The theory of graph (and structure) convergence gained recently a substantial attention. Various notions of convergence were proposed, adapted to different contexts, including Lovasz et al. theory of dense graph limits based on the notion of left convergence and Benjamini–Schramm theory of bounded degree graph limits based on the notion of local convergence. The latter approach can be extended into a notion of local convergence for graphs (stronger than left convegence) as follows: A sequence of graphs is local convergent if, for every local first-order formula, the probability that the formula is satisfied for a random (uniform independent) assignment of the free variables converge as n grows to infinity. In this talk, we show that the local convergence of a sequence of graphs allows to decompose the graphs in the sequence in a coherent way, into concentration clusters (intuitively corresponding to the limit non-zero measure connected components), a residual cluster, and a negligible set. Also, we mention that if we consider a stronger notion of local-global convergence extending Bollobas and Riordan notion of local-global convergence for graphs with bounded degree, we can further refine our decomposition by exhibiting the expander-like parts.

graphs - structural limit - graph limit - asymptotic connectivity[-]