Age of Gossip in Ring Networks in the Presence of Jamming Attacks
Priyanka Kaswan, Şennur Ulukuş
Abstract
We consider a system with a source which maintains the most current version of a file, and a ring network of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> user nodes that wish 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 this network in the presence of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\tilde{n}$</tex> jammers that disrupt inter-node communications. To this purpose, 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 that when the number of jammers scales as a fractional power of the network size, i.e., <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\tilde{n}=cn^{\alpha}$</tex> , the version age scales as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\sqrt{n}$</tex> when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha < \frac{1}{2}$</tex> , and as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n^{\alpha}$</tex> when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha\geq\frac{1}{2}$</tex> . As the version age of a ring network without any jammers scales as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\sqrt{n}$</tex> , our result implies that the version age with gossiping is robust against up to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\sqrt{n}$</tex> jammers in a ring network.