On mining approximate and exact fault-tolerant frequent itemsets

Abstract

Robust frequent itemset mining has attracted much attention due to the necessity to find frequent patterns from noisy data in many applications. In this paper, we focus on a variant of robust frequent itemsets in which a small amount of “faults” is allowed in each item and each supporting transaction. This problem is challenging since computing fault-tolerant support count is NP-hard and the anti-monotone property does not hold when the amount of allowable faults is proportional to the size of the itemset. We develop heuristic methods to solve an approximation version of the problem and propose speedup techniques for the exact problem. Experimental results show that our heuristic algorithms are substantially faster than the state-of-the-art exact algorithms while the error is acceptable. In addition, the proposed speedup techniques substantially improve the efficiency of the exact algorithms.

Appendix

There are many applications of FTFI mining. We present an example in World Wide Web (WWW) as follows. An interesting and important topic in the study of WWW is the discovery of implicit local structures, i.e., cyber-communities, in order to better understand the sociological behavior and ever-increasing phenomena in the Web.

To this end, we commonly model the WWW as a directed Web graph G(V, E) where each vertex represents a web page and each arc represents a hyperlink from one web page to another. Then a cyber-community is a subgraph with dense connections from a subset of vertices (representing webpages) to another subset (representing interests, etc.). Dourisboure et al. [10] showed that those relatively dense subgraphs instead of complete subgraphs capture larger and more meaningful communities in the Web.

Fig. 8

A cyber-community in the Political Blogs Dataset modeled as a collection of transactions. There are 7 columns representing the itemset $X = \{a,b,c,d,e,f,g\}$ and 111 rows representing the set of supporting transactions Y. A mark indicates the absence of an item in the corresponding transaction. With lower minimum support threshold, frequent itemsets $X_1 = \{a,b,c,d,e\}$ and $X_2 = \{d,e,f,g\}$ together with their supporting transactions are marked as rectangles at the top-left corner and top/bottom-right corners, respectively

In the language of frequent itemset mining, one can view a vertex v as an item $i_v$ and the set of out-neighbors of v as a transaction $t_v$, i.e., $t_v=\{i_u| u \in U \text { and } (v,u) \in E\}$. Then a fault-tolerant frequent pattern (i.e., an itemset together with its supporting transactions where each item appears in most of the transactions and each transaction contains most of the items) represents a cyber-community in which most (but not necessarily all) of a set of web pages share many (but not necessarily all) of a set of common interests or authorities.

We tested our algorithms on the Political Blogs Dataset⁵ to demonstrate the power of fault-tolerant frequent itemset mining. The Political Blogs dataset is a directed network of 1490 webblogs on US politics with 19,090 hyperlinks between these webblogs. Each blog in the dataset has an attribute describing its political leaning as either liberal or conservative.

After modeling it as a transactional database as described above and removing the empty transactions (corresponding to vertices with no out-going edges), we are left with 1065 transactions. We applied our exact and (approximate) iterative insertion-based FTFI mining algorithms (described in Sects. 4, 3.2.2) with a minimum support threshold of $\sigma = 10\%$ and proportional relaxation parameters $(\alpha _p, \beta _p) = (0.3, 0.2)$. Both algorithms extracted 198 FTFIs. Using the classic frequent itemset definition (i.e., without any fault tolerance), only 59 frequent itemsets can be found.

To further illustrate the advantage of FTFIs over standard frequent itemset, we look at a randomly selected fault-tolerant frequent pattern (X, Y) where $X = \{a,b,c,d,e,f,g\}$ and $|Y| = 111$, see Fig. 8. It turns out that all webblogs from X belong to the same political leaning, i.e., liberal, and 108 out of 111 transactions from Y share the political leaning of liberal.

On the other hand, the frequent itemset mining algorithm (with no fault tolerance) was unable to discover X at the same minimum support of 10%. Even when the minimum support is lowered to 5%, it can only identify some fragments of X such as $X_1 = \{a,b,c,d,e\}$ and $X_2 = \{d,e,f,g\}$. See Fig. 8.