Open Access
2014 Vulnerability of robust preferential attachment networks
Maren Eckhoff, Peter Mörters
Author Affiliations +
Electron. J. Probab. 19: 1-47 (2014). DOI: 10.1214/EJP.v19-2974


Scale-free networks with small power law exponent are known to be robust, meaning that their qualitative topological structure cannot be altered by random removal of even a large proportion of nodes. By contrast, it has been argued in the science literature that such networks are highly vulnerable to a targeted attack, and removing a small number of key nodes in the network will dramatically change the topological structure. Here we analyse a class of preferential attachment networks in the robust regime and prove four main results supporting this claim: After removal of an arbitrarily small proportion $\varepsilon > 0$ of the oldest nodes (1) the asymptotic degree distribution has exponential instead of power law tails; (2) the largest degree in the network drops from being of the order of a power of the network size $n$ to being just logarithmic in $n;$ (3) the typical distances in the network increase from order log log $n$ to order log $n$ and (4) the network becomes vulnerable to random removal of nodes. Importantly, all our results explicitly quantify the dependence on the proportion $\varepsilon$ of removed vertices. For example, we show that the critical proportion of nodes that have to be retained for survival of the giant component undergoes a steep increase as $\varepsilon$ moves away from zero, and a comparison of this result with similar ones for other networks reveals the existence of two different universality classes of robust network models. The key technique in our proofs is a local approximation of the network by a branching random walk with two killing boundaries, and an understanding of the particle genealogies in this process, which enters into estimates for the spectral radius of an associated operator.


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Maren Eckhoff. Peter Mörters. "Vulnerability of robust preferential attachment networks." Electron. J. Probab. 19 1 - 47, 2014.


Accepted: 5 July 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1300.05282
MathSciNet: MR3238777
Digital Object Identifier: 10.1214/EJP.v19-2974

Primary: 05C80
Secondary: 60J85 , 60K35 , 90B15

Keywords: Barab\'asi-Albert model , diameter , killed branching random walk , maximal degree , multitype branching process , network distance , percolation , power law , preferential attachment , robustness , Scale-free network , small world , vulnerability

Vol.19 • 2014
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