Edge exchangeable models

Abstract

Exchangeable models for countable vertex-labeled graphs cannot replicate the large sample behaviors of sparsity and power law degree distribution observed in many network datasets. Out of this mathematical impossibility emerges the question of how network data can be modeled in a way that reflects known empirical behaviors and respects basic statistical principles. We address this question by observing that edges, not vertices, act as the statistical units in networks constructed from interaction data, making a theory of edge-labeled networks more natural for many applications. In this context we introduce the concept of edge exchangeability, which unlike its vertex exchangeable counterpart admits models for networks with sparse and/or power law structure. Our characterization of edge exchangeable networks gives rise to a class of nonparametric models, akin to graphon models in the vertex exchangeable setting. Within this class, we identify a tractable family of distributions with a clear interpretation and suitable theoretical properties, whose significance in estimation, prediction, and testing we demonstrate.Is there a notion of probabilistic symmetry whose ergodic measures [...] describe useful statistical models for sparse graphs with network properties? (Orbanz and Roy, 2015, p. 459) We address this question as part of our broader development of edge exchangeable network models, which are most appropriate when the edges are the statistical units, as they are for the interaction processes we study. We define this setup more precisely in Sections 2-3 and go on to establish basic properties of edge exchangeable models throughout Sections 4-8. Interaction dataDefinition 2.1 (Interaction data). For a set P, we write fin(P) to denote the set of all finite (ordered) multisets of P. An interaction process for a population P is a correspondence I : I → fin(P) between a set I indexing interactions and finite multisets of P. P :#P =#P {I : S → fin(P ) : ρI = I for some bijection ρ : P → P }.The equivalence class E I corresponds to an edge-labeled network structure, as in Figure 1(c). Note that E I does not depend on the specific set P, and so we may disregard P, or treat it implicitly as P = N, in our discussion. We write E S for the set of networks with edges labeled in S ⊆ N.For any S ⊆ S, we define the restriction of E ∈ E S to E S by E| S , the edgelabeled network obtained by removing any edges labeled in S \ S . If E = E I for some interaction process I : S → fin(P), then E| S is the edge-labeled network induced by the restricted process I| S : S → fin(P), s → I(s).Remark 2.2. For clarity we reserve the term graph to specifically refer to a vertexlabeled structure, such as the objects given in (1), (2), and Figure 1(b). We use the term network for the generic unlabeled structure in Figure 1(a) and edge-labeled network for the object defined in (5) and shown in Figure 1(c).

On Edge Exchangeable Random Graphs

“…Random hypergraphs with a given number of edges. We give a minor modification of the original definition by Crane and Dempsey [11,12]; we will see att the end of this subsection that our definition is equivalent to the original one. Definition 4.2.…”

“…by a random label with a continuous distribution in some set, for example U (0, 1), making independent choices for each Y i . Equivalently, and more directly, we obtain G * m from the probability measure µ ′ on Z * by the following construction, which is the original definition by Crane and Dempsey [11,12]. Definition 4.5 (Crane and Dempsey [11,12]).…”

“…Equivalently, and more directly, we obtain G * m from the probability measure µ ′ on Z * by the following construction, which is the original definition by Crane and Dempsey [11,12]. Definition 4.5 (Crane and Dempsey [11,12]). Given a probability measure µ on Z * , we define a sequence of finite random (multi)hypergraphs (G * m ) ∞ m=1 as in Definition 4.2 with the modification that in every edge Y i = {y i1 , .…”

“…A model for edge exchangeable random graphs and hypergraphs was recently introduced by Crane and Dempsey [11,12], who also gave a representation theorem showing that every infinite edge exchangeable random hypergraph can be constructed by this model. An equivalent model, using somewhat different formulations, was given by Broderick and Cai [7] and Campbell, Cai and Broderick [8], see Remark 4.7.…”

“…Section 6 studies the numbers of vertices and edges in the graphs. Section 7 considers an important special case of the model, called rank 1 ; we study two multigraph examples previously considered by Crane and Dempsey [11] and Pittel [36] and show that they are of this type.…”