Efficient and secure decentralized network size estimation
IFIP'12 Proceedings of the 11th international IFIP TC 6 conference on Networking - Volume Part I
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We consider a model for monitoring the connectivity of a network subject to node or edge failures. In particular, we are concerned with detecting $(\epsilon,k)$-failures: events in which an adversary deletes up to $k$ network elements (nodes or edges), after which there are two sets of nodes $A$ and $B$, each at least an $\epsilon$ fraction of the network, that are disconnected from one another. We say that a set $D$ of nodes is an $(\epsilon,k)$-detection set if, for any $(\epsilon,k)$-failure of the network, some two nodes in $D$ are no longer able to communicate; in this way, $D$ “witnesses” any such failure. Recent results show that for any graph $G$, there is an $(\epsilon,k)$-detection set of size bounded by a polynomial in $k$ and $\epsilon$, independent of the size of $G$. In this paper, we expose some relationships between bounds on detection sets and the edge-connectivity $\lambda$ and node-connectivity $\kappa$ of the underlying graph. Specifically, we show that detection set bounds can be made considerably stronger when parameterized by these connectivity values. We show that for an adversary that can delete $k \lambda$ edges, there is always a detection set of size $O(\frac{k}{\epsilon}\log\frac{1}{\epsilon})$ which can be found by random sampling. Moreover, an $(\epsilon,\lambda)$-detection set of minimum size (which is at most $\frac{1}{\epsilon}$) can be computed in polynomial time. A crucial point is that these bounds are independent not just of the size of $G$ but also of the value of $\lambda$. Extending these bounds to node failures is much more challenging. The most technically difficult result of this paper is that a random sample of $O(\frac{1}{\epsilon}\log\frac{1}{\epsilon})$ nodes is a detection set for adversaries that can delete a number of nodes up to $\kappa$, the node-connectivity. For the case of edge-failures we use VC-dimension techniques and the cactus representation of all minimum edge-cuts of a graph; for node failures, we develop a novel approach for working with the much more complex set of all minimum node-cuts of a graph.