The advent of online social networks has facilitated fast and wide spread of information. However, some users, especially members of minority groups, may be less likely to receive information spreading on the network, due to their disadvantaged network position. We study the optimization problem of adding new connections to a network to enhance fairness in information access among different demographic groups. We provide a concrete formulation of this problem where information access is measured in terms of resistance distance, {offering a new perspective that emphasizes global network structure and multi-path connectivity.} The problem is shown to be NP-hard. We propose a simple greedy algorithm which turns out to output accurate solutions, but its run time is cubic, which makes it undesirable for large networks. As our main technical contribution, we reduce its time complexity to linear, leveraging several novel approximation techniques. In addition to our theoretical findings, we also conduct an extensive set of experiments using both real-world and synthetic datasets. We demonstrate that our linear-time algorithm can produce accurate solutions for networks with millions of nodes.

Random walk centrality is a fundamental metric in graph mining for quantifying node importance and influence, defined as the weighted average of hitting times to a node from all other nodes. Despite its ability to capture rich graph structural information and its wide range of applications, computing this measure for large networks remains impractical due to the computational demands of existing methods. In this paper, we present a novel formulation of random walk centrality, underpinning two scalable algorithms: one leveraging approximate Cholesky factorization and sparse inverse estimation, while the other sampling rooted spanning trees. Both algorithms operate in near-linear time and provide strong approximation guarantees. Extensive experiments on large real-world networks, including one with over 10 million nodes, demonstrate the efficiency and approximation quality of the proposed algorithms.

Graph Neural Networks (GNNs) have demonstrated strong representation learning capabilities for graph-based tasks. Recent advances on GNNs leverage geometric properties, such as curvature, to enhance their representation capabilities by modeling complex connectivity patterns and information flow within graphs. However, most existing approaches primarily focus on discrete graph topology, overlooking diffusion dynamics and task-specific dependencies essential for effective learning. To address this, we propose a learnable integration of Bakry-Émery curvature, which captures both structural and diffusion aspects of information propagation. We develop an efficient, learnable approximation strategy, making curvature computation scalable for large graphs. Furthermore, we introduce an adaptive depth mechanism that dynamically adjusts message-passing layers per vertex based on its curvature, ensuring efficient propagation. Our theoretical analysis establishes a link between curvature and feature distinctiveness, showing that high-curvature vertices require fewer layers, while low-curvature ones benefit from deeper propagation. Extensive experiments on diverse downstream tasks validate the effectiveness of our approach, showing that the proposed depth-adaptive mechanism consistently uplifts the performance of a wide range of GNN architectures.

We study how the stubbornness of social network users influences opinion polarization and disagreement. Our work is in the context of the popular Friedkin-Johnson opinion formation model, where users update their opinion as a function of the opinion of their connections and their own innate opinion. Stubbornness then is formulated in terms of the stress a user puts on its innate opinion.

The Kirchhoff index, which is the sum of the resistance distance between every pair of nodes in a network, is a key metric for gauging network performance, where lower values signify enhanced performance. In this paper, we study the problem of minimizing the Kirchhoff index by adding edges. We first provide a greedy algorithm for solving this problem and give an analysis of its quality based on the bounds of the submodularity ratio and the curvature. Then, we introduce a gradient-based greedy algorithm as a new paradigm to solve this problem. To accelerate the computation cost, we leverage geometric properties, convex hull approximation, and approximation of the projected coordinate of each point. To further improve this algorithm, we use pre-pruning and fast update techniques, making it particularly suitable for large networks. Our proposed algorithms have nearly-linear time complexity. We provide extensive experiments on ten real networks to evaluate the quality of our algorithms. The results demonstrate that our proposed algorithms outperform the state-of-the-art methods in terms of efficiency and effectiveness. Moreover, our algorithms are scalable to large graphs with over 5 million nodes and 12 million edges.

To obtain a foundational understanding of timeline algorithms and viral content in shaping public opinions, computer scientists started to study augmented versions of opinion formation models from sociology. In this paper, we generalize the popular Friedkin--Johnsen model to include the effects of external media sources on opinion formation. Our goal is to mathematically analyze the influence of biased media, arising from factors such as manipulated news reporting or the phenomenon of false balance. Within our framework, we examine the scenario of two opposing media sources, which do not adapt their opinions like ordinary nodes, and analyze the conditions and the number of periods required for radicalizing the opinions in the network. When both media sources possess equal influence, we theoretically characterize the final opinion configuration. In the special case where there is only a single media source present, we prove that media sources which do not adapt their opinions are significantly more powerful than those which do. Lastly, we conduct the experiments on real-world and synthetic datasets, showing that our theoretical guarantees closely align with experimental simulations.

We study resistance eccentricity, a fundamental metric in network science for measuring the structural significance of a node. For a node in a graph, the resistance eccentricity is its maximum resistance distance to all other nodes. Fast computation of resistance eccentricity for a given subset of nodes is essential for a wide range of applications. However, a naive computation, requiring the pseudoinverse of the graph Laplacian, takes cubic time and is thus infeasible for huge networks with millions of nodes. In this paper, we devise a near-linear time algorithm to approximate the resistance eccentricity for one or multiple given nodes, accompanied by a theoretically guaranteed error bound. Furthermore, we investigate the problem of minimizing the resistance eccentricity for a given node by adding$k$missing edges to the graph, for a budget$k$. We show that while the objective function is monotone, it does not possess the submodularity property, ruling out the classical hill-climbing algorithm with theoretical guarantees. Instead, we propose two fast heuristic algorithms to approximately solve this problem. Then, we conduct extensive experiments on different networks with sizes up to several million nodes, demonstrating the superiority of our algorithms in terms of efficiency and effectiveness.

Thebiharmonic distance (BD) is a fundamental metric that measures the distance of two nodes in a graph. It has found applications in network coherence, machine learning, and computational graphics, among others. In spite of BD's importance, efficient algorithms for the exact computation or approximation of this metric on large graphs remain notably absent. In this work, we provide several algorithms to estimate BD, building on a novel formulation of this metric. These algorithms enjoy locality property (that is, they only read a small portion of the input graph) and at the same time possess provable performance guarantees. In particular, our main algorithms approximate the BD between any node pair with an arbitrarily small additive error ε in time O(1/ε2 poly(log n/ε)). Furthermore, we perform an extensive empirical study on several benchmark networks, validating the performance and accuracy of our algorithms.

Consider a graph G and suppose that initially each node is colored either black or white. In the majority model, in each round all nodes simultaneously update their color to the most frequent color among their neighbors. Experiments on the graph data from the real world social networks (SNs) suggest that if an extremely small set of high-degree nodes, often referred to as the elites, all agree on a color, that color becomes the dominant color at the end of the process. We propose two countermeasures that can be adopted by individual nodes relatively easily and guarantee that the elites will not have this disproportionate power to engineer the dominant output color. The first countermeasure essentially requires each node to make some new connections at random, while the second one demands the nodes to be more reluctant towards changing their color. We verify their effectiveness and correctness both theoretically and experimentally. We also investigate the majority model and a variant of it when the initial coloring is random on the real world SNs and several random graph models. In particular, our results on the Erdős-Rényi and regular random graphs confirm or support several theoretical findings or conjectures by the prior work regarding the threshold behavior of the process.

Critical nodes in networks are extremely vulnerable to malicious attacks to trigger negative cascading events such as the spread of misinformation and diseases. Therefore, effective moderation of critical nodes is very vital for mitigating the potential damages caused by such malicious diffusions. The current moderation methods are computationally expensive. Furthermore, they disregard the fundamental metric of information centrality, which measures the dissemination power of nodes. We investigate the problem of removing$k$edges from a network to minimize the information centrality of a target node$v$while preserving the network's connectivity. We prove that this problem is computationally challenging: it is NP-complete and its objective function is not supermodular. However, we propose three approximation greedy algorithms using novel techniques such as random walk-based Schur complement approximation and fast sum estimation. One of our algorithms runs in nearly linear time in the number of edges. To complement our theoretical analysis, we conduct a comprehensive set of experiments on synthetic and real networks with over one million nodes. Across various settings, the experimental results illustrate the effectiveness and efficiency of our proposed algorithms.