Connectedness in Graph Limits

We define direct sums and a corresponding notion of connectedness for graph limits. Every graph limit has a unique decomposition as a direct sum of connected components. As is well-known, graph limits may be represented by symmetric functions on a probability space; there are natural definitions of direct sums and connectedness for such functions, and there is a perfect correspondence with the corresponding properties of the graph limit. Similarly, every graph limit determines an infinite random graph, which is a.s. connected if and only if the graph limit is connected. There are also characterizations in terms of the asymptotic size of the largest component in the corresponding finite random graphs, and of minimal cuts in sequences of graphs converging