基于布尔网络断层扫描的缺陷节点计数与定位
Counting and localizing defective nodes by Boolean network tomography
摘要：
在大集合中识别缺陷项是现实生活中许多应用的主要问题。我们考虑通过基于布尔网络断层扫描（BNT）的方法定位网络中的缺陷节点的问题，该方法基于从端到端测量路径的布尔结果推断信息。路径集合上的{ \em Identifiability}条件当然是非常相关的，该条件保证明确地发现或计数缺陷节点。我们研究旧的和新的可识别性条件，从理论和应用的角度都有助于这个问题。(1)节点数和路径数之间的精确权衡是什么，以便最多可以明确地识别$k$节点？答案是已知的只有$k=1$和我们回答的问题，任何$k$，设置一个问题，隐含地在以前的作品中留下了开放。(2)我们研究了无歧义可识别节点数的上界和下界，引入了新的可识别性条件，这些条件严格地隐含着无歧义可识别性，并且严格地隐含着无歧义可识别性;（3）我们一方面利用这些新的条件设计算法，以细粒度的方式计算缺陷节点，另一方面证明了BNT识别网络中缺陷节点问题的第一复杂性困难性。(4)我们引入了一个随机模型，在这个模型中，我们研究了可明确识别的缺陷节点数量的下限，并使用这个模型通过最大似然估计方法来估计真实网络中的缺陷节点数量
英文摘要：
Identifying defective items in larger sets is a main problem with many applications in real life situations. We consider the problem of localizing defective nodes in networks through an approach based on boolean network tomography (BNT), which is grounded on inferring informations from the boolean outcomes of end-to-end measurements paths. {\em Identifiability} conditions on the set of paths which guarantee discovering or counting unambiguously the defective nodes are of course very relevant. We investigate old and introduce new identifiability conditions contributing this problem both from a theoretical and applied perspective. (1) What is the precise tradeoff between number of nodes and number of paths such that at most $k$ nodes can be identified unambiguously ? The answer is known only for $k=1$ and we answer the question for any $k$, setting a problem implicitly left open in previous works. (2) We study upper and lower bounds on the number of unambiguously identifiable nodes, introducing new identifiability conditions which strictly imply and are strictly implied by unambiguous identifiability; (3) We use these new conditions on one side to design algorithmic heuristics to count defective nodes in a fine-grained way, on the other side to prove the first complexity hardness results on the problem of identifying defective nodes in networks via BNT. (4) We introduce a random model where we study lower bounds on the number of unambiguously identifiable defective nodes and we use this model to estimate that number on real networks by a maximum likelihood estimate approach
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
