Preventing Unraveling in Social Networks Gets Harder

Rajesh Hemant Chitnis, Fedor V. Fomin, Petr A. Golovach

Abstract: The behavior of users in social networks is often observed to be affected by the actions of their friends. Bhawalkar et al. (Bhawalkar et al. 2012) 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 k the equilibrium for this model is a maximal subgraph whose minimum degree is ≥ k. However the dropping out of individuals with degrees less than k 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. (Bhawalkar et al. 2012) considered the ANCHORED k-CORE problem: Given a graph G and integers b,k and p do there exist a set of vertices B ⊆ H ⊆ V (G) such that |B| ≤ b, |H| ≥ p and every vertex v e H \ B has degree at least k is the induced subgraph G[H]. They showed that the problem is NP-hard for k ≥ 2 and gave some inapproximability and fixed-parameter intractability results. In this paper we give improved hardness results for this problem. In particular we show that the ANCHORED k-CORE problem is W[1]-hard parameterized by p, even for k = 3. This improves the result of Bhawalkar et al. (Bhawalkar et al. 2012) (who show W[2]-hardness parameterized by b) as our parameter is always bigger since p ≥ b Then we answer a question of Bhawalkar et al. (Bhawalkar et al. 2012) by showing that the ANCHORED k-CORE problem remains NP-hard on planar graphs for all k ≥ 3, even if the maximum degree of the graph is k + 2. Finally we show that the problem is FPT on planar graphs parameterized by b for all k ≥ 7.