PrimeTime Protocol for Finite-Time Consensus

• PrimeTime Protocol is a finite-time consensus algorithm that uses combinatorial prime encoding to uniquely represent each agent's data and its origin.
• The protocol enables agents in dynamic, open networks to update their data using maximal exponent extraction, reducing the communication overhead compared to traditional tabular methods.
• Leveraging unique prime factorization ensures reliable consensus, though its efficiency is best for small networks with low-range integer data due to rapid integer growth.

PrimeTime is a finite-time consensus protocol designed for open distributed networks, specifically in contexts where agents must reach agreement on the complete set of individual data alongside explicit knowledge of data provenance (origin IDs). Unlike average consensus algorithms, PrimeTime abandons the assumption of static membership and tabular data transmission, introducing a combinatorial encoding scheme leveraging the arithmetic properties of prime numbers. For sufficiently small networks or limited integer data, PrimeTime enables agents to communicate all relevant data and origins more efficiently—in bits—than the standard approach of tabularizing ID-data pairs.

1. Motivation and Problem Statement

Distributed consensus often requires both access to the full set of agent data and explicit traceability of data origins (IDs). Applications such as intersection management for autonomous vehicles necessitate not only what data are present but also precise knowledge of which agent supplied each datum. In traditional schemes, communicating large tables that map IDs to data leads to rapid message size growth with network scale. Alternatively, sending only partial tables can result in slow convergence. Existing consensus protocols frequently lack robustness to dynamic agent membership (joining and leaving).

PrimeTime addresses these limitations by allowing the transmission of all agent-originated data in a single integer encoding that implicitly tags each data value with its origin. The protocol is robust to open network membership and achieves finite-time consensus.

2. Protocol Design and Mechanism

Encoding Scheme

PrimeTime assigns each agent a unique prime number $p_i$ as an identifier and encodes its local data $d_i$ as the exponent of that prime. At any round, the collective state over $N$ agents is encoded in a message:

$M = \prod_{i=1}^N p_i^{d_i}$

where $d_i$ are small nonnegative integer values (e.g., representing state, measurement, or signal). The unique factorization theorem guarantees that $M$ unambiguously encodes every agent's ID-data pair.

Transmission and Decoding

Agents transmit the encoded value $M$ to their neighbors. Upon receipt, each agent performs prime factorization of $M$, extracting each $d_i$ from the exponent of the corresponding $p_i$. This process obviates the need for tabular mapping and explicit transmission of IDs.

Iterative Update

Consensus iteration proceeds as agents aggregate received messages, updating their local $d_i$ to the maximal exponent observed for each $p_i$. After each round, agents recompute $M$ for the next transmission. Termination occurs when all messages reflect the maximal global dataset, achieving consensus in finite time.

3. Mathematical Properties and Consensus Dynamics

The protocol relies fundamentally on unique prime factorization in the ring of integers. Given the set of possible primes (the agents' public identifiers), factorization is computationally efficient and trivial for small data values.

When agents receive multiple messages, aggregation is performed by selecting, for each prime, the highest exponent observed—corresponding to the maximal piece of data available for that agent across all received messages. Unlike average consensus or sum aggregation, this maximum operation ensures that the global dataset propagates through the network monotonically until consensus.

$$\forall i \in [1,N]:\quad d_i^{(\text{next})} = \max_{j \in \mathcal{N}}(d_i^{(j)})$$

where $\mathcal{N}$ indexes neighbors at the current iteration.

4. Communication Efficiency and Comparison with Tabular Methods

PrimeTime enables agents to transmit the complete data-origin set via a single integer, dramatically reducing communication overhead for small networks and low-range data. The bit cost for message transmission is approximately:

$$\log_2(M) = \log_2 \left( \prod_{i=1}^N p_i^{d_i} \right ) = \sum_{i=1}^N d_i \cdot \log_2(p_i)$$

For the worst-case scenario with all data at their maximum $D$, the bit count is at most $D \cdot \sum_{i=1}^N \log_2(p_i)$.

Comparatively, explicit tables require

$N \cdot (\text{ID bits} + \text{data bits})$

which grows linearly with network size and data width. For small $N$ and modest $D$, PrimeTime offers significant reductions in message size.

Method	Bits Required	Robust to Membership Change?
Tabular	$N \cdot (\text{ID bits} + \text{data bits})$	No
PrimeTime	$\log_2(\prod p_i^{d_i})$	Yes

For large-scale networks or high-range data, the integer $M$ grows super-polynomially, suggesting a limitation in scalability and practical deployment.

5. Robustness to Network Dynamics

PrimeTime is tailored for "open networks"—where agents may join or leave at arbitrary times.

• Agent Joining: Newly added agents are assigned a new unique prime $p_{N+1}$, and $d_{N+1}$ is included in subsequent encodings.
• Agent Leaving: Data for missing agents are represented by setting their exponents to zero; such primes simply do not appear in the factorization.
• Consensus remains robust, as agreement on the set of agents and their prime IDs is externally maintained.

As a result, PrimeTime seamlessly adapts to changing network memberships without requiring protocol restarts, reinitialization, or complex bookkeeping.

6. Applicability and Limitations

PrimeTime is most effective under the following constraints:

• Small Networks: Because the size of the encoded integer grows rapidly with $N$ and data range $D$, PrimeTime is suited to networks with modest agent counts.
• Small Integer Data: Exponent values must remain reasonably constrained to avoid computational and transmission inefficiencies.
• Immediate Data Provenance: Scenarios demanding knowledge of both values and origins, rather than only aggregated states.

The protocol is not recommended for large-scale deployments without further augmentation. The paper notes the conceptual "Incremental PrimeTime" algorithm to mitigate message size expansion but does not provide operational details. Extension to directed communications and dynamic topologies remains an open research direction.

Limitations include:

• Doublings in network size or data range yield multiplicative (not additive) increases in encoded integer length.
• Agents must maintain correct prime-to-agent assignments for meaningful decoding.
• While factorization is simple for known small primes, cryptographic-sized primes and data render the protocol impractical.

7. Context and Related Work

PrimeTime diverges from consensus protocols that focus on average consensus, sum propagation, or vector tagging, by using combinatorial encoding to minimize communication. It is especially relevant to situations where each agent must maintain full knowledge of both data and its origins. Classical methods sacrifice either communication efficiency or speed of consensus in open network topologies.

For comparative algorithms and technical background, see works such as [Wang & Xiao, 2010], [Sundaram & Hadjicostis, 2007], and [Sandryhaila et al., 2014], as referenced in the originating publication (Abrahamson et al., 2023). These works explore consensus under static membership, directed topology constraints, and varying data fusion objectives.

PrimeTime’s design demonstrates that combinatorial coding via prime numbers can be leveraged for efficient finite-time consensus in networks with dynamic membership, with the caveat that scalability is bounded by combinatorial growth of message integers.