This note attempts to understand graph limits as defined by Lovasz and Szegedy (2006)} in terms of harmonic analysis on semigroups. is done representing probability distributions random exchangeable graphs mixtures characters the semigroup unlabeled with node-disjoint union, thereby providing an alternative derivation de Finetti's theorem for graphs.

We consider the classic Moran process modeling the spread of genetic mutations, as extended to structured populations by Lieberman et al. (Nature, 2005). In this process, individuals are the vertices of a connected graph G. Initially, there is a single mutant vertex, chosen uniformly at random. In each step, a random vertex is selected for reproduction with a probability proportional to its fit...

We consider an edge-isoperimetric problem (EIP) on the cartesian powers of graphs. One of our objectives is to extend the list of graphs for whose cartesian powers the lexicographic order provides nested solutions for the EIP. We present several new classes of such graphs that include as special cases all presently known graphs with this property. Our new results are applied to derive best poss...

We introduce nite relational structures called sketches, that represent edge crossings in drawings of nite graphs. We consider the problem of characterizing sketches in Monadic Second-Order logic. We answer positively the question for framed sketches, i.e., for those representing drawings of graphs consisting of a planar connected spanning subgraph (the frame) augmented with additional edges th...

Let c : E(G) → [k] be an edge-coloring of a graph G, not necessarily proper. For each vertex v, let c̄(v) = (a1, . . . , ak), where ai is the number of edges incident to v with color i. Reorder c̄(v) for every v in G in nonincreasing order to obtain c∗(v), the color-blind partition of v. When c∗ induces a proper vertex coloring, that is, c∗(u) 6= c∗(v) for every edge uv in G, we say that c is col...

It is exponentially unlikely that a sparse random graph or hypergraph is connected, but such graphs occur commonly as the giant components of larger random graphs. This simple observation allows us to estimate the number of connected graphs, and more generally the number of connected d-uniform hypergraphs, on n vertices with m = O(n) edges. We also estimate the probability that a binomial rando...

Estimating a covariance matrix efficiently and discovering its structure are important statistical problems with applications in many fields. This article takes a Bayesian approach to estimate the covariance matrix of Gaussian data. We use ideas from Gaussian graphical models and model selection to construct a prior for the covariance matrix that is a mixture over all decomposable graphs, where...

This paper has dual aims. First is to develop practical universal coding methods for unlabeled graphs. Second is to use these for graph anomaly detection. The paper develops two coding methods for unlabeled graphs: one based on the degree distribution, the second based on the triangle distribution. It is shown that these are efficient for different types of random graphs, and on real-world grap...