Controversy, disagreement, conflict, polarization and opinion divergence in social networks have been the subject of much recent research. In particular, researchers have addressed the question of how such concepts can be quantified given people's prior opinions, and how they can be optimized by influencing the opinion of a small number of people or by editing the network's connectivity. Here, rather than optimizing such concepts given a specific set of prior opinions, we study whether they can be optimized in the average case and in the worst case over all sets of prior opinions. In particular, we derive the worst-case and average-case conflict risk of networks, and we propose algorithms for optimizing these. For some measures of conflict, these are non-convex optimization problems with many local minima. We provide a theoretical and empirical analysis of the nature of some of these local minima, and show how they are related to existing organizational structures. Empirical results show how a small number of edits quickly decreases its conflict risk, both average-case and worst-case. Furthermore, it shows that minimizing average-case conflict risk often does not reduce worst-case conflict risk. Minimizing worst-case conflict risk on the other hand, while computationally more challenging, is generally effective at minimizing both worst-case as well as average-case conflict risk.

Finding out whether a word occurs significantly more often in one text or corpus than in another is an important question in analysing corpora. As noted by Kilgarriff (2005), the use of the  2 and log-likelihood ratio tests is problematic in this context, as they are based on the assumption that all samples are statistically independent of each other. However, words within a text are not independent. As pointed out in Kilgarriff (2001) and Paquot & Bestgen (2009), it is possible to represent the data differently and employ other tests, such that we assume independence at the level of texts rather than individual words. This allows us to account for the distribution of words within a corpus. In this article we compare the significance estimates of various statistical tests in a controlled resampling experiment and in a practical setting, studying differences between texts produced by male and female fiction writers in the British National Corpus. We find that the choice of the test, and hence data representation, matters. We conclude that significance testing can be used to find consequential differences between corpora, but that assuming independence between all words may lead to overestimating the significance of the observed differences, especially for poorly dispersed words. We recommend the use of the t-test, Wilcoxon rank-sum test, or bootstrap test for comparing word frequencies across corpora.

Hypothesis testing using constrained null models can be used to compute the significance of data mining results given what is already known about the data. We study the novel problem of finding the smallest set of patterns that explains most about the data in terms of a global p value. The resulting set of patterns, such as frequent patterns or clusterings, is the smallest set that statistically explains the data. We show that the newly formulated problem is, in its general form, NP-hard and there exists no efficient algorithm with finite approximation ratio. However, we show that in a special case a solution can be computed efficiently with a provable approximation ratio. We find that a greedy algorithm gives good results on real data and that, using our approach, we can formulate and solve many known data-mining tasks. We demonstrate our method on several data mining tasks. We conclude that our framework is able to identify in various settings a small set of patterns that statistically explains the data and to formulate data mining problems in the terms of statistical significance

Dimensionality reduction and manifold learning methods such as t-distributed stochastic neighbor embedding (t-SNE) are frequently used to map high-dimensional data into a two-dimensional space to visualize and explore that data. Going beyond the specifics of t-SNE, there are two substantial limitations of any such approach: (1) not all information can be captured in a single two-dimensional embedding, and (2) to well-informed users, the salient structure of such an embedding is often already known, preventing that any real new insights can be obtained. Currently, it is not known how to extract the remaining information in a similarly effective manner. We introduce conditional t-SNE (ct-SNE), a generalization of t-SNE that discounts prior information in the form of labels. This enables obtaining more informative and more relevant embeddings. To achieve this, we propose a conditioned version of the t-SNE objective, obtaining an elegant method with a single integrated objective. We show how to efficiently optimize the objective and study the effects of the extra parameter that ct-SNE has over t-SNE. Qualitative and quantitative empirical results on synthetic and real data show ct-SNE is scalable, effective, and achieves its goal: it allows complementary structure to be captured in the embedding and provided new insights into real data.