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With the arrival of modern internet era, large public networks of varioustypes have come to existence to benefit the society as a whole and severalresearch areas such as sociology, economics and geography in particular.However, the societal and research benefits of these networks have also givenrise to potentially significant privacy issues in the sense that maliciousentities may violate the privacy of the users of such a network by analyzingthe network and deliberately using such privacy violations for deleteriouspurposes. Such considerations have given rise to a new active research areathat deals with the quantification of privacy of users in large networks andthe corresponding investigation of computational complexity issues of computingsuch quantified privacy measures. In this paper, we formalize three suchprivacy measures for large networks and provide non-trivial theoreticalcomputational complexity results for computing these measures. Our results showthe first two measures can be computed efficiently, whereas the third measureis provably hard to compute within a logarithmic approximation factor.Furthermore, we also provide computational complexity results for the case whenthe privacy requirement of the network is severely restricted, including anefficient logarithmic approximation.

On the Computational Complexities of Three Privacy Measures for Large Networks Under Active Attack