Dominant Cycle Overlap

• Dominant cycle overlap is a family of concepts that defines how particular cycles, selected by criteria such as ratios, weights, or persistence, determine key system behaviors.
• It spans diverse fields—from combinatorial optimization and non-equilibrium Markov processes to helioseismic solar cycle analysis and orbital dynamics—by quantifying how cycle interactions constrain outcomes.
• The analytical framework not only sharpens theoretical bounds for approximation algorithms but also underpins efficient computational heuristics in transport and curvature estimation.

Dominant cycle overlap is not a single standardized invariant but a family of technically specific notions that arise when a system admits many cycles and only a subset governs approximation loss, steady-state transport, temporal persistence, or spectral forcing. The common structure is a two-stage selection: first, one identifies “dominant” cycles by a domain-specific criterion such as overlap-to-length ratio, cycle weight, residual activity, or spectral amplitude; second, one studies how those cycles overlap, interfere, or constrain one another. In combinatorial optimization, the dominant cycles are the cycle-cover components whose ratios exceed a threshold and determine the upper bound on total closing-edge overlap; in non-equilibrium Markov processes they are the largest-weight cycles in a flux decomposition; in helioseismology they are the lingering low-latitude bands of an outgoing solar cycle overlapping the onset of a new one; and in long-term celestial mechanics they are the normally dominant $g_2-g_5$ eccentricity oscillations that are suppressed during secular-resonance episodes [(Englert et al., 2021); (Altaner et al., 2011); (Simoniello et al., 2016); (Zeebe et al., 2024)].

1. Cross-disciplinary meanings of dominance and overlap

Across the literature, “dominance” and “overlap” are defined by different operators, and conflating them obscures the underlying mathematics. In the shortest-superstring setting, overlap is a string overlap carried by arcs of a complete directed overlap graph, and dominance is attached to cycles whose overlap-to-length ratio $R(c)=o(c)/w(c)$ lies above a threshold. In non-equilibrium steady states, overlap is measured by shared vertices or directed edges between cycles, while dominance is determined by the cycle weights $m^*_\alpha$ produced by an exact flux decomposition. In solar-cycle analysis, overlap is temporal and latitudinal: it is the interval during which old-cycle low-latitude activity persists after new-cycle activity has already begun at mid-latitudes. In orbital dynamics, overlap is dynamical interference between a secular resonance and the otherwise dominant $405\ \mathrm{kyr}$ eccentricity cycle. In permutation generation, overlap is the exact matching of suffixes and prefixes in an $s$-overlap cycle, and “dominant” refers to choosing $s$ as large as possible without destroying connectivity. In graph curvature approximation, cycle overlap is used algorithmically through short cycles that minimize transport cost [(Englert et al., 2021); (Altaner et al., 2011); (Simoniello et al., 2016); (Zeebe et al., 2024); (Horan, 2013); (Zhou et al., 2 Jun 2026)].

Domain	Dominance criterion	Overlap notion
Shortest superstring	$\alpha < R(c) \le 2$ large cycles dominate the bound on $O$	String overlap on cycle-cover arcs
NESS cycle decomposition	Largest cycle weights $m^*_\alpha$	Node-overlap, edge-overlap, normalized edge similarity
Solar cycle progression	Lingering old-cycle activity below $15^\circ$	Temporal overlap of successive cycles
Earth orbital eccentricity	Dominant $R(c)=o(c)/w(c)$0 term in Earth’s eccentricity	Suppression during $R(c)=o(c)/w(c)$1 resonance episodes
Permutation $R(c)=o(c)/w(c)$2-ocycles	Maximal feasible $R(c)=o(c)/w(c)$3 near $R(c)=o(c)/w(c)$4	Exact suffix/prefix overlap
CCOM curvature	Short cycles prioritized in transport	Overlap of 3-, 4-, and 5-cycles around an edge

A recurring misconception is that overlap always means direct geometric intersection of cycles. The cited work shows otherwise. Depending on the field, overlap may be concatenative, probabilistic, temporal, spectral, or algorithmic, and dominance may refer to ratios, weights, persistence, or forcing amplitude rather than mere size.

2. Ratio-dominant cycles in shortest-superstring approximation

In the shortest-superstring problem, the input is a set of strings $R(c)=o(c)/w(c)$5 with no string a substring of another. The complete directed overlap graph $R(c)=o(c)/w(c)$6 has one vertex per string and an arc profit $R(c)=o(c)/w(c)$7 equal to the length of the longest suffix of $R(c)=o(c)/w(c)$8 that is also a prefix of $R(c)=o(c)/w(c)$9. For a directed cycle

$m^*_\alpha$0

the distance from $m^*_\alpha$1 to $m^*_\alpha$2 is

$m^*_\alpha$3

the total length of the cycle in the distance graph is

$m^*_\alpha$4

the overlap of the cycle-closing edge is

$m^*_\alpha$5

and the overlap-to-length ratio is

$m^*_\alpha$6

Englert–Matsakis–Vesel classify the cycles of a maximum-overlap cycle-cover using a parameter $m^*_\alpha$7 into three types: small cycles with $m^*_\alpha$8, large cycles with $m^*_\alpha$9, and extra-large cycles with $405\ \mathrm{kyr}$0. Their key observation is that the large cycles with $405\ \mathrm{kyr}$1 are the dominant ones: they are exactly the cycles whose leftover overlap beyond $405\ \mathrm{kyr}$2 forces $405\ \mathrm{kyr}$3 upward in the inequality $405\ \mathrm{kyr}$4, where $405\ \mathrm{kyr}$5, $405\ \mathrm{kyr}$6, and $405\ \mathrm{kyr}$7 (Englert et al., 2021).

The central contribution is a pair of incomparable upper bounds on $405\ \mathrm{kyr}$8:

$405\ \mathrm{kyr}$9

and

$s$0

where $s$1. A positive linear combination,

$s$2

is chosen so that all $s$3-sums acquire coefficient exactly $s$4. Solving the resulting linear system yields

$s$5

and hence

$s$6

This ratio classification directly sharpens approximation guarantees. For GREEDY, the analysis gives

$s$7

where $s$8 and $s$9 are restricted to the culprit cycles isolated in the classical proof. Using $s$0 yields

$s$1

and therefore

$s$2

For the general shortest-superstring approximation scheme, MGREEDY is also a $s$3-approximation, and combining it with a $s$4-approximation for MaxATSP with $s$5 gives

$s$6

These improve the earlier $s$7 bound for GREEDY due to Kaplan and Shafrir and the $s$8-approximation algorithm of Mucha (Englert et al., 2021).

The conceptual significance is that not all high-overlap cycles are equally obstructive. Small cycles force the optimal superstring to respect the same cyclic order, large cycles yield only a weaker extracted cost, and extra-large cycles are overlap-poor enough to be absorbed into the $s$9 term. The dominant-cycle overlap, in this sense, is not simply the largest overlap present in the instance; it is the overlap surplus contributed by the specific ratio band $\alpha < R(c) \le 2$0.

3. Dominant cycles and overlap measures in non-equilibrium steady states

For a finite irreducible Markov process with directed graph $\alpha < R(c) \le 2$1, one-way steady-state fluxes are $\alpha < R(c) \le 2$2, and the node condition

$\alpha < R(c) \le 2$3

holds at every node. A directed cycle $\alpha < R(c) \le 2$4 is an equivalence class of self-avoiding closed paths

$\alpha < R(c) \le 2$5

with cyclic rotations identified. The indicator $\alpha < R(c) \le 2$6 equals $\alpha < R(c) \le 2$7 if edge $\alpha < R(c) \le 2$8 belongs to $\alpha < R(c) \le 2$9 and $O$0 otherwise, while $O$1 if the cycle visits node $O$2 (Altaner et al., 2011).

Altaner and collaborators describe an iterative decomposition algorithm for the steady-state flux. After enumerating all cycles $O$3 in some order, one initializes $O$4, sets

$O$5

updates

$O$6

and at the end obtains the exact decomposition

$O$7

The non-negativity of the cycle weights and exact reconstruction follow from telescoping sums and preservation of the node condition.

Dominant cycles are then selected by their weights. One typical rule is to choose the smallest subset $O$8 such that

$O$9

with, for example, $m^*_\alpha$0 or $m^*_\alpha$1. Overlap between two cycles $m^*_\alpha$2 and $m^*_\alpha$3 is quantified in two basic ways:

$m^*_\alpha$4

which counts common vertices, and

$m^*_\alpha$5

which counts shared directed edges. A normalized similarity index is

$m^*_\alpha$6

The paper’s 2-particle, 4-site TASEP example makes the role of dominant-cycle overlap explicit. There are six configurations, all forward-hop rates are $m^*_\alpha$7 except hops over the periodic boundary, which occur at rate $m^*_\alpha$8, and the four non-trivial self-avoiding cycles of length $m^*_\alpha$9 are

$15^\circ$0

With enumeration $15^\circ$1, the weights $15^\circ$2 are piecewise linear. For $15^\circ$3, $15^\circ$4, $15^\circ$5, $15^\circ$6, and $15^\circ$7; at $15^\circ$8, only two cycles survive; for $15^\circ$9, $R(c)=o(c)/w(c)$00, $R(c)=o(c)/w(c)$01, $R(c)=o(c)/w(c)$02, and $R(c)=o(c)/w(c)$03. At $R(c)=o(c)/w(c)$04, the dominant set changes abruptly, and the derivatives $R(c)=o(c)/w(c)$05 have a kink there. For $R(c)=o(c)/w(c)$06, one possible dominant active set is $R(c)=o(c)/w(c)$07, and for the pair $R(c)=o(c)/w(c)$08,

$R(c)=o(c)/w(c)$09

This setting shows that dominant-cycle overlap can itself change discontinuously even when the underlying steady-state fluxes vary continuously.

4. Overlap of successive solar cycles in helioseismic observations

In helioseismic studies of the solar cycle, overlap is defined operationally as the interval during which the old cycle’s low-latitude bands remain active after the new cycle has already begun at mid-latitudes. If $R(c)=o(c)/w(c)$10 denotes the epoch of minimum helioseismic activity at latitude $R(c)=o(c)/w(c)$11, then

$R(c)=o(c)/w(c)$12

with $R(c)=o(c)/w(c)$13 and $R(c)=o(c)/w(c)$14. The same paper also writes

$R(c)=o(c)/w(c)$15

but uses the $R(c)=o(c)/w(c)$16 formulation because the minimum is taken as the turn-around between old and new cycles (Simoniello et al., 2016).

Using GONG helioseismic data, the study finds that the progression below $R(c)=o(c)/w(c)$17 differs materially from that at higher latitudes. In the high-frequency global $R(c)=o(c)/w(c)$18-mode band $R(c)=o(c)/w(c)$19–$R(c)=o(c)/w(c)$20, which probes the very near-surface layers, the measured overlap durations are approximately $R(c)=o(c)/w(c)$21 months at $R(c)=o(c)/w(c)$22–$R(c)=o(c)/w(c)$23, about $R(c)=o(c)/w(c)$24 months at $R(c)=o(c)/w(c)$25–$R(c)=o(c)/w(c)$26, and less than $R(c)=o(c)/w(c)$27 month above $R(c)=o(c)/w(c)$28. High-degree ring-diagram modes show hemispheric asymmetry: in the $R(c)=o(c)/w(c)$29–$R(c)=o(c)/w(c)$30 band, the South reaches minimum in October 2010 and the North in September 2009, implying $R(c)=o(c)/w(c)$31 months; in the $R(c)=o(c)/w(c)$32–$R(c)=o(c)/w(c)$33 band, the offset is about $R(c)=o(c)/w(c)$34 months.

Depth dependence is resolved by separating intermediate-degree modes into three frequency ranges with upper turning points $R(c)=o(c)/w(c)$35: low-$R(c)=o(c)/w(c)$36 ($R(c)=o(c)/w(c)$37–$R(c)=o(c)/w(c)$38, $R(c)=o(c)/w(c)$39–$R(c)=o(c)/w(c)$40), medium-$R(c)=o(c)/w(c)$41 ($R(c)=o(c)/w(c)$42–$R(c)=o(c)/w(c)$43, $R(c)=o(c)/w(c)$44–$R(c)=o(c)/w(c)$45), and high-$R(c)=o(c)/w(c)$46 ($R(c)=o(c)/w(c)$47–$R(c)=o(c)/w(c)$48, $R(c)=o(c)/w(c)$49–$R(c)=o(c)/w(c)$50). In the $R(c)=o(c)/w(c)$51–$R(c)=o(c)/w(c)$52 band, the overlap durations are approximately $R(c)=o(c)/w(c)$53 months, $R(c)=o(c)/w(c)$54 months, and $R(c)=o(c)/w(c)$55 months respectively, indicating that the tail-like overlap is present at all depths down to about $R(c)=o(c)/w(c)$56 and is slightly longer in the deeper layers.

The physical picture is that the cycle starts at mid-latitudes and migrates equatorward and poleward, but sunspot eruptions of the old cycle continue below $R(c)=o(c)/w(c)$57. That prolonged low-latitude activity delays the onset of the new cycle and produces a tail-like attachment between successive cycles. The low-latitude frequency-shift curve decays more slowly after the second maximum than the $R(c)=o(c)/w(c)$58–$R(c)=o(c)/w(c)$59 curve, and the extra residual shift of roughly one year delays the epoch of minimum by approximately $R(c)=o(c)/w(c)$60–$R(c)=o(c)/w(c)$61 months. The low-latitude signal also shows a single-peak structure, whereas latitudes above $R(c)=o(c)/w(c)$62 exhibit a double peak.

These observations are used as dynamo constraints. The reported interpretation is that $R(c)=o(c)/w(c)$63 thin-shell dynamo-wave models generically produce too-long, high-latitude overlaps, whereas flux-transport dynamo models with a surface Babcock–Leighton $R(c)=o(c)/w(c)$64-effect and a deep equatorward return flow naturally give a short $R(c)=o(c)/w(c)$65–$R(c)=o(c)/w(c)$66 year overlap confined to $R(c)=o(c)/w(c)$67. To match $R(c)=o(c)/w(c)$68–$R(c)=o(c)/w(c)$69 yr and its depth profile, the equatorward return-flow speed at the base of the convection zone must be $R(c)=o(c)/w(c)$70–$R(c)=o(c)/w(c)$71, and the surface poloidal-source strength must be $R(c)=o(c)/w(c)$72–$R(c)=o(c)/w(c)$73 (Simoniello et al., 2016).

5. Resonance-induced overlap with the dominant $R(c)=o(c)/w(c)$74 eccentricity cycle

In long-term solar-system dynamics, Earth’s eccentricity can be represented in the Laplace–Lagrange framework by

$R(c)=o(c)/w(c)$75

where $R(c)=o(c)/w(c)$76 are apsidal precession rates and $R(c)=o(c)/w(c)$77 are nodal precession rates. The principal long-eccentricity cycle in Earth’s eccentricity is the beat frequency

$R(c)=o(c)/w(c)$78

with recent-period value

$R(c)=o(c)/w(c)$79

This is the familiar long eccentricity cycle dominated by Venus’ and Jupiter’s orbits (Zeebe et al., 2024).

The paper identifies a secular resonance

$R(c)=o(c)/w(c)$80

tracked through the resonant angle

$R(c)=o(c)/w(c)$81

Band-pass filtering is applied to $R(c)=o(c)/w(c)$82 using rectangular windows of $R(c)=o(c)/w(c)$83 about $R(c)=o(c)/w(c)$84 and $R(c)=o(c)/w(c)$85 about $R(c)=o(c)/w(c)$86. In the nonresonant regime, $R(c)=o(c)/w(c)$87 circulates monotonically through $R(c)=o(c)/w(c)$88; during resonance, it librates about a fixed value. The effective half-width is set by the filter bandwidths, so libration occurs when

$R(c)=o(c)/w(c)$89

with $R(c)=o(c)/w(c)$90 in the implementation.

The dominant-cycle overlap here is not a direct intersection of two cycles but a resonance episode that overlaps with, weakens, or erases the otherwise dominant $R(c)=o(c)/w(c)$91 signal. In an ensemble of $R(c)=o(c)/w(c)$92 independent $R(c)=o(c)/w(c)$93-Gyr integrations generated by $R(c)=o(c)/w(c)$94 au perturbations to Earth’s initial position, about $R(c)=o(c)/w(c)$95 of the solutions enter one or more $R(c)=o(c)/w(c)$96 episodes. During such episodes, the sharp $R(c)=o(c)/w(c)$97 peak in FFT spectra of Earth’s eccentricity splits into two nearby peaks of greatly reduced power, the $R(c)=o(c)/w(c)$98 beat loses coherence and amplitude, and the power of the $R(c)=o(c)/w(c)$99 cycle often drops below that of the short $m^*_\alpha$00 eccentricity cycle. Time-frequency plots then resemble neither the modern pattern nor typical stratigraphic Milanković signals.

The resonance episodes typically last from a few Myr up to several tens of Myr, and multiple entries and exits often occur on chaotic diffusion timescales of approximately $m^*_\alpha$01–$m^*_\alpha$02 yr. The paper reports weakened or absent long-eccentricity-cycle intervals at times such as $m^*_\alpha$03, $m^*_\alpha$04, and $m^*_\alpha$05, without pushing Mercury’s eccentricity beyond $m^*_\alpha$06. The mechanism is that chaotic diffusion in $m^*_\alpha$07 intermittently realizes $m^*_\alpha$08; when $m^*_\alpha$09, the resonant angle switches from circulation to libration, the $m^*_\alpha$10 mode enters a mixed state, and clean beating with $m^*_\alpha$11 is suppressed. Exiting the resonance restores a clean $m^*_\alpha$12 mode and the $m^*_\alpha$13 cycle re-emerges.

The climatic implication is a weakened eccentricity forcing. Because insolation amplitude scales as $m^*_\alpha$14, a reduction of maximal Earth eccentricity from approximately $m^*_\alpha$15 to approximately $m^*_\alpha$16 reduces eccentricity-forcing amplitude by a factor of about $m^*_\alpha$17. The astrochronological implication is more disruptive: the long-held assumption that the $m^*_\alpha$18 cycle is a stable metronome requires revision, and deep-time chronologies that lock onto a continuous $m^*_\alpha$19 signal must permit $m^*_\alpha$20 Myr intervals in which the dominant cycle is interrupted or absent (Zeebe et al., 2024).

6. Maximal-overlap cycles for permutations

For permutations, an $m^*_\alpha$21-overlap cycle or $m^*_\alpha$22-ocycle is a cyclic listing of all $m^*_\alpha$23 permutations of $m^*_\alpha$24,

$m^*_\alpha$25

such that the last $m^*_\alpha$26 symbols of $m^*_\alpha$27 agree exactly with the first $m^*_\alpha$28 symbols of $m^*_\alpha$29. Writing

$m^*_\alpha$30

and

$m^*_\alpha$31

the overlap condition is simply $m^*_\alpha$32 (Horan, 2013).

Horan’s main theorem gives a necessary and sufficient condition for the existence of such cycles. If $m^*_\alpha$33, then an $m^*_\alpha$34-ocycle on all permutations of $m^*_\alpha$35 exists if and only if

$m^*_\alpha$36

Equivalently,

$m^*_\alpha$37

This immediately excludes universal cycles for permutations, since the universal-cycle case $m^*_\alpha$38 gives $m^*_\alpha$39.

The proof proceeds through an Eulerian transition digraph. The vertices are length-$m^*_\alpha$40 strings that occur as prefixes or suffixes of permutations, and each permutation defines a directed edge from its $m^*_\alpha$41-prefix to its $m^*_\alpha$42-suffix. An Eulerian tour corresponds bijectively to an $m^*_\alpha$43-ocycle. The graph is balanced because every permutation beginning with a given $m^*_\alpha$44-tuple corresponds uniquely to one ending in that tuple by cyclic shift. Connectivity when $m^*_\alpha$45 is obtained by partitioning a permutation into consecutive blocks of size $m^*_\alpha$46 and showing, using rotations by $m^*_\alpha$47 and adjacent transpositions within the current $m^*_\alpha$48-suffix, that any two adjacent symbols in the $m^*_\alpha$49-prefix can be swapped. Since adjacent transpositions generate the full symmetric group, the graph is weakly connected. When $m^*_\alpha$50, the suffix stays inside a single block of length $m^*_\alpha$51, block order cannot change, and the digraph splits into disconnected components.

Two small examples illustrate the boundary sharply. For $m^*_\alpha$52, $m^*_\alpha$53 and $m^*_\alpha$54, so a $m^*_\alpha$55-ocycle exists:

$m^*_\alpha$56

For $m^*_\alpha$57, $m^*_\alpha$58 and $m^*_\alpha$59, so no $m^*_\alpha$60-ocycle exists; the transition digraph splits into three disjoint components.

The paper explicitly uses “dominant” in a comparative sense: an $m^*_\alpha$61-ocycle is the next best structure to a universal cycle and retains as large an overlap as possible, with $m^*_\alpha$62 close to $m^*_\alpha$63, without forcing disconnection. In this combinatorial setting, dominant-cycle overlap means maximal feasible overlap subject to global traversability of the state space.

7. Cycle overlap as an algorithmic transport principle

A related but distinct use of cycle overlap appears in the approximation of Ollivier–Ricci curvature. For a simple, undirected, unweighted graph $m^*_\alpha$64 and an edge $m^*_\alpha$65, one considers neighbor measures

$m^*_\alpha$66

with lazy-mass parameter $m^*_\alpha$67, computes the $m^*_\alpha$68-Wasserstein distance

$m^*_\alpha$69

subject to the marginal constraints, and defines

$m^*_\alpha$70

The CCOM method reformulates the transport around short cycles containing $m^*_\alpha$71 (Zhou et al., 2 Jun 2026).

If one restricts attention to paths through short cycles involving $m^*_\alpha$72, then for $m^*_\alpha$73 and $m^*_\alpha$74 the distance $m^*_\alpha$75 can only be $m^*_\alpha$76 or $m^*_\alpha$77. For a transport plan $m^*_\alpha$78, define

$m^*_\alpha$79

Then

$m^*_\alpha$80

Mass moved along a 3-cycle contributes to $m^*_\alpha$81, along a 4-cycle to $m^*_\alpha$82, and along a 5-cycle to $m^*_\alpha$83. The CCOM heuristic therefore fully exploits all 3-cycles, then greedily assigns as much mass as possible to 4-cycles and 5-cycles in order of “weakest overlap first,” and finally routes any remainder at distances $m^*_\alpha$84 or $m^*_\alpha$85.

The method enumerates 3-, 4-, and 5-cycles containing the edge, constructs source and destination sets, and uses a greedy/pruning procedure that sorts sources by the number of available destinations and destinations by the number of available sources. Theoretical support is given by an optimal-transport principle: ordering any feasible plan by the lexicographically descending vector

$m^*_\alpha$86

shows that maximizing $m^*_\alpha$87 also minimizes $m^*_\alpha$88. A coarse uniform error bound reported for the approximation is

$m^*_\alpha$89

The final closed form for the approximate curvature when $m^*_\alpha$90 is

$m^*_\alpha$91

where $m^*_\alpha$92 are the masses transported through 3-, 4-, and 5-cycles. Empirically, the method is evaluated on small graphs and large scale-free networks. The reported findings are that exact LP becomes infeasible beyond roughly $m^*_\alpha$93k nodes, CCOM scales linearly and runs in seconds on $m^*_\alpha$94-million-edge graphs, MAE on Holme–Kim scale-free random graphs up to $m^*_\alpha$95 nodes remains below $m^*_\alpha$96, and curvature-based community detection improves relative to several baselines. This is not a theory of dominant cycles in the same sense as the superstring or NESS work, but it shows that cycle overlap can be elevated from a descriptive quantity to an organizing computational principle (Zhou et al., 2 Jun 2026).

Taken together, these literatures show that dominant cycle overlap is best understood as a structural theme rather than a single formal object. Its precise meaning depends on whether cycles are selected by ratio, weight, persistence, spectral prominence, or admissible overlap length, but the analytical role is consistent: overlap among the dominant cycles is what determines the hardest combinatorial cases, the effective transport channels, the observable switching behavior, or the failure of a previously assumed metronome.