In this document, we introduce a notion of entropy for stochastic processes on marked rooted graphs. For this, we employ the framework of local weak limit theory for sparse marked graphs, also known as the objective method, due to Benjamini, Schramm, Aldous, Steele and Lyons [BS01, AS04, AL07]. Our contribution is a generalization of the notion of entropy introduced by Bordenave and Caputo [BC15] to graphs which carry marks on their vertices and edges.The theory of time series is the engine driving an enormous range of applications in areas such as control theory, communications, information theory and signal processing. It is to be expected that a theory of stationary stochastic processes indexed by combinatorial structures, in particular graphs, would eventually have a similarly wide-ranging impact. * ⊂ Ḡ * consist of isomorphism classes of rooted marked graphs where all the vertices of the connected component of the root are at distance at most h from the root. For instance, forFor a Polish space X, we say that a sequence of Borel probability measures (µ n ∈ P(X) : n ∈ N) converges weakly to µ ∈ P(X), and write µ n ⇒ µ, if, for any bounded continuous function f : X → R, we have f dµ n → f dµ. If X is Polish, then weak * to have the same mark component as g, i.e. g k [m] := g[m], and subgraph component the truncation of the subgraph component of g up to depth k, i.e. g k [s] := (g[s]) k . For a marked graph G, two adjacent vertices u, v in G, and h ≥ 1, we define the depth h type of the edge (u, v) as).(4)Note that we have employed the convention that the first component on the right hand * , we define, where (G, o) is an arbitrary member of [G, o]. This notation is well-defined, since E h (g, g ′ )(G, o), thought of as a function of (G, o) for fixed integer h ≥ 1 and g, g ′ ∈ Ξ × Ḡh−1 *, is invariant under rooted isomorphism.For h ≥ 1, P ∈ P( Ḡh * ), and g, g ′ ∈ Ξ × Ḡh−1 *, define e P (g, g ′ ) :Here, (G, o) is a member of the isomorphism class [G, o] that has law P . This notation is well-defined for the same reason as above.< ∞ and e P (g, g ′ ) = e P (g ′ , g) for all g, g ′ ∈ Ξ × Ḡh−1 * .The following simple lemma indicates the importance of the concept of admissibility.Lemma 1. Let h ≥ 1, and let µ ∈ P u ( Ḡ * ) be a unimodular probability measure with deg(µ) < ∞. Let P := µ h . Then P is admissible.

Local state transformation is the problem of transforming an arbitrary number of copies of a bipartite resource state to a bipartite target state under local operations. That is, given two bipartite states, is it possible to transform an arbitrary number of copies of one of them to one copy of the other state under local operations only? This problem is a hard one in general since we assume that the number of copies of the resource state is arbitrarily large. In this paper we prove some bounds on this problem using the hypercontractivity properties of some super-operators corresponding to bipartite states. We measure hypercontractivity in terms of both the usual super-operator norms as well as completely bounded norms.Comment: 27 page

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Graphical data is comprised of a graph with marks on its edges and vertices. The mark indicates the value of some attribute associated to the respective edge or vertex. Examples of such data arise in social networks, molecular and systems biology, and web graphs, as well as in several other application areas. Our goal is to design schemes that can efficiently compress such graphical data without making assumptions about its stochastic properties. Namely, we wish to develop a universal compression algorithm for graphical data sources. To formalize this goal, we employ the framework of local weak convergence, also called the objective method, which provides a technique to think of a marked graph as a kind of stationary stochastic processes, stationary with respect to movement between vertices of the graph. In recent work, we have generalized a notion of entropy for unmarked graphs in this framework, due to Bordenave and Caputo, to the case of marked graphs. We use this notion to evaluate the efficiency of a compression scheme. The lossless compression scheme we propose in this paper is then proved to be universally optimal in a precise technical sense. It is also capable of performing local data queries in the compressed form. * This paper was presented in part at 2017 IEEE International Symposium on Information Theory