We continue the study of the communication complexity of gap cycle counting problems. These problems have been introduced by Verbin and Yu [SODA 2011] and have found numerous applications in proving streaming lower bounds. In the noisy gap cycle counting problem (NGC), there is a small integer $k \geq 1$ and an $n$-vertex graph consisted of vertex-disjoint union of either $k$-cycles or $2k$-cycles, plus $O(n/k)$ disjoint paths of length $k-1$ in both cases (``noise''). The edges of this graph are partitioned between Alice and Bob whose goal is to decide which case the graph belongs to with minimal communication from Alice to Bob. We study the robust communication complexity -- `a la Chakrabarti, Cormode, and McGregor [STOC 2008] -- of NGC, namely, when edges are partitioned randomly between the players. This is in contrast to all prior work on gap cycle counting problems in adversarial partitions. While NGC can be solved trivially with zero communication when $k < \log{n}$, we prove that when $k$ is a constant factor larger than $\log{n}$, the robust (one-way) communication complexity of NGC is $\Omega(n)$ bits. As a corollary of this result, we can prove several new graph streaming lower bounds for random order streams. In particular, we show that any streaming algorithm that for every $\varepsilon > 0$ estimates the number of connected components of a graph presented in a random order stream to within an $\varepsilon \cdot n$ additive factor requires $2^{\Omega(1/\varepsilon)}$ space, settling a conjecture of Peng and Sohler [SODA 2018]. We further discuss new implications of our lower bounds to other problems such as estimating size of maximum matchings and independent sets on planar graphs, random walks, as well as to stochastic streams.

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