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

Emergent behavior in collective dynamics

Eitan Tadmor University of Maryland1

“Advances in Applied Mathematics” — in memoriam of Saul Abarbanel Tel-Aviv U., Dec. 18, 2018

1Center 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 {vi(t)} are velocities, orientations, opinions, ....

• Alignment

d dt vi(t) = λτ

• j∈Ni

φij(vj − vi) + λ

• j

∇ψ(|xj − xi|)

• Communication protocol:

φij = φ(xi, xj) 0, ˙ xi = vi

• The structure of φ is context-dependent; in general – not known

Approximate shape — derived empirically, learned from the data2,

• r postulated based on phenomenological arguments

⋆ Study how different classes of φ’s affect the collective dynamics

2Lu, 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:

• Alignment (self-organize)

d dt vi(t) = λ N

• j∈Ni

φij(vj − vi)

• Key role — Geometric neighborhoods3

φij := φ(|xi − xj|) Ni = BR0{xi}, R0 = diamxSupp {φ} Repulsion, Alignment, Attraction (cohesion)

3Aoki (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 model4 — long-range alignment of velocities {vi(t)}N

i=1

d dt vi(t) = λ N

• j

φ(|xi − xj|)

• vj − vi
• ,

φ(r) = 1 (1 + r2)β

• Singular kernels4b — emphasize near-by neighbors

d dt vi(t) = λ N

• j

vj − vi |xi − xj|β , φ(r) = r−β

• Vicsek model4c for flocking — short-range alignment of orientations

vi(t + ∆t) = s

• |xj−xi|R0 vj

|

|xj−xi|R0 vj| + noise

• φ(r) = 1R0(r)
• 4F. Cucker & S. Smale, Emergent Behavior in Flocks (2007)

4bCarrillo, Mucha, Peszek, Soler... 4cVicsek, Czir´

• k, 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: vi(t) − → u∞, xi(t) ∼ xi∞ + tu∞?

• Alignment as a diffusion process on graphs:

d dt v(t)= λ N

• φij(vj − vi),

φij = φ(xi, xj) Graph G = (V , E): ✿✿✿✿✿✿✿ vertices V ={vi} ⊂ Rn;

✿✿✿✿✿

edges Eφ ={eij} ⊂ Rn×Rn

✿✿✿✿

grad ∇φ(v)ij :=

• φij(vi − vj);

✿✿✿✿✿✿✿✿✿✿

divergence divφ(e)i :=

• j
• φij(eij − eji)

and ✿✿✿✿✿✿✿✿✿ Laplacian: ∆φ := −1 2divφ ◦ ∇φ, ∆φ(v)i =

• j

φij(vi − vj) ⋆ Symmetric protocol: (Aφ)i=j = φ(xi, xj), (degφ)ii =

• j

φ(xi, xj) d dt v(t)=− λ N ∆φ(v(t)), ∆φ := degφ − Aφ (positive!)

Emergent behavior in collective dynamics 7

Emergent behavior as t → ∞ (cont’d)

d dt v(t) = λ N

• φij(vj − vi)
• − λ

N ∆φ(v(t))

• Poincare inequality:

(∆(v), v) µ

• i,j

|vi − vj|2, µ = κ2(∆) > 0

• d

dt

• v(t)
• − λ

N µ(t)

• v(t)
• with
• v
• 2 :=

i,j

|vi − vj|21/2

• Dictated by Fiedler #: µ(t) = κ2(∆φ(x(t))) > 0,

∆φ := degφ − Aφ Flocking depends on propagation of

✿✿✿✿✿✿✿✿✿✿

connectivity of the graph G(v(t)) ⋆ Long range interactions — unconditional flocking5: ∞ φ(r)dr = ∞

• 1

N

• i

|vi − v|2 → 0, v = average(v(0)) ⋆ Short range interactions — instabilities in discrete dynamics Interplay between dynamics on graph and graph driven by the dynamics

5Ha & 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) = a1{r 1

√ 2 } + b1{ 1 √ 2 r<1}

a = b = 1 : φ(r) = 1{0<r<1} (a, b) = (0.1, 1)

2 4 6 8 10 5 10 15 20 25 30 35 40

1

φ

1

2 4 6 8 10 5 10 15 20 25 30 35 40

1 1

φ

.1

2 4 6 8 10 5 10 15 20 25 30 35 40

φ

2 4 6 8 10 5 10 15 20 25 30 35 40

φ

2 4 6 8 10 5 10 15 20 25 30 35 40

φ

2 4 6 8 10 5 10 15 20 25 30 35 40

φ

• Homophilious dynamics: align with those that think alike (a ≫ b) vs.
• Heterophilious dynamics: ”bonding with the different” (a ≪ b)
• Heterophilious dynamics enhances connectivity5a:

• f

Emergent behavior in collective dynamics 9

Large crowd dynamics

• Empirical distribution: 1

N

• j

δxj(t)(x) ⊗ δvj(t)(v) f (t, x, v), N ≫ 1 Hydrodynamic description in terms of (ρ, ρu) :=

• (1, v)f (t, x, v)dv

     mass : ∂tρ + ∇x · (ρu) = 0 momentum : ∂t(ρu) + ∇x · (ρu ⊗ u + P(f )) = ρ Aρ(u)

• Alignment

Aρ(u)= λ

• Rnφ(x, y)
• u(t, y) − u(t, x)
• ρ(t, y)dy
• Stress tensor6

Pij(f ) =

• Rn(vi − ui)(vj − uj)f (t, x, v)dv
• Transport+Alignment:

ut + u · ∇xu + 1 ρPij(f ) = Aρ(u)

6Ha & 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. enstrophy7 Aρ(u)= λ
• φ(x, y)
• u(y) − u(x)
• dρ(y)

d dt

• R2n |u(y) − u(x)|2dρ(x)dρ(y)=−λ
• R2n
• Aρ(u(y)), u(y)
• dρ(y)
• Fluctuations:
• u(t)
• 2,ρ :=
• R2n |u(t, y) − u(t, x)|2dρ(t, x)dρ(t, y)

and since φ(|x − y|)x,y∈Supp ρ(t,·) φ(

• u0
• t)

d dt

• u(t)
• 2,ρ −λ µ(t)
• u(t)
• 2,ρ,

µ(t) φ(

• u0
• t)
• Again — long-range interactions imply unconditional flocking7b:
• φ(r)dr = ∞
• Rn |u(t, x) − u|2ρ(t, x)dx −

→ 0

• Existence of smooth solution, u(t, ·) ∈ C 1: n = 1, 2; n > 2 is open

— dependence on critical thresholds in initial configurations7c

7Independent of the closure relation! S.-Y. Ha & ET KRM (2008) 7bET & C. Tan, Proc. Roy. Soc. A (2014) 7cY.-P. Choi, Carrillo, ET., Tan (2015) Emergent behavior in collective dynamics 12

Short range interactions I. R0 := diamxSupp{φ} ≪ diamxSupp{ρ0}

• So far — flocking behavior for ✿✿✿✿✿✿✿✿✿✿

long-range interactions What about more realistic ✿✿✿✿✿✿✿✿✿✿ short-range interactions?

• Key estimate — energy vs. enstrophy

d dt

• R2n

|u(y) − u(x)|2dρ(x)dρ(y)=−λ

• R2n

φ(|x − y|)|u(y) − u(x)|2dρ(x)dρ(y)

• J. Morales
• J. Peszek

✿✿✿✿✿✿✿✿

Theorem (multiD)8 Assume that Supp{ρ0} is chain connected — connected by sequence of overlapping balls ∀(x, y) ∈ Fx,y = ∪K

α=1Br(xα),

xα ∈ Supp{ρ0} If λ is large enough, then smooth solutions must flock: If λ #K(r)

m3(Br) then covering of scale r persists in time

λ

• R2n

φ(|x−y|)|u(y)−u(x)|2dρ(x)dρ(y)

• R2n

|u(y)−u(x)|2dρ(x)dρ(y)

8Morales, Peszek, ET (2018), Flocking with short-range interactions Emergent behavior in collective dynamics 13

Short range interactions II — regular kernels

No threshold restriction on λ:      ρt + ∇x · (ρu) = 0 ut + u · ∇xu = λ

• y

φ(|x − y|)(u(y) − u(x))ρ(y)dy subject to 1 Λ1R0(r) φ(r) Λ12R0(r), R0 ≪ diamxSupp ρ(t, ·)

✿✿✿✿✿✿✿

Regular✿✿✿✿✿✿✿ kernels9 (

✿✿✿✿✿✿✿

multiD): φ(| · |) : Tn → R+,

• rR0

φ(r) = 1 If density fluctuations, ρ±(t) :=

• max

min

• ρ(t, ·), are not too large:

1 M {ρ+(t) − ρ−(t)} min

k=0{1 −

φ(k)} smooth solutions must flock

• Flocking for short-range interactions with sub-critical configurations
• When smooth solutions exist? — open

9Emergence with short-range interactions (2019) Emergent behavior in collective dynamics 14

Short range interactions III — a new paradigm

• R. Shvydkoy

New paradigm: interaction between agents depends on...

✿✿✿✿

how ✿✿✿✿✿✿✿✿ crowded✿✿ is✿✿✿✿✿ their✿✿✿✿✿✿✿✿✿✿✿✿ intermediate✿✿✿✿✿✿ region

✿✿✿

• f ✿✿✿✿✿✿✿✿✿✿✿✿✿✿

communication10

✿✿✿✿✿✿

Region✿✿

• f

✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿

communication Ω(x, y) Set a ✿✿✿✿✿✿✿✿✿✿ topological✿✿✿✿✿✿✿✿ distance: distρ(x, y) :=

• Ω(x,y)

ρ(t, z)dz 1/n Short-range kernel : φ(x, y)

topological distance

• φ(distρ(x, y)) ×1R0(|x − y|)

Topological distance: φ(distρ) = 1 distn

ρ(x, y) =

• Ω(x,y)

ρ(t, z)dz −1 Why ‘topological’? — dependence on ‘nearest neighbors’11

10Shvydkoy-ET, Topological models for emergent dynamics w/short-range... (2018) 11Cavagna et. al., Starflag project (2008-2010) Emergent behavior in collective dynamics 15

Short-range interactions III — topological interactions

ut + u · ∇xu = λ

• φ(x, y)
• u(y) − u(x)
• ρ(y)dρ(y)
• Communication protocol:

φ(x, y) = φ1(|x − y|) × 1 distn

ρ(x, y)

. . . making pointwise sense of Lφ(g) :=

• φ(x, y)(g(y) − g(x))dy

✿✿✿✿✿✿✿✿

Theorem1 (multiD)12. Smooth non-vacuous solutions must flock: φ(x, y) = φ1(|x − y|) × 1 distn

ρ(x, y),

ρ(t, ·) 1 1 + t

• Key estimate — local (Campanato) variations by o(1) enstrophy:
• B r

10 (x)

|u(y) − ux,r|2dρ(y)

• φ(x, y)|u(x) − u(y)|2dρ(x)dρ(y)

Propagation of information through sliding averages:

b b b b

x− x1 xK x+ B0 B1 BK BK+1 n 12Shvydkoy-ET, Topological models for emergent dynamics w/short-range... (2018) Emergent behavior in collective dynamics 16

THANK YOU

Emergent behavior in collective dynamics 17