Pressure Threshold Model (PT)

• Pressure Threshold Model (PT) is a framework capturing threshold-driven dynamics induced by accumulated pressure, applied in both clarinet acoustics and social-network diffusion.
• In clarinet acoustics, PT employs iterated maps and bifurcation delay to predict oscillation onset under time-varying mouth pressure, distinguishing dynamic from static thresholds.
• In network science, PT extends the Linear Threshold model with adaptive influence amplification, resulting in non-submodular spread behavior during influence maximization.

Pressure Threshold Model (PT) designates threshold-driven dynamics in which state transitions depend on an accumulated notion of pressure, but the term is used for two technically distinct model families. In clarinet acoustics, PT denotes a dynamic-threshold framework for predicting the onset of self-sustained oscillations in an iterated-map model under time-varying mouth pressure, extending the static oscillation threshold to regimes with bifurcation delay (Bergeot et al., 2012). In social-network diffusion, PT denotes a graph process that extends the Linear Threshold (LT) model by increasing a newly activated node’s outgoing influence proportionally to the pressure it received at activation (Stutsman et al., 16 Sep 2025). The common vocabulary of pressure and threshold therefore spans two separate mathematical settings: a nonlinear delayed map for reed–bore interaction, and a progressive synchronous diffusion process on weighted directed graphs.

1. Terminological scope and domain-specific meanings

In the clarinet literature, the central problem is the onset of oscillation when the musician’s mouth pressure is not constant but increases through time. Simple clarinet models based on iterated maps successfully estimate the threshold of oscillation as a function of a constant blowing pressure, yet when the blowing pressure gradually increases, oscillations appear at a much higher value than in the static case; this is the dynamic oscillation threshold associated with bifurcation delay (Bergeot et al., 2014). In that usage, PT is effectively a pressure-threshold framework for onset prediction in a non-autonomous nonlinear dynamical system.

In network science, PT is explicitly introduced as the “Pressure Threshold model,” a diffusion model for influence propagation on social networks. It preserves the LT activation rule but adds a feedback mechanism: outgoing influence from a newly activated node is amplified according to the influence pressure that caused its activation (Stutsman et al., 16 Sep 2025). Here PT is not an onset threshold in an acoustic oscillator but a threshold diffusion process with adaptive edge weights.

A common misconception is that PT denotes a single standardized model across disciplines. The available literature instead uses the same label for unrelated systems with different state variables, update operators, and analytical objectives. In the clarinet case, the state is the outgoing acoustic wave $p_n^+$ and the threshold concerns a flip bifurcation; in the network case, the state is a node activation indicator $s_v(t)$ and the threshold concerns cumulative in-neighbor influence.

2. Clarinet PT as an iterated-map onset model

The clarinet PT model is built on a generator–resonator decomposition. The exciter is the reed–mouthpiece system, modeled by a nonlinear characteristic $F$ linking mouthpiece pressure $p$ to volume flow $u$. The resonator is a lossless cylindrical bore with one-delay sign-inverting reflection, so that in discrete time $p_n^-=-r p_{n-1}^+$, with $r=1$ in the lossless case treated in the paper (Bergeot et al., 2014).

The normalization is centered on the reed-closing pressure $P_M=kH$, where $k$ is reed stiffness per unit displacement and $H$ is the reed tip opening at rest. The dimensionless control parameter is

$s_v(t)$0

with analogous normalization for mouthpiece pressure and flow. The reed opening parameter is

$s_v(t)$1

where $s_v(t)$2 is the characteristic impedance, $s_v(t)$3 the effective reed width, and $s_v(t)$4 air density (Bergeot et al., 2014).

Using the wave decomposition

$s_v(t)$5

and the lossless reflection law, the clarinet becomes a one-step non-autonomous iterated map

$s_v(t)$6

The nonlinear reed characteristic is expressed in terms of the pressure drop $s_v(t)$7. In the standard Bernoulli-based piecewise law used in the paper,

$s_v(t)$8

$s_v(t)$9

$F$0

The factor $F$1 captures the reduction of effective opening area as the reed approaches closure, while the reverse-flow branch omits that factor in the standard formulation used by the paper (Bergeot et al., 2014).

For constant $F$2, the non-oscillating regime is a fixed point $F$3 satisfying

$F$4

Under the assumptions of a lossless resonator, ideal spring reed, and $F$5, the fixed point loses stability through a flip bifurcation at the static threshold

$F$6

beyond which the steady regime is a 2-cycle (Bergeot et al., 2012).

3. Static and dynamic oscillation thresholds under a linear pressure ramp

For a linear mouth-pressure ramp,

$F$7

the clarinet map becomes non-autonomous: $F$8 The key phenomenon is bifurcation delay: the orbit continues to track the unstable fixed-point branch past $F$9, so the actual onset occurs at a dynamic threshold $p$0 strictly above the static threshold (Bergeot et al., 2012).

The dynamic analogue of the fixed point is the invariant curve $p$1, defined by

$p$2

For small $p$3, the invariant curve admits a perturbation expansion, with $p$4. Bergeot et al. derive the threshold condition by examining the growth of deviations from this invariant curve. The deterministic dynamic threshold $p$5 is given by

$p$6

In the deterministic regime, this threshold is largely independent of $p$7 provided $p$8 is small enough and the initial state is close to the stable branch (Bergeot et al., 2014).

The same analysis also yields a first-order approximation to the invariant curve in the linear-ramp case: $p$9 with

$u$0

This formulation makes explicit that the onset delay is governed by the derivative $u$1 evaluated along the invariant curve rather than by the static fixed-point criterion alone (Bergeot et al., 2012).

Finite numerical precision or physical noise suppresses bifurcation delay. Modeling perturbations as additive white noise with variance $u$2, Bergeot et al. obtain the sweep-dominant estimator

$u$3

where $u$4 depends on the local slope of $u$5 near the static bifurcation (Bergeot et al., 2014). In this regime, larger $u$6 produces larger overshoot above $u$7, while larger noise yields earlier onset and smaller delay. Numerical experiments also show strong sensitivity to precision: at low precision the delay collapses toward $u$8, whereas very high precision recovers the deterministic prediction (Bergeot et al., 2012).

4. Exponential stabilization of mouth pressure and note-attack dynamics

A linear ramp is analytically convenient but physically incomplete because a musician does not increase mouth pressure indefinitely during a note attack. The 2014 extension therefore studies an exponential approach to a target pressure,

$u$9

equivalently

$p_n^-=-r p_{n-1}^+$0

with $p_n^-=-r p_{n-1}^+$1 in the paper’s examples (Bergeot et al., 2014).

The analysis is reduced to a linear sweep by introducing the auxiliary variable

$p_n^-=-r p_{n-1}^+$2

so that

$p_n^-=-r p_{n-1}^+$3

Defining $p_n^-=-r p_{n-1}^+$4, the dynamics becomes

$p_n^-=-r p_{n-1}^+$5

The invariant-curve condition in $p_n^-=-r p_{n-1}^+$6-space is

$p_n^-=-r p_{n-1}^+$7

and the corresponding deterministic threshold satisfies

$p_n^-=-r p_{n-1}^+$8

The mouth-pressure threshold is then recovered as

$p_n^-=-r p_{n-1}^+$9

A sweep-dominant expression of the same form is obtained for $r=1$0, with a constant $r=1$1 defined by the linearization of $r=1$2 near the bifurcation (Bergeot et al., 2014).

The comparison between linear and exponential profiles is one of the main substantive results. In deterministic simulations, representative thresholds are around $r=1$3 for a linear ramp and near $r=1$4 for the exponential profile when $r=1$5, $r=1$6, and $r=1$7. The paper interprets this as a consequence of the slowing of $r=1$8’s rate as it approaches the target, which reduces the bifurcation overshoot (Bergeot et al., 2014). The numerical study identifies two regimes: DReg, with weak dependence on $r=1$9, and SDReg, with strong dependence on $P_M=kH$0. Using a common rise-time measure $P_M=kH$1 to reach $P_M=kH$2 of $P_M=kH$3, the exponential profile reaches threshold in fewer time steps even when, in SDReg, its threshold in pressure may exceed that of the linear profile.

The numerical methodology also differs across the two clarinet papers. In the exponential-rise study, the dynamic threshold is the first time the orbit’s distance to the invariant curve exceeds $P_M=kH$4, with simulations run in arbitrary precision arithmetic using mpmath in Python and effective noise level $P_M=kH$5 (Bergeot et al., 2014). In the earlier linear-ramp study, $P_M=kH$6 is detected when the second-order difference of $P_M=kH$7 changes sign between successive samples (Bergeot et al., 2012). Both analyses underscore the same point: static thresholds alone are insufficient for note-attack transients.

5. PT as a pressure-amplified diffusion model on social networks

In social networks, the Pressure Threshold model is defined on a directed graph $P_M=kH$8 with edge weights $P_M=kH$9 and node thresholds $k$0. Under the weighted-cascade initialization used in experiments,

$k$1

so that initially $k$2 (Stutsman et al., 16 Sep 2025).

Each node has an activation state $k$3, with active set

$k$4

The process is progressive: once active, a node remains active. The incoming pressure at node $k$5 is

$k$6

and the activation rule is LT-style: $k$7 The PT extension is the influence adjustment performed when a node activates. If $k$8 activates at time $k$9, let

$H$0

For each outgoing neighbor $H$1 with $H$2 and $H$3,

$H$4

where $H$5 is the amplification parameter (Stutsman et al., 16 Sep 2025).

The update schedule is synchronous and two-phase. First, all inactive nodes are tested for activation by comparing $H$6 to $H$7. Second, outgoing edges of the newly activated nodes are amplified toward inactive neighbors. The process halts at a fixed point when the newly activated set is empty. There is no global re-normalization after amplification: per-edge clamping ensures $H$8, but the sum of in-neighbor weights at a recipient may exceed $H$9 after updates (Stutsman et al., 16 Sep 2025).

A recurrent misunderstanding is to treat PT as merely LT with different notation. The reduction

$s_v(t)$00

shows instead that LT is a special case of PT. PT adopts the LT activation criterion but adds a state-dependent feedback loop through adaptive outgoing influence.

6. Influence maximization, structural properties, and empirical behavior

The influence-maximization problem under PT is to choose a seed set $s_v(t)$01 with $s_v(t)$02 maximizing expected final spread,

$s_v(t)$03

where $s_v(t)$04 is the final active set under PT diffusion (Stutsman et al., 16 Sep 2025). Because $s_v(t)$05 recovers LT, influence maximization under PT is NP-hard via reduction from LT. The spread function is monotone: if $s_v(t)$06, then $s_v(t)$07. However, PT is not submodular in general when $s_v(t)$08.

The non-submodularity is exhibited by a four-node counterexample with $s_v(t)$09, edges $s_v(t)$10 and $s_v(t)$11 each of weight $s_v(t)$12, edge $s_v(t)$13 of weight $s_v(t)$14, thresholds $s_v(t)$15 and $s_v(t)$16, and $s_v(t)$17. For $s_v(t)$18, only $s_v(t)$19 remains active and $s_v(t)$20. For $s_v(t)$21, the spread is $s_v(t)$22. For $s_v(t)$23, node $s_v(t)$24 activates with pressure $s_v(t)$25, its edge to $s_v(t)$26 is amplified to $s_v(t)$27, and the final spread is $s_v(t)$28. For $s_v(t)$29, seeded $s_v(t)$30 does not adjust its outgoing weight, so $s_v(t)$31 remains inactive and the spread is $s_v(t)$32. Hence

$s_v(t)$33

violating submodularity (Stutsman et al., 16 Sep 2025). Greedy procedures such as CELF and CELF++ remain usable as heuristics, but the classical $s_v(t)$34 guarantee does not apply.

The experiments use Monte Carlo spread estimation with $s_v(t)$35 simulations per evaluation. On Facebook with $s_v(t)$36, CELF produces distinct seed sequences under LT and PT; the two agree early but diverge later, and PT introduces vertices such as $s_v(t)$37, $s_v(t)$38, $s_v(t)$39, $s_v(t)$40, and $s_v(t)$41 that never appear under LT (Stutsman et al., 16 Sep 2025). Average influence also increases with $s_v(t)$42. Selected endpoints reported in the paper include: Facebook at $s_v(t)$43, with LT spread $s_v(t)$44, PT spread $s_v(t)$45 for $s_v(t)$46, and $s_v(t)$47 for $s_v(t)$48; Bitcoin at $s_v(t)$49, with LT spread $s_v(t)$50 and PT spread $s_v(t)$51 for $s_v(t)$52; Wikipedia, where PT with $s_v(t)$53 reaches full coverage $s_v(t)$54 by approximately $s_v(t)$55, and PT with $s_v(t)$56 reaches full coverage by $s_v(t)$57; and an Erdős–Rényi network, where PT with $s_v(t)$58 reaches full coverage $s_v(t)$59 by $s_v(t)$60 (Stutsman et al., 16 Sep 2025).

The density effect is explained directly in terms of aggregate amplification. If $s_v(t)$61 is the set of nodes newly activated at round $s_v(t)$62, the total added incoming weight across their outgoing edges to inactive neighbors is

$s_v(t)$63

Approximating the inactive-neighbor count by $s_v(t)$64 early in the process gives

$s_v(t)$65

This explains why higher edge–node ratios, such as Facebook $s_v(t)$66 and Wikipedia $s_v(t)$67, display stronger PT amplification than the sparser Bitcoin network with edge–node ratio $s_v(t)$68 (Stutsman et al., 16 Sep 2025).

7. Assumptions, limitations, and interpretive boundaries

The clarinet PT framework is intentionally idealized. The reed is treated as an ideal spring; reed motion-induced flow is neglected in the 2012 analysis; no reed mass, inertia, or contact dynamics are included; the bore is a lossless straight cylinder with perfect reflection; and the Bernoulli nonlinearity is quasi-steady (Bergeot et al., 2012). In the 2014 exponential-rise study, losses are ignored by setting $s_v(t)$69, and threshold predictions depend on the piecewise map $s_v(t)$70 and especially on $s_v(t)$71 near the fixed point or invariant curve (Bergeot et al., 2014). The sweep-dominant regime further shows that the dynamic threshold depends strongly on precision or noise, so measured onset can approach the static threshold even when the deterministic theory predicts substantial delay.

The network PT model also imposes restrictive assumptions. Diffusion is progressive, thresholds are fixed per run and independent across nodes, the graph is stationary during diffusion, and amplification is controlled by a single scalar $s_v(t)$72. Because updates are clamped edgewise rather than renormalized, large $s_v(t)$73 can produce saturated edges $s_v(t)$74 and may violate the initial in-weight normalization at recipients (Stutsman et al., 16 Sep 2025). The lack of submodularity for $s_v(t)$75 removes standard approximation guarantees for greedy influence maximization, although the paper notes that approximate submodularity can hold empirically for small $s_v(t)$76 on large networks.

These limitations clarify the scope of the term. In clarinet acoustics, PT is a predictive framework for dynamic onset under prescribed mouth-pressure profiles, with the central distinction between static and dynamic oscillation thresholds. In network science, PT is an adaptive-threshold diffusion model in which activation pressure feeds back into subsequent influence transmission. The two usages share threshold logic and path dependence, but the literature treats them as separate model classes rather than as instances of a single unified formalism (Bergeot et al., 2014).