The achievable rate region and degrees of freedom are studied for interference channels with confidential messages under strong secrecy constraints. The problem is studied based on the framework of Han and Verdú’s channel resolvability theory. It is shown that if the random binning rate for securing a confidential message is slightly above the resolution of its corresponding wiretapped channel,...

In this note we consider intervals and convex sets of strong product. Vertices of an arbitrary interval of G H are classified with shortest path properties of one factor and a walk properties of a slightly modified second factor. The convex sets of the strong product are characterized by convexity of projections to both factors and three other local properties, one of them being 2-convexity.

Calculating a product of multiple graphs has been studied in mathematics, engineering, computer science, and more recently in network science, particularly in the context of multilayer networks. One of the important questions to be addressed in this area is how to characterize spectral properties of a product graph using those of its factor graphs. While several such characterizations have alre...

In this note we recall some recent progress in understanding the representation theory of ∗-algebras over rings C = R(i) where R is ordered and i2 = −1. The representation spaces are modules over auxiliary ∗-algebras with inner products taking values in this auxiliary ∗-algebra. The ring ordering allows to implement positivity requirements for the inner products. Then the representations are re...

This work is concerned with the prime factor decomposition (PFD) of strong product graphs. A new quasi-linear time algorithm for the PFD with respect to the strong product for arbitrary, finite, connected, undirected graphs is derived. Moreover, since most graphs are prime although they can have a product-like structure, also known as approximate graph products, the practical application of the...

The practical application of graph prime factorization algorithms is limited in practice by unavoidable noise in the data. A first step towards error-tolerant “approximate” prime factorization, is the development of local approaches that cover the graph by factorizable patches and then use this information to derive global factors. We present here a local, quasi-linear algorithm for the prime f...