Reproduce “Collective Hysteresis and Multistability in Threshold Networks” on OpenResearch
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About this paper
Moses Boudourides
A network extension of threshold-driven collective hysteresis shows that row-stochastic connectivity preserves the scalar model’s homogeneous cusp and stability threshold, while modular organization creates additional heterogeneous equilibria. Weakly coupled equitable blocks can support at least 2^m stable configurations for m bistable modules; in the two-block case, mixed states disappear through symmetry-related heterogeneous cusps with coupling and activity-contrast scalings validated numerically.
Key insights
- Under row-stochastic coupling, the homogeneous branch, scalar folds, and cusp depend on the threshold distribution and feedback parameters—not on detailed network topology. Heterogeneous modes cannot destabilize first.
- Network topology still controls multistability: m weakly coupled bistable equitable blocks retain at least 2^m stable product states, including mixed low–high configurations.
- The disappearance of unequal two-block states is governed by heterogeneous cusps distinct from the scalar cusp; near bistability onset, critical coupling and stimulus displacement scale linearly with \(\beta-\beta_c\), while activity contrast scales as \((\beta-\beta_c)^{1/2}\).
Summary by alphaXiv.
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