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

In a peer-to-peer sensor or communication network, information spreading can be categorized into single-piece dissemination (when one node shares its information) and multicast dissemination (when all nodes share their information) [1]. But some applications lie between two categories, in which a node needs to collect a multitude of messages or observations generated at the same time to construct meaningful information. If any $k+1$ out of a total of $n$ messages are sufficient to construct the information in a system, it is called a $k$-out-of-$n$ system. Such systems find applications in various domains including robotics, cryptography, and communication systems.

For example, in cryptography, the information source can apply $(k,n)$-Threshold Signature Scheme (TSS) [2] on the information to put it into $n$ distinct keys such that any subset of $n$ keys with $k+1$ cardinality would be sufficient to decode the encrypted message. Furthermore, $k$-out-of-$n$ systems enhance error correction performance in data transmission. For instance, the relative positioning of the target object using the Time Difference of Arrival (TDoA) technique [3] requires at least three simultaneously generated Time of Flight measurements while more measurement increase the accuracy of estimation of relative position of the object. Another common example is $(n,k+1)$-MDS error correction codes [4], in which any $k+1$ codeword suffices for message decoding, with additional redundant codewords that facilitate error correction.

Motivated by these applications, we consider in this work an information source that generates updates and then encrypts (encodes) them by using a $k$-out-of-$n$ systems, e.g. $(k,n)$-TSS. For the sake of simplicity, we consider the source using $(k,n)$-TSS, but it is worth noting that our results are applicable to any $k$-out-of-$n$ system, some of examples are discussed above. The source is able to send the encrypted messages to the $n$ receiver nodes instantaneously. Upon receiving these updates, the nodes start to share their local messages with their neighboring nodes to decrypt the source’s update. Each node determines the required number of keys $k$ to achieve a given precision rate $\alpha$ on the information for a given precision-rate function $D(k,n,\beta)$, which will be introduced later. The nodes that get $k$ different messages of the same update from their neighbors can decode source’s information. We study two different settings where ($i$) the nodes have memory, in which case they can hold the current and also the previous keys received from the source, and ($ii$) the nodes do not have memory, in which case they can only hold the keys from the most current update. For both of these settings, we study the information freshness achieved by the receivers as a result of applying $(k,n)$-TSS. Age of information (AoI) has been introduced as a new performance metric to measure the freshness of information in communication networks [5]. Inspired by recent studies, advancements in Age of Information (AoI) have been made in various gossip network scenarios, including those examining scalability, optimization, and security [6, 7, 8, 9, 10, 11, 12, 13]. In all these aforementioned works, the source sends its information to gossip nodes without using any encryption.

For the first time, in this work, we consider the version age of information in a gossip network where the source encrypts the information. Then, we have the following contributions:

• •

  We derive closed-form expressions for the time average of $k$-keys version age for an arbitrary non-homogeneous network (e.g., independently activated channels).

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  We show that the time average of the $k$-keys version age for a node in both schemes decreases as the edge activation rate increases, or as the number of keys required to decode the information decreases, or as the number of gossip pairs in the network increases in the numerical results,

• •

  We show that a memory scheme yields a lower time average of $k$-keys version age compared to a memoryless scheme. However, the difference between the two schemes diminishes with infrequent source updates, frequent gossip between nodes, or a decrease in $k$ for a fixed number of nodes.

We consider an information updating system consisting of a single source, which is labeled as node $0$, and $n$ receiver nodes. The information at the source is updated at times distributed according to a Poisson counter, denoted by $N_{0}(t)$, with rate $\lambda_{s}$. We refer to the time interval between $\ell$th and $(\ell+1)$th information updates (messages) as the $\ell$th version cycle and denote it by $U^{\ell}$. Each update is stamped by the current value of the process $N_{0}(t)=\ell$ and the time of the $\ell$th update is labelled $\tau_{\ell}$ once it is generated. The stamp $\ell$ is called version-stamp of the information.

We assume that the source is able to instantaneously encrypt the information update by using $(k,n)$-TSS once it is generated. To be more precise, we assume that the source puts the information update into $n$ distinct keys and sends one of the unique keys to each receiver node at the time $\tau_{\ell}$, instantaneously. Once a node gets a unique key from the source at $\tau_{\ell}$ for version $\ell$, it is aware of that there is new information at the source. Each node wishes its knowledge of the source to be as timely as possible. The timeliness is measured for an arbitrary node $j$ by the difference between the latest version of the message at the source node, $N_{0}(t)$, and the latest version of the message which can be decrypted at node $j$, denoted by $N^{k}_{j}(t)$. This metric has been introduced as version age of information in [6, 14]. We call it $k$-keys version age of node $j$ at time $t$ and denote it as

Recall that in the $(k,n)$-TSS, a node needs to have $k+1$ keys with the version stamp $\ell$ in order to decrypt the information at the source generated at $\tau_{\ell}$. Since the source sends a unique key to all receiver nodes, a node needs $k$ additional distinct keys with version $\ell$ to decrypt the $\ell$th message. We denote by $\vec{G}=(V,\vec{E})$ the directed graph with node set $V$ and edge set $\vec{E}$. We let $\vec{G}$ represent the communication network according to which nodes exchange information. If there is a directed edge $e_{ij}\in\vec{E}$, we call node $j$ gossiping neighbor of node $i$. We consider a precision-rate function given by $D(k,n,\beta):\mathbb{N}\times\mathbb{N}\times\mathbb{R}^{+}\to[0,1]$ that quantifies how precisely a node decodes the status update if it collects $k+1$ out of $n$ symbols, where $\beta$ represents a relevant system parameter. For a given precision rate $\alpha$, the required number of keys for node $j$ is defined as $k_{j}(\beta,\alpha)\!:=\!\inf\{k\in\mathbb{N}:D(k,n,\beta)\!\geq\!\alpha\}$. For simplicity, we assume $D(k,n,\beta)\!:=\!\sum_{i=0}^{\lfloor k\rfloor}{n\choose i}\beta^{i}(1\!-\!\beta)^{n-i}$. However, any function that is monotonically increasing in $k\leq n$ for fixed $n$ can serve as a precision-rate function. Here, the rate $\beta$ represents the amount of noise in each keys. We call a communication network $(k,n)$-TSS feasible for rate $(\beta,\alpha)$ if the node $0$ has out-bound connections to all other nodes and the smallest in-degree of the receiver nodes is greater than the smallest $k_{j}(\beta,\alpha)$ among all nodes.¹¹1In this work, all the nodes may have different $k_{j}(\beta,\alpha)$s. Our results are directly applicable for any arbitrary selection of $k_{j}(\beta,\alpha)$s. For that, we may omit the user index $j$ and instead use $k$, directly.

We consider a $(k,n)$-TSS feasible network, in which, nodes are allowed to communicate and share only the keys that are received from the source with their gossiping neighbor. The edge $e_{ij}$ is activated at times distributed according to the Poisson counter $N^{ij}(t)$, which has a rate $\lambda_{ij}$ and once activated, node $i$ sends a message to node $j$, instantaneously. All counters are pairwise independent. This process occurs under two distinct schemes: with memory and memoryless.

In the memory scheme, nodes can store (and send) the keys of the previous updates. For example, if the edge $e_{ij}$ is activated at $t$, node $i$ sends node $j$ all the keys that the source has sent to node $i$ since the last activation of $N^{ij}(t)$ before $t$. For the illustration in Fig. 1, node $i$ sends the set of keys with the versions $\{\ell,\ell+1,\ell+2\}$ to node $j$ in the memory scheme. Note that this can be implemented by finite memory in a finite node network with probability $1$; we will provide below the distribution of the number of keys in the message. In the memoryless scheme, nodes have no memory and only store the latest key obtained from the source. If the edge $e_{ij}$ is activated at $t$, node $i$ sends node $j$ only the last key that the source sent to node $i$ before $t$. Referring again to the illustration in Fig. 1, node $i$ in this case sends only the key with the version $\{\ell+2\}$ to node $j$.

Fig. 2(a) and Fig. 2(b) depict the sample path of the $k$-keys version age process ${A}^{k}(t)$ (resp.  $\bar{A}^{k}(t)$) for a node with memory (resp. without memory). It is worth noting that we associate the notation $\bar{\cdot}$ with the memoryless scheme. It is assumed that edge activations and source updates occur at the same time in both schemes in the figures. In the memory scheme, we define the service time of information with version $\ell$ to an arbitrary node $j$, denoted by $S^{\ell}_{j}$, as the duration between $\tau_{\ell}$ and the time when node $j$ can decrypt the information with version $\ell$, as shown in Fig. 2(a). In the memoryless scheme, a node can miss an information update with version $\ell$ if it cannot get $k$ more distinct keys before the next update arrives at $\tau_{\ell+1}$. Thus, for a node without memory, we define $S^{\ell}_{j}$ as the duration between $\tau_{\ell}$ and the earliest time when the node can decrypt information with a version of at least $\ell$. In Fig. 2(b), the node could only decode the information with version $\ell\!+\!2$ while missing $\ell$ and $\ell\!+\!1$. Therefore, the service times $S^{\ell}$ and $S^{\ell+1}$ end at the same time as the service time $S^{\ell+2}$ ends.

Let $\Delta^{k}_{j}(t)$ be the total $k$-keys version age of node $j$, defined as the integrated $k$-keys version age of node $j$, $A^{k}_{j}(\tau)$, until time $t$. For both schemes, the time average of $k$-keys version age process of node $j$ is defined as follows:

We interchangeably call $\Delta^{k}_{j}$ the version age of $k$-keys for node $j$ on $({\beta},\alpha)$. If nodes in the network have no memory, we denote the version age of $k$-keys for node $j$ by $\bar{\Delta}^{k}_{j}$.

In this section, we first introduce the main concepts that will be useful to derive age expressions and then provide closed-form expressions for the version age of $k$-keys with memory, $\Delta^{k}_{j}$; and without memory, $\bar{\Delta}^{k}_{j}$.

Consider a set of random variables $\mathcal{Y}=\{Y_{i}\}_{i=1}^{n}$. We denote the $k^{th}$ smallest variable in the set $\mathcal{Y}$ by $Y_{(k:n)}$. We call $Y_{(k:n)}$ the $k$th order statistic of $n$-samples ($k=1,2,\cdots,n$) in the set $\mathcal{Y}$. Let $\mathcal{N}^{+}_{j}=\{i\in V,e_{ij}\in\vec{E}\}$ be the set of nodes with out-bound connections to node $j$; denote its cardinality by $n_{j}$. Let $X_{ij}$ be the times between successive activations of the edge $e_{ij}$. Then, $X_{ij}$ is an exponential random variable with mean $1/\lambda_{ij}$. Let $\mathcal{X}_{j}$ be the set of random variables $X_{ij}$, $\forall i\in\mathcal{N}^{+}_{j}$. We denote the $k$th order statistic of the set $\mathcal{X}_{j}$ by $\mathcal{X}_{(k:n_{j})}$.

In this section, we compare empirical results obtained from simulations to our analytical results.

We consider a SHN. Fig. 3 depicts the simulation and the theoretical results for both $\Delta^{k}$ and $\bar{\Delta}^{k}$ as a function of gossip rate $\lambda_{e}$ when $\lambda_{s}=10$. The simulation results for $\Delta^{k}$ and $\bar{\Delta}^{k}$ align closely with the theoretical calculations provided in Theorems 1 and 2, respectively. In both schemes, we observe that the version age of $k$-keys for a node decreases with the rise in the gossip rate $\lambda_{e}$, while keeping the network size $n$ and $\lambda_{s}$ constant. We observe, in Fig. 3(a) and Fig. 3(b), that both $\Delta^{k}$ and $\bar{\Delta}^{k}$ increase with the growth of $k$ for a fixed $\lambda_{e}$ and $n$ and they decreases as $n$ increases for fixed $\lambda_{e}$ and $k$. Also, we observe, in Fig. 3, that $\Delta^{k}$ is less than $\bar{\Delta}^{k}$ for the same values of $\lambda_{e},\lambda_{s}$ and $(k,n)$. Fig. 4 depicts $\Delta^{k}$ as a function of the number of the nodes $n$ for various gossip rates $\lambda_{e}$. We observe, in Fig. 4, that $\Delta^{k}$ converges to $\frac{k\lambda_{s}}{\lambda_{e}}$ as $n$ grows. It aligns with Cor. 2. Fig. 5(b) depicts $\bar{\Delta}^{k}$ and $\Delta^{k}$ as functions of $(\beta,\alpha)$ for the given $D(k,n,\beta)$ in the introduction. We observe that in Fig. 5(b), both $\bar{\Delta}^{k}$ and $\Delta^{k}$ increase as the required precision $\alpha$ increases. Additionally, for the same precision rate $\alpha$, higher noise in the measurement $\beta$ results in greater $\bar{\Delta}^{k}$ and $\Delta^{k}$.

To quantify the value of memory in a network, we define the memory critical gossip rate of a $(k,n)$-TSS network for a margin $\varepsilon$, denoted by $\lambda^{\varepsilon}(k,n)$, as the smallest gossip rate $\lambda_{e}$ such that $|\Delta^{k}-\bar{\Delta}^{k}|\leq\varepsilon$. We first observe, in Fig. 5(a), that $\lambda^{\varepsilon}(k,n)$ xponentially increases as $k$ increases for fixed $n$. Now, consider the event $E\!:=\!\{\mathcal{X}_{(k:n-1)}\leq U\}$, one can easily see that $Pr(E)$ converges to $1$ as $\lambda_{e}$ increases (frequent gossipping between nodes) or $k$ decreases for fixed $n$. In this case, the expectation $\mathbb{E}[\min(\mathcal{X}_{(k:n-1)},U)]\!\to\!\mathbb{E}[\mathcal{X}_{(k:n-1)}]$ in (7). It implies that $\bar{\Delta}^{k}$ approach to $\Delta^{k}$ as $Pr(E)$ goes to $1$. These observations show that $\bar{\Delta}^{k}$ approaches ${\Delta}^{k}$ as $\lambda_{e}$ increases or $k$ decreases.