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 decryption in a $(k, n)$-Threshold Signature Scheme, each receiver node requires at least $k+1$ different keys with the same version, shared over peer-to-peer connections. 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 version age of information. Our work focuses on determining closed-form expressions for the time average age of information in a heterogeneous random graph. Our work not only allows to verify the expected outcome that a memory scheme results in a lower average age compared to a memoryless scheme, but also provides the quantitative difference between the two. In our numerical results, we quantify the value of memory and demonstrate that the advantages of memory diminish with infrequent source updates, frequent gossipping between nodes, or a decrease in $k$ for a fixed number of nodes.

翻译：我们考虑基于八卦网络的信息更新系统，该系统包含单个信源和$n$个接收节点。信源将信息加密为带有版本戳的$n$个不同密钥，并向每个节点发送唯一密钥。在$(k,n)$-门限签名方案中，每个接收节点需要通过点对点连接获取至少$k+1$个具有相同版本的不同密钥才能完成解密。我们研究了两种方案：记忆方案（节点保留信源当前及历史加密消息）和无记忆方案（节点仅允许保留信源当前消息）。通过采用版本信息时效性度量信息更新的"及时性"，重点推导了异质随机图中平均信息时效的时间平均闭合表达式。本研究不仅验证了预期结论（记忆方案相比于无记忆方案具有更低的平均时效），还量化了两者之间的差异。数值结果表明，记忆价值随着信源更新频率降低、节点间八卦交互稀疏化或固定节点数量下$k$值减小而减弱。