Basic Cycle Ratio in Complex Networks

• Basic Cycle Ratio (BCR) is a cycle-centric node ranking measure that quantifies both local cycle participation and global connectivity in complex networks.
• BCR refines traditional cycle metrics by normalizing joint cycle participation, thereby achieving superior individuation and cost-efficient identification of influential spreaders.
• BCR enhances network resilience and optimizes intervention strategies by leveraging fundamental cycle structures to outperform classical centrality measures.

The Basic Cycle Ratio (BCR) is a quantitative network-theoretic indicator developed to address the challenge of identifying influential nodes for spreading processes, information diffusion, and maintaining cohesion in complex networks. By leveraging fundamental cycle structures, BCR incorporates both local (neighborhood cycle participation) and global (network-wide cycle connectivity) perspectives, extending beyond traditional star-based centrality measures or purely path-based approaches. The BCR metric adapts notions from previously established cycle number and cycle ratio indices, refining their methodology to improve the individuation, cost-effectiveness, and practical performance of node ranking, particularly in social network applications.

1. Mathematical Foundations of Cycle-Based Metrics

Cycle-based analysis departs from conventional network centrality paradigms by focusing on closed paths (cycles) as the elementary organizational units of complex networks. A basic cycle is defined as a minimal set of links forming a closed loop; mathematically, for an undirected graph, each basic cycle is generated via a spanning tree where the addition of a non-tree edge (s, t) combines with the unique path $P_{st}$ in the tree to form $c_k = \{ (s, t) \cup P_{st} \}$ (Zheng et al., 30 Sep 2025).

To structure cycle information, one constructs the cycle number matrix $C$ of size $N \times N$ for $N$ nodes, where:

• $c_{ii}$ denotes the count of basic cycles including node $i$;
• $c_{ij}$ (for $i \neq j$) counts the cycles that include both nodes $i$ and $j$.

As formalized in (Fan et al., 2019) and (Fan et al., 2020), this cycle number matrix underlies the cycle ratio and BCR computations, integrating the redundancy and distributed connectivity intrinsic to cycles.

2. Definition and Calculation of the Basic Cycle Ratio (BCR)

The BCR for a node $i$ aggregates its joint participation in cycles with other nodes, normalized by the total cycle involvement of each partner node. The explicit formula for $i$ (when $c_{ii} > 0$) is:

$$\mathrm{BCR}_i = \sum_{j,\; c_{ij}>0} \frac{c_{ij}}{c_{jj}}$$

If a node does not participate in any cycle ($c_{ii}=0$), $\mathrm{BCR}_i=0$. Each summand quantifies the relative prominence of node $i$ in the cycle repertoire of node $j$.

This formulation distinguishes BCR from prior cycle-based metrics:

• Cycle Number (CN): Counts only the raw number of cycles passing through $i$ without normalization.
• Cycle Ratio (CR): Focuses on the shortest cycles and their distribution, but may lack broader representational context.

BCR synthesizes these perspectives, yielding a measure sensitive to both dense local cycle integration and widespread network connectivity (Zheng et al., 30 Sep 2025).

3. Comparative Analysis with Classical and Cycle-Oriented Centralities

Traditional centrality measures—degree, H-index, coreness, betweenness—predominantly emphasize star-based (immediate-neighbor) or shortest-path structures. They often exhibit high mutual correlation and can overlook nodes embedded in alternative cyclic routes or possessing distributed influence (Fan et al., 2020).

Cycle-based approaches such as CN or CR enhance the detection of nodes with potential for feedback, redundancy, and robust connectivity. However, CN lacks normalization and CR may neglect broader structural participation. BCR advances these by explicitly considering both the strength and range of a node’s co-cyclic engagements.

Empirical studies reveal that:

Centrality	Local Info	Global Info	Cycle Awareness	Spreading Performance	Cost-Efficiency
Degree	High	Low	None	Limited	Moderate
Betweenness	Low	High	None	Competitive	Variable
CN	High	Variable	Basic Cycles	Moderate	Moderate
CR	Variable	Variable	Shortest Cycles	Good	Moderate
BCR	High	High	Basic Cycles	Superior	High

BCR achieves superior spreading efficiency, assigns nearly unique scores (high individuation ability), and offers high cost-effectiveness under typical resource constraints (Zheng et al., 30 Sep 2025).

4. Practical Applications and Empirical Validation

BCR is constructed to optimize the selection of spreading agents in the context of viral marketing, epidemic control, and information dissemination in complex networks. Applications detailed in (Zheng et al., 30 Sep 2025) and (Fan et al., 2020) include:

• Influential Spreaders: BCR identifies nodes that accelerate contagion under the SIR model, often resulting in wider final reach and greater early-stage infection than nodes ranked by degree or betweenness.
• Cost-Effective Interventions: BCR enables selection of top spreaders with lower degree or resource requirements, maximizing return in scenarios where activation cost matters.
• Distributed Coverage: Nodes with high BCR tend to be spatially dispersed, ensuring broader network influence than clusters of degree hubs.
• Network Resilience: BCR’s recognition of cycle-centric connectivity aids in planning robust infrastructure and defending against targeted attacks (Fan et al., 2019).

Multiple real-world social networks (e.g., Ia-facebook, Soc-epinions, Soc-hamsterster) have been evaluated; BCR consistently outperforms classical and cycle-based centralities in spreading effectiveness and cost reduction.

5. Transformation to Hypernetwork Representations

Cycle-centric investigation enables the reinterpretation of ordinary networks as hypernetworks, wherein each basic cycle becomes a hyperedge. For a hypernetwork $H=(V, E)$:

• The incidence matrix $M(H)$ has $m_{ij}=1$ if node $i$ is part of cycle $e_j$ (hyperedge), else $0$ (Fan et al., 2019).
• The derived cycle number matrix, $B(G) = M(H) M(H)^\top$, encodes co-participation in cycles.
• Normalization produces the cycle ratio matrix, with elements $r_{ij} = b_{ij}/b_{ii}$.

This representation enriches the analysis of mesoscale structures and reveals nontrivial functional dependencies among nodes, facilitating deeper insights into network organization (Fan et al., 2019).

6. Extensions, Robustness, and Future Research Directions

The initial BCR methodology is defined for undirected, unweighted networks. Potential future refinements, as suggested in (Zheng et al., 30 Sep 2025), include:

• Generalization: Application to directed and weighted graphs by adapting cycle definitions and matrix computations.
• Hybrid Approaches: Integration of BCR with star-based or acyclic structural features to extend utility in tree-like networks.
• Robustness: Empirical studies demonstrate that BCR is robust to spanning tree selection randomness, maintaining stable performance across realizations.
• Dynamic Analysis: BCR’s dual local/global sensitivity may inform research into the interplay between local redundancy and global connectivity during time-evolving processes.

A plausible implication is that the adoption of BCR could stimulate further developments in metric design incorporating cycle-based redundancy, which has hitherto been underexplored in classical network science.

7. Summary and Contextual Significance

The Basic Cycle Ratio is a refined cycle-centric node ranking metric, mathematically grounded in the cycle number matrix and its normalization. BCR effectively balances local neighborhood participation and global connective role, demonstrating empirically superior ability to select powerful and cost-efficient spreaders. By revealing novel information poorly captured by classical centrality indices, BCR broadens the analytical toolkit for network scientists and practitioners engaged in resilience engineering, intervention planning, and the paper of complex contagion. Its flexibility and demonstrated performance on real social networks position BCR as a significant advancement in centrality research, with direct implications for practical applications and future theoretical extensions (Zheng et al., 30 Sep 2025, Fan et al., 2019, Fan et al., 2020).