Hysteron Transition Graphs

• Hysteron Transition Graphs are directed graphs that represent irreversible state changes in collections of bistable hysterons with state-dependent switching thresholds.
• They capture complex behaviors such as avalanches, cascading transitions, and bifurcation-induced cascades, bridging the classical Preisach model with interacting generalizations.
• These graphs enable programmable metamaterial designs and finite-state computational architectures by mapping switching events and addressing inverse design challenges.

Searching arXiv for the cited papers on hysteron transition graphs and related interacting hysteron models. arxiv_search(query="Hysteron Transition Graphs interacting hysterons transition graphs catastrophe theory", max_results=10, sort_by="relevance") Hysteron transition graphs, often abbreviated as t-graphs, are directed graphs that encode the irreversible state changes of collections of hysteretic bistable elements under monotonic or cyclic driving. In the contemporary interacting-hysteron literature, nodes denote metastable or stable fixed-point states of the full system, and directed edges denote transitions induced when the control parameter crosses state-dependent switching thresholds or when a stable branch loses stability at a bifurcation. In this sense, a t-graph is a compact representation of pathway memory: it records which transitions are available, which are inaccessible, and how interactions, avalanches, and parameter variation reorganize the admissible response of driven disordered media and programmable metamaterials (Teunisse et al., 2024, Muhaxheri et al., 11 Aug 2025).

1. Preisach origin and the interacting generalization

The elementary hysteron is a bistable unit with two states and distinct up and down switching thresholds. In a discrete description, hysteron $i$ flips from $-$ to $+$ when $\gamma > \gamma_i^+$, flips from $+$ to $-$ when $\gamma < \gamma_i^-$, and otherwise retains its state; in the absence of interactions, macroscopic hysteresis can then be represented as a superposition of such independent two-state elements (Lindeman et al., 2023). This is the classical Preisach setting, in which each hysteron is characterized by fixed switching fields and the system response is determined by their ordering.

Within the Preisach model, the associated transition graphs are highly constrained. For $n$ hysterons, only $n!$ distinct t-graph topologies occur; the graphs are tree-like, transitions are single-hysteron flips, avalanches are absent, and return-point memory is strictly satisfied (Hecke, 2021). The older hysteron-based magnetic literature also connects this independent-hysteron picture to the Preisach distribution $\rho(H_c,H_u)$ and to the mixed second derivative of the magnetization measured in first order reversal curve protocols, but this interpretation presupposes that the system is truly a hysteron ensemble with negligible correlations (Ruta et al., 2016).

The interacting generalization replaces fixed thresholds by state-dependent switching fields. A broad formulation is

$-$0

where $-$1 is the collective binary state and $-$2 captures interactions that may be pairwise, reciprocal, or higher-order (Teunisse et al., 2024). In the linear pairwise form used extensively in t-graph studies,

$-$3

so the next switching event depends not only on the external drive but also on the current configuration of all other hysterons (Hecke, 2021). This state dependence is the essential source of scrambling, avalanches, broken return-point structure, and the combinatorial growth of graph topologies.

2. Graph structure, avalanches, and realizability

The full t-graph is organized around the notion of a scaffold. For each state $-$4, one defines the critical hysterons

$-$5

and the corresponding state switching fields

$-$6

The scaffold therefore records, for every state, which hysteron is encountered first on an up-sweep or down-sweep and at which field this first instability occurs (Teunisse et al., 2024). In the Preisach model, the scaffold already determines the graph. For interacting hysterons, it is only the skeleton.

The discrepancy arises because an initial instability can trigger a multi-step avalanche. After the critical hysteron flips, the new state may itself be unstable at the same value of the drive, producing a cascade of additional flips before the trajectory terminates in a stable state. The available avalanche passages can be organized as finite binary trees rooted at a state and a driving direction, and the actual t-graph is obtained by following these passages until a stable endpoint is reached (Teunisse et al., 2024). This yields composite transitions that can skip intermediate states and generates graph-theoretic features absent in the Preisach case.

A constructive inverse theory accompanies this description. Given a desired t-graph or transition subgraph, one can generate linear design inequalities that enforce stability of the initial state, correct choice of critical hysteron, appropriate instability of intermediate avalanche states, and stability of the final state. These inequalities define a partial order on the switching fields. Realizability is then equivalent to consistency of that partial order: one computes its transitive closure, checks for contradictions such as cyclic inequalities, and, if the system is satisfiable, constructs explicit switching fields by a linear extension or topological sort (Teunisse et al., 2024). In this formulation, realizability is not assumed; candidate graphs can fail because of self-loops, irresolvable race conditions, or incompatible orderings.

This framework also sharpens the distinction between the combinatorics of candidate graphs and the subset that are physically consistent. For $-$7, one enumeration cited 117 candidate graphs, of which 13, excluding Garden of Eden states, are realizable (Teunisse et al., 2024). A different enumeration focused on pairwise interacting hysterons reported that the number of distinct t-graphs jumps from two in the Preisach case to 11 as soon as interactions are introduced (Hecke, 2021). Taken together, these results show that even the two-hysteron problem already contains nontrivial issues of classification, realizability, and statistical weight.

3. Catastrophe-theoretic construction from continuous dynamics

A complementary formulation treats interacting hysterons as a continuous, dissipative gradient dynamical system,

$-$8

with $-$9 the continuous hysteron coordinates, $+$0 the global driving field, and $+$1 the system parameters (Muhaxheri et al., 11 Aug 2025). In the bistable construction discussed there, each hysteron has local energy

$+$2

interactions may be represented by

$+$3

and the total energy is

$+$4

Within this gradient setting, t-graphs are constructed from the loss of stability of equilibria. The generic codimension-1 event is the fold, or saddle-node, bifurcation, characterized by

$+$5

together with the nondegeneracy conditions from Sotomayor’s theorem,

$+$6

where $+$7 and $+$8 are the right and left null eigenvectors of $+$9 (Muhaxheri et al., 11 Aug 2025). At such a fold, a stable node and a saddle collide and annihilate. The transition-graph edge is determined by the escape route: for $\gamma > \gamma_i^+$0, one integrates the flow from near the bifurcation and records which stable state is ultimately reached.

Higher-codimension catastrophes organize changes in graph topology. Standard cusps and dual cusps occur when fold curves coalesce tangentially; crossings of fold curves correspond to simultaneous transitions of different states and are associated with avalanches. Near a cusp, center-manifold reduction leads to an expansion

$\gamma > \gamma_i^+$1

and sign changes of coefficients such as $\gamma > \gamma_i^+$2 and $\gamma > \gamma_i^+$3 demarcate the cusp structure (Muhaxheri et al., 11 Aug 2025). By varying parameters such as $\gamma > \gamma_i^+$4 and $\gamma > \gamma_i^+$5, one obtains curves in parameter space along which cusps and crossings occur, and these codimension-2 loci partition parameter space into regions of distinct t-graph topology. In this language, inaccessible states appear naturally as Garden of Eden states.

4. Topological diversity, scrambling, and asymptotic regimes

Interactions induce a profusion of transition pathways. For three interacting hysterons, more than 15,000 distinct t-graph topologies have been reported, in contrast to the six Preisach graphs for three independent hysterons (Hecke, 2021). The new topological features include scrambling, avalanches, dissonance, multigraph structure, subharmonic cycles, and breakdown of loop return-point memory.

Scrambling means that the order of switching becomes state-dependent: a hysteron that flips before another in one state may flip after it in a different state. Avalanches can be vertical, horizontal, or pseudo-avalanches, depending on the interaction signs and the geometry of the transition. Dissonance denotes transitions in which increasing the drive can decrease magnetization, or decreasing the drive can increase it. Multigraph structure refers to multiple directed edges between the same pair of states, for example both an up and a down transition. Subharmonic cycles, or S-cycles, are drive cycles for which the state repeats only after more than one period (Hecke, 2021). By contrast, ferromagnetic-only interactions preserve loop return-point memory through the no-passing rule.

Two asymptotic limits clarify which of these features are generic and which require finite hysteron span. When the hysteron span $\gamma > \gamma_i^+$6 dominates all other scales, the scaffold is unchanged by uniform span rescaling, all states are locally stable for some interval of the drive, and avalanches are strongly suppressed: only monotonic avalanches persist, while mixed avalanches and non-monotonic multi-hysteron transitions become impossible. For negative interactions, all avalanches can be suppressed, and the diversity of the t-graph population collapses (Teunisse et al., 17 Dec 2025).

In the opposite limit $\gamma > \gamma_i^+$7, hysterons reduce to interacting binary spins. Independent spins have a trivial t-graph, and for interacting spins all nontrivial t-graphs require avalanches. This establishes a sharp statement: in the spin limit, nontrivial pathway memory is impossible without collective multi-flip events (Teunisse et al., 17 Dec 2025). The same work further shows that a single hysteron can be mimicked by a pair of strongly interacting spins, so that $\gamma > \gamma_i^+$8 interacting hysterons can be mapped to $\gamma > \gamma_i^+$9 interacting spins. Two constructions are discussed there: Mapping I with symmetric pairs, which suffices without avalanches but can generate race conditions when avalanches occur, and Mapping II with asymmetric pairs, which avoids external race conditions for sufficiently large internal coupling $+$0.

5. Memory, latching, and design for computation

The memory significance of t-graphs is most transparent in the contrast between return-point memory and frustration-enabled alternatives. In non-interacting or cooperatively interacting systems, revisiting a past turning point restores the system to a previous state, and symmetric cyclic protocols yield multiple nested memories. Under asymmetric driving, by contrast, the classical return-point mechanism stores only the largest previous amplitude (Lindeman et al., 2023).

Frustrated interactions can break this restriction. In the hysteron-latch construction, two interacting hysterons with frustration violate return-point memory and thereby realize multiple memories of asymmetric driving. The switching thresholds are state-dependent,

$+$1

and a pair acts as a latch when inequalities such as

$+$2

are satisfied (Lindeman et al., 2023). In this regime, the pair remembers the last input amplitude that placed it into a latching configuration and resets only when a larger amplitude is applied. The reported probability that a random pair supports a latch configuration scales as $+$3, and the number of storable memories scales as the square root of the system size in random ensembles but linearly with the number of latches in designed systems (Lindeman et al., 2023).

At a more general design level, t-graphs have been proposed as finite-state architectures for embodied computation. Explicit examples discussed in the interacting-hysteron literature include analog-to-digital conversion, accumulators, sequence recognition, and other state-machine-like behaviors obtained by tuning $+$4, $+$5, or the corresponding design inequalities (Hecke, 2021, Teunisse et al., 2024). The catastrophe-theoretic framework refines this by treating codimension-2 bifurcation curves as handles for metamaterial programming: changing parameters can enable or disable escape routes and thereby reconfigure the t-graph itself (Muhaxheri et al., 11 Aug 2025).

These design ambitions come with a documented complexity cost. The number of transition graphs and bifurcation boundaries grows quasi-exponentially with system size, the accessible parameter space fragments rapidly as $+$6 increases, and desirable graphs may occupy exceedingly small, fine-tuned regions of parameter space (Muhaxheri et al., 11 Aug 2025). The resulting inverse design problem is therefore not merely combinatorial; it is also geometrically delicate.

6. Experimental embodiments and methodological limits

A concrete experimental embodiment of hysteron networks has been realized in electronic networks of bistable negative differential resistance elements. In that system, each hysteron is an NDR device constructed from MOSFETs and Op-Amps, with two stable states set by a non-monotonic current-voltage characteristic $+$7. The device is tunable by $+$8 and $+$9, and its switching thresholds are the current at the peak and valley of the IV curve, $-$0 and $-$1 (Altman et al., 8 Feb 2025).

In the main experimental configuration, a 9-element series network was tuned so that interaction effects were negligible, yielding a physical realization close to the non-interacting Preisach regime. Collective memory states were represented as binary strings of edge states; switching order was imposed by

$-$2

and, for a three-hysteron example with strict ordering, the increasing-drive sequence is $-$3 (Altman et al., 8 Feb 2025). The same platform also exhibited avalanche-like events in some runs and was presented as a testbed for more strongly interacting, non-Preisachian graphs with avalanches and multiperiodic orbits.

A separate limitation arises in magnetic characterization, where hysteron-based descriptions can be overextended beyond their regime of validity. The FORC method and Preisach-style extraction of switching-field distributions rely on microscopic wiping out and minor-loop congruency. In correlated systems, these assumptions fail: switching becomes collective, cluster reversals replace individual grain thresholds, de-shearing or symmetrizing the FORC diagram is inadequate, and the inverse problem becomes ill-posed because the same macroscopic FORC shape can arise from many different combinations of intrinsic and interaction parameters (Ruta et al., 2016). In that literature, FORC-based extraction is reported to be valid only in cases of weak spatial correlation of the magnetization, and a more general approach is direct fitting of realistic microscopic models, such as kinetic Monte Carlo simulations, to experimental data.

Taken together, these developments position hysteron transition graphs as a unifying language for hysteresis, pathway memory, and programmable sequential response. They generalize the Preisach picture from fixed thresholds to state-dependent switching and from tree-like return-point structure to avalanche-dressed, parameter-sensitive networks of transitions. At the same time, the same body of work emphasizes that graph richness, inverse design difficulty, and the breakdown of hysteron-based inference in correlated systems are intrinsic parts of the subject rather than peripheral complications.