arXiv subjects

Juan A. Rodriguez-Velazquez

Publications and source records attributed to Juan A. Rodriguez-Velazquez.

23 records · Page 2Linked to original sources

arXiv2010

Partitioning a graph into defensive k-alliances

A defensive $k$-alliance in a graph is a set $S$ of vertices with the property that every vertex in $S$ has at least $k$ more neighbors in $S$ than it has outside of $S$. A defensive $k$-alliance $S$ is called global if it forms a dominating set. In this paper we study the problem of partitioning the vertex set of a graph into (global) defensive $k$-alliances. The (global) defensive $k$-alliance partition number of a graph $Γ=(V,E)$, ($ψ_{k}^{gd}(Γ)$) $ψ_k^{d}(Γ)$, is defined to be the maximum number of sets in a partition of $V$ such that each set is a (global) defensive $k$-alliance. We obtain tight bounds on $ψ_k^{d}(Γ)$ and $ψ_{k}^{gd}(Γ)$ in terms of several parameters of the graph including the order, size, maximum and minimum degree, the algebraic connectivity and the isoperimetric number. Moreover, we study the close relationships that exist among partitions of $Γ_1\times Γ_2$ into (global) defensive $(k_1+k_2)$-alliances and partitions of $Γ_i$ into (global) defensive $k_i$-alliances, $i\in \{1,2\}$.

math.CO↗

arXiv2009

On global offensive k-alliances in graphs

We investigate the relationship between global offensive $k$-alliances and some characteristic sets of a graph including $r$-dependent sets and $τ$-dominating sets. As a consequence of the study, we obtain bounds on the global offensive $k$-alliance number in terms of several parameters of the graph.

math.CO↗

arXiv2005

Complex Networks as Hypergraphs

The representation of complex systems as networks is inappropriate for the study of certain problems. We show several examples of social, biological, ecological and technological systems where the use of complex networks gives very limited information about the structure of the system. We propose to use hypergraphs to represent these systems by introducing the concept of the complex hyper-network. We define several structural measures for complex hyper-networks. These measures characterize hyper-network structures on the basis of node participation in different hyper-edges (groups) and sub-hypergraphs. We also define two clustering coefficients, one characterizing the transitivity in the hyper-network through the proportion of hyper-triangles to paths of length two and the other characterizing the formation of triples of mutually adjacent groups in the hyper-network. All of these characteristics are studied in two different hyper-networks; a scientific collaboration hyper-network and an ecological competence hyper-network.

physics.soc-ph↗

arXiv2005

Subgraph Centrality in Complex Networks

We introduce a new centrality measure that characterizes the participation of each node in all subgraphs in a network. Smaller subgraphs are given more weight than larger ones, which makes this measure appropriate for characterizing network motifs. We show that the subgraph centrality (SC) can be obtained mathematically from the spectra of the adjacency matrix of the network. This measure is better able to discriminate the nodes of a network than alternate measures such as degree, closeness, betweenness and eigenvector centralities. We study eight real-world networks for which SC displays useful and desirable properties, such as clear ranking of nodes and scale-free characteristics. Compared with the number of links per node, the ranking introduced by SC (for the nodes in the protein interaction network of S. cereviciae) is more highly correlated with the lethality of individual proteins removed from the proteome.

cond-mat.stat-mech↗

arXiv2005

Spectral Measures of Bipartivity in Complex Networks

We introduce a quantitative measure of network bipartivity as a proportion of even to total number of closed walks in the network. Spectral graph theory is used to quantify how close to bipartite a network is and the extent to which individual nodes and edges contribute to the global network bipartivity. It is shown that the bipartivity characterizes the network structure and can be related to the efficiency of semantic or communication networks, trophic interactions in food webs, construction principles in metabolic networks, or communities in social networks.

cond-mat.stat-mech↗