedure:

68 Random graphs First model of the topology of a complex system Interesting theoretical results Baseline for comparison with other topologies Strictly speaking, a random graph model is defined in terms of an ensemble of graphs generated through a given procedure: vertices are positioned by choosing two vertices at random (i.e., on the basis of a uniform distribution) degree distribution is Poissonian (Gaussian in the limit case)

69 Random graphs

70 Features of interest Vertex degree (in- and out-degree if edges are oriented) Diameter, characteristic path length et similia Clustering coefficient

71 Characteristic length L(G) Informally: average path length between any pair of vertices.

72 Characteristic length L(G) Informally: average path length between any pair of vertices. Random graphs short L(G) Grid graphs long L(G)

73 Clustering γ b a c Informally: γ quantifies the probability that, given vertex a connected to b and c, there is an edge between b and c.

74 Clustering γ b a c Informally: γ quantifies the probability that, given vertex a connected to b and c, there is an edge between b and c. Random graphs low γ Grid graphs high γ

75 Scale-free nets Scale-free networks can represent the topology of: Social relations (e.g.,scientific collaborations) Web-pages The Internet...

76 Scale-free nets Degree distribution: number of vertices with degree k k γ Few vertices with many connections (hubs) and many vertices with few connections Robust against accidental damages Fragile w.r.t. specific attacks

77 Scale-free nets

78 Scale-free nets Dynamics dramatically different from random and regular topologies Implications in medicine (e.g., epidemics), society, Internet Related to small-world phenomena (low length, high clustering)

79 Scale-free nets development A net evolves to a scale-free topology if the following two conditions hold (sufficient condition): Growth: older vertices have a higher number of connections Preferential attachment: new vertices tend to be attached to vertices with many connections (prob. is proportional to the number of links) Model variants taking also into account vertex fitness

80 References Serra, R., Zanarini, G.: Complex Systems and Cognitive Processes. Springer, Berlin, Germany (1990) Bar Yam, Y.: Dynamics of Complex Systems. Studies in nonlinearity, Addison Wesley, Reading, MA (1997) Kauffman, S.: The Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press, UK (1993) Newman, M.E.J.: Networks. An Introduction. Oxford University Press, UK (2010)