  Subgraph counting is a fundamental primitive in graph processing, with
applications in social network analysis (e.g., estimating the clustering
coefficient of a graph), database processing and other areas. The space
complexity of subgraph counting has been studied extensively in the literature,
but many natural settings are still not well understood. In this paper we
revisit the subgraph (and hypergraph) counting problem in the sketching model,
where the algorithm's state as it processes a stream of updates to the graph is
a linear function of the stream. This model has recently received a lot of
attention in the literature, and has become a standard model for solving
dynamic graph streaming problems.
  In this paper we give a tight bound on the sketching complexity of counting
the number of occurrences of a small subgraph $H$ in a bounded degree graph $G$
presented as a stream of edge updates. Specifically, we show that the space
complexity of the problem is governed by the fractional vertex cover number of
the graph $H$. Our subgraph counting algorithm implements a natural vertex
sampling approach, with sampling probabilities governed by the vertex cover of
$H$. Our main technical contribution lies in a new set of Fourier analytic
tools that we develop to analyze multiplayer communication protocols in the
simultaneous communication model, allowing us to prove a tight lower bound. We
believe that our techniques are likely to find applications in other settings.
Besides giving tight bounds for all graphs $H$, both our algorithm and lower
bounds extend to the hypergraph setting, albeit with some loss in space
complexity.
