The RRS conjecture for constrained multi-relational graphons in the non-extremal regime is resolved, proving that entropy-maximizing solutions are step functions with finitely many blocks under the condition the subgraph density constraints are analytically independent and for almost all feasible combinations of sufficient statistics.
Abstract
The principle of maximum entropy provides a fundamental framework for characterizing typical structures of large random networks subject to observable constraints. In their pioneering numerical experiments \cite{radin2014asymptotics}, Radin, Ren, and Sadun conjectured that entropy-maximizing graphons satisfying subgraph density constraints are stochastic block models a conjecture we term the RRS conjecture. While several special cases have been proven for single-relation graphs with specific constraint families, the general problem has remained open, particularly for multi-relational networks. We resolve the RRS conjecture for constrained multi-relational graphons in the non-extremal regime, proving that entropy-maximizing solutions are step functions with finitely many blocks under the condition the subgraph density constraints are analytically independent and for almost all feasible combinations of sufficient statistics. Our proof employs a differential geometric technique to study solutions of constrained optimization problems in function space via functions with a finite parametrization (step functions). The two cornerstones of this work are: the generalization of subgraph density notion to $h$-subgraph density and the proof that manifolds that define the constrained region for the solutions maintain topological stability without developing new connected components under refinement. Together, these enable proving that no new global optima emerge in higher-dimensional spaces.
We develop a unified framework for constructing combinatorial structures under local constraints. Our approach extends the configuration model for random graphs with a prescribed degree sequence, and covers many special cases, including bipartite graphs, directed graphs, oriented graphs, edge-colored (bipartite) graphs, and (directed) hypergraphs. By reformulating half-edge matching as an independent set problem in an auxiliary graph, we identify 2-uniformity, a property characterising when greedy sampling preserves asymptotic uniformity. We classify all 2-uniform graphs and show that only two classes, the configuration space and the bipartite configuration space, have unbounded independence number, enabling the asymptotic regime. Our main theorem then gives the asymptotic sampling distribution and enumeration formulae for configurations, with error terms of order $O(d_{\max}^4\log m/m+d_{\max}^2(\log m)^2/m)$ as the number of edges $m$ tends to infinity with maximum degree $d_{\max}=O(m^{1/4}/\log m)$. This settles the long-standing $O(m^{1/4-\tau})$ bound (for some fixed $\tau>0$), making the critical exponent explicit. Furthermore, our theorem accommodates forbidden edges, provided that each vertex participates in at most $O(m^{1/4}/\log m)$ of them. In particular, this enables the sampling of edge-colored graphs with prescribed degree sequences for each color class by constructing the colored subgraphs one at a time.
I. Kryven, Rik Versendaal, Mike de Vries· 0 citations
Correlation clustering is a fundamental unsupervised learning problem. On complete graphs, both the min-disagreement and min-max objectives admit constant-factor approximations, yet on general (non-complete) graphs, the best guarantees blow up to $O(\log n)$ and $O(\sqrt{n})$. This gap between the two regimes motivates the following question: are there classes of incomplete graphs that circumvent the lower bounds on general graphs and admit approximation guarantees approaching those attainable on complete graphs? We study a natural class of graphs obtained by randomly subsampling a complete signed graph $G$, where each edge is independently deleted with probability $q$. For such graph instances both for the min-max and the min-disagreement objectives, we prove approximation guarantees (depending on $q$) that are substantially better than the bounds achievable for general graphs. We supplement our theoretical results with experiments that also suggest that the approximation ratios of our algorithm are close to those of the complete graph and better than the worst-case bounds for general (non-complete) graphs.
N. RajathRaoK., Jens Schlöter, Sami Davies et al.· 0 citations
This paper investigates the relationship between coding theory and extremal combinatorics by representing codes in general metric spaces as independent sets in proximity graphs. We provide a generalized framework for the Gilbert-Varshamov (GV) bound applicable to codes over any finite metric space and explore the conditions under which global combinatorial parameters can force the existence of codes exceeding this bound. Central to our analysis is the introduction of Ramsey-Sidorenko and independence-forcing graphs. We establish density thresholds for various graph families and utilize the Karush--Kuhn--Tucker conditions to analyze entropy optimization in the Hamming case. Furthermore, we derive upper bounds on code sizes using fractional packings in vertex-transitive and nonedge-transitive graphs. Our findings demonstrate that local subgraph statistics alone are insufficient to surpass the GV bound in the Hamming case, suggesting that improvements must stem from large-scale structural properties of the space.
We consider multipartite random graphs with given degree sequences, within and across different partitions. Under general assumptions, we prove the local limit of this graph is a multi-type branching process, establish that a giant component exists only when the local limit survives, and deduce that the typical distance is of logarithmic order in probability in the supercritical regime. Our analysis removes two major assumptions from Gamarnik and Misra (2015), where the giant component problem for this model was first considered. In particular, we do not assume irreducibility of the local limit, and provide a general framework to extract giant components even when the limiting branching process is reducible, which we hope to be useful in other contexts. We also provide a new simpler survival criterion of multi-type branching processes, which we hope to be useful when direct calculation of the spectral radius of the offspring matrix may prove to be difficult.
The joint asymptotic distribution of any finite collection of network moments in random graphs sampled from a graphon, which includes both the nondegenerate case as well as the degenerate case, provides the higher-order fluctuation theory for subgraph counts in the graphon model.
Anirban Chatterjee, S. Dan, B. Bhattacharya· Annals of Statistics· 0 citations
Comparing graphs for structural similarity is one of the most important problems in graph analytics. However, due to the nonlinear nature of graphs, this problem is not straightforward to solve. Most existing graph comparison methods either lack expressiveness, do not provide interpretable measures of similarity, or incur high computational costs, limiting their applicability to large graphs. In this article, we propose novel graph kernels based on quantum Rényi $\alpha $ -entropies of different orders, computed from both the unnormalized and normalized Laplacian matrices. We investigate the properties of these entropies and show that they are determined by the frequencies and degree statistics of substructures of different types and sizes, such as simple paths and cycles ofdifferentlengths. By utilizing quantum Rényi $\alpha $ -entropies of different orders, our approach defines efficient, theoretically grounded, and interpretable graph kernels capable of characterizing the structure of unlabeled graphs. Through extensive experiments on benchmark datasets, we demonstrate that our methods achieve competitive or superior performance compared with state-of-the-art techniques, including deep learning approaches, while remaining computationally efficient.
Furqan Aziz· IEEE Transactions on Neural...· 0 citations