Triangle counting is a fundamental and widely studied problem on static graphs, and recently on temporal graphs, where edges carry information on the timings of the associated events. Streaming processing and resource efficiency are crucial requirements for counting triangles in modern massive temporal graphs, with millions of nodes and up to billions of temporal edges. However, current exact and approximate algorithms are unable to handle large-scale temporal graphs. To fill such a gap, we introduce STEP, a scalable and efficient algorithm to approximate temporal triangle counts from a stream of temporal edges. STEP combines predictions to the number of triangles a temporal edge is involved in, with a simple sampling strategy, leading to scalability, efficiency, and accurate approximation of all eight temporal triangle types simultaneously. We analytically prove that, by using a sublinear amount of memory, STEP obtains unbiased and very accurate estimates. In fact, even noisy predictions can significantly reduce the variance of STEP ’s estimates. Our extensive experiments on massive temporal graphs with up to billions of edges demonstrate that STEP outputs high-quality estimates and is more efficient than state-of-the-art methods.

Characterizing graph properties is fundamental to the analysis and to our understanding of real-world networked systems. The local clustering coefficient, and the more-recent, local closure coefficient, capture powerful properties that are essential in a large number of applications, ranging from graph embeddings to graph partitioning. Such coefficients capture the local density of the neighborhood of each node, considering incident triangle structures and paths of size 2. For this reason, we refer to these coefficients collectively as local triadic coefficients. In this work, we consider the novel and fundamental problem of efficiently computing the average of local triadic coefficients, over a given partition of the nodes of the input graph into a set of disjoint buckets. The average local triadic coefficients of the nodes in each bucket provide a better insight into the interplay of graph structure and the properties of the nodes associated to each bucket. Unfortunately, exact computation, which requires listing all triangles in a graph, is infeasible for large networks. Hence, we focus on obtaining highly-accurate probabilistic estimates. We develop Triad, an adaptive algorithm based on sampling, which can be used to estimate the average local triadic coefficients for a partition of the nodes into buckets. Triad is based on a new class of unbiased estimators, and non-trivial bounds on its sample complexity, enabling the efficient computation of highly accurate estimates. Finally, we show how Triad can be efficiently used in practice on large networks, and we present a case study showing that average local triadic coefficients can capture highorder patterns over collaboration networks.

Content feeds provided by platforms such as X (formerly Twitter) and TikTok are consumed by users on a daily basis. In this paper, we revisit the native advertising problem in content feeds, initiated by Ieong et al. Given a sequence of organic items (e.g., videos or posts) relevant to a user’s interests or to an information search, the goal is to place ads within the organic content so as to maximize a reward function (e.g., number of clicks), while accounting for two considerations: (1) an ad can only be inserted after a relevant content item; (2) the users’ attention decays after consuming content or ads. These considerations provide a natural model for capturing both the advertisement effectiveness and the user experience. In this paper, we design fast and practical 2-approximation greedy algorithms for the associated optimization problem, improving over the best-known practical algorithm that only achieves an approximation factor of 4. Our algorithms exploit a counter-intuitive observation, namely, while top items are seemingly more important due to the decaying attention of the user, taking good care of the bottom items is key for obtaining improved approximation guarantees. We then provide the first comprehensive empirical evaluation on the problem, showing the strong empirical performance of our methods.

In World Wide Web (WWW) systems, networks (or graphs) serve as a fundamental tool for representing, analyzing, and understanding linked data, providing significant insights into the underlying systems. Naturally, most real-world systems have inherent temporal information, e.g., interactions in social networks occur at specific moments in time and last for a certain period. Temporal networks, i.e., network data modeling temporal information, enable novel and fundamental discoveries about the underlying systems they model, otherwise not captured by static networks that ignore such temporal information. In this tutorial, we present state-of-the-art models and algorithmic techniques for mining temporal networks that can provide precious insights into a plethora of web-related applications. We present how temporal networks can be used to extract novel information, especially in web-related network data, and highlight the challenges that arise when modeling temporal information compared to traditional static network-based approaches. We first overview different temporal network models. We then show how such powerful models can be leveraged to extract novel insights through suitable mining primitives. In particular, we present recent advances addressing most foundational problems for temporal network mining—ranging from the computation of temporal centrality measures, temporal motif counting, and temporal communities to bursty events and anomaly detection.

Finding dense subnetworks, with density based on edges or more complex structures, such as subgraphs or k-cliques, is a fundamental algorithmic problem with many applications. While the problem has been studied extensively in static networks, much remains to be explored for temporal networks. In this work we introduce the novel problem of identifying the temporal motif densest subnetwork, i.e., the densest subnetwork with respect to temporal motifs, which are high-order patterns characterizing temporal networks. Identifying temporal motifs is an extremely challenging task, and thus, efficient methods are required. To address this challenge, we design two novel randomized approximation algorithms with rigorous probabilistic guarantees that provide high-quality solutions. We perform extensive experiments showing that our methods outperform baselines. Furthermore, our algorithms scale on networks with up to billions of temporal edges, while baselines cannot handle such large networks. We use our techniques to analyze a financial network and show that our formulation reveals important network structures, such as bursty temporal events and communities of users with similar interests.