For example in this paper, they refer to a "convex combination of trees" (pg. 2, first paragraph), and also, more generally, to "convex combination of graphs" (pg. 2, footnote).
-> "For instance, when K = V , the results of Racke [Rac08] give a convex combination of trees H with a loss of O(log n). We call this a tree-based flow-sparsifier, meaning that it is a convex combination of trees."
-> "More generally, for a class F of graphs, we define an F-flow-sparsifier to be a sparsifier that uses a single graph from F, and an F-based flow-sparsifier to be a sparsifier that uses a convex combination of graphs from F."
Any examples, intuition/connection to the classical ideas of convexity would be much appreciated.
It means a convex combination of characteristic vectors of graphs. A characteristic vector is a $0$-$1$ valued vector, indexed by node pairs, in which a component of $1$ indicates that the corresponding edge is in the graph.