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Collective Hysteresis and Multistability in Threshold Networks

Moses Boudourides

arXiv:2609.10580v1cs.SImath.DSnlin.AO

TL;DR

The paper asks which features of Kaye’s scalar hysteresis persist when feedback is transmitted through a network. It analyzes a deterministic threshold-response network using equitable quotients, proving that weak interblock coupling preserves stable block assignments and produces heterogeneous cusps with characteristic near-onset scaling.

  • Problem

    Prior results do not determine how row-stochastic network coupling changes Kaye’s scalar cusp, when weakly coupled equitable blocks yield multistability, or how heterogeneous cusps approach the scalar cusp.

    Source: §1 · p. 2

  • Method

    The paper studies a deterministic network of continuous node activities with row-stochastic weighted feedback and uses equitable partitions to derive exact quotient systems.

    Source: §1 · p. 2, §5 · p. 9

  • Results

    Near scalar bistability onset, heterogeneous-cusp coupling is of order δ while the block-activity difference is of order δ^1/2, with numerical continuation confirming limiting exponents across gamma, logistic, and normal distributions.

    Source: §6 · p. 11, §7 · p. 15

  • Takeaways & Limitations

    Weak coupling between equitable blocks preserves a combinatorial family of stable equilibria that lift to stable full-network states, while row-stochastic coupling leaves the homogeneous branch unchanged.

    Source: §7 · p. 15

    The deterministic response model is not derived from a stochastic network of binary threshold agents; that derivation would require a separate limit theorem.

    Source: §1 · p. 2, §7 · p. 15

Abstract

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In a mechanistic model of the dawn chorus, Kaye showed that heterogeneous activation thresholds and a shared feedback signal determined by the population's active fraction can produce abrupt collective activation and hysteresis. We extend this mechanism to a network of interacting agents. Each node has a continuous activation level, and a nonnegative row-stochastic matrix determines how node activities contribute to individual feedback. We prove that sufficiently weak feedback yields a unique globally attracting equilibrium. For all feedback strengths, the homogeneous dynamics exactly reproduce Kaye's scalar equation; consequently, network topology does not alter the folds or cusp of the homogeneous branch, and no heterogeneous mode becomes unstable before the homogeneous mode. For equitable partitions, the network admits an exact quotient system in which nodes within a block receive the same aggregate input from every block. When blocks are uncoupled, the quotient reduces to independent copies of Kaye's scalar equation. We show that every assignment of stable scalar equilibria to blocks persists under sufficiently weak interblock coupling, producing a combinatorial family of stable quotient equilibria that lift to stable full-network equilibria and remain under small perturbations that break exact equitability. For two symmetrically coupled blocks, branches with unequal block activities terminate at a pair of symmetry-related cusp bifurcations. Near the onset of bistability, we derive scaling laws for the interblock coupling at which these bifurcations occur, the activity difference between the blocks at bifurcation, and the corresponding shift of the external stimulus from the scalar cusp. Numerical continuation confirms the scaling laws for gamma, logistic, and normal threshold distributions.

1 Introduction

Prior work links heterogeneous thresholds and shared feedback to collective hysteresis, while network threshold models examine topology and cascades. This paper studies which scalar-cusp features persist under row-stochastic network coupling and when equitable blocks support multistability.

1 Introduction

  • Kaye’s dawn-chorus model derives collective hysteresis from heterogeneous bird thresholds and feedback set by the overall active fraction.Unlike that shared-feedback baseline, this paper specifies interactions through row-stochastic matrices and uses equitable partitions for exact quotients.Source: §1 · pp. 1–2
  • Prior results do not determine how row-stochastic coupling changes Kaye’s model, including which scalar-cusp features persist and when weak interblock coupling produces multistability.Existing work covers network threshold cascades [2, 3], equitable reductions [10] [11] [12], and multistability in weakly coupled cusp systems.Source: §1 · p. 2
  • The paper proves that stable scalar equilibria assigned independently to equitable blocks persist under weak interblock coupling, and that unequal-activity branches in two symmetrically coupled blocks end at symmetry-related cusps.Near scalar bistability onset, it derives scaling laws for cusp coupling, block-activity difference, and stimulus shift.Source: §1 · p. 2
  • The paper extends Kaye’s mean-field equation deterministically: nodes have continuous activation levels and receive weighted activity feedback, without deriving the equation from stochastic binary-agent dynamics.A stochastic derivation would require a separate limit theorem.Source: §1 · p. 2

2 Threshold dynamics on a row-stochastic network

The model is a cooperative threshold-response system on a row-stochastic network, with a unique globally attracting equilibrium under sufficiently weak feedback. Equitable partitions yield exact lower-dimensional dynamics because nodes in each block receive equal aggregate input from every block.

2 Threshold dynamics on a row-stochastic network

  • Each node’s activation responds to external stimulus and a weighted average of network activities, with nonnegative row-stochastic weights giving every node unit total incoming feedback weight.The state remains in the unit cube, and the flow is order preserving.Source: §2 · p. 3
  • When βLf < 1, the network has a unique equilibrium, and contraction with Gronwall’s inequality gives convergence of solutions.The condition requires neither irreducibility nor an equitable partition.Source: §2 · p. 3

2.1 Equitable partitions and exact quotients

  • An equitable partition groups nodes whose aggregate weight from each block is identical, producing a unique quotient matrix of block-level weights.The quotient matrix is also row-stochastic when the original network is row-stochastic.Source: §2.1 · p. 4
  • The block-constant subspace is invariant and follows the exact quotient dynamics; this holds for directed or weighted networks and does not require symmetry or equal block sizes.In the 15-node illustration with block sizes 4, 5, and 6, full-network trajectories agree with lifted quotient trajectories to numerical integration tolerance.Source: §2.1 · p. 4, Figure 1 · p. 5

3 Homogeneous dynamics and spectral stability

Every row-stochastic network preserves the scalar equation on its homogeneous subspace, so topology leaves that branch’s folds and cusp unchanged. Along the branch, heterogeneous modes cannot become unstable before the homogeneous mode.

3 Homogeneous dynamics and spectral stability

  • For every row-stochastic network, the homogeneous set carries exactly the scalar equation, so its equilibrium branch, folds, and cusp are independent of network topology.Source: §3 · p. 5
  • No heterogeneous mode becomes unstable before the scalar fold; for reducible networks, additional heterogeneous neutral directions can occur at the fold without becoming unstable earlier.Irreducibility makes the critical kernel uniquely homogeneous; without row-stochastic normalization, homogeneous invariance and this spectral ordering can fail.Source: §3 · pp. 5–6
  • For the Gamma benchmark with shape 4 and scale 0.8, the folds meet at βc = 3.570762; above βc, the branch has a bistable interval and hysteresis.The density has a nondegenerate maximum at zc = 2.4.Source: Figure 2 · p. 6

4 Combinatorial multistability under weak interblock coupling

When uncoupled, equitable blocks follow independent copies of the scalar model, so assignments of stable scalar equilibria persist as a combinatorial family under sufficiently weak interblock coupling. These quotient equilibria lift to stable full-network states and persist locally under small perturbations that break equitability.

Uncoupled blocks

  • At zero interblock coupling, the quotient system consists of independent copies of Kaye’s scalar equation.The persistence result starts from product equilibria of these uncoupled blocks.Source: §4 · p. 7

Persistence and counting

  • r stable scalar equilibria yield at least r^m stable quotient equilibria under sufficiently weak coupling, or at least 2^m within a scalar bistable interval.Each assignment of scalar equilibria to blocks extends to a stable branch.Source: §4 · p. 7

Full-network stability

  • The lifted quotient equilibria are asymptotically stable in the full network, including against perturbations transverse to the equitable subspace.The stability condition bounds the spectral radius using the maximum block-level feedback.Source: §4 · p. 7

Persistence after breaking equitability

  • Each hyperbolic stable equilibrium persists under sufficiently small row-stochastic perturbations that break equitability, though node activities need not remain equal within blocks.This is local continuation near an exactly equitable matrix, not an approximate quotient or a global reduction bound.Source: §4 · p. 8

First-order displacement

  • As interblock coupling increases from zero, a low–high state initially moves inward: its low activity rises while its high activity falls.The first-order displacement is derived by differentiating the quotient equilibrium equation at zero coupling.Source: §4 · p. 8

Two-block branches

  • For the Gamma benchmark at β = 4, increasing coupling narrows the stable mixed branch, which is absent at ε = 0.05 above its heterogeneous-cusp value.Exchanging blocks gives a reflected state with the same mean.Source: Figure 3 · p. 9

5 Heterogeneous cusps for two symmetrically coupled blocks

The two-block parameter map yields symmetry-related heterogeneous cusp bifurcations near the scalar cusp. Their coupling and stimulus shifts vanish linearly with distance from onset, while block-activity contrast vanishes with its square root.

5 Heterogeneous cusps for two symmetrically coupled blocks

  • The map (a, s) → (u, ε) parameterizes heterogeneous equilibria; its singular points are folds, and its terminal mixed-state cusps are Whitney cusps.The equilibrium equations are invariant under exchanging the two blocks.Source: §5 · p. 9
  • For each β just above βc, two exchange-related generic Whitney cusps occur at the same parameter pair, with opposite block contrasts.The cusps terminate branches with unequal block activities and do not lie on the homogeneous branch.Source: §5 · pp. 10–11
  • As β approaches βc, cusp coupling and stimulus displacement vanish linearly in β − βc, while block-activity contrast vanishes as (β − βc)^1/2.The heterogeneous cusps converge to the scalar cusp.Source: §5 · pp. 10–11
  • Taylor expansion and rescaling produce two branches for small positive t, exchanged by s → −s, and the rank-one singularities satisfy the generic cusp conditions.The implicit-function theorem yields the branches; the blow-up analysis preserves cusp type.Source: §5 · pp. 10–11

6 Numerical results

Numerical continuation tests the predicted two-block cusp scalings across threshold distributions and tracks stable product states as coupling grows. Perturbation tests also show local persistence of a stable equilibrium when exact equitability is broken.

6 Numerical results

  • The primary benchmark used a gamma distribution with shape 4 and scale 0.8, with α = 1; the two-block and scalar cusps were computed at β = 4.Branch tracing, exact-Jacobian stability checks, and independent cusp-genericity checks were deterministic.Source: §6 · p. 11
  • At zero coupling, there are 2^m stable product equilibria; increasing coupling removes tracked heterogeneous branches until only two homogeneous attractors remain.For m = 3, all eight branches remained stable down to ε = 10−4.Source: §6 · p. 11, Figure 6 · p. 14
  • At β = 4, the Gamma mixed-state fold curves meet at a heterogeneous cusp; its equilibrium residual was 3.33 × 10−16.The Jacobian determinant and quadratic cusp coefficient had magnitudes 2.36 × 10−17 and 1.98 × 10−15, respectively.Source: §6 · p. 11, Figure 4 · p. 13
  • For gamma, fitted exponents were 0.9989, 0.4991, and 1.0003; logistic and normal distributions matched the limiting exponents within 1.3 × 10−3.The computed-to-asymptotic ratios for coupling, block contrast, and stimulus shift approach one as β approaches βc.Source: §6 · p. 11, Figure 5 · p. 13
  • Under deterministic perturbations, equilibrium displacement and within-block dispersion were linear to leading order; all computed equilibria remained stable.Fits over the six smallest perturbations gave slopes 1.034 and 0.988, respectively; these tests do not establish a global robustness threshold.Source: §6 · p. 12, Figure 7 · p. 14

7 Discussion

The discussion separates threshold-driven homogeneous hysteresis from network-supported heterogeneous multistability. It states the conditions and boundaries of the results, including the scope of perturbation persistence and open questions about approximate and asymmetric quotients.

7 Discussion

  • When β sup_z f(z) < 1, every row-stochastic network has a unique globally attracting equilibrium; at stronger feedback, heterogeneous modes do not destabilize before the homogeneous mode.The threshold distribution sets the scalar cusp, while density curvature at its maximum determines local scaling near that cusp.Source: §7 · p. 15
  • A hyperbolic equilibrium from an exact equitable partition persists under small interaction-matrix perturbations, although within-block activities may diverge and the exact quotient is lost.The result does not establish that empirically identified groups satisfy exact equitability.Source: §7 · p. 15
  • Near scalar bistability onset, heterogeneous-cusp coupling is O(δ) while the block-activity difference is O(δ^1/2), so a small coupling increase can eliminate visibly unequal activity states.Here δ = β − βc.Source: §7 · p. 15
  • The results assume row-stochastic interactions and a deterministic response model; unequal row sums can break homogeneous invariance, and no stochastic binary-agent derivation is provided.Persistence under small violations of equitability applies to individual hyperbolic equilibria, not to an approximately invariant quotient or uniform error bounds along a full bifurcation diagram.Source: §7 · p. 15
  • Open questions include uniform error bounds for approximately equitable networks away from bifurcations and how asymmetric quotient matrices alter the symmetric two-block heterogeneous cusps.Both questions retain the distinction between threshold-driven homogeneous hysteresis and network-supported heterogeneous stable states.Source: §7 · p. 15

8 Conclusion

The conclusion extends Kaye’s threshold mechanism to row-stochastic networks and distinguishes the homogeneous hysteresis inherited from the scalar model from additional multistability supported by equitable blocks. It summarizes the weak-feedback condition, stable-state persistence, and heterogeneous cusp scaling results.

8 Conclusion

  • If β sup_z f(z) < 1, the network has a unique globally attracting equilibrium; at any feedback strength, its homogeneous dynamics reproduce Kaye’s scalar equation without changing its folds or cusp.No heterogeneous mode becomes unstable before the homogeneous mode.Source: §8 · p. 16
  • Stable scalar equilibria assigned independently to equitable blocks persist as stable full-network equilibria under sufficiently weak interblock coupling.For two symmetrically coupled blocks, unequal-activity branches terminate at heterogeneous cusp bifurcations.Source: §8 · p. 16
  • Near scalar bistability onset, the heterogeneous-cusp coupling, block-activity difference, and external-stimulus shift follow the scaling laws derived in Theorem 5.1.Source: §8 · p. 16

A Explicit fold and cusp equations

This appendix presents fold and cusp equations used in the computations and gives the parameter-recovery sequence after solving them.

A Explicit fold and cusp equations

B Independent Lyapunov–Schmidt check

The appendix independently checks the two-block cusp conditions with normalized left and right null vectors, a cubic coefficient, and a transversality test. It reports benchmark values for the cusp coefficient and determinant.

B Independent Lyapunov–Schmidt check

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