Dominance Switching in Multidisciplinary Systems

• Dominance switching is a phenomenon where the dominant element in a system changes over time due to dynamics, feedback, or external control.
• It spans diverse fields—such as control systems, optimization, social networks, ecology, and physics—each formalizing the concept with methods like cone contraction, solution preorders, or invasion hierarchies.
• This concept informs practical approaches in designing robust feedback systems, optimizing network interventions, and understanding state transitions in both engineered and natural domains.

In the cited literature, dominance switching does not denote a single formalism. It is used for technically distinct phenomena in which the dominant object changes under dynamics, feedback, learning, constraints, or external control: a low-dimensional dominant mode in switched systems, a preferred solution in a constraint problem, a dominant market regime, a social leader, an invasion direction in a cyclic game, a dominant species set in a community, or a dominant physical state in a nanoscale or antiferromagnetic device (Berger et al., 2018, Guns et al., 2018, Li et al., 2018, Szolnoki et al., 2014, Mezei et al., 2020, Wu et al., 2024). Across these usages, the common structure is a shift in which component governs long-run behavior, instantaneous transitions, or stationary occupation.

Domain	Dominant object	Typical switching driver
Switched and feedback systems	Dominant modes, attractors, or dynamics	Automaton-constrained switching, feedback gain, complementarity, self-switching times (Berger et al., 2018, Padoan et al., 2019, Miranda-Villatoro et al., 2018, Gallo et al., 2020)
Constraint and decision systems	Preferred or non-dominated solutions	Dominance relations and dominance nogoods (Guns et al., 2018)
Strategic and social systems	Market leader, socially dominant actor, effective power	Timing options, learning, network rewiring, interpersonal weights (Li et al., 2018, Ye et al., 2017, Ganzfried, 2022)
Animal and ecological systems	Dominance relation, invasion hierarchy, common species set	Assessment, interaction range, hedging cost, stochasticity (Grewal et al., 2013, Hall et al., 2019, Szolnoki et al., 2014, Kessler et al., 2024)
Physical systems	Dominant conductance state or AFM domain orientation	Displacement, voltage, current density (Mezei et al., 2020, Wu et al., 2024)

1. Conceptual scope

In control and dynamical-systems theory, dominance switching concerns changes in the number or identity of dominant modes, often formalized through $p$-dominance, cone contraction, or state-dependent switching laws. In optimization and constraint programming, it refers to a change in which solutions are preferred under a preorder on feasible assignments. In strategic and social models, it refers to changes in which player, agent, or subgroup has the effective advantage. In ecological and evolutionary models, it denotes reversals in invasion direction or transitions between dominance and more even coexistence. In physical systems, it refers to externally controlled changes in which metastable state has larger stationary occupation or which switching symmetry is realized (Berger et al., 2018, Guns et al., 2018, Li et al., 2018, Szolnoki et al., 2014, Mezei et al., 2020).

A recurrent ambiguity is that dominance may mean spectral dominance, asymptotic attractor dimension, solution preference, social power, ecological abundance, or stationary occupancy. Accordingly, switching may describe either abrupt transitions between states, slow transitions after assessment or learning, or parameter-driven changes in the asymptotically dominant regime. The literature therefore treats dominance switching less as a single theorem than as a family of structurally related phenomena.

2. Dynamical-systems and control formulations

A central formalization appears in switching linear systems. "Path-complete $p$-dominance for switching linear systems" introduces path-dominance, defined with respect to a path-complete automaton $Aut=(Q,\Sigma,\delta)$, a family of quadratic $p$-cones $K(P_q)$, and edgewise cone-contraction conditions of the form

$$A_\sigma^\top P_{q_2}A_\sigma - \gamma_d^2 P_{q_1} \preceq -\varepsilon I .$$

The key consequence is a dominated splitting with $p$-dimensional forward-invariant subspaces $H(t)$ and $(n-p)$-dimensional subspaces $V(t)$, so that admissible trajectories become asymptotically $p$0-dimensional. This extends the LTI notion of dominant or slow modes to constrained switching and yields an SDP-based feasibility test for path-dominance (Berger et al., 2018).

The robustness counterpart is developed by "Dominance margins for feedback systems", which generalizes classical gain and phase margins to $p$1-gain, $p$2-phase, and $p$3-disk margins. In that framework, strict $p$4-dominance characterizes low-dimensional attractors rather than mere convergence to an equilibrium: $p$5 supports multistability, while $p$6 supports simple attractors including limit cycles. The proposed margins quantify how much perturbation a feedback interconnection can tolerate while preserving the same dominant attractor dimension, especially in Lur’e systems exhibiting switching and oscillatory behavior (Padoan et al., 2019).

An allied nonsmooth formulation appears in "Dominance analysis of linear complementarity systems". There, incremental $p$7-dominance and $p$8-dissipativity are extended to linear complementarity systems, where switching is induced by complementarity relations rather than a prescribed external signal. The asymptotic interpretation parallels the smooth theory: $p$9 implies convergence to a unique equilibrium, $Aut=(Q,\Sigma,\delta)$0 permits multiple equilibria, and $Aut=(Q,\Sigma,\delta)$1 permits limit cycles. The paper illustrates these regimes with an op-amp model, a Schmitt trigger, and a relaxation oscillator, thereby treating dominance switching as a design transition between stable, multistable, and oscillatory circuit behavior (Miranda-Villatoro et al., 2018).

A probabilistic analogue appears in "Self-Switching Markov Chains: emerging dominance phenomena". There, the law of the location process changes only when a target set is hit, so switching times are trajectory-dependent. The long-run occupation measure of the state process is proportional to the mean inter-switching time $Aut=(Q,\Sigma,\delta)$2, and asymptotic concentration occurs on the subset of states with maximal large-$Aut=(Q,\Sigma,\delta)$3 episode durations. The paper gives finite-support and Laplace-type conditions under which only a strict subset of dynamical states is observed in the limit, and it shows that the switching times can exhibit a metastability-like property (Gallo et al., 2020).

Taken together, these works place dominance switching within a rigorous language of invariant cones, indefinite Lyapunov or dissipativity inequalities, regenerative switching times, and asymptotic low-dimensional structure. A plausible implication is that the phrase is most mathematically mature when the dominant set can be characterized by explicit inequalities, spectral separation, or occupation-measure asymptotics.

3. Preference, optimization, and strategic decision systems

In constraint programming, dominance switching is formulated at the level of solution preorders rather than trajectories. "Solution Dominance over Constraint Satisfaction Problems" defines a dominance relation $Aut=(Q,\Sigma,\delta)$4 as a preorder on the solution set of a CSP and introduces the Constraint Dominance Problem $Aut=(Q,\Sigma,\delta)$5. The central operational device is the dominance nogood: after finding a solution $Aut=(Q,\Sigma,\delta)$6, one posts a constraint such as $Aut=(Q,\Sigma,\delta)$7, thereby excluding future solutions dominated by $Aut=(Q,\Sigma,\delta)$8. Because the underlying CSP is unchanged while only the dominance relation is altered, the same feasible set can be explored under single-objective, lexicographic, multi-objective, minimal-model, Max-CSP, MSS, or CP-net dominance. In this setting, dominance switching means changing the order relation that determines which feasible solutions count as better, and the framework makes that switch algorithmically explicit (Guns et al., 2018).

A related but game-theoretic notion appears in "Stochastic Switching Games". There, two firms control a discrete market regime $Aut=(Q,\Sigma,\delta)$9, interpreted as relative dominance, while switching decisions are driven by a continuous stochastic factor $p$0. Player 1 can increase $p$1 by $p$2; player 2 can decrease it by $p$3. Threshold-type Feedback Nash Equilibria determine when each player exercises this timing option. With an Ornstein–Uhlenbeck $p$4, the induced dominance process is recurrent, so market leadership switches repeatedly in the long run. With a Geometric Brownian Motion $p$5 having positive drift, the equilibrium eventually becomes absorbed in a state where one player gains permanent advantage. Here dominance switching is neither purely exogenous nor purely learning-based; it is the equilibrium outcome of competing impulse controls under stochastic state dependence (Li et al., 2018).

These two lines of work use very different mathematics—incremental CSP solving versus coupled optimal stopping—but share a common abstraction: a dominant object is selected by a comparison rule, and switching occurs when the rule or its state-dependent evaluation changes.

4. Social and behavioral hierarchies

Animal contest models make dominance switching explicitly contingent on assessment and incomplete information. "Formation of Dominance Relationships via Strategy Updating in an Asymmetric Hawk-Dove Game" models two animals with unknown RHP asymmetry $p$6, Bayesian updating of beliefs from fight outcomes, and a responsiveness parameter $p$7 controlling how Hawk probabilities are adjusted. Dominance relations are not imposed from the start; they emerge as estimates of $p$8 change. Because fight outcomes are stochastic and beliefs can be temporarily wrong, the model permits transient dominance by the objectively weaker animal and later reversal as further evidence accumulates. The paper identifies such reversals as most plausible when $p$9 is small, when the system is near payoff sign changes, or when learning is slow relative to outcome noise (Grewal et al., 2013).

A complementary analysis appears in "Dominance, Sharing, and Assessment in an Iterated Hawk--Dove Game". There, extensive simulations show that sharing is stable only when the cost of fighting is low and the animals in a contest have similar RHPs, whereas dominance relationships are stable in most other situations. Under incomplete information, the best-performing strategies involve a limited assessment phase followed by a stable relationship in which fights are avoided; the duration of assessment depends on both the costliness of fighting and the RHP difference. In this literature, dominance switching is the transition from mutual aggression or sharing to an asymmetric stable relationship once information about asymmetry becomes precise enough (Hall et al., 2019).

In social-network models, dominance switching concerns the redistribution of persistent social power. "Modification of Social Dominance in Social Networks by Selective Adjustment of Interpersonal Weights" studies the DeGroot–Friedkin model on a star topology, where all social power accumulates at the center individual. The paper then gives several local intervention strategies—adding new individuals, adding new interpersonal links, or adjusting trust weights—and derives necessary and sufficient conditions under which the center no longer has the greatest social power. The switching threshold is therefore encoded in the modified relative interaction matrix rather than in transient behavior (Ye et al., 2017).

The same idea appears in a different dynamical form in "Stable Relationships". There, the dominance parameters $K(P_q)$0 and $K(P_q)$1 are fixed traits, but effective power is the time-varying quantity $K(P_q)$2. Since the nontrivial eigenvalue of the update matrix is $K(P_q)$3, regimes with $K(P_q)$4 generate oscillatory changes in the sign of $K(P_q)$5, so effective dominance can alternate even though underlying dominance preferences do not change. Stable relationships are possible when both are dominant but $K(P_q)$6, or when one is dominant and the other submissive under analogous bounds; both-submissive pairs are unstable (Ganzfried, 2022).

Dominance can also act as a communication architecture rather than a coercive hierarchy. "Social insect colony as a biological regulatory system: Information flow in dominance networks" analyzes dominance networks in Ropalidia marginata, reports overrepresentation of the feed-forward loop motif, and shows through Boolean modeling that these networks are stable under small fluctuations yet more efficient at information transfer than randomized counterparts. Although the paper does not study temporal reversals directly, a plausible implication is that changes in dominance links would rewire the colony’s information-flow pathways and thereby alter regulatory roles (Nandi et al., 2014).

5. Ecological, evolutionary, and community-level reversals

In ecological and evolutionary games, dominance switching often means a reversal in who invades whom. "From pairwise to group interactions in games of cyclic dominance" shows that in spatial rock–paper–scissors systems, enlarging the interaction range from pairwise to von Neumann or Moore neighborhoods changes not only stationary fractions but also the relation of dominance itself: group interactions can decelerate and even reverse invasion fronts. The same microscopic invasion rates $K(P_q)$7 therefore produce different effective macroscopic dominance relations once indirect multipoint interactions are introduced (Szolnoki et al., 2014).

"A novel route to cyclic dominance in voluntary social dilemmas" studies a four-strategy system of cooperators, defectors, loners, and risk-averse hedgers. For sufficiently small hedging cost $K(P_q)$8, the dominant non-transitive loop is $K(P_q)$9, with loners disappearing. For large $$A_\sigma^\top P_{q_2}A_\sigma - \gamma_d^2 P_{q_1} \preceq -\varepsilon I .$$0, hedgers disappear and the system reverts to the classical voluntary-participation cycle $$A_\sigma^\top P_{q_2}A_\sigma - \gamma_d^2 P_{q_1} \preceq -\varepsilon I .$$1. In the intermediate region, all four strategies coexist and the invasion-rate equalities imply a more complex four-way balance. Here dominance switching is literally a switch between two different cyclic dominance mechanisms as a single cost parameter is varied (Guo et al., 2020).

At the scale of whole communities, the term is used for transitions between abundance structures rather than pairwise contests. "Dominance to egalitarian transition in diverse communities" defines $$A_\sigma^\top P_{q_2}A_\sigma - \gamma_d^2 P_{q_1} \preceq -\varepsilon I .$$2 as the number of species needed to constitute at least half of all individuals and studies the evenness ratio $$A_\sigma^\top P_{q_2}A_\sigma - \gamma_d^2 P_{q_1} \preceq -\varepsilon I .$$3. In the dominance phase, $$A_\sigma^\top P_{q_2}A_\sigma - \gamma_d^2 P_{q_1} \preceq -\varepsilon I .$$4 grows sublinearly with richness $$A_\sigma^\top P_{q_2}A_\sigma - \gamma_d^2 P_{q_1} \preceq -\varepsilon I .$$5, so a vanishing fraction of species carries most of the abundance. In the egalitarian phase, $$A_\sigma^\top P_{q_2}A_\sigma - \gamma_d^2 P_{q_1} \preceq -\varepsilon I .$$6 grows linearly with $$A_\sigma^\top P_{q_2}A_\sigma - \gamma_d^2 P_{q_1} \preceq -\varepsilon I .$$7, so a finite fraction of species is needed. The transition is controlled by the tail exponent $$A_\sigma^\top P_{q_2}A_\sigma - \gamma_d^2 P_{q_1} \preceq -\varepsilon I .$$8 of the stationary species-abundance distribution $$A_\sigma^\top P_{q_2}A_\sigma - \gamma_d^2 P_{q_1} \preceq -\varepsilon I .$$9, with the critical condition $p$0. This makes dominance switching a stochastic phase transition between hyperdominance and broad abundance sharing (Kessler et al., 2024).

Across these ecological papers, the dominant object may be an invasion direction, a cyclic loop, or an abundance-bearing subset. A common theme is that dominance need not be a static property of pairwise payoffs; it can be reorganized by interaction range, added strategies, or stochastic abundance dynamics.

6. Physical and materials realizations

At the nanoscale, dominance switching appears as controlled changes in stationary occupation between metastable physical states. "Voltage-controlled binary conductance switching in gold-4,4'-bipyridine-gold single-molecule nanowires" resolves two conductance states, HighG at approximately $p$1 and LowG at approximately $p$2, with exponential dwell-time distributions characteristic of a two-state Poisson process. The relative dominance of the two states is quantified by the mean dwell times $p$3 and $p$4. Mechanical displacement changes which state dominates by tilting the effective double-well landscape, whereas voltage causes an exponential speedup of both switching directions without typically inverting the dominant state. In that system, dominance switching is mechanical rather than electrical (Mezei et al., 2020).

An antiferromagnetic realization is reported in "Current-density-modulated Antiferromagnetic Domain Switching Revealed by Optical Imaging in Pt/CoO(001) Bilayer". There, magneto-optical birefringence imaging shows that at moderate current densities the Néel vector switches to a state with $p$5, with a critical current density that remains nearly constant across CoO thicknesses. The same switching is observed in Pt/Al$p$6O$p$7/CoO, which excludes spin current injection as the primary cause and points instead to a thermomagnetoelastic mechanism. At higher current density, a second regime appears in which the switched state satisfies $p$8. The dominant switching symmetry therefore changes with current density, and the paper treats this as a transition between distinct current-driven AFM switching regimes (Wu et al., 2024).

These physical examples sharpen an important distinction. In some systems the control parameter primarily changes the ratio of transition rates and hence the dominant state occupation, as with electrode displacement in a molecular junction. In others it changes the dominant mechanism or symmetry class of the switching process itself, as in the two current-density regimes of Pt/CoO. A plausible implication is that dominance switching in physical systems is especially useful when different external controls separately tune stationary bias and switching timescale.

Dominance switching is therefore best understood as a cross-disciplinary family of state-selection phenomena. What changes from field to field is the identity of the dominant object and the formal apparatus used to describe it: cones and LMIs in switched control, preorders and nogoods in constraint solving, threshold equilibria in stochastic games, Bayesian assessment in animal conflict, invasion-rate balances in cyclic games, abundance exponents in ecology, and two-state kinetics or symmetry-changing control in condensed-matter systems. The unifying feature is that dominance is not fixed; it is contingent on dynamics, information, architecture, and control.