Emergent behavior in collective dynamics Eitan Tadmor University of - PowerPoint PPT Presentation

Emergent behavior in collective dynamics Eitan Tadmor University of Maryland 1 “Advances in Applied Mathematics” — in memoriam of Saul Abarbanel Tel-Aviv U., Dec. 18, 2018 1 Center for Scientific Computation and Mathematical Modeling (CSCAMM) Department of Mathematics and Institute for Physical Science & Technology Emergent behavior in collective dynamics 1

Fundamentals A fascinating aspect of self-organized dynamics: How short-range interactions lead to the emergence of higher-order structures (patterns, action)? Short range interactions : Different rules of engagement; a notion of a ✿✿✿✿✿✿✿✿✿✿✿✿ neighborhood ◮ Fundamentals on short-range interactions: environmental averaging: alignment, synchronization, attraction-repulsion, phase transition, ... ◮ Despite the variety – living agents, “thinking” or mechanical agents: Coherent structures emerge for large time t ≫ 1, large crowds N ≫ 1: � emergence of flocks, swarms, colonies, parties, consensus, ... Global effect : Emergence on ✿✿✿✿✿✿✿✿✿ long-range scales ✿✿✿✿✿✿ Emergent behavior in collective dynamics 2

A basic paradigm in collective dynamics — alignment • Dynamics of N particles � Crowd dynamics (of birds, human, robots): in different contexts { v i ( t ) } are velocities, orientations, opinions, .... � � d d t v i ( t ) = λτ φ ij ( v j − v i ) + λ ∇ ψ ( | x j − x i | ) • Alignment j ∈N i j φ ij = φ ( x i , x j ) � 0 , x i = v i ˙ • Communication protocol: • The structure of φ is context-dependent; in general – not known Approximate shape — derived empirically, learned from the data 2 , or postulated based on phenomenological arguments ⋆ Study how different classes of φ ’s affect the collective dynamics 2 Lu, Maggioni Tang & M. Zhong, Discovering laws of interaction from observations Emergent behavior in collective dynamics 3

Geometric neighborhoods • Time scale: τ = 1 N ; external potential: repulsion/attraction ∇ ψ �→ 0: � d t v i ( t ) = λ d φ ij ( v j − v i ) • Alignment (self-organize) N j ∈N i • Key role — Geometric neighborhoods 3 φ ij := φ ( | x i − x j | ) � N i = B R 0 { x i } , R 0 = diam x Supp { φ } Repulsion, Alignment, Attraction (cohesion) 3 Aoki (1982) Reynolds (1987) — 1998 Academy Scientific and Technical Award for “ pioneering contributions ... 3D computer animation” Emergent behavior in collective dynamics 4

Examples (w/geometric neighborhoods) • Cucker-Smale model 4 — long-range alignment of velocities { v i ( t ) } N i =1 � � � d t v i ( t ) = λ d 1 φ ( | x i − x j | ) v j − v i , φ ( r ) = (1 + r 2 ) β N j • Singular kernels 4 b — emphasize near-by neighbors � d t v i ( t ) = λ d v j − v i φ ( r ) = r − β | x i − x j | β , N j • Vicsek model 4 c for flocking — short-range alignment of orientations � | x j − x i | � R 0 v j v i ( t + ∆ t ) = s | � | x j − x i | � R 0 v j | + noise φ ( r ) = 1 R 0 ( r ) � 4 F. Cucker & S. Smale, Emergent Behavior in Flocks (2007) 4 b Carrillo, Mucha, Peszek, Soler... 4 c Vicsek, Czir´ ok, Ben-Jacob, Cohen, Shochet (1995) Emergent behavior in collective dynamics 5

Self-organized dynamics — different questions/tools arise in different fields Biology — The role of empirical data Flocks, swarms, colonies, ... — how are they formed? Since there is no Newton’s law — what are the rules of engagement? ⋆ Are the observed patterns system specific? Physics — Order and disorder in complex systems Models are different but deep analogies in patterns of equilibrium Stability near ”thermal equilibrium” — statistical mechanics ⋆ Ensembles act similarly–can we classify collective patterns? Computer Science — The role of discrete geometry Agents form networks – large-time large-crowd network dynamics ⋆ Clustering and spectral theory of graphs Engineering — Design features - control and synchronization Can we control collective dynamics – optimize traffic, improve safety? Mathematics — Agent-based models; non-local PDEs Agent-based � kinetic models � macroscopic models ⋆ Numerical and analytical studies of ‘social hydrodynamics’ Emergent behavior in collective dynamics 6

First limit — emergent behavior as t → ∞ Does “averaging” lead to flocking: v i ( t ) − → u ∞ , x i ( t ) ∼ x i ∞ + t u ∞ ? • Alignment as a diffusion process on graphs: � d t v ( t )= λ d φ ij ( v j − v i ) , φ ij = φ ( x i , x j ) N vertices V = { v i } ⊂ R n ; edges E φ = { e ij } ⊂ R n × R n Graph G = ( V , E ): ✿✿✿✿✿✿✿ ✿✿✿✿✿ � � � grad ∇ φ ( v ) ij := φ ij ( v i − v j ); φ ij ( e ij − e ji ) divergence div φ ( e ) i := ✿✿✿✿ ✿✿✿✿✿✿✿✿✿✿ j � Laplacian: ∆ φ := − 1 and ✿✿✿✿✿✿✿✿✿ 2div φ ◦ ∇ φ , ∆ φ ( v ) i = φ ij ( v i − v j ) j � ⋆ Symmetric protocol: ( A φ ) i � = j = φ ( x i , x j ) , (deg φ ) ii = φ ( x i , x j ) j d t v ( t )= − λ d N ∆ φ ( v ( t )) , ∆ φ := deg φ − A φ ( positive !) Emergent behavior in collective dynamics 7

Emergent behavior as t → ∞ (cont’d) � d t v ( t ) = λ d − λ φ ij ( v j − v i ) � N ∆ φ ( v ( t )) N � | v i − v j | 2 , µ = κ 2 (∆) > 0 • Poincare inequality: (∆( v ) , v ) � µ i , j � � | v i − v j | 2 � 1 / 2 � � � � � � d � − λ � v ( t ) N µ ( t ) v ( t ) with v 2 := d t i , j • Dictated by Fiedler #: µ ( t ) = κ 2 (∆ φ ( x ( t )) ) > 0 , ∆ φ := deg φ − A φ � Flocking depends on propagation of connectivity of the graph G ( v ( t )) ✿✿✿✿✿✿✿✿✿✿ ⋆ Long range interactions — unconditional flocking 5 : � ∞ � 1 | v i − v | 2 → 0 , φ ( r )d r = ∞ v = average( v (0)) � N i ⋆ Short range interactions — instabilities in discrete dynamics Interplay between dynamics on graph and graph driven by the dynamics 5 Ha & ET (2008); Ha & Liu (2009); Motsch & ET (2014) Emergent behavior in collective dynamics 8

Short range interactions: The emergence of many clusters • 100 uniformly distributed opinions: φ ( r ) = a 1 { r � 1 2 } + b 1 { 1 2 � r < 1 } √ √ a = b = 1 : φ ( r ) = 1 { 0 < r < 1 } ( a , b ) = (0 . 1 , 1) φ φ 1 1 10 10 . 1 0 1 0 1 8 8 s x i s x i 6 6 n n Opinio Opinio 4 4 2 2 0 0 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 ( t ( t time ) time ) φ th b / a = . 1 φ th b / a = 1 φ w ith b / a = 2 φ w ith b / a = 10 w i w i 10 10 10 10 8 8 8 8 s x i s x i s x i s x i 6 6 6 6 n n n n Opinio Opinio Opinio Opinio 4 4 4 4 2 2 2 2 0 0 0 0 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 ( t ( t ( t ( t time ) time ) time ) time ) • Homophilious dynamics: align with those that think alike ( a ≫ b ) vs. • Heterophilious dynamics: ”bonding with the different” ( a ≪ b ) Emergent behavior in collective dynamics 9 • Heterophilious dynamics enhances connectivity 5 a : lusters of er Numb ratio

Large crowd dynamics � • Empirical distribution: 1 δ x j ( t ) ( x ) ⊗ δ v j ( t ) ( v ) � f ( t , x , v ) , N ≫ 1 N � j Hydrodynamic description in terms of ( ρ, ρ u ) := (1 , v ) f ( t , x , v )d v  ∂ t ρ + ∇ x · ( ρ u ) mass : = 0   ∂ t ( ρ u ) + ∇ x · ( ρ u ⊗ u + P ( f )) = ρ A ρ ( u ) momentum :   � � � • Alignment A ρ ( u )= λ u ( t , y ) − u ( t , x ) R n φ ( x , y ) ρ ( t , y )d y � • Stress tensor 6 P ij ( f ) = R n ( v i − u i )( v j − u j ) f ( t , x , v )d v u t + u · ∇ x u + 1 • Transport+Alignment: ρ P ij ( f ) = A ρ ( u ) 6 Ha & ET(2008); Carrillo et. al.(2012); Karper, Mellet, Trivisa (2013) Emergent behavior in collective dynamics 10

Hydrodynamic vs. agent-base description S. Motsch Vicsek model: agent-base model vs. hydrodynamic description Emergent behavior in collective dynamics 11

Smooth solutions must flock � � � • Energy fluctuations vs. enstrophy 7 A ρ ( u )= λ φ ( x , y ) u ( y ) − u ( x ) d ρ ( y ) � � � � d R 2 n | u ( y ) − u ( x ) | 2 d ρ ( x )d ρ ( y )= − λ A ρ ( u ( y )) , u ( y ) d ρ ( y ) d t R 2 n � � � R 2 n | u ( t , y ) − u ( t , x ) | 2 d ρ ( t , x )d ρ ( t , y ) • Fluctuations: u ( t ) 2 ,ρ := � � and since φ ( | x − y | ) x , y ∈ Supp ρ ( t , · ) � φ ( t ) � u 0 � � � � � � d u ( t ) 2 ,ρ � − λ µ ( t ) u ( t ) 2 ,ρ , µ ( t ) � φ ( u 0 t ) d t • Again — long-range interactions imply unconditional flocking 7 b : � � R n | u ( t , x ) − u | 2 ρ ( t , x )d x − φ ( r )d r = ∞ � → 0 • Existence of smooth solution, u ( t , · ) ∈ C 1 : n = 1 , 2; n > 2 is open — dependence on critical thresholds in initial configurations 7 c 7 Independent of the closure relation! S.-Y. Ha & ET KRM (2008) 7 b ET & C. Tan, Proc. Roy. Soc. A (2014) 7 c Y.-P. Choi, Carrillo, ET., Tan (2015) Emergent behavior in collective dynamics 12