Josep M. Brunat

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 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.

On Cayley line digraphs

Abstract Given a colouring Δ of a d -regular digraph G and a colouring Π of the symmetric complete digraph on d vertices with loops, the uniformly induced colouring L Π Δ of the line digraph LG is defined. It is shown that the group of colour-preserving automorphisms of ( LG , L Π Δ ) is a subgroup of the group of colour-permuting automorphisms of ( G , Δ ). This result is then applied to prove that if ( G , Δ ) is a d -regular coloured digraph and ( LG , L Π Δ ) is a Cayley digraph, then ( G , Δ ) is itself a Cayley digraph Cay (Ω, Δ) and Π is a group of automorphisms of Ω. In particular, a characterization of those Kautz digraphs which are Cayley digraphs is given. If d =2, for every arc-transitive digraph G , LG is a Cayley digraph when the number k of orbits by the action of the so-called Rankin group is at most 5. If k ⩾ 3 the arc-transitive k -generalized cycles for which LG is a Cayley digraph are characterized.

Symmetries in Steinhaus Triangles and in Generalized Pascal Triangles

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

Put Three and Three Together: Triangle-Driven Community Detection

Community detection has arisen as one of the most relevant topics in the field of graph data mining due to its applications in many fields such as biology, social networks, or network traffic analysis. Although the existing metrics used to quantify the quality of a community work well in general, under some circumstances, they fail at correctly capturing such notion. The main reason is that these metrics consider the internal community edges as a set, but ignore how these actually connect the vertices of the community. We propose the Weighted Community Clustering (WCC), which is a new community metric that takes the triangle instead of the edge as the minimal structural motif indicating the presence of a strong relation in a graph. We theoretically analyse WCC in depth and formally prove, by means of a set of properties, that the maximization of WCC guarantees communities with cohesion and structure. In addition, we propose Scalable Community Detection (SCD), a community detection algorithm based on WCC, which is designed to be fast and scalable on SMP machines, showing experimentally that WCC correctly captures the concept of community in social networks using real datasets. Finally, using ground-truth data, we show that SCD provides better quality than the best disjoint community detection algorithms of the state of the art while performing faster.

The Power-Compositions Determinant and Its Application to Global Optimization

Let C(n,p) be the set of p-compositions of an integer n, i.e., the set of p-tuples

A polynomial generalization of the power-compositions determinant

\alpha=(\alpha_1,\ldots,\alpha_p)

Cayley Digraphs from Complete Generalized Cycles

of nonnegative integers such that

Endo-circulant digraphs: connectivity and generalized cycles

\alpha_1+\cdots+\alpha_p=n

Digraphs on permutations

. The main result of this paper is an explicit formula for the determinant of the matrix whose entries are

Galois Graphs: Walks, Trees and Automorphisms

\alpha^{\beta}=\alpha_1^{\beta_1}\cdots\alpha_p^{\beta_p}