Concrete mathematics, a foundation for computer science, by Ronald L. Graham, Donald E. Knuth and Oren Patashnik. Pp 625. £24·95. 1989. ISBN 0-201-14236-8 (Addison-Wesley)

This book provides the most basic problems, concepts, and well-established results from the topological structure and analysis of interconnection networks in the graph-theoretic language. It covers the basic principles and methods of network design, several well-known networks such as hypercubes, de Bruijn digraphs, Kautz digraphs, double loop, and other networks, and the newest parameters to measure performance of fault-tolerant networks such as Menger number, Rabin number, fault-tolerant diameter, wide-diameter, restricted connectivity, and (l,w)-dominating number. Audience: The book is suitable for those readers who are working on or intend to start research in design analysis of the topological structure of interconnection networks, particularly undergraduates and postgraduates specializing in computer science and applied mathematics.

Topological Structure and Analysis of Interconnection Networks

http://www.mittag-leffler.se/sites/default/files/IML-0405s-18.pdf

Advanced Determinant Calculus: A Complement

Due to recent developments of parallel and distributed computing, the design and analysis of various interconnection networks has been a main topic of research for the past few years and is still stimulated by the new technologies of communication networks such as optic fibers. There are many advantages in using Cayley (di)graphs as models for interconnection networks. This work first surveys some classes of Cayley graphs which are well studied as models of interconnection networks. Results and problems related to routings in networks are then presented, with emphasis on loads of nodes and links in routings.

Cayley graphs and interconnection networks

Algebraic coding theory is a new and rapidly developing subject, popular for its many practical applications and for its fascinatingly rich mathematical structure. This book provides an elementary yet rigorous introduction to the theory of error-correcting codes. Based on courses given by the author over several years to advanced undergraduates and first-year graduated students, this guide includes a large number of exercises, all with solutions, making the book highly suitable for individual study.

A first course in coding theory, by Raymond Hill. Pp 251. £12·50. 1986. ISBN 0-19-853803-0 (Oxford University Press)

Community detection has arisen as one of the most relevant topics in the field of graph data mining due to its importance in many fields such as biology, social networks or network traffic analysis. The metrics proposed to shape communities are too lax and do not consider the internal layout of the edges in the community, which lead to undesirable results. We define a new community metric called WCC. The proposed metric meets a minimum set of basic properties that guarantees communities with structure and cohesion. We experimentally show that WCC correctly quantifies the quality of communities and community partitions using real and synthetic datasets, and compare some of the most used community detection algorithms in the state of the art.

Shaping communities out of triangles

Community detection has arisen as one of the most relevant topics in the field of graph data mining due to its importance in many fields such as biology, social networks or network traffic analysis. The metrics proposed to shape communities are generic and follow two approaches: maximizing the internal density of such communities or reducing the connectivity of the internal vertices with those outside the community. However, these metrics take the edges as a set and do not consider the internal layout of the edges in the community. We define a set of properties oriented to social networks that ensure that communities are cohesive, structured and well defined. Then, we propose the Weighted Community Clustering (WCC), which is a community metric based on triangles. We proof that analyzing communities by triangles gives communities that fulfill the listed set of properties, in contrast to previous metrics. Finally, we experimentally show that WCC correctly captures the concept of community in social networks using real and syntethic datasets, and compare statistically some of the most relevant community detection algorithms in the state of the art.

Shaping Communities out of Triangles

We give explicit formulae for obtaining the binary sequences which produce Steinhaus triangles and generalized Pascal triangles with rotational and dihedral symmetries.

/pdf/symmetries-in-steinhaus-triangles-and-in-generalized-pascal-19ad1y841t.pdf

Symmetries in Steinhaus triangles and in generalized Pascal triangles

On Cayley line digraphs

Let C(n,p) be the set of p-compositions of an integer n, i.e., the set of p-tuples $\alpha=(\alpha_1,\ldots,\alpha_p)$ of nonnegative integers such that $\alpha_1+\cdots+\alpha_p=n$. The main result of this paper is an explicit formula for the determinant of the matrix whose entries are $\alpha^{\beta}=\alpha_1^{\beta_1}\cdots\alpha_p^{\beta_p}$ where $\alpha,\beta\in C(n,p)$. The formula shows that the determinant is positive and has a nice factorization. As an application, it is shown that the polynomials $p_\alpha(x)=(\alpha_1x_1+\cdots+\alpha_px_p)^n$ with $\alpha\in C(n,p)$ form a basis of the vector space Hn[x1, . . . ,xp] of homogeneous polynomials of degree n in p variables. The result is of interest in the context of global optimization because it allows an explicit representation of polynomials as a difference of convex functions.

Josep M. Brunat

Papers

Shaping communities out of triangles

Shaping Communities out of Triangles

Symmetries in Steinhaus triangles and in generalized Pascal triangles

On Cayley line digraphs

The Power-Compositions Determinant and Its Application to Global Optimization