Defensive alliances in graphs: a survey

A set $S$ of vertices of a graph $G$ is a defensive $k$-alliance in $G$ if every vertex of $S$ has at least $k$ more neighbors inside of $S$ than outside. This is primarily an expository article surveying the principal known results on defensive alliances in graph. Its seven sections are: Introduction, Computational complexity and realizability, Defensive $k$-alliance number, Boundary defensive $k$-alliances, Defensive alliances in Cartesian product graphs, Partitioning a graph into defensive $k$-alliances, and Defensive $k$-alliance free sets.Comment: 25 page

Robustness of airline alliance route networks

The aim of this study is to analyze the robustness of the three major airline alliances’ (i.e., Star Alliance, oneworld and SkyTeam) route networks. Firstly, the normalization of a multi-scale measure of vulnerability is proposed in order to perform the analysis in networks with different sizes, i.e., number of nodes. An alternative node selection criterion is also proposed in order to study robustness and vulnerability of such complex networks, based on network efficiency. And lastly, a new procedure – the inverted adaptive strategy – is presented to sort the nodes in order to anticipate network breakdown. Finally, the robustness of the three alliance networks are analyzed with (1) a normalized multi-scale measure of vulnerability, (2) an adaptive strategy based on four different criteria and (3) an inverted adaptive strategy based on the efficiency criterion. The results show that Star Alliance has the most resilient route network, followed by SkyTeam and then oneworld. It was also shown that the inverted adaptive strategy based on the efficiency criterion – inverted efficiency – shows a great success in quickly breaking networks similar to that found with betweenness criterion but with even better results.Peer ReviewedPostprint (author’s final draft

Common adversaries form alliances: modelling complex networks via anti-transitivity

Anti-transitivity captures the notion that enemies of enemies are friends, and arises naturally in the study of adversaries in social networks and in the study of conflicting nation states or organizations. We present a simplified, evolutionary model for anti-transitivity influencing link formation in complex networks, and analyze the model's network dynamics. The Iterated Local Anti-Transitivity (or ILAT) model creates anti-clone nodes in each time-step, and joins anti-clones to the parent node's non-neighbor set. The graphs generated by ILAT exhibit familiar properties of complex networks such as densification, short distances (bounded by absolute constants), and bad spectral expansion. We determine the cop and domination number for graphs generated by ILAT, and finish with an analysis of their clustering coefficients. We interpret these results within the context of real-world complex networks and present open problems