The steady growth of graph data in various applications has resulted in wide-spread research in finding significant sub-structures in a graph. In this paper, we address the problem of finding statistically significant connected subgraphs where the nodes of the graph are labeled. The labels may be either discrete where they assume values from a pre-defined set, or continuous where they assume values from a real domain and can be multi-dimensional. We motivate the problem citing applications in spatial co-location rule mining and outlier detection. We use the chi-square statistic as a measure for quantifying the statistical significance. Since the number of connected subgraphs in a general graph is exponential, the naive algorithm is impractical. We introduce the notion of contracting edges that merge vertices together to form a super-graph. We show that if the graph is dense enough to start with, the number of super-vertices is quite low, and therefore, running the naive algorithm on the super-graph is feasible. If the graph is not dense, we provide an algorithm to reduce the number of super-vertices further, thereby providing a trade-off between accuracy and time. Empirically, the chi-square value obtained by this reduction is always within 96% of the optimal value, while the time spent is only a fraction of that for the optimal. In addition, we also show that our algorithm is scalable and it significantly enhances the ability to analyze real datasets.

The problem of identification of statistically significant patterns in a sequence of data has been applied to many domains such as intrusion detection systems, financial models, web-click records, automated monitoring systems, computational biology, cryptology, and text analysis. An observed pattern of events is deemed to be statistically significant if it is unlikely to have occurred due to randomness or chance alone. We use the chi-square statistic as a quantitative measure of statistical significance. Given a string of characters generated from a memoryless Bernoulli model, the problem is to identify the substring for which the empirical distribution of single letters deviates the most from the distribution expected from the generative Bernoulli model. This deviation is captured using the chi-square measure. The most significant substring (MSS) of a string is thus defined as the substring having the highest chi-square value. Till date, to the best of our knowledge, there does not exist any algorithm to find the MSS in better than O ( n 2 ) time, where n denotes the length of the string. In this paper, we propose an algorithm to find the most significant substring, whose running time is O ( n 3/2 ) with high probability. We also study some variants of this problem such as finding the top-t set, finding all substrings having chi-square greater than a fixed threshold and finding the MSS among substrings greater than a given length. We experimentally demonstrate the asymptotic behavior of the MSS on varying the string size and alphabet size. We also describe some applications of our algorithm on cryptology and real world data from finance and sports. Finally, we compare our technique with the existing heuristics for finding the MSS.