We consider the problem of controlling the spread of harmful items in networks, such as the contagion proliferation of diseases or the diffusion of fake news. We assume the linear threshold model of diffusion where each node has a threshold that measures the node resistance to the contagion. We study the parameterized complexity of the problem: Given a network, a set of initially contaminated nodes, and two integers $k$ and $\ell$, is it possible to limit the diffusion to at most $k$ other nodes of the network by immunizing at most $\ell$ nodes? We consider several parameters associated to the input, including: the bounds $k$ and $\ell$, the maximum node degree $\Delta$, the treewidth, and the neighborhood diversity of the network. We first give $W[1]$ or $W[2]$-hardness results for each of the considered parameters. Then we give fixed-parameter algorithms for some parameter combinations.

中文翻译：

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阈值模型中免疫的参数化复杂度

我们考虑了控制有害项目在网络中传播的问题，例如疾病的传播扩散或假新闻的传播。我们假设扩散的线性阈值模型，其中每个节点都有一个阈值，该阈值可测量节点抵抗传染的能力。我们研究了问题的参数化复杂性：给定一个网络，一组最初受污染的节点以及两个整数$ k $和$ \ ell $，是否有可能将扩散限制为最多$ k $个网络的其他节点通过最多免疫$ \ ell $个节点？我们考虑与输入相关的几个参数，包括：边界$ k $和$ \ ell $，最大节点度$ \ Delta $，树宽和网络的邻域多样性。我们首先为每个考虑的参数给出$ W [1] $或$ W [2] $-硬度结果。