Age of $k$-out-of-$n$ Systems on a Gossip Network

TL;DR

This work addresses timely information dissemination in gossip networks where updates are encrypted via a $(k,n)$-Threshold Signature Scheme and decoding requires at least $k+1$ matching-version keys. It develops two node operation modes—memory and memoryless—and derives closed-form, time-average expressions for the $k$-keys version age in heterogeneous networks using a precision-rate function $D(k,n,\beta)$ and target rate $\alpha$. Key results include exact formulas for $\Delta^{k}_j$ under memory, $\bar{\Delta}^{k}_j$ under memoryless, SHN special cases, and insights into how edge activation rates, network size, and $k$ affect information freshness. The findings quantify the benefit of memory in encrypted gossip systems and provide design guidance for maintaining fresh updates in distributed cryptographic networks.

Abstract

We consider information update systems on a gossip network, which consists of a single source and $n$ receiver nodes. The source encrypts the information into $n$ distinct keys with version stamps, sending a unique key to each node. For decoding the information in a $k$-out-of-$n$ system, each receiver node requires at least $k+1$ different keys with the same version, shared over peer-to-peer connections. Each node determines $k$ based on a given function, ensuring that as $k$ increases, the precision of the decoded information also increases. We consider two different schemes: a memory scheme (in which the nodes keep the source's current and previous encrypted messages) and a memoryless scheme (in which the nodes are allowed to only keep the source's current message). We measure the ''timeliness'' of information updates by using the $k$-keys version age of information. Our work focuses on determining closed-form expressions for the time average age of information in a heterogeneous random graph under both with memory and memoryless schemes.

Figures (5)

• Figure 1: Sample timeline of the source update and the edge $e_{ij}$ activation. The last activation of $e_{ij}$ is marked by ($\bullet)$ and the previous activations of $e_{ij}$ are marked by ($\circ$).
• Figure 2: Sample path of the $k$-keys version age (a) $A^k(t)$ for a node with memory and (b) $\bar{A}^k(t)$ for a node without memory.
• Figure 3: (a) $\Delta^k$ and (b) $\bar{\Delta}^k$ as a function of $\lambda_e$ on a SHN when $\lambda_s=10$. Solid lines in Fig. \ref{['fig:theoric_experimental']}(a) and Fig. \ref{['fig:theoric_experimental']}(b) show theoretical $\Delta^k$ and theoretical $\bar{\Delta}^k$, respectively. Simulation results for $(2,6)$,$(2,8)$,$(4,6)$,$(4,8)$ TSS are marked by $\blacklozenge,\bullet,\blacksquare,\times$, respectively.
• Figure 4: $\Delta^k$ as a function of $n$ when $k=10$ and $\lambda_s=15$. Solid lines show the theoretical $\Delta^k$ while simulation results for $\lambda_e=\{50,100,150\}$ selections are marked by $\blacklozenge,\bullet,\times$, respectively. Dashed lines show the theoretical asymptotic value of $\Delta^k$ on $n$.
• Figure 5: (a) The memory critical gossip rate $\lambda^\varepsilon(k,30)$ as a function of $k$ for $\varepsilon \in \{10^{-2},10^{-1},1\}$ and (b) $\bar{\Delta}^k$,${\Delta}^k$ and $k(\beta,\alpha)$ as a function of $\alpha \in[0,1]$ for $\beta\in \{0.2,0.5,0.8\}$ when $\lambda_s=15$ and $n=30$.

Theorems & Definitions (7)

• Theorem 1
• Lemma 1
• Corollary 1
• Corollary 2
• Theorem 2
• Lemma 2
• Corollary 3