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.

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.

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 network analysis, the betweenness centrality of a node informally captures the fraction of shortest paths visiting that node. The computation of the betweenness centrality measure is a fundamental task in the analysis of modern networks, enabling the identification of the most central nodes in such networks. Additionally to being massive, modern networks also contain information about the time at which their events occur. Such networks are often called temporal networks. The temporal information makes the study of the betweenness centrality in temporal networks (i.e., temporal betweenness centrality) much more challenging than in static networks (i.e., networks without temporal information). Moreover, the exact computation of the temporal betweenness centrality is often impractical on even moderately-sized networks, given its extremely high computational cost. A natural approach to reduce such computational cost is to obtain high-quality estimates of the exact values of the temporal betweenness centrality. In this work we present , the first sampling-based approximation algorithm for estimating the temporal betweenness centrality values of the nodes in a temporal network, providing rigorous probabilistic guarantees on the quality of its output. is able to compute the estimates of the temporal betweenness centrality values under two different optimality criteria for the shortest paths of the temporal network. In addition, outputs high-quality estimates with sharp theoretical guarantees leveraging on the empirical Bernstein bound, an advanced concentration inequality. Finally, our experimental evaluation shows that significantly reduces the computational resources required by the exact computation of the temporal betweenness centrality on several real world networks, while reporting high-quality estimates with rigorous guarantees.

Counting the number of occurrences of small connected subgraphs, called temporal motifs, has become a fundamental primitive for the analysis of temporal networks, whose edges are annotated with the time of the event they represent. One of the main complications in studying temporal motifs is the large number of motifs that can be built even with a limited number of vertices or edges. As a consequence, since in many applications motifs are employed for exploratory analyses, the user needs to iteratively select and analyze several motifs that represent different aspects of the network, resulting in an inefficient, time-consuming process. This problem is exacerbated in large networks, where the analysis of even a single motif is computationally demanding. As a solution, in this work we propose and study the problem of simultaneously counting the number of occurrences of multiple temporal motifs, all corresponding to the same (static) topology (e.g., a triangle). Given that for large temporal networks computing the exact counts is unfeasible, we propose odeN, a sampling-based algorithm that provides an accurate approximation of all the counts of the motifs. We provide analytical bounds on the number of samples required by odeN to compute rigorous, probabilistic, relative approximations. Our extensive experimental evaluation shows that odeN enables the approximation of the counts of motifs in temporal networks in a fraction of the time needed by state-of-the-art methods, and that it also reports more accurate approximations than such methods.

The identification and counting of small graph patterns, called network motifs, is a fundamental primitive in the analysis of networks, with application in various domains, from social networks to neuroscience. Several techniques have been designed to count the occurrences of motifs in static networks, with recent work focusing on the computational challenges provided by large networks. Modern networked datasets contain rich information, such as the time at which the events modeled by the networks edges happened, which can provide useful insights into the process modeled by the network. The analysis of motifs in temporal networks, called temporal motifs, is becoming an important component in the analysis of modern networked datasets. Several methods have been recently designed to count the number of instances of temporal motifs in temporal networks, which is even more challenging than its counterpart for static networks. Such methods are either exact, and not applicable to large networks, or approximate, but provide only weak guarantees on the estimates they produce and do not scale to very large networks. In this work we present an efficient and scalable algorithm to obtain rigorous approximations of the count of temporal motifs. Our algorithm is based on a simple but effective sampling approach, which renders our algorithm practical for very large datasets. Our extensive experimental evaluation shows that our algorithm provides estimates of temporal motif counts which are more accurate than the state-of-the-art sampling algorithms, with significantly lower running time than exact approaches, enabling the study of temporal motifs, of size larger than the ones considered in previous works, on billion edges networks.