Preventing Unraveling in Social Networks Gets Harder
Rajesh Chitnis, Fedor V. Fomin, Petr A. Golovach

TL;DR
This paper advances the understanding of the computational complexity of the Anchored $k$-Core problem in social networks, establishing new hardness results and fixed-parameter tractability in planar graphs.
Contribution
It improves hardness bounds for the problem, showing W[1]-hardness parameterized by p for k=3, and proves NP-hardness on planar graphs for all k≥3, also providing FPT results for certain parameters.
Findings
W[1]-hardness for p parameterization at k=3
NP-hardness on planar graphs for all k≥3
FPT algorithm on planar graphs parameterized by b for k≥7
Abstract
The behavior of users in social networks is often observed to be affected by the actions of their friends. Bhawalkar et al. \cite{bhawalkar-icalp} introduced a formal mathematical model for user engagement in social networks where each individual derives a benefit proportional to the number of its friends which are engaged. Given a threshold degree the equilibrium for this model is a maximal subgraph whose minimum degree is . However the dropping out of individuals with degrees less than might lead to a cascading effect of iterated withdrawals such that the size of equilibrium subgraph becomes very small. To overcome this some special vertices called "anchors" are introduced: these vertices need not have large degree. Bhawalkar et al. \cite{bhawalkar-icalp} considered the \textsc{Anchored -Core} problem: Given a graph and integers and do there exist a…
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