Introductory concepts Boolean networks Complex networks An Introduction to Complex Systems Science Andrea Roli DEIS, Campus of Cesena Alma Mater Studiorum Universita` di Bologna [email protected] Andrea Roli An Introduction to Complex Systems Science Introductory concepts Boolean networks Complex networks 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. Andrea Roli An Introduction to Complex Systems Science Introductory concepts Boolean networks Complex networks Outline 1 Introductory concepts Complex systems Main concepts 2 Boolean networks Basics Random Boolean Networks Applications of Boolean Networks 3 Complex networks Andrea Roli An Introduction to Complex Systems Science Introductory concepts Complex systems Boolean networks Main concepts Complex networks Outline 1 Introductory concepts Complex systems Main concepts 2 Boolean networks Basics Random Boolean Networks Applications of Boolean Networks 3 Complex networks Andrea Roli An Introduction to Complex Systems Science Introductory concepts Complex systems Boolean networks Main concepts Complex networks Complex systems science 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, wholes and relationships. Andrea Roli An Introduction to Complex Systems Science Introductory concepts Complex systems Boolean networks Main concepts Complex networks Complex systems Examples of complex systems are: The brain The society The ecosystem The cell The ant colonies The stock market ::: Andrea Roli An Introduction to Complex Systems Science Introductory concepts Complex systems Boolean networks Main concepts Complex networks Complex systems science CSS is interdisciplinary and it involves: Mathematics Physics Computer science Biology Economy Philosophy ...just to mention some. Andrea Roli An Introduction to Complex Systems Science Introductory concepts Complex systems Boolean networks Main concepts Complex networks Complex systems science 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 Andrea Roli An Introduction to Complex Systems Science Introductory concepts Complex systems Boolean networks Main concepts Complex networks Complex systems science Some prominent research topics in CSS: Evolution & emergence Systems biology Information & computation Complex networks Physics of Complexity Andrea Roli An Introduction to Complex Systems Science Introductory concepts Complex systems Boolean networks Main concepts Complex networks Reductionism vs. Holism 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. Andrea Roli An Introduction to Complex Systems Science Introductory concepts Complex systems Boolean networks Main concepts Complex networks Complex vs. Complicated Complex: from Latin (cum + plexere); it means “intertwined”. Complicated: from Latin (cum + plicare); it means “folded together”. Andrea Roli An Introduction to Complex Systems Science Introductory concepts Complex systems Boolean networks Main concepts Complex networks Properties of complex systems 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 Andrea Roli An Introduction to Complex Systems Science Introductory concepts Complex systems Boolean networks Main concepts Complex networks Outline 1 Introductory concepts Complex systems Main concepts 2 Boolean networks Basics Random Boolean Networks Applications of Boolean Networks 3 Complex networks Andrea Roli An Introduction to Complex Systems Science Introductory concepts Complex systems Boolean networks Main concepts Complex networks Emergence Emergence refers to understanding how collective properties arise from the properties of parts. A common case of emergence is self-organisation Andrea Roli An Introduction to Complex Systems Science Introductory concepts Complex systems Boolean networks Main concepts Complex networks Self-organisation 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 Andrea Roli An Introduction to Complex Systems Science Introductory concepts Complex systems Boolean networks Main concepts Complex networks Example: Benard´ cells Andrea Roli An Introduction to Complex Systems Science Introductory concepts Complex systems Boolean networks Main concepts Complex networks Further examples Andrea Roli An Introduction to Complex Systems Science Introductory concepts Complex systems Boolean networks Main concepts Complex networks Model of a system 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 Andrea Roli An Introduction to Complex Systems Science Introductory concepts Complex systems Boolean networks Main concepts Complex networks Properties of a model 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 Andrea Roli An Introduction to Complex Systems Science Introductory concepts Complex systems Boolean networks Main concepts Complex networks Example: the logistic map xt+1 = rxt (1 − xt ) xi 2 [0; 1] r 2 [0; +1[ Simple model of population growth Different kinds of behaviour depending on the values of r Andrea Roli An Introduction to Complex Systems Science Introductory concepts Complex systems Boolean networks Main concepts Complex networks Logistic map: steady states r ≤ 3 ! single value 3 < r < 3:57 ! repeated sequence of values r ≥ 3:57 ! sequence of values without apparent structure Andrea Roli An Introduction to Complex Systems Science Introductory concepts Complex systems Boolean networks Main concepts Complex networks Attractors Attractor Portion of the state space towards which a dynamical system evolves over time. Fixed point (Limit) Cycle Strange attractor Andrea Roli An Introduction to Complex Systems Science Introductory concepts Complex systems Boolean networks Main concepts Complex networks Logistic map: attractors r ≤ 3 ! fixed point 3 < r < 3:57 ! cycle r ≥ 3:57 ! strange attractor Andrea Roli An Introduction to Complex Systems Science Introductory concepts Complex systems Boolean networks Main concepts Complex networks Deterministic chaos Deterministic model Sensitivity to initial conditions In practice, it is impossible to make long term predictions The attractor is a strange attractor Andrea Roli An Introduction to Complex Systems Science Introductory concepts Complex systems Boolean networks Main concepts Complex networks Strange attractor Non-integer dimension A strange attractor is a fractal Self-similarity Andrea Roli An Introduction to Complex Systems Science Introductory concepts Complex systems Boolean networks Main concepts Complex networks Complexity Complexity lies at the edge of order and chaos Andrea Roli An Introduction to Complex Systems Science Introductory concepts Complex systems Boolean networks Main concepts Complex networks Complexity Statistical complexity of a system Complexity = Entropy × Disequilibrium Andrea Roli An Introduction to Complex Systems Science Introductory concepts Basics Boolean networks Random Boolean Networks Complex networks Applications of Boolean Networks Outline 1 Introductory concepts Complex systems Main concepts 2 Boolean networks Basics Random Boolean Networks Applications of Boolean Networks 3 Complex networks Andrea Roli An Introduction to Complex Systems Science Introductory concepts Basics Boolean networks Random Boolean Networks Complex networks Applications of Boolean Networks Boolean networks Andrea Roli An Introduction to Complex Systems Science Introductory concepts Basics Boolean networks Random Boolean Networks Complex networks Applications of Boolean Networks Boolean networks Introduced by Stuart Kauffman in 1969 as a genetic regulatory network model Discrete-time / discrete-state dynamical system Non trivial (complex) dynamics Andrea Roli An Introduction to Complex Systems Science - Boolean value xi - Boolean function fi Node i: Boolean function arguments are variables associated to input nodes of i Node state (i.e., Boolean variable) updated as a function of fi Introductory concepts Basics Boolean networks Random Boolean Networks Complex networks Applications of Boolean Networks Structure Oriented graph of N nodes Andrea Roli An Introduction to Complex Systems Science - Boolean value xi - Boolean function fi Boolean function arguments are