Linear embeddings of graphs and graph limits

Many real-life networks can be modelled by stochastic processes with a spatial embedding. In such processes, the link probability decreases with distance. Using the theory of graph limits, we show how to recognize graph sequences produced by random graph processes with a linear embedding (a natural embedding into R). We define an operator Γ which applies to graph limits, which assumes the value zero precisely for graph limits with a linear embedding. Moreover we introduce a corresponding graph parameter Γ∗ and show that, for graph sequences which converge to a graph limit under the cut-norm, the Γ∗-values converge to the Γ-value of the limit.