FreSCo：单纯形复合体中频繁模式的挖掘
FreSCo: Mining Frequent Patterns in Simplicial Complexes
摘要：
单纯形复形是对高阶关系进行建模的图的推广。在本文中，我们介绍单纯模式-我们称之为simplets -和推广的频繁模式挖掘的任务，从图的领域的单纯复合体。我们的任务是特别具有挑战性的，由于巨大的搜索空间和高阶同构的需要。我们发现，在一个复杂的单形的出现可以减少到一个二分图同构问题，在线性时间和最多的二次空间。然后，我们提出了一个反单调的频率措施，使我们能够开始探索从小simplets和停止扩大一个simplet，只要它的频率低于最低频率阈值。配备了这些想法和一个聪明的数据结构，我们开发了一个内存意识的算法，通过仔细利用之间的关系，简单的复杂和之间的simplets，实现效率和可扩展性，我们复杂的挖掘任务。我们的算法FreSCo有两种形式：它可以计算单形的确切频率，或者更快地，它可以确定单形是否频繁，而不必计算确切的频率。实验结果证明了FreSCo挖掘复杂的各种大小和维度的频繁的Simplets的能力，和相对于传统的图形模式的Simplets的意义。
英文摘要：
Simplicial complexes are a generalization of graphs that model higher-order relations. In this paper, we introduce simplicial patterns -- that we call simplets -- and generalize the task of frequent pattern mining from the realm of graphs to that of simplicial complexes. Our task is particularly challenging due to the enormous search space and the need for higher-order isomorphism. We show that finding the occurrences of simplets in a complex can be reduced to a bipartite graph isomorphism problem, in linear time and at most quadratic space. We then propose an anti-monotonic frequency measure that allows us to start the exploration from small simplets and stop expanding a simplet as soon as its frequency falls below the minimum frequency threshold. Equipped with these ideas and a clever data structure, we develop a memory-conscious algorithm that, by carefully exploiting the relationships among the simplices in the complex and among the simplets, achieves efficiency and scalability for our complex mining task. Our algorithm, FreSCo, comes in two flavors: it can compute the exact frequency of the simplets or, more quickly, it can determine whether a simplet is frequent, without having to compute the exact frequency. Experimental results prove the ability of FreSCo to mine frequent simplets in complexes of various size and dimension, and the significance of the simplets with respect to the traditional graph patterns.
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