Cycle Ratio: Concepts and Applications

• Cycle ratio is a normalized quantity derived from cycles, characterizing node importance, graph cyclicity, and optimization ratios across different domains.
• It encompasses distinct formulations in complex networks, directed optimization, facility location, Hamiltonian chordal graphs, and solar physics.
• Empirical evaluations and algorithmic advancements highlight its unique advantages over classical metrics by leveraging cycle-based normalization.

Cycle ratio is not a single invariant. Across the literature, the term denotes several distinct normalized quantities built from cycles: a node-importance measure based on shared participation in shortest or basic cycles in complex networks, a graph-level fraction of nodes that lie on at least one cycle, a directed-cycle objective such as cost-to-time or gradient-to-length ratio in combinatorial optimization, an approximation ratio for strategyproof facility location on a cycle, a ratio of non-extendible cycle length to graph size in Hamiltonian chordal graphs, and a solar-physics ratio between sunspot-number amplitude and sunspot-group area at cycle maximum (Fan et al., 2019, Fan et al., 2020, Zhang et al., 2021, Bringmann et al., 2017, Chen et al., 2023, Rogowski et al., 19 May 2025, Lafond et al., 2013, Javaraiah, 2024).

1. Terminological scope

The meaning of cycle ratio is discipline-dependent. In graph and network research, the common theme is normalization of a cycle-derived quantity, but the object being normalized differs substantially.

Domain	Quantity	Source
Complex networks	Node importance from shared shortest or basic cycles	(Fan et al., 2019, Fan et al., 2020, Zheng et al., 30 Sep 2025)
Network classification	Fraction of nodes belonging to cycles	(Zhang et al., 2021)
Directed-graph optimization	Minimum cost-to-time or gradient-to-length ratio of a cycle	(Bringmann et al., 2017, Chen et al., 2023)
Facility location on a cycle	Approximation ratio of a strategyproof mechanism	(Rogowski et al., 19 May 2025)
Hamiltonian chordal graphs	Ratio of non-extendible cycle length to graph size	(Lafond et al., 2013)
Solar physics	Ratio of sunspot-number amplitude to sunspot-group area	(Javaraiah, 2024)

Because these usages are non-equivalent, statements about cycle ratio are only meaningful relative to the underlying model class, the cycle family being counted or optimized over, and the normalization convention.

2. Cycle ratio as a node-importance measure in complex networks

In complex-network analysis, cycle ratio was introduced as a cycle-based alternative to star-based node centralities. One formulation defines cycle ratio as “the sum of the proportions of the node $i$ appearing in the basic cycles of the nodes contained in basic cycles of node $i$.” In the worked example for node $1$, the value is computed as

$4/4 + 3/4 + 2/4 + 2/3 + 1/1 = 47/12.$

This construction is tied to a cycle-number matrix $B(G)=[b_{ij}]_{|V|\times|V|}$, where $b_{ii}$ is the cycle number of node $i$ and $b_{ij}$ for $i\neq j$ is the number of co-cycles of $i$ and $i$0. When basic cycles are regarded as hyperedges in a hypernetwork $i$1, the matrix is

$i$2

and row-normalization yields a cycle-ratio matrix whose column sums equal the cycle ratio of the corresponding node (Fan et al., 2019).

A later formulation restricted attention to shortest cycles. Let $i$3 be the set of shortest cycles associated with node $i$4, and let $i$5. The cycle number matrix $i$6 is defined so that $i$7 is the number of cycles in $i$8 containing node $i$9, while $1$0 for $1$1 is the number of cycles in $1$2 passing through both $1$3 and $1$4. The resulting cycle ratio $1$5 sums the fractions $1$6 over all nodes $1$7 with $1$8, and is set to $1$9 when $4/4 + 3/4 + 2/4 + 2/3 + 1/1 = 47/12.$0. The interpretation given is that $4/4 + 3/4 + 2/4 + 2/3 + 1/1 = 47/12.$1 measures how strongly node $4/4 + 3/4 + 2/4 + 2/3 + 1/1 = 47/12.$2 participates in other nodes’ associated shortest cycles (Fan et al., 2020).

This shortest-cycle version was motivated by two considerations. First, longer cycles were treated as less relevant to the connectivity and dynamical roles under study. Second, exhaustive enumeration of all cycles is computationally infeasible in most networks because the number of cycles grows exponentially with length. Supplementary results reported in that line of work state that including second-shortest and third-shortest cycles does not necessarily improve performance, and may reduce discriminability because many nodes’ cycle sets overlap too strongly (Fan et al., 2020).

Empirically, the shortest-cycle-based index was contrasted with degree, H-index, coreness, betweenness, and articulation ranking. Using Kendall’s Tau, degree, H-index, and coreness were found to be highly correlated with one another, while cycle ratio was much less correlated with those measures. In node-percolation experiments, betweenness performed best overall, cycle ratio was close to the best, degree, H-index, and coreness were weaker, and articulation ranking was worst. In pinning control, betweenness and articulation ranking were often better than cycle ratio, but cycle ratio was better than degree, H-index, and coreness. In early-stage epidemic spreading across $4/4 + 3/4 + 2/4 + 2/3 + 1/1 = 47/12.$3 comparisons, cycle ratio ranked first $4/4 + 3/4 + 2/4 + 2/3 + 1/1 = 47/12.$4 times, second $4/4 + 3/4 + 2/4 + 2/3 + 1/1 = 47/12.$5 times, and third only $4/4 + 3/4 + 2/4 + 2/3 + 1/1 = 47/12.$6 times (Fan et al., 2020).

The same literature emphasized limitations. The method does not apply to trees or tree-like networks with no cycles, and it assigns cycle ratio $4/4 + 3/4 + 2/4 + 2/3 + 1/1 = 47/12.$7 to all nodes not belonging to any shortest cycle. A proposed hybrid correction was

$4/4 + 3/4 + 2/4 + 2/3 + 1/1 = 47/12.$8

where $4/4 + 3/4 + 2/4 + 2/3 + 1/1 = 47/12.$9 is degree and $B(G)=[b_{ij}]_{|V|\times|V|}$0 is a tunable parameter (Fan et al., 2020).

A recent extension, Basic Cycle Ratio (BCR), replaces shortest cycles with a basic cycle set induced by a spanning tree. For every non-tree edge $B(G)=[b_{ij}]_{|V|\times|V|}$1, the basic cycle is

$B(G)=[b_{ij}]_{|V|\times|V|}$2

where $B(G)=[b_{ij}]_{|V|\times|V|}$3 is the unique path between $B(G)=[b_{ij}]_{|V|\times|V|}$4 and $B(G)=[b_{ij}]_{|V|\times|V|}$5 in the spanning tree. The basic-cycle number matrix counts how many basic cycles contain node $B(G)=[b_{ij}]_{|V|\times|V|}$6 and how many contain both $B(G)=[b_{ij}]_{|V|\times|V|}$7 and $B(G)=[b_{ij}]_{|V|\times|V|}$8, and the BCR score uses the same ratio pattern as earlier cycle-ratio definitions. On six real-world social networks, BCR was reported to have the lowest average Kendall correlation with other methods, around $B(G)=[b_{ij}]_{|V|\times|V|}$9, to achieve the best or near-best individuation in five out of six cases, and to remain robust across $b_{ii}$0 random spanning-tree realizations (Zheng et al., 30 Sep 2025). This suggests a shift from shortest local loops to a fundamental cycle basis as the operative representation of cyclic structure.

3. Cycle-Nodes-Ratio as a graph-level measure of cyclicity

A distinct usage is Cycle Nodes Ratio (CNR), a graph-level quantity designed to measure how close a network is to a tree network. For an undirected connected graph $b_{ii}$1 with $b_{ii}$2 nodes and $b_{ii}$3 edges, if $b_{ii}$4 is the number of cycle nodes, then

$b_{ii}$5

A cycle node is defined as “the node sites along one or more cycle(s) path in a network.” Under this normalization, a cycle graph has $b_{ii}$6 and a tree or star graph has $b_{ii}$7 (Zhang et al., 2021).

The computational procedure is bridge-based. A node is a cycle node if and only if it is connected by at least one no-cut edge. The proposed CDFS algorithm uses Tarjan-style DFS with discovery times $b_{ii}$8 and low-link values $b_{ii}$9, with an edge $i$0 classified as a cut edge when

$i$1

Cycle nodes are then identified from the set of nodes incident to no-cut edges. The reported complexity is $i$2 with adjacency lists and $i$3 with adjacency matrices (Zhang et al., 2021).

For Erdős–Rényi networks, the paper gave approximate analytical solutions for CNR and emphasized three phenomena: at fixed average degree, CNR is essentially independent of network size; at fixed network size, CNR increases with average degree; and there is a critical turning point at average degree $i$4. The critical phenomenon was linked to the giant component: for $i$5, all nodes are in tree components and the number of cycle nodes is $i$6; for $i$7, the giant component emerges and drives the appearance of cycle nodes (Zhang et al., 2021).

CNR was explicitly compared with two-core ratio (TCR). In ER networks the two are very similar, but in Watts–Strogatz networks and some real and fungal networks, TCR can be much larger than CNR because the $i$8-core can include cycle connect nodes that are not actually on any cycle. The distinction is therefore not merely terminological: CNR counts only nodes on actual cycle paths, whereas TCR may retain nodes that persist under leaf pruning without belonging to any cycle (Zhang et al., 2021).

The same study reported that real networks are generally lower in CNR than ER networks with the same mean degree. When microscopic CNR saturates near $i$9, improved spectral coarse-graining was used to expose macro-level differences: two networks could both have CNR $b_{ij}$0 before coarse-graining, yet after coarse-graining one remained at $b_{ij}$1 while another dropped to $b_{ij}$2. In machine-learning-based network recognition using K-Nearest Neighbor and five features—density $b_{ij}$3, modularity $b_{ij}$4, average degree $b_{ij}$5, global clustering coefficient $b_{ij}$6, and CNR $b_{ij}$7—removing $b_{ij}$8 caused the largest performance drop, from precision $b_{ij}$9, recall $i\neq j$0, F1 $i\neq j$1 to precision $i\neq j$2, recall $i\neq j$3, F1 $i\neq j$4 (Zhang et al., 2021).

4. Minimum-ratio cycles in directed-graph optimization

In combinatorial optimization, a closely related but distinct notion is the minimum-ratio cycle. One classical problem assigns each directed edge a cost $i\neq j$5 and transit time $i\neq j$6, and seeks a directed cycle $i\neq j$7 minimizing

$i\neq j$8

This is the minimum cost-to-time ratio cycle problem. A standard reduction reweights edges by

$i\neq j$9

so that if $i$0, the reweighted graph has a negative cycle; if $i$1, the minimum cycle weight is $i$2; and if $i$3, the graph has no negative cycle. The reported strongly polynomial algorithm runs in

$i$4

described as the first improvement over Megiddo’s $i$5 algorithm for sparse graphs. The same work also gave a general-graph bound of

$i$6

with fast matrix multiplication, and a constant-treewidth bound of $i$7, more specifically $i$8 (Bringmann et al., 2017).

A more recent dynamic formulation considered a directed graph $i$9 with edge lengths $i$00, edge gradients $i$01, and circulations $i$02. The ratio objective is

$i$03

A dynamic algorithm for the min-ratio cycle problem must, after each update, output a cycle $i$04 satisfying

$i$05

The principal approximation factor reported for the dynamic data structure is

$i$06

The same framework gave the first almost-linear time algorithms for incremental cycle detection, strongly connected component maintenance, $i$07-$i$08 shortest path, maximum flow, and minimum-cost flow (Chen et al., 2023).

The dynamic construction proceeds through a deterministic $i$09-oblivious routing, a hierarchical routing graph, decomposition into monotone cycles, reduction to fundamental cycles on a small set of trees, and a portal-routing plus dynamic-spanner recursion. One derived guarantee states that the approximate minimum-ratio cycle can be represented as a fundamental cycle in one of $i$10 trees, and the same machinery yields deterministic incremental cycle detection in total time

$i$11

In that application, every edge is given capacity $i$12, cost $i$13, and threshold $i$14, so the existence of a directed cycle is converted into a feasible negative-cost circulation problem (Chen et al., 2023).

5. Ratios on cyclic domains and cycle-extendibility

In strategyproof facility location on a cycle, cycle ratio refers to an approximation ratio rather than a structural graph invariant. For a profile $i$15 and mechanism $i$16, the social cost is

$i$17

the optimum is

$i$18

and the mechanism’s approximation ratio is the smallest $i$19 such that

$i$20

For any set of agents with an odd cardinality and a cyclic graph $i$21 of length $i$22, a strategyproof mechanism was shown to achieve approximation ratio at most $i$23, improving the previous upper bound $i$24 for $i$25. The mechanism, RD+PCD, is a $i$26–$i$27 mixture of Random Dictator and Proportional Circle Distance, and its analysis uses a cycle-cutting technique that replaces the cycle distance

$i$28

with the line-like distance

$i$29

after normalization (Rogowski et al., 19 May 2025).

A separate graph-theoretic use concerns the ratio of a non-extendible cycle to the total number of vertices in a Hamiltonian chordal graph. Hendry’s conjecture that every Hamiltonian chordal graph is cycle extendible was disproved by a base counterexample $i$30 on $i$31 vertices, with heavy edges $i$32, $i$33, $i$34, $i$35, and $i$36, a Hamiltonian cycle

$i$37

and a non-Hamiltonian cycle

$i$38

By pasting cliques onto heavy edges, counterexamples were constructed on every $i$39 vertices. A further clique pasted onto edge $i$40 yields a graph on $i$41 vertices while preserving a non-extendible cycle of length $i$42, so the ratio becomes

$i$43

which tends to $i$44 as $i$45 grows. Consequently, for any real number $i$46, there exists a Hamiltonian chordal graph $i$47 with a non-extendible cycle $i$48 satisfying

$i$49

This is a cycle-length ratio rather than a node-importance or optimization ratio, but it shows another established use of cycle-based normalization (Lafond et al., 2013).

6. Solar-cycle ratio in heliophysics

In heliophysics, the relevant cycle is the solar cycle. The quantity studied is the ratio of sunspot-number amplitude to whole-sphere sunspot-group area at the maximum epoch of each cycle. If $i$50 is the maximum $i$51-month smoothed monthly mean total sunspot number $i$52, and $i$53 is the $i$54-month smoothed monthly mean whole-sphere sunspot-group area WSGA at the same epoch, then the ratio of interest is

$i$55

The associated annual mean relations were fitted, cycle by cycle for Solar Cycles $i$56–$i$57, by both

$i$58

and

$i$59

The nonlinear fit was reported to be better than the linear fit for Solar Cycles $i$60, $i$61, and $i$62 (Javaraiah, 2024).

A secular decreasing trend was found in the linear slope,

$i$63

where $i$64 is the solar-cycle number. For the nonlinear fit, the first-order coefficient was

$i$65

and the quadratic coefficient was

$i$66

The ratio $i$67 had mean $i$68, standard deviation $i$69, and a long-term modulation of amplitude $i$70. A $i$71-year variation was reported, with crests near Solar Cycles $i$72 and $i$73, and this pattern was used to infer that Solar Cycle $i$74 should be larger than Solar Cycle $i$75 and that Solar Cycle $i$76 should be smaller than Solar Cycle $i$77 (Javaraiah, 2024).

The same work coupled the ratio analysis with earlier precursor relations and a $i$78-year periodicity in cycle amplitudes. The resulting numerical predictions for Solar Cycle $i$79 were

$i$80

with the $i$81-year similarity argument giving the preferred estimate $i$82. The predicted maximum epoch was $i$83 (March $i$84) $i$85 months, the ending epoch $i$86 (March $i$87) $i$88 months, and the terminal $i$89 about $i$90 (Javaraiah, 2024).

The term cycle ratio therefore spans several mathematically unrelated constructs. In some areas it is a node score derived from shared cycle participation; in others it is a graph-level cyclicity fraction, a cycle-optimization objective, a performance ratio on a cyclic metric space, a cycle-length fraction, or a solar-cycle diagnostic. The unifying idea is normalization by a cycle-associated quantity, but the semantics, algorithms, and interpretations are field-specific.