#### Online Algorithms for Network Robustness under Connectivity Constraints

##### Deepan Muthirayan, Pramod P. Khargonekar

In this paper, we present algorithms for designing networks that are robust to node failures with minimal or limited number of links. We present algorithms for both the static network setting and the dynamic network setting; setting where new nodes can arrive in the future. For the static setting, we present algorithms for constructing the optimal network in terms of the number of links used for a given node size and the number of nodes that can fail. We then consider the dynamic setting where it is disruptive to remove any of the older links. For this setting, we present online algorithms for two cases: (i) when the number of nodes that can fail remains constant and (ii) when only the proportion of the nodes that can fail remains constant. We show that the proposed algorithm for the first case saves nearly $3/4$th of the total possible links at any point of time. We then present algorithms for various levels of the fraction of the nodes that can fail and characterize their link usage. We show that when $1/2$ the number of nodes can fail at any point of time, the proposed algorithm saves nearly $1/2$ of the total possible links at any point of time. We show that when the number of nodes that can fail is limited to the fraction $1/(2m)$ ($m \in \mathbb{N}$), the proposed algorithm saves nearly as much as $(1-1/2m)$ of the total possible links at any point of time. We also show that when the number of nodes that can fail at any point of time is $1/2$ of the number of nodes plus $n$, $n \in \mathbb{N}$, the number of links saved by the proposed algorithm reduces only linearly in $n$. We conjecture that the saving ratio achieved by the algorithms we present is optimal for the dynamic setting.

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