How Robust are Timely Gossip Networks to Jamming Attacks?

We consider a semantics-aware communication system, where timeliness is the semantic measure, with a source which maintains the most current version of a file, and a network of $n$ user nodes with the goal to acquire the latest version of the file. The source gets updated with newer file versions as a point process, and forwards them to the user nodes, which further forward them to their neighbors using a memoryless gossip protocol. We study the average version age of the network in the presence of $\tilde{n}$ jammers that disrupt inter-node communications, for the connectivity-constrained ring topology and the connectivity-rich fully connected topology. For the ring topology, we construct an alternate system model of mini-rings and prove that the version age of the original model can be sandwiched between constant multiples of the version age of the alternate model. We show in a ring network that when the number of jammers scales as a fractional power of the network size, i.e., $\tilde n= cn^\alpha$, the version age scales as $\sqrt{n}$ when $\alpha < \frac{1}{2}$, and as $n^{\alpha}$ when $\alpha \geq \frac{1}{2}$. As version age of a ring network without any jammers scales as $\sqrt{n}$, our result implies that version age with gossiping is robust against upto $\sqrt{n}$ jammers in a ring network. We then study the connectivity-rich fully connected topology, where we derive a greedy approach to place $\tilde{n}$ jammers to maximize age of the resultant network, which uses jammers to isolate as many nodes as possible, thereby consolidating all links into a single mini-fully connected network. We show in this network that version age scales as $\log{n}$ when $\tilde{n}=cn\log{n}$ and as $n^{\alpha-1}$, $1<\alpha\leq2$ when $\tilde{n}=cn^{\alpha}$, implying the network is robust against $n\log{n}$ jammers, since the age in a fully connected network without jammers scales as $\log{n}$.