Structural properties of the line-graphs associated to directed networks

Regino Criado; Julio Flores; Alejandro J. García del Amo; Miguel Romance; Regino Criado; Julio Flores; Alejandro J. García del Amo; Miguel Romance

doi:10.3934/nhm.2012.7.373

Networks and Heterogeneous Media

2012, Volume 7, Issue 3: 373-384. doi: 10.3934/nhm.2012.7.373

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Structural properties of the line-graphs associated to directed networks

1.
Departamento de Matemática Aplicada, Universidad Rey Juan Carlos, 28933 Móstoles (Madrid)
2.
Departamento de Matemáatica Aplicada, Universidad Rey Juan Carlos, 28933 Móstoles (Madrid)
3.
Departamento de Matemática Aplicada, Universidad Rey Juan Carlos, 28933 Móstoles (Madrid), Spain

Received: 01 December 2011 Revised: 01 May 2012
Primary: 05C90, 05C75; Secondary: 68M10, 94C15, 90B18.

The centrality and efficiency measures of an undirected network $G$ were shown by the authors to be strongly related to the respective measures on the associated line graph $L(G)$. In this note we extend this study to a directed network $\vec{G}$ and its associated directed network $\vec{L}(\vec{G})$. The Bonacich centralities of these two networks are shown to be related in a surprisingly simpler manner than in the non directed case. Efficiency is also considered and the corresponding relations established. In addition, an estimation of the clustering coefficient of $\vec{L}(\vec{G})$ is given in terms of the clustering coefficient of $\vec{G}$, and by means of an example we show that a reverse estimation cannot be expected.
Given a non directed graph $G$, there is a natural way to obtain from it a directed line graph, namely $\vec{L}(D(G))$, where the directed graph $D(G)$ is obtained from $G$ in the usual way. With this approach the authors estimate some parameters of $\vec{L}(D(G))$ in terms of the corresponding ones in $L(G)$. Particularly, we give an estimation of the norm difference between the centrality vectors of $\vec{L}(D(G))$ and $L(G)$ in terms of the Collatz-Sinogowitz index (which is a measure of the irregularity of $G$). Analogous estimations are given for the efficiency measures. The results obtained strongly suggest that for a given non directed network $G$, the directed line graph $\vec{L}(D(G))$ captures more adequately the properties of $G$ than the non directed line graph $L(G)$.
- Complex networks,
- dual graph,
- line graph,
- directed line graph
Citation: Regino Criado, Julio Flores, Alejandro J. García del Amo, Miguel Romance. Structural properties of the line-graphs associated to directed networks[J]. Networks and Heterogeneous Media, 2012, 7(3): 373-384. doi: 10.3934/nhm.2012.7.373

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Abstract

The centrality and efficiency measures of an undirected network $G$ were shown by the authors to be strongly related to the respective measures on the associated line graph $L(G)$. In this note we extend this study to a directed network $\vec{G}$ and its associated directed network $\vec{L}(\vec{G})$. The Bonacich centralities of these two networks are shown to be related in a surprisingly simpler manner than in the non directed case. Efficiency is also considered and the corresponding relations established. In addition, an estimation of the clustering coefficient of $\vec{L}(\vec{G})$ is given in terms of the clustering coefficient of $\vec{G}$, and by means of an example we show that a reverse estimation cannot be expected.
Given a non directed graph $G$, there is a natural way to obtain from it a directed line graph, namely $\vec{L}(D(G))$, where the directed graph $D(G)$ is obtained from $G$ in the usual way. With this approach the authors estimate some parameters of $\vec{L}(D(G))$ in terms of the corresponding ones in $L(G)$. Particularly, we give an estimation of the norm difference between the centrality vectors of $\vec{L}(D(G))$ and $L(G)$ in terms of the Collatz-Sinogowitz index (which is a measure of the irregularity of $G$). Analogous estimations are given for the efficiency measures. The results obtained strongly suggest that for a given non directed network $G$, the directed line graph $\vec{L}(D(G))$ captures more adequately the properties of $G$ than the non directed line graph $L(G)$.

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