Introducing Complexity Science James Rising Topics W Topic Tools Applica+on 2 Systems Dynamics, Economy 3 Non-linearity and Chaos Stas@cal Iden@ficaon of chaos 4 Monte Carlo methods, Climate Stochas@city Markov Chains 5 Informaon Theory Cellular Automata Ecology 6 Es@maon methods SOC (sand piles) 7 Fractals and Scaling 8 Agent-based models Conflict Space and History Gene@c Algorithms 9 Networks Social Interac@ons, 10 Graph Theory Measuring culture 11 Agent-based models Disease dynamics Dynamic Networks (again) 12 Bigraphs, Es@mang Scale Mul@-level Governance Cross-Scales Separaon What is Complexity? • The opposite of simplicity? • Computer-driven research? • Non-linearity explored with agents or networks? Complexity Science: The study of systems capable of mul8-level emergence. Problems with the Field • Computer scien@sts run amok – Ar@ficial intelligence? • Incomplete work, slopping thinking – Simplis@c approaches, disconnected from reality • Dogmac complexity-mongers • Lack of credibility, understanding • A focus on techniques – SOC, networks, ABM • Ill-defined terms: – “Systems that are Complex” What are Systems? • 18th c.: Integrity of wholes – Music: meaning of parts – Cosmology: circular causality • Homeostasis and overdeterminism – Gender (roles in a system) – Family (more than the sum) – Evolu@on (the construc@on of wholes) Founders • Ludwig von Bertalanffy – Sets of elements standing in interac@on – Isomorphisms between contexts • Biological, ecological, social, economic, poli@cal, par@cle – Informaon theory • Jay Forrester – Social system dynamics – Management, communicaon, understanding – Counterintui@ve effects Complexity • As opposed to complicatedness? • Heterogeneous – Spaality maers, and most interes@ng systems don’t tend toward equilibrium. • Historical – History maers, in socie@es and ecosystems; you can’t get places but by a path • Endogenous – A complex system’s basic proper@es are much more likely to be a func@on of its contents than its context. • Incompleteness – You can’t know everything, but what you don’t know can tell you a lot. Complexity Makes a Difference • Beer models • Connec@ons across contexts • Opportuni@es for inves@gaon and change • Scales and Networks Levels of Complexity Classical System Studies Complexity Science Linearity Non-linearity, chaos Catastrophe Memoryless “State” and evolu@on Data and history Causality Cybernecs, Emergence equifinality, teleology Unorganizaon Structures and Fractal complexity (chance) feedback Closed systems Open systems Mul@-level systems Themodynamic drive Con@nued dynamics Drive toward to equilibrium heterogeneity Gaussians (missing) Fat-tailed distribu@ons Complexity Done Right • Testable hypotheses – Communicaon at core • Applying many tools – Judicious use of agents and systems and networks • Calibraon and validaon – Connec@ons to reality • Evolu@onary, cross-scale – Understanding both elements and whole The Coming Revolu@on • How do we make networks dynamic? • How do we make systems precise? • How do we understand complex drivers? • A new language – Networked systems, incomplete systems, overlapping systems, hierarchical systems • New tools – For research, communicaon, data analysis, es@mators 12 Oct 2004 11:54 AR AR229-ES35-20.tex AR229-ES35-20.sgm LaTeX2e(2002/01/18) P1: GJB 568 FOLKE ETRegime Shijs AL. Figure 2 Alternate states in a diversity of ecosystems (1, 4) and the causes (2) and triggers (3) behind loss of resilience and regime shifts. For more examples, see Thresholds Database on the Web site www.resalliance.org. by University of Colorado - Boulder on 09/16/09. For personal use only. Humans have, over historical time but with increased intensity after the indus- trial revolution, reduced the capacity of ecosystems to cope with change through a Annu. Rev. Ecol. Evol. Syst. 2004.35:557-581. Downloaded from arjournals.annualreviews.org combination of top-down (e.g., overexploitation of top predators) and bottom-up impacts (e.g., excess nutrient influx), as well as through alterations of disturbance regimes including climatic change (e.g., prevention of fire in grasslands and for- est or increased bleaching of coral reefs because of global warming) (Nystr¨om et al. 2000, Paine et al. 1998, Worm et al. 2002). The result of those combined impacts tends to be leaking, simplified, and “weedy” ecosystems characterized by unpredictability and surprise in their capacity to generate ecosystem services. The likelihood that an ecological system will remain within a desired state is related to slowly changing variables that determine the boundaries beyond which disturbances may push the system into another state (Scheffer & Carpenter 2003). Iterave Maps Iterative Maps: • A simple tool for studying nonlinear dynamics xt+1 = f(xt) 1 01adBARYAM_29412 3/10/02 10:15 AM Page 22 22 I n t r o du c tIterave Maps 1 i o n a n d P r e l i m i n a r i e s (a) s(t) s(t) s(t)=c s(t)=c 0 1 2 3 4 5 6 7 s(t–1) t (b) s(t)=s(t–1) +v s(t) s(t)=s(t–1)+v s(t) 0 1 2 3 4 5 6 7 s(t–1) t Figure 1.1.1 Th e left panels show the time- d ep e n den t value of the system variable s(t) re- su l t i n g from iterative map s . The first panel (a) shows the result of iterat i n g the const a n t map ; (b) shows the result of addi n g v to the prev i ous value dur i n g each time interval; (c)–(f) show th e result of mul t i p l y i n g by a const a n t g, wher e each figu r e shows the behavior for a differen t ran ge of g values: (c) g > 1, (d) 0 < g < 1, (e) 1 < g < 0, and (f) g < 1. The righ t panels are a differen t way of show i n g grap h i cally the results of iterat i on s and are const r u cted as fol l o w s . First plot the funct i on f(s) (solid line), wher e s(t) f(s(t 1)). This can be thou g h t of as plot- ti n g s(t) vs. s(t 1). Second, plot the iden tity map s(t) s(t 1) (das h ed line). Mark the ini- ti al value s(0) on the hor i z o n tal axis, and the point on the graph of s(t) that corres p o n ds to th e point whose abscissa is s(0), i.e. the point (s(0), s(1)). These are shown as squar es . From th e point (s(0), s(1)) draw a hor i z o n tal line to intersect the iden tity map. The inte r s e c t i on po i n t is (s(1), s(1)). Draw a vertical line back to the iterative map. This is the point (s(1 ) , s(2)). Successive values of s(t) are obtained by iterat i n g this grap h i cal proc e d ur e. ❚ Fixed points,not surprisingly, play an important role in iterative maps. They help us describe the state and behavior of the system after many iterations. There are two kinds of fixed p oints—stable and unstable. Stable fixed points are characterized by “attracting” the result of iteration of points that are nearby. More precisely, there exists # 29412 Cust: AddisonWesley Au: Bar-Yam Pg. No. 22 Title: Dynamics Complex Systems Short / Normal / Long 01adBARYAM_29412 3/10/02 10:15 AM Page 23 Iterave Maps 2 (c) s(t)=gs(t–1) s(t) s(t)=gs(t–1) s(t) g>1 g>1 0 1 2 3 4 5 6 7 s(t–1) t s(t)=gs(t–1) (d) 0<g<1 s(t) s(t)=gs(t–1) s(t) 0<g<1 0 1 2 3 4 5 6 7 s(t–1) t (e) s(t)=gs(t–1) s(t) s(t)=gs(t–1) s(t) -1<g<0 -1<g<0 0 1 2 3 4 5 6 7 s(t–1) t (f) s(t)=gs(t–1) s(t) s(t)=gs(t–1) s(t) g<-1 g<-1 0 1 2 3 4 5 6 7 s(t–1) t 23 # 29412 Cust: AddisonWesley Au: Bar-Yam Pg. No. 23 Title: Dynamics Complex Systems Short / Normal / Long 01adBARYAM_29412 3/10/02 10:15 AM Page 23 (c) s(t)=gs(t–1) s(t) s(t)=gs(t–1) s(t) g>1 g>1 0 1 2 3 4 5 6 7 s(t–1) t s(t)=gs(t–1) (d) 0<g<1 s(t) s(t)=gs(t–1) s(t) 0<g<1 0 1 2 3 4 5 6 7 s(t–1) t Iterave Maps 3 (e) s(t)=gs(t–1) s(t) s(t)=gs(t–1) s(t) -1<g<0 -1<g<0 0 1 2 3 4 5 6 7 s(t–1) t (f) s(t)=gs(t–1) s(t) s(t)=gs(t–1) s(t) g<-1 g<-1 0 1 2 3 4 5 6 7 s(t–1) t 23 # 29412 Cust: AddisonWesley Au: Bar-Yam Pg. No. 23 Title: Dynamics Complex Systems Short / Normal / Long Iterave Maps 4 REVIEWS NATURE Vol 461 3 September 2009 j j High resilience Box 1 Critical transitions in the fold catastrophe model j a The equilibrium state of a system can respond in different ways to Basin of attraction changes in conditions such as exploitation pressure or temperature rise (Box 1 Figure a, b, c).