A logical approach to concentration

Concentration results say that a sequence of random variables becomes progressively concentrated around the mean. Such results are common in the study of functions of random graphs. We introduce a real-valued logic with various aggregate operators on graphs, including summation, and prove that every term in the language, seen as a random variable on random graphs within the classical Erdős-Rényi random graph model, is concentrated. We prove this for dense and sparse variants of Erdős-Rényi graphs. On the one hand, our results extend the line of work originating with Fagin and Glebskii et al. on zero-one laws for dense random graphs, as well as the zero-one law of Shelah and Spencer for sparse random graphs. On the other hand, they can be seen as a meta-theorem for inferring concentration results on random graphs, and we give examples of such applications.

💡 Research Summary

The paper introduces a real‑valued logical language, denoted Agg F, that is capable of expressing a wide range of graph statistics through the use of aggregate operators such as summation (Σ), maximum (Max), minimum (Min), global average (Avg) and local average (LAvg). The atomic building blocks of the language are constants, the edge‑predicate E(u,v), and equality of vertices. In addition, the language is closed under a family F of real‑valued functions, called connectives, which serve as the analogue of Boolean connectives in classical logic. By repeatedly applying these connectives and the aggregate operators, one can write terms that compute, for a given graph G, quantities such as the number of copies of a fixed subgraph, the maximum number of extensions of a rooted pattern, the proportion of vertices satisfying a first‑order formula, or the total sum of a local property over all vertices.

The authors study the behaviour of closed terms (terms with no free variables) when the underlying graph is drawn from the Erdős–Rényi model G(n,p). Two regimes are considered: (i) dense graphs where p is a constant independent of n, and (ii) sparse graphs where p=n^{‑α} with 0<α<1 and α irrational. The main result is a meta‑theorem: every closed term of Agg F concentrates around its mean in both regimes. Concentration here means that for any fixed power p>0, the ratio of the term to its expectation converges to 1 in L_p, i.e. the term is sharply peaked around its expected value.

To obtain this result the paper defines two important classes of connectives. The class F_relip consists of Lipschitz functions (or more generally functions with bounded “sensitivity” to changes in their arguments). The class F_rlpoly consists of functions that grow at most polynomially in their arguments. The authors prove that any connective from either class preserves concentration: if a vector of random variables is already concentrated, then applying a function from F_relip or F_rlpoly yields another concentrated variable. This is proved by a careful combination of martingale difference bounds, Chebyshev’s inequality, and a “smoothness” argument that controls how the aggregate operators amplify fluctuations.

In the dense case, the paper shows that for any term built from F_rlpoly together with Σ and Max, the expectation can be approximated by a polynomial in n and p, and if the term is bounded it converges in probability to a constant. This subsumes classical results such as concentration of subgraph counts (e.g., triangles) and of maximum extensions (e.g., maximum degree). Moreover, because the language can simulate any first‑order formula via Boolean connectives and Max (existential quantification), the meta‑theorem implies the classical FO zero‑one law for G(n,p) as a corollary.

In the sparse case, the situation is more delicate because the FO zero‑one law fails for rational α. The authors restrict to irrational α and prove that Agg F with Lipschitz connectives and the Max operator still yields concentration and convergence to a constant. This extends the Shelah–Spencer zero‑one law to the richer real‑valued setting and shows that the presence of Max does not destroy convergence when α is irrational.

The paper also discusses related work. Prior studies on convergence of real‑valued logics (e.g.,