[Paper Review] Random Graphons and a Weak Positivstellensatz for Graphs

László Lovász, Balázs Szegedy|ArXiv.org|Feb 8, 2009

Advanced Graph Theory Research1 references4 citations

TL;DR

This paper establishes a weak Positivstellensatz for graph limits by showing that any linear inequality between homomorphism densities holding asymptotically for all graphs can be approximated arbitrarily closely by sums of squares in the algebra of partially labeled graphs. It introduces relaxed versions of graph limit structures—such as ergodic measures on countable graphs and non-multiplicative, isolate-indifferent graph parameters—proving their equivalence and enabling a formal proof system for asymptotic inequalities using Cauchy-Schwarz-like identities.

ABSTRACT

In an earlier paper the authors proved that limits of convergent graph sequences can be described by various structures, including certain 2-variable real functions called graphons, random graph models satisfying certain consistency conditions, and normalized, multiplicative and reflection positive graph parameters. In this paper we show that each of these structures has a related, relaxed version, which are also equivalent. Using this, we describe a further structure equivalent to graph limits, namely probability measures on countable graphs that are ergodic with respect to the group of permutations of the nodes. As an application, we prove an analogue of the Positivstellensatz for graphs: We show that every linear inequality between subgraph densities that holds asymptotically for all graphs has a formal proof in the following sense: it can be approximated arbitrarily well by another valid inequality that is a "sum of squares" in the algebra of partially labeled graphs.

Motivation & Objective

• To extend the theory of graph limits by introducing relaxed structural equivalents to graphons, random graph models, and graph parameters.
• To establish the equivalence of these relaxed structures, including ergodic probability measures on countable graphs under permutation invariance.
• To provide a formal proof system for asymptotic linear inequalities between subgraph densities using sums of squares in the algebra of partially labeled graphs.
• To confirm that extremal graph theory inequalities—like those in Mantel-Turán—can be derived via Cauchy-Schwarz-type identities in the limit.

Proposed method

• Introduces a relaxed version of graph parameters by replacing multiplicativity with invariance under deletion of isolated nodes.
• Defines a new class of random graph models without the locality condition, allowing for distributions over graphons.
• Uses the algebra of partially labeled graphs (quantum graphs) to formalize inequalities and define sums of squares.
• Applies duality and limit arguments in matrix spaces to show that non-negative graph parameters are limits of sums of squares.
• Establishes equivalence between ergodic measures on countable graphs and relaxed graph limit objects via the cut distance topology.
• Employs the cut norm and cut distance to define convergence and compactness in the space of graphons and their relaxations.

Experimental results

Research questions

• RQ1 Can the multiplicativity condition in graph parameters be relaxed while preserving equivalence to graph limit structures?
• RQ2 Is there a natural probabilistic structure—specifically, an ergodic measure on countable graphs—that characterizes graph limits?
• RQ3 Can every asymptotic linear inequality between subgraph densities be formally derived from sums of squares in the graph algebra?
• RQ4 To what extent do Cauchy-Schwarz-type identities capture all valid asymptotic inequalities in extremal graph theory?
• RQ5 How do relaxed graph limit objects (e.g., distributions over graphons) relate to classical graphons and random graph models?

Key findings

• Every linear inequality between homomorphism densities that holds asymptotically for all graphs can be approximated arbitrarily well by a sum of squares in the algebra of partially labeled graphs.
• The space of sums of squares is dense in the cone of valid linear inequalities between homomorphism densities, establishing a weak Positivstellensatz for graphs.
• Ergodic probability measures on countable graphs are equivalent to relaxed graph limit structures, including distributions over graphons and isolate-indifferent graph parameters.
• The equivalence between relaxed graph limit objects (e.g., non-local random graph models, non-multiplicative parameters) is established via duality and limit arguments in matrix spaces.
• The dual cone of non-negative, isolate-indifferent, flatly reflection-positive parameters is generated by sums of squares, confirming the formal proof system.
• The cut norm topology ensures compactness of the space of graphons, enabling convergence and limit analysis essential for the main results.