Burning some myths on privacy properties of social networks against active attacks

TL;DR

This work challenges the blanket view that social networks inherently lack privacy under active attacks by examining the $k$-metric antidimension and the $(k,\ell)$-anonymity framework. It introduces the polynomial-time ADIM-1 recognition tool, explores structural conditions for $k$-ARS graphs with $k\ge2$, and demonstrates that $1$-metric antidimensional graphs are far rarer in both random and real networks than widely assumed. The study also investigates how graph products can be used to embed $1$-metric antidimensional graphs into larger, less vulnerable structures, and it provides several theoretical results tying $k$-ARS to vertex connectivity, modular decompositions, and diameter. Collectively, the results suggest that privacy in social networks may be more robust than the myth suggests, while offering concrete methods to quantify and potentially enhance privacy against active attacks. The findings have implications for privacy assessment and anonymization strategies in networked data, guiding future work on graph-structural privacy measures.

Abstract

This work focuses on showing some arguments addressed to dismantle the extended idea about that social networks completely lacks of privacy properties. We consider the so-called active attacks to the privacy of social networks and the counterpart $(k,\ell)$-anonymity measure, which is used to quantify the privacy satisfied by a social network against active attacks. To this end, we make use of the graph theoretical concept of $k$-metric antidimensional graphs for which the case $k=1$ represents those graphs achieving the worst scenario in privacy whilst considering the $(k,\ell)$-anonymity measure. As a product of our investigation, we present a large number of computational results stating that social networks might not be as insecure as one often thinks. In particular, we develop a large number of experiments on random graphs which show that the number of $1$-metric antidimensional graphs is indeed ridiculously small with respect to the total number of graphs that can be considered. Moreover, we search on several real networks in order to check if they are $1$-metric antidimensional, and obtain that none of them are such. Along the way, we show some theoretical studies on the mathematical properties of the $k$-metric antidimensional graphs for any suitable $k\ge 1$. In addition, we also describe some operations on graphs that are $1$-metric antidimensional so that they get embedded into another larger graphs that are not such, in order to obscure their privacy properties against active attacks.

Figures (4)

• Figure 1: The ten biconnected graphs $G$ having ${\rm Adim}(G)=1$.
• Figure 2: Results about the second kind of experiments with the $G(n,p)$ model, where $n\in [11,50]$ and $p = \frac{ 1.001\ln n }{n}$. Notice that the second chart reports the same data but as a line chart with log scale on the vertical axis.
• Figure 3: The six geodetic graphs $G$ of order $9$ having ${\rm Adim}(G)=1$.
• Figure 4: The fifteen geodetic graphs $G$ of order $10$ having ${\rm Adim}(G)=1$.

Theorems & Definitions (31)

• Lemma 4.1
• proof
• Corollary 4.2
• Corollary 4.3
• proof
• Proposition 4.4
• proof
• Theorem 4.5
• proof
• Definition 5.1: $u$-branches and $\epsilon$-equivalence in trees