On the $$(k,\ell )$$ ( k , ℓ ) -anonymity of networks via their k-metric antidimension

Abstract This work focuses on the $$(k,\ell )$$ ( k , ℓ ) -anonymity of some networks as a measure of their privacy against active attacks. Two different types of networks are considered. The first one consists of graphs with a predetermined structure, namely cylinders, toruses, and 2-dimensional Hamming graphs, whereas the second one is formed by randomly generated graphs. In order to evaluate the $$(k,\ell )$$ ( k , ℓ ) -anonymity of the considered graphs, we have computed their k-metric antidimension. To this end, we have taken a combinatorial approach for the graphs with a predetermined structure, whereas for randomly generated graphs we have developed an integer programming formulation and computationally tested its implementation. The results of the combinatorial approach, as well as those from the implementations indicate that, according to the $$(k,\ell )$$ ( k , ℓ ) -anonymity measure, only the 2-dimensional Hamming graphs and some general random dense graphs are achieving some higher privacy properties.