Preventing unraveling in social networks: the anchored k-core problem

Authors:
Kshipra Bhawalkar;Jon Kleinberg;Kevin Lewi;Tim Roughgarden;Aneesh Sharma
Affiliations:
Stanford University, Stanford, CA;Cornell University, Ithaca, NY;Stanford University, Stanford, CA;Stanford University, Stanford, CA;Twitter, Inc.
Venue:
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part II
Year:
2012

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Abstract

We consider a model of user engagement in social networks, where each player incurs a cost to remain engaged but derives a benefit proportional to the number of engaged neighbors. The natural equilibrium of this model corresponds to the k-core of the social network -- the maximal induced subgraph with minimum degree at least k. We study the problem of "anchoring" a small number of vertices to maximize the size of the corresponding anchored k-core -- the maximal induced subgraph in which every non-anchored vertex has degree at least k. This problem corresponds to preventing "unraveling" -- a cascade of iterated withdrawals. We provide polynomial-time algorithms for general graphs with k=2, and for bounded-treewidth graphs with arbitrary k. We prove strong inapproximability results for general graphs and k≥3.