Katz centrality is an important metric for measuring the relative importance of nodes in a graph, commonly used in social networks. This measure is prohibitively expensive to compute exactly, as it requires up to cubic-time complexity. Although several approximation methods exist, most fail to exploit parallelism, limiting their scalability to large graphs. To address this limitation, we propose two highly parallelizable Monte Carlo algorithms based on random walks named MC-Katz and MC-KatzP, which approximate the truncated Katz centrality and its original form, respectively. Empirical results on nine real-world datasets show that they achieve low mean relative error and high accuracy in ranking the most influential nodes in the networks, while delivering two to three orders of magnitude speedups over the baselines.

Due to the growing concern about unsavory behaviors of machine learning models toward certain demographic groups, the notion of 'fairness' has recently drawn much attention from the community, thereby motivating the study of fairness in graph clustering. Fair graph clustering aims to partition the set of nodes in a graph into k disjoint clusters such that the proportion of each protected group within each cluster is consistent with the proportion of that group in the entire dataset. It is, however, computationally challenging to incorporate fairness constraints into existing graph clustering algorithms, particularly for large graphs. To address this problem, we propose FairAD, a computationally efficient fair graph clustering method. It first constructs a new affinity matrix based on the notion of algebraic distance such that fairness constraints are imposed. A graph coarsening process is then performed on this affinity matrix to find representative nodes that correspond to k clusters. Finally, a constrained minimization problem is solved to obtain the solution of fair clustering. Experiment results on the modified stochastic block model and six public datasets show that FairAD can achieve fair clustering while being up to 40 times faster compared to state-of-the-art fair graph clustering algorithms.

Patching nodes is an effective network defense strategy for malware control at early stages, and its performance is primarily dependent on how accurately the infection propagation is characterized. In this paper, we aim to design a novel patching policy based on the susceptible-infected epidemic network model by incorporating the influence of patching delay–the type of delay that has been largely overlooked in designing patching policies in the literature, while being prevalent in practice. We first identify ‘critical edges’ that form a boundary to separate the most likely infected nodes from the nodes which would still remain healthy after the patching delay. We next leverage the critical edges to determine which nodes to be patched in light of limited patching resources at early stages. To this end, we formulate a constrained graph partitioning problem and use its solution to identify a set of nodes to patch or vaccinate under the limited resources, to effectively prevent malware propagation from getting through the healthy region. We numerically validate that our patching policy significantly outperforms other baseline policies in protecting the healthy nodes under limited patching resources and in the presence of patching delay.