Continuous Connectivity Ratio (CCR)

• CCR is a unified reliability metric designed for RIS-assisted, ISAC-enabled V2X systems, capturing continuous V2I connectivity and latency-bound V2V delivery.
• It employs a sliding-window approach for V2I links and a probabilistic success condition for V2V transmissions to ensure consistent network performance.
• Leveraged in frameworks like VariSAC, CCR optimizes resource allocation using advanced techniques such as graph neural networks and Soft Actor-Critic.

Searching arXiv for the cited paper and related CCR usages to ground the article in the current literature. Continuous Connectivity Ratio (CCR) is a unified reliability metric introduced for RIS-assisted, ISAC-enabled vehicle-to-everything systems to characterize two distinct reliability requirements within a single formalism: the sustained temporal reliability of vehicle-to-infrastructure (V2I) links and the probabilistic, latency-bound reliability of vehicle-to-vehicle (V2V) links. In the VariSAC framework, CCR is both an evaluation metric and an optimization target, designed to support assured, time-continuous connectivity under highly dynamic and heterogeneous vehicular network topologies (Tang et al., 8 Sep 2025).

1. Conceptual definition and problem setting

CCR was devised to address the coexistence of distinct V2I and V2V connectivity requirements in vehicular networks. The underlying setting combines Reconfigurable Intelligent Surfaces (RIS) and Integrated Sensing and Communication (ISAC), where dynamic spatial resource management and real-time adaptation to environmental changes are central, but unified reliability modeling is difficult because V2I and V2V links obey different service semantics (Tang et al., 8 Sep 2025).

The metric is defined as a unified reliability measure that characterizes both the temporal reliability of V2I links and the probabilistic delivery guarantees of V2V links. For V2I, the emphasis is on the proportion of vehicles that maintain continuous connectivity over $N$ consecutive time slots while satisfying communication and, for sensing vehicles, sensing thresholds. For V2V, the emphasis is on the success probability of completing a payload transmission within a strict latency deadline. In this form, CCR jointly captures time-extended and task-specific connectivity requirements across heterogeneous V2X services (Tang et al., 8 Sep 2025).

A central feature of the definition is that it does not treat reliability as an instantaneous snapshot. The V2I component is explicitly windowed over time, while the V2V component is expressed as a delivery success probability under a deadline. This suggests a metric aimed at persistent service continuity rather than isolated per-slot feasibility.

2. Mathematical structure

CCR is presented through complementary V2I and V2V components, together with a total or unified form (Tang et al., 8 Sep 2025).

For a non-sensing vehicle $v$, the sliding-window connectivity indicator is

$$\Psi_{v}^{t} = \begin{cases} 1, & \text{if } \sum_{n=t-N+1}^{t} \mathbb{I}\{R_{v}^{n} \geq R_{\text{th}}\} = N \ 0, & \text{otherwise.} \end{cases}$$

For a sensing vehicle $j$, the indicator additionally enforces an SNR constraint:

$$\Psi_{j}^{t} = \begin{cases} 1, & \text{if } \sum_{n=t-N+1}^{t} \mathbb{I}\{R_{j}^{n} \geq R_{\text{th}},\, \mathrm{SNR}_{j}^{n} \geq \mathrm{SNR}_{\text{th}}\} = N \ 0, & \text{otherwise.} \end{cases}$$

The aggregate V2I component is

$$\mathrm{CCR}_{\mathrm{V2I}} = \frac{1}{V(T-N+1)} \sum_{v=1}^{V} \sum_{t=N}^{T} \Psi_{v}^{t}.$$

For V2V transmission, CCR is defined as a success probability:

$$\mathrm{CCR}_{\mathrm{V2V}} = \operatorname{Pr}\left\{\sum_{t=1}^{T} c_{v,d}^{t} \cdot R_{d}^{t} \cdot \Delta t \geq K \right\},$$

where $c_{v,d}^{t} \in \{0,1\}$ indicates channel selection, $R_d^t$ is the achievable rate, $\Delta t$ is the slot duration, and $v$0 is the required payload in bits.

The total form is written as

$v$1

Component	Expression	Semantics
V2I	$v$2, $v$3, $v$4	Continuous connectivity over a sliding window
V2V	$v$5	Probabilistic payload delivery within latency
Unified	$v$6	Joint optimization target

The paper also gives a joint optimization form:

$v$7

with constraints enforcing per-slot rate and SNR thresholds together with resource feasibility (Tang et al., 8 Sep 2025).

3. Unified reliability semantics

The principal significance of CCR lies in the way it unifies heterogeneous reliability semantics. For V2I links, the $v$8-slot indicator captures temporal continuity: a vehicle is counted as connected only if threshold-satisfying performance persists across the entire sliding window. For sensing vehicles, this continuity is conditioned jointly on rate and SNR thresholds, so communication and sensing reliability are coupled in the same indicator (Tang et al., 8 Sep 2025).

For V2V links, the semantics are different. The relevant event is not uninterrupted service over a window, but successful completion of a payload transmission within a strict latency budget. CCR therefore uses a probabilistic success condition based on accumulated delivered bits over time. This matches the sporadic but critical one-shot reliability requirement associated in the paper with safety messages (Tang et al., 8 Sep 2025).

The unification is therefore not a reduction of V2I and V2V to the same physical event. Rather, both are cast into a single optimization framework under shared resource constraints. The sliding-window formulation gives V2I a temporally persistent notion of reliability, while the V2V component preserves the stochastic delivery semantics typical of latency-sensitive links. A common misconception would be to read CCR as merely a per-slot coverage indicator; the formalism instead encodes persistence for V2I and deadline-constrained success probability for V2V.

4. Measurement and modeling procedure

CCR is operationalized through distinct measurement procedures for the two link classes. For V2I, the procedure maintains a window of the latest $v$9 slots for each vehicle; if every slot in that window satisfies the required threshold conditions, the corresponding indicator is set to $$\Psi_{v}^{t} = \begin{cases} 1, & \text{if } \sum_{n=t-N+1}^{t} \mathbb{I}\{R_{v}^{n} \geq R_{\text{th}}\} = N \ 0, & \text{otherwise.} \end{cases}$$0, otherwise to $$\Psi_{v}^{t} = \begin{cases} 1, & \text{if } \sum_{n=t-N+1}^{t} \mathbb{I}\{R_{v}^{n} \geq R_{\text{th}}\} = N \ 0, & \text{otherwise.} \end{cases}$$1. The aggregate V2I CCR is then formed by counting the proportion over vehicles and time (Tang et al., 8 Sep 2025).

For V2V, the procedure sums the actual delivered bits over the time horizon $$\Psi_{v}^{t} = \begin{cases} 1, & \text{if } \sum_{n=t-N+1}^{t} \mathbb{I}\{R_{v}^{n} \geq R_{\text{th}}\} = N \ 0, & \text{otherwise.} \end{cases}$$2 and evaluates whether the full load is delivered within the deadline. The success probability can then be interpreted as the fraction of samples for which the payload threshold is crossed, yielding an empirical estimate of delivery reliability (Tang et al., 8 Sep 2025).

This modeling choice makes CCR explicitly time-continuous in the sense used by the paper. The V2I term depends on sustained threshold satisfaction over a sliding window rather than on isolated slots, while the V2V term depends on deadline-constrained cumulative delivery rather than on instantaneous rate alone. A plausible implication is that CCR is intended to suppress policies that optimize short-term throughput at the expense of continuity.

5. Role in VariSAC and system optimization

In VariSAC, CCR functions as the direct optimization objective. The framework uses a graph neural network with residual adapters to encode complex, high-dimensional system states and to capture spatial dependencies among vehicles, base stations, and RIS nodes. The resulting state representations are then processed by a Soft Actor-Critic agent, which jointly optimizes channel allocation, power control, and RIS configurations in order to maximize CCR-driven long-term rewards (Tang et al., 8 Sep 2025).

The reward design directly incorporates the CCR indicators:

$$\Psi_{v}^{t} = \begin{cases} 1, & \text{if } \sum_{n=t-N+1}^{t} \mathbb{I}\{R_{v}^{n} \geq R_{\text{th}}\} = N \ 0, & \text{otherwise.} \end{cases}$$3

In the description accompanying this reward, actions are rewarded when sliding-window connectivity is maintained for V2I and penalized for undelivered V2V payloads (Tang et al., 8 Sep 2025).

Within this formulation, CCR is not simply a reporting statistic after training. It is the quantity that aligns the reinforcement learning objective with continuous V2I ISAC connectivity and high-probability V2V delivery within deadline. Extensive experiments on real-world urban datasets are reported to show that VariSAC consistently outperforms existing baselines in terms of continuous V2I ISAC connectivity and V2V delivery reliability, enabling persistent connectivity in highly dynamic vehicular environments (Tang et al., 8 Sep 2025).

6. Terminological scope and disambiguation

The acronym “CCR” is used in several unrelated arXiv literatures, and the Continuous Connectivity Ratio should therefore be distinguished from homonymous constructs in other fields.

In operator algebra and $$\Psi_{v}^{t} = \begin{cases} 1, & \text{if } \sum_{n=t-N+1}^{t} \mathbb{I}\{R_{v}^{n} \geq R_{\text{th}}\} = N \ 0, & \text{otherwise.} \end{cases}$$4-semigroup theory, “CCR flows” denotes Canonical Commutation Relation flows; examples include "Arveson's characterisation of CCR flows: the multiparameter case" (Sundar, 2019), "An asymmetric multiparameter CCR flow" (Sundar, 2020), and "Examples of Multiparameter CCR flows with non-trivial index" (Sarkar et al., 2021). In set theory, “CCR” denotes the countable choice axiom for sets of reals in "TD implies CCR" (Peng et al., 2020). In continuous-variable quantum information, "Introducing the Correlation Concentration Ratio (CCR): Quantitative Framework for Comparing Quantum Cluster States" defines CCR as a covariance-matrix-based topology metric for cluster states (Ahadi et al., 25 Apr 2026). In mobile ad-hoc network analysis, the normalized connection-time fraction $$\Psi_{v}^{t} = \begin{cases} 1, & \text{if } \sum_{n=t-N+1}^{t} \mathbb{I}\{R_{v}^{n} \geq R_{\text{th}}\} = N \ 0, & \text{otherwise.} \end{cases}$$5 is described as CCR in "Connection times in large ad-hoc mobile networks" (Döring et al., 2013).

These usages are mathematically and conceptually distinct. The Continuous Connectivity Ratio in VariSAC belongs specifically to V2X reliability modeling, where it unifies sustained V2I connectivity and probabilistic V2V delivery guarantees under a single time-continuous objective (Tang et al., 8 Sep 2025).