From Simplified Particle Models to Complex Social Agent Simulations
Watch how the MOSH simulation exhibits emergent behaviors: moshers spontaneously form pits with stronger cohesion and alignment than MASHer, listeners defend good spots near the stage, exhausted moshers retreat to rest, and organize with coordinated rotation. The physical barrier below the stage prevents agents from entering restricted areas. Non-convertible listeners (low aggression + high shyness) actively flee from the mosh pit center and exhibit 2.5× stronger avoidance. These patterns emerge naturally from agent interactions without explicit programming of collective behaviors. The angular momentum graph below tracks the rotational dynamics of the mosh pit, with sustained non-zero values indicating stable pit formation.
Collective human behavior in extreme social settings has long fascinated researchers across physics, sociology, and complex systems. Heavy metal concerts represent a unique environment where highly energized crowds exhibit emergent collective motion patterns, most notably mosh pits and circle pits. These phenomena provide a rare opportunity to study human crowd dynamics in a controlled, repeatable, and ethically observable setting.
Original Inspiration:
Silverberg, J. L., Bierbaum, M., Sethna, J. P., & Cohen, I. (2013). Collective Motion of Moshers at Heavy Metal Concerts. Physical Review Letters, arXiv:1302.1886v1The 2013 study by Silverberg et al. pioneered the application of flocking models to mosh pit dynamics using the MASHer (Mobile Active Simulated Humanoid) framework, demonstrating that human collective behavior could be reproduced using simplified particle-based simulations. Their work identified two primary states: a disordered gas-like state (mosh pit) and an ordered vortex-like state (circle pit).
While groundbreaking, the MASHer simulation made several simplifying assumptions that limited the emergence of complex dynamics:
This work extends the MASHer model by introducing sophisticated social agent behaviors that capture the nuanced dynamics of real mosh pits. The MOSH (Modelling Organized Self-Propelled Humans) simulation incorporates individual personality traits, stamina systems, spatial satisfaction, dynamic type conversion, and emergent social structures. These enhancements enable stable, long-lasting emergent patterns across a wide range of parameters without requiring carefully tuned initial conditions.
The MASHer model combines four fundamental forces acting on each agent i:
$\vec{F}_{\text{repulsion}}^i = \begin{cases} \varepsilon\left(1 - \frac{r_{ij}}{2r_0}\right)^{5/2} \hat{r}_{ij} & \text{if } r_{ij} < 2r_0 \\ 0 & \text{otherwise} \end{cases}$
where $\varepsilon$ is the repulsion strength, $r_{ij}$ is the distance between agents $i$ and $j$, $r_0$ is the agent radius, and $\hat{r}_{ij}$ is the unit vector pointing from $j$ to $i$.
$\vec{F}_{\text{propulsion}}^i = \mu(v_0 - v_i)\hat{v}_i$
where $\mu$ is the damping coefficient, $v_0$ is the preferred speed ($v_0 = 1$ for active MASHers, $v_0 = 0$ for passive MASHers), $v_i$ is the current speed, and $\hat{v}_i$ is the velocity direction.
$\vec{F}_{\text{flocking}}^i = \alpha \frac{\sum_{j\in N_i} \vec{v}_j}{\left|\sum_{j\in N_i} \vec{v}_j\right|}$
where $\alpha$ is the flocking strength, $N_i$ is the set of neighbors within radius $r_{\text{flock}} = 4r_0$, and the summation aligns the agent's velocity with nearby active MASHers.
$\vec{F}_{\text{noise}}^i = \vec{\eta}_i$
where $\vec{\eta}_i$ is a vector with components drawn from a Gaussian distribution with zero mean and variable standard deviation (applied only to active MASHers).
$\vec{F}_{\text{total}}^i = \vec{F}_{\text{repulsion}}^i + \vec{F}_{\text{propulsion}}^i + \vec{F}_{\text{flocking}}^i + \vec{F}_{\text{noise}}^i$
$\vec{v}_i(t + \Delta t) = \vec{v}_i(t) + \vec{F}_{\text{total}}^i \Delta t$
$\vec{r}_i(t + \Delta t) = \vec{r}_i(t) + \vec{v}_i(t) \Delta t$
Positions and velocities are updated using the Newton-Störmer-Verlet algorithm with periodic boundary conditions.
The MOSH model extends MASHer with additional social forces and individual agent properties. Each agent $i$ has personality traits (aggression $a_i$, shyness $s_i$, assertiveness $\alpha_i$, mobility $m_i$), stamina $\sigma_i$, exhaustion $e_i$, and spot satisfaction $S_i$ (for listeners).
$\rho_i = \frac{N_i}{\pi R_{\text{perception}}^2}$
Local density around agent $i$, where $N_i$ is the number of neighbors within $R_{\text{perception}} = 3 \times \text{personalSpace}$.
$\vec{F}_{\text{density}}^i = \sum_{j\in N_i} \left[ f_{\text{collision}}(r_{ij}) + f_{\text{pressure}}(\rho_i, r_{ij}) \right]$
For listeners: $f_{\text{pressure}} = (\rho_i - \rho_{\text{desired}}) \times 200 \times k_{\text{avoid}} \times m_i \times (1 - \alpha_i \times \text{discomfort})$
For moshers: Different density preferences ($\rho_{\text{desired}}^{\text{mosher}} \in [0.006, 0.010]$, $\rho_{\text{desired}}^{\text{listener}} \in [0.002, 0.004]$)
$\vec{F}_{\text{cohesion}}^i = 0.03 \times (\bar{r}_{\text{moshers}} - \vec{r}_i) \quad \text{[moshers only]}$
$\vec{F}_{\text{alignment}}^i = 0.12 \times (\bar{v}_{\text{moshers}} - \vec{v}_i) \quad \text{[moshers only]}$
Stronger cohesion (2× increase) and alignment (1.5× increase) compared to MASHer model, promoting tighter mosh pit formation.
$S_i = 0.4 \times S_{\text{stage}} + 0.4 \times S_{\text{density}} + 0.2 \times S_{\text{mosh}}$
$S_{\text{stage}} = 1 - \min(d_{\text{stage}}/300, 1)$
$S_{\text{density}} = 1 - \min(|\rho_i - \rho_{\text{desired}}|/0.01, 1)$
$S_{\text{mosh}} = 1 - \frac{N_{\text{moshers}}}{N_i}$
Satisfied listeners ($S_i > 0.6$) reduce movement: $v_{\max} \leftarrow v_{\max} \times (1 - 0.85S_i)$ and increase friction: $f_{\text{friction}} = 0.7 + 0.1(1 - S_i)$
$e_i(t + \Delta t) = e_i(t) + [(v_i/4 \times 0.0005) + (\text{slams}_i \times 0.001)] \times k_{\text{exhaustion}}^i$
Exhaustion increases with activity. Stamina distribution: 15% tireless ($\sigma = \infty$), 20% high ($\sigma \in [1.5, 2.0]$), 35% average ($\sigma \in [1.0, 1.3]$), 30% low ($\sigma \in [0.5, 0.8]$). Resting triggered when $e_i > 0.7 + 0.1\sigma_i$.
$\text{Recovery: } e_i(t + \Delta t) = \max(0, e_i(t) - [0.003 + \sigma_i/100]) \quad \text{[while resting]}$
$\vec{F}_{\text{radial}}^i = -0.3 \times \text{sign}(r_i - R_{\text{pit}}) \times \hat{r}_i$
$\vec{F}_{\text{tangential}}^i = 0.8 \times d_i \times \hat{\theta}_i$
where $r_i$ is distance from pit center, $R_{\text{pit}}$ is average radius, $d_i \in \{-1, +1\}$ is rotation direction, and $\hat{\theta}_i$ is tangent vector. Spacing control: adjust tangential force $\pm 0.2$ based on distance to nearest mosher ahead.
$P(\text{listener} \rightarrow \text{mosher}) = 0.1 \times (1 - s_i) \times a_i \times \frac{N_{\text{mosh}}}{N_{\text{mosh}} + N_{\text{listen}} + 1} + \beta_{\text{probe}}$
$P(\text{mosher} \rightarrow \text{listener}) = 0.05 \times \frac{(1 - E_i) + \min(\text{slams}_i/10, 1) + e_i + I_{\text{isolated}}}{4}$
where $\beta_{\text{probe}} = 0.3$ if probed by initiator, $I_{\text{isolated}} = 0.3$ if $N_{\text{mosh}} < 3$. Initiators (30% of moshers) actively recruit through probing bumps.
$C_i = (1 - s_i) \times a_i \quad \text{[convertibility]}$
$\vec{F}_{\text{flee}}^i = 0.15 \times (1 - d_{\text{mosh}}/80) \times \frac{\vec{r}_i - \bar{r}_{\text{mosh}}}{d_{\text{mosh}}} \quad \text{if } C_i < 0.15$
Listeners with low convertibility ($C_i < 0.15$) exhibit enhanced avoidance: 2.5× stronger separation from moshers and active fleeing from mosh pit center within 80 units.
$\text{if } y_i < y_{\text{barrier}}: \quad y_i \leftarrow y_{\text{barrier}}, \quad v_y^i \leftarrow \max(0, v_y^i)$
Hard constraint preventing agents from entering the stage area ($y_{\text{barrier}} = 60$). Vertical velocity component is clipped to prevent upward motion.
$f_{\text{friction}} = \begin{cases} 0.75 & \text{if } N_{\text{moshers}}^{\text{nearby}} = 0 \\ 0.95 & \text{otherwise} \end{cases}$
When no active moshers within 100 units, friction increases dramatically (0.75 vs 0.95), causing listeners to exhibit solid-like crystalline packing with minimal movement.
Each agent has unique traits: aggression, shyness, assertiveness, mobility, and energy levels that influence their behavior and interactions.
Moshers have varying stamina levels (15% tireless, 30% tire easily). Exhausted agents seek safe resting spots among listeners before returning to the pit.
Listeners evaluate their position based on stage proximity, crowd density, and mosh avoidance. Satisfied listeners defend their spots and resist movement.
Agents can convert between listener and mosher types based on energy, proximity to moshers, exhaustion, and social influence from initiators.
30% of moshers are initiators who actively recruit listeners through probing interactions and lead pit formation even when isolated.
When enough Moshers gather, they spontaneously form coordinated circular motion with self-organization and spacing.
When no moshers are nearby, listeners exhibit solid-like crystalline packing with dramatically increased friction and resistance to motion.
Each agent has individual density preferences. Moshers seek higher density (0.006-0.010), listeners prefer lower density (0.002-0.004).
| Feature | MASHer (2013) | MOSH (2025) |
|---|---|---|
| Passive Agents | Completely stationary (v₀ = 0) | Dynamic with spot satisfaction and defensive behaviors |
| Energy System | Constant energy, no exhaustion | Stamina variations, exhaustion, resting cycles |
| Type Conversion | Fixed at initialization | Dynamic probabilistic conversion based on local conditions |
| Social Structure | Uniform flocking only | Initiators, probing, recruitment, personality-driven interactions |
| Spatial Behavior | Simple position-based forces | Spot evaluation, satisfaction tracking, territory defense |
| Pit Formation | Parameter-dependent, fragile | Self-organizing with detection, synchronization, and stability |
| Emergence Stability | Requires specific initial conditions | Robust across wide parameter ranges |
The MASHer model uses Mobile Active Simulated Humanoids with Hertzian soft-body repulsion, self-propulsion, flocking interactions, and Gaussian noise. Passive MASHers are completely stationary (v₀ = 0), while active MASHers seek their preferred speed. The model successfully reproduces Maxwell-Boltzmann velocity distributions in the gas-like mosh pit regime. The vorticity graph displays angular momentum fluctuations, revealing the transition between disordered (mosh pit) and ordered (circle pit) states.
MASHer Model: Pattern formation is highly sensitive to parameter tuning. pits require careful balance of flocking strength and noise. Mosh pits exist in limited parameter regions. Transitions between states are abrupt and often unstable.
MOSH Model: Emergent patterns are robust across wide parameter ranges. Pits form spontaneously when sufficient moshers gather, even from random initial conditions. The system exhibits natural phase transitions driven by agent interactions rather than global parameters. Enhanced cohesion (2×) and alignment (1.5×) promote stable mosh pit formation.
MASHer Model: Captures aggregate statistics (velocity distributions, spatial correlations) but lacks individual behavioral diversity. All active agents behave identically; all passive agents are frozen.
MOSH Model: Reproduces observed phenomena like exhaustion-induced retreat, spot defense by listeners, initiator-driven pit formation, non-convertible listener fleeing behavior, and the 95% counter-clockwise bias in circle pit rotation. Individual differences lead to heterogeneous crowd dynamics. Physical barriers model real venue constraints.
Both MASHer and MOSH models demonstrate that complex collective human behavior can emerge from relatively simple local interaction rules. The MASHer work established that mosh pit dynamics follow physical principles analogous to gas kinetics. MOSH extends this by showing that incorporating social psychological factors like personality, fatigue, satisfaction, and influence produces richer emergent structures that more closely match real concert observations. The addition of non-convertible listener behavior and physical barriers further enhances realism without sacrificing computational efficiency.
Several phenomena remain unexplored: wall of death dynamics (two opposing groups charging), collective jumping patterns, injury risk modeling, and architectural safety design. The MOSH model provides a foundation for investigating these scenarios. Additionally, machine learning techniques could identify optimal crowd management strategies by training on simulation data. Further validation against real concert footage could refine individual parameters and test predictive accuracy.
Understanding crowd dynamics in extreme social settings has applications beyond concert venues. The principles discovered here inform evacuation planning, theme park design, and public gathering safety. By studying humans under controlled extreme conditions, we gain insights applicable to emergencies where panic and high energy create similar collective behaviors.
References
Silverberg, J. L., Bierbaum, M., Sethna, J. P., & Cohen, I. (2013). Collective Motion of Moshers at Heavy Metal Concerts. Physical Review Letters. arXiv:1302.1886v1 [physics.soc-ph]
Original MASHer simulation available at: mattbierbaum.github.io/moshpits.js
This work presents MOSH (Modelling Organized Self-Propelled Humans), an extension that builds directly on the MASHer framework with enhanced social dynamics, individual personality traits, stamina systems, and emergent behavioral patterns.