Collective Dynamics and the Emergence of Patterns

MATH 858J, Fall 2026

Course Information

 

Course description

We will survey recent mathematical developments in collective dynamics of interacting agents. A main question of practical interest is how fundamental protocols of interaction — attraction, alignment, and repulsion — facilitate the emergence of high-order patterns in non-equilibrium systems.
Different models based on such protocols go back to the influential works of Kuramoto, Reynolds, Vicsek and Cucker & Smale. We will discuss their broad range of applications, from synchronization and opinion dynamics to swarming, robotics and the dynamics of transformers in LLM.
Three levels of descriptions will be considered: microscopic description of agent-based dynamics on graphs, mesoscopic mean-field description, and large-crowd hydrodynamic description. .





Syllabus

1     Agent-based models: slides [ 10 examples ]

1.1  First-order models for aggregation . . . . . . . . . . . . . . . . . . . .
   • Emergence of consensus . . . . . . . . . . . . . . . . . . . . . . . .
      R. Hegselmann & U. Krause, The bounded confidence model (2002) (2019)
      J. Lorenz A survey (2007)
      V. Blondel, J. Hendricks & J. Tsitsiklis, On Krause’s multi-agent consensus (2009)
      C. Brugna & G. Toscani, Opinions and convictions (2015)
   • Bearing only model . . . . . . . . . . . . . . . . . . .
      L. Bellaiche1 & A. Bruckstein, Limited visibility (2017)
   • Elo rating . . . . . . . . . . . . . . . . . . . . . . . .
      P.-E. Jabin and S. Junca, Continuous model (2015)
   • Clustering . . . . . . . . . . . . . . . . . . . . . . . .
   K. Fukunaga & L. Hostetler, Pattern recognition (1975)
      D. Comaniciu & P. Meer, Consensus-based mean-shift (2002)
      E. Arias-Castro, D. Mason & B. Pelletier Estimation of the Gradient Lines (2016)
   • Transformers and the geometry of emerging clusters . . . . . . . . . . . . . . . .
      B. Geshkovski et. al. Mathematical Perspective (2025)
      P. Rigollet, Mean-field dynamics (2026)

1.2  Second-order models for alignment
   1.2.1 Flocking and Swarming . . . . . . . . . . . .
   • Cucker-Smale (CS) model: F. Cucker & S. Smale, Emergent behavior (2007a) (2007b)
      Extensions:
   • Motch-Tadmor (MT) model: S. Motsch & E. Tadmor, New model (2010)
      A. Kotil, CS and MT probablistic method (2022)
      Y.-J. Gao & J.-G. Dong, CS with cone of vision (2026)
   • p-alignment
      S.-Y Ha, T. Ha & J-H. Kim, CS with p-alignment, (2010)(p>1)
      J. A. Carrillo, Y.-P. Choi & M Hauray, CS with p-alignment (2014) Sec 4. (p<1)
      E. Tadmor, p-alignment with pressure (2022) Sec. 5.3 (p>1)
   • PTWA model: P. Degond & S. Motsch, Swarming agents using curvature (2011)
   1.2.2 Communication kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
   • Metric vs. topological
   • Matrix communication
   • Short- vs. long-range tails
   • Regular vs. singular head
1.3  3Zone model: Attraction • Repulsion • Alignment . . . . . . . . . . . . . . . . . . .
       Anticipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4  Extensions • External forcing • Rayleigh-type friction • Time delay . . . . . . .
• Vicsek model . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5  Synchronization • Kuramoto and related models . . . . . . . . . . . . . . . . . . . .

Lecture notes [ Part I. Agent-based models ]

2    Large-time emerging behavior

2.1  First-order models • Consensus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2  Second-order models • Flocking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
      ℓ^∞-diameter -- coefficient of ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . .

Lecture notes [ Part II.1 Coefficient of ergodicity ]

      ℓ²-diameter -- graph connectivity • spectral gap . . . . . . . . . . . . . . . . . . . .
      Short-range kernels: propagation of connectivity . . . . . . . . . . . . . . . . . . .
      Homophilious vs. heterophilious dynamics. . . . . . . . . . . . . . . . . . . . . . . .

3    Large-crowd behavior -- hydrodynamic description

3.1  Mean-field limit -- First-order aggregation models . . . . . . . . . . . . . . . . . . .
3.2  Mean-field limit -- second-order models . . . . . . . . . . . . . . . . . .

4    Large-crowd behavior -- hydrodynamic description

4.1 The closure of entropic pressure
4.2 Regularity and critical thresholds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3  Large-time emerging behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
       Spectral gap • Energy Flctuations • Entropy . . . . . . . . . . . . . . . . . . . . . .
4.4  Multi-species • Anticipation • External forcing . . . . . . . . . . . . . . . . . . . . .

5     Multi-scale descriptions

5.1  Multi-Flocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2  Multi-species. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


References:

• N. Bellomo, J. Carrillo, P. Degond & E. Tadmor (eds), ``Active Particles'', Vol 1 (2017), Vol 2 (2019)
  Vol 3 (2022), Vol 4 (2024), Birkhäuser.
• I. Couzin & N. Franks, Self-organized lane formation and optimized traffic flow in army ants,
  Proc. R. Soc. Lond. B, 270 (2003) 139-146.
• F. Cucker & S. Smale, On the math. of emergence, Japan. J. Math. 2 (2007) 197-227.
• F. Cucker & S. Smale, Emergent behavior in flocks. IEEE Trans. Automat. Control 52 (2007).
• S.-Y. Ha & E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking,
  Kinetic and Related Models 1(3) (2008) 415-435.
• R. Hegselmann & U. Krause, Opinion dynamics and bounded confidence: models, analysis simulation,
  J. Artificial Soc. and Social Simul. 5(3) (2002).
• S. Motsch & E. Tadmor, A new model for self-organized dynamics and its flocking behavior,
  JSP 144(5) (2011) 923-947.
• C. Reynolds, Flocks, herds and schools: A distributed behavioral model,
  ACM SIGGRAPH 21 (1987) 25-34.
• R. Shu & E. Tadmor, Flocking hydrodynamics with external potentials, ARMA 238 (2020) 347-381.
• R. Shu & E. Tadmor, Anticipation breeds alignment, ARMA 240 (2021) 203-241.
• R. Shvydkoy, Dynamics and Analysis of Alignment Models of Collective Behavior, Springer, 2021
  (lecture notes).
• R. Shvydkoy & E. Tadmor, Topologically-based fractional diffusion and emergent dynamics with
  short-range interactions, SIMA 52(6) (2020) 5792-5839.
• E. Tadmor, Swarming: hydrodynamic alignment with pressure, Bulletin AMS 60(3) (2023) 285-325.
• T. Vicsek, Czirók, E. Ben-Jacob, I. Cohen & O. Schochet, Novel type of phase transition in a system of
  self-driven particles, PRL 75 (1995) 1226-1229.
• T. Vicsek & A. Zefeiris, Collective motion, Physics Reprints, 517 (2012) 71-140.


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Eitan Tadmor