Monotone graph limits and quasimonotone graphs
The recent theory of graph limits gives a powerful framework for understanding the properties of suitable (convergent) sequences (Gn) of graphs in terms of a limiting object that may be represented by a symmetric function W on [0, 1]2, i.e., a kernel or graphon. In this context it is natural to wish to relate specific properties of the sequence to specific properties of the kernel. Here we show that the kernel is monotone (i.e., increasing in both variables) if and only if the sequence satisfies a quasimonotonicity property defined by a certain functional tending to zero. As a tool we prove an inequality relating the cut and L1 norms of kernels of the form W1 −W2 with W1 and W2 monotone that may be of interest in its own right; no such inequality holds for general kernels.
Bollobás, B., Janson, S., & Riordan, O. (2012). Monotone graph limits and quasimonotone graphs. Internet Mathematics, 8 (3), 187-231. https://doi.org/10.1080/15427951.2012.687243