We extend the classical edge-triangle Exponential Random Graph Model (ERGM) to an inhomogeneous setting in which vertices carry types determined by an underlying partition. This leads to a block-structured ERGM where interaction parameters depend on vertex types. We establish a large deviation principle for the associated sequence of measures and derive the corresponding variational formula for the limiting free energy. In the ferromagnetic regime, where the parameters governing triangle densities are nonnegative, we reduce the variational problem to a scalar optimization problem, thereby identifying the natural block counterpart of the replica symmetric regime. Under additional restrictions on the parameters, comparable to the classical Dobrushin's uniqueness region, we prove uniqueness of the maximizer and derive a law of large numbers for the edge density.
Social networks, like many biological and technological networks, are known to exhibit a high degree of clustering, also referred to as transitivity. Informally, if two nodes share a common neighbor, they are more likely to be linked to each other as well. A central modelling challenge is to relate such local features, captured by the frequency of small subgraphs and therefore easily measurable, to the global structure of the network. Among probabilistic models for social networks (see e.g. [26]), Exponential Random Graph Models (ERGMs) have a particularly transparent statistical justification. More precisely, if one prescribes the empirical values of finitely many network statistics (such as the number of edges, triangles, or other small subgraphs), then among all probability distributions on graphs consistent with these constraints, the entropy-maximizing distribution is a Gibbs measure [20]. Focusing on unconstrained, undirected ERGMs, a substantial body of rigorous probabilistic results has been developed. These include the derivation of the limiting free energy and the analysis of its phase diagram, as well as more general results on the asymptotic structure and typical behavior of such graphs (see, e.g., [8,7,25,31,2,13]). A complementary line of research investigates fluctuation phenomena and limit theorems for subgraph densities, obtained both through statistical mechanics techniques (see, e.g., [4,22,3,23,24]) and via Stein's method, the latter also providing quantitative normal approximations [15,27,30].
Classical ERGMs are typically homogeneous, in the sense that their law is invariant under relabellings of the vertex set. Real networks, however, often exhibit structural heterogeneity: vertices carry attributes (or types) that significantly influence link formation. In this work we study inhomogeneous ERGMs in which heterogeneity is introduced through an underlying partition of the vertex set into finitely many blocks. Each vertex is assigned a type (or color) according to the block to which it belongs. Thus, in the underlying reference measure, the probability to create an edge between two vertices depends on their types. Models of this kind were originally introduced in the physics literature [17] to describe community organization, where types may represent political orientation, scientific field, thematic similarity of webpages, or other attributes.
From a probabilistic point of view, inhomogeneous ERGMs can be seen as exponential tilts of dense inhomogeneous Erdős-Rényi graphs, in the same way that classical ERGMs arise as tilts of the homogeneous Erdős-Rényi model. Vertex types enter the Hamiltonian through the interaction parameters that weight the various subgraph densities, depending on the types of the vertices involved in those subgraphs. For instance, one may assign a higher weight to triangles formed by vertices of the same type than to mixed-type configurations, thereby reflecting an underlying notion of proximity, such as shared interests or geographical closeness.
Among recent developments on block-structured graph models, we mention the derivation of a Large Deviation Principle (LPD) in [18,6], together with advances in the closely related theory of probability graphons [1,12], which extend the classical graphon framework to settings where edges carry additional decorations.
The main novelty of this paper is to provide a step further towards the understanding of inhomogeneous ERGMs. We show that the variational principle available for the free energy for homogeneous ERGMs can be transferred to this richer setting, yielding a well-posed variational problem under minimal structural assumptions on the underlying partition. This leads to an explicit characterization of the maximizers and to a natural block-structured counterpart of the replica symmetric regime. Outside this regime, we expect effects associated with symmetry breaking and phase transitions, whose analysis we defer to future work. The techniques of the proofs rely on classical tools, suitably adapted to the block structure of the model. In what follows, we briefly list our main results.
In Thm. 2.12, we prove a large deviation principle for the associated sequence of measures. From the corresponding rate function, we derive the variational formula for the free energy; an alternative direct derivation is provided in Appendix A.3.
In Theorem 2.13, we derive the Euler-Lagrange equations that characterize any optimizer of the variational formula, without imposing restrictions on the parameter regime.
In Thm. 2.14, assuming nonnegativity of parameters tuning the triangle densities, we reduce the variational problem for the free energy to a scalar optimization problem, leading to the fixed-point system (2.24). In Thm. 2.15, we identify a restricted parameter regime (comparable to the so-called Dobrushin uniqueness region) under which this system admits a unique solution. Combining these results, we obtain our
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