Social and information networks may become polarized, leading to echo chambers and political gridlock. Accurately measuring this phenomenon is a critical challenge. Existing measures often conflate genuine structural division with random topological features, yielding misleadingly high polarization scores on random networks, and failing to distinguish real-world networks from randomized null models. We introduce DSP, a Diffusion-based Structural Polarization measure designed from first principles to correct for such biases. DSP removes the arbitrary concept of 'influencers' used by the popular Random Walk Controversy (RWC) score, instead treating every node as a potential origin for a random walk. To validate our approach, we introduce a set of desirable properties for polarization measures, expressed through reference topologies with known structural properties. We show that DSP satisfies these desiderata, being near-zero for non-polarized structures such as cliques and random networks, while correctly capturing the expected polarization of reference topologies such as monochromatic-splittable networks. Our method applied to U.S. Congress datasets uncovers trends of increasing polarization in recent years. By integrating a null model into its core definition, DSP provides a reliable and interpretable diagnostic tool, highlighting the necessity of statistically-grounded metrics to analyze societal fragmentation.

We present MaNIACS , a sampling-based randomized algorithm for computing high-quality approximations of the collection of the subgraph patterns that are frequent in a single, large, vertex-labeled graph, according to the Minimum Node Image-based (MNI) frequency measure. The output of MaNIACS comes with strong probabilistic guarantees, obtained by using the empirical Vapnik–Chervonenkis (VC) dimension, a key concept from statistical learning theory, together with strong probabilistic tail bounds on the difference between the frequency of a pattern in the sample and its exact frequency. MaNIACS leverages properties of the MNI-frequency to aggressively prune the pattern search space, and thus to reduce the time spent in exploring subspaces that contain no frequent patterns. In turn, this pruning leads to better bounds to the maximum frequency estimation error, which leads to increased pruning, resulting in a beneficial feedback effect. The results of our experimental evaluation of MaNIACS on real graphs show that it returns high-quality collections of frequent patterns in large graphs up to two orders of magnitude faster than the exact algorithm.

“One side will make you grow taller, and the other side will make you grow shorter – The Caterpillar, Alice in Wonderland We introduce a novel null model for assessing the results obtained by analyzing an observed transactional dataset (e.g., significant frequent itemsets) using statistical hypothesis testing. Our null model maintains more properties of the observed dataset than existing models. Specifically, we preserve the Bipartite Joint Degree Matrix of the bipartite graph corresponding to the dataset, which ensures that the number of caterpillars, i.e., paths of length three, is preserved, in addition to the item supports and the transaction lengths, which are the properties considered by previous works. We describe ALICE, a suite of two Markov-Chain Monte-Carlo algorithms for sampling datasets from our null model, based on a carefully defined set of states and efficient operations to move between them. The results of our experimental evaluation show that ALICE mixes fast and scales well, and that our null model finds different significant results than ones previously considered in the literature.

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.

We present MaNIACS, a sampling-based randomized algorithm for computing high-quality approximations of the collection of the subgraph patterns that are frequent in a single, large, vertex-labeled graph, according to the Minimum Node Image-based (MNI) frequency measure. The output of MaNIACS comes with strong probabilistic guarantees, obtained by using the empirical Vapnik-Chervonenkis (VC) dimension, a key concept from statistical learning theory, together with strong probabilistic tail bounds on the difference between the frequency of a pattern in the sample and its exact frequency. MaNIACS leverages properties of the MNI-frequency to aggressively prune the pattern search space, and thus to reduce the time spent in exploring subspaces containing no frequent patterns. In turn, this pruning leads to better bounds to the maximum frequency estimation error, which leads to increased pruning, resulting in a beneficial feedback effect. The results of our experimental evaluation of MaNIACS on real graphs show that it returns high-quality collections of frequent patterns in large graphs up to two orders of magnitude faster than the exact algorithm.

A simplicial complex is a generalization of a graph: a collection of n-ary relationships (instead of binary as the edges of a graph), named simplices. In this paper, we develop a new tool to study the structure of simplicial complexes: we generalize the graph notion of truss decomposition to complexes, and show that this more powerful representation gives rise to different properties compared to the graph-based one. This power, however, comes with important computational challenges derived from the combinatorial explosion caused by the downward closure property of complexes.

Graph pattern mining aims at identifying structures that appear frequently in large graphs, under the assumption that frequency signifies importance. In real life, there are many graphs with weights on nodes and/or edges. For these graphs, it is fair that the importance (score) of a pattern is determined not only by the number of its appearances, but also by the weights on the nodes/edges of those appearances. Scoring functions based on the weights do not generally satisfy the apriori property, which guarantees that the number of appearances of a pattern cannot be larger than the frequency of any of its sub-patterns, and hence allows faster pruning. Therefore, existing approaches employ other, less efficient, pruning strategies. The problem becomes even more challenging in the case of multiple weighting functions that assign different weights to the same nodes/edges. In this work we propose a new family of scoring functions that respects the apriori property, and thus can rely on effective pruning strategies. We provide efficient and effective techniques for mining patterns in multi-weighted graphs, and we devise both an exact and an approximate solution. In addition, we propose a distributed version of our approach, which distributes the appearances of the patterns to examine among multiple workers. Extensive experiments on both real and synthetic datasets prove that the presence of edge weights and the choice of scoring function affect the patterns mined, and the quality of the results returned to the user. Moreover, we show that, even when the performance of the exact algorithm degrades because of an increasing number of weighting functions, the approximate algorithm performs well and with fairly good quality. Finally, the distributed algorithm proves to be the best choice for mining large and rich input graphs.