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We study the fundamental problem of counting the number of nodes in a sparse network (of unknown size) under the presence of a large number of Byzantine nodes. We assume the full information model where the Byzantine nodes have complete knowledge about the entire state of the network at every round (including random choices made by all the nodes), have unbounded computational power, and can deviate arbitrarily from the protocol. Essentially all known algorithms for fundamental Byzantine problems (e.g., agreement, leader election, sampling) studied in the literature assume the knowledge (or at least an estimate) of the size of the network. It is nontrivial to design algorithms for Byzantine problems that work without knowledge of the network size, especially in boundeddegree (expander) networks where the local views of all nodes are (essentially) the same and limited, and Byzantine nodes can quite easily fake the presence/absence of non-existing nodes. To design truly local algorithms that do not rely on any global knowledge (including network size), estimating the size of the network under Byzantine nodes is an important first step. Our main contribution is a randomized distributed algorithm that estimates the size of a network under the presence of a large number of Byzantine nodes. In particular, our algorithm estimates the size of a sparse, &#x201C;small-world&#x201D;, expander network with up to O(n<sup>1-&#x03B4;</sup>) Byzantine nodes, where n is the (unknown) network size and &#x03B4; &gt; 0 can be be any arbitrarily small (but fixed) constant. Our algorithm outputs a (fixed) constant factor estimate of log(n) with high probability; the correct estimate of the network size will be known to a large fraction ((1 - &#x03F5;)fraction, for any fixed positive constant &#x03F5;) of the honest nodes. Our algorithm is fully distributed, lightweight, and simple to implement, runs in O(log<sup>3</sup> n) rounds, and requires nodes to send and receive messages of only small-sized messages per round; any node's local computation cost per round is also small.

Network Size Estimation in Small-World Networks Under Byzantine Faults