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This paper considers the problem of a leader that seeks to optimally influence the opinions of agents in a directed network through connecting with a limited number of the agents (“direct followers”), possibly in the presence of a fixed competing leader. The settings involving a single leader and two competing leaders are unified into a general combinatoric optimization problem, for which two heuristic approaches are developed. The first approach is based on a convex relaxation scheme, possibly in combination with the $\ell _1$ -norm regularization technique, and the second is based on a greedy selection strategy. The main technical novelties of this work are in the establishment of supermodularity of the objective function and convexity of its continuous relaxation. The greedy approach is guaranteed to have a lower bound on the approximation ratio sharper than $(1-1/e)$ , while the convex approach can benefit from efficient (customized) numerical solvers to have practically comparable solutions possibly with faster computation times. The two approaches can be combined to provide improved results. In numerical examples, the approximation ratio can be made to reach ${\text{90}\%}$ or higher depending on the number of direct followers. [ABSTRACT FROM AUTHOR]

The recent literature indicates that preserving global pairwise sample similarity is of great importance for feature selection and that many existing selection criteria essentially work in this way. In this paper, we argue that besides global pairwise sample similarity, the local geometric structure of data is also critical and that these two factors play different roles in different learning scenarios. In order to show this, we propose a global and local structure preservation framework for feature selection (GLSPFS) which integrates both global pairwise sample similarity and local geometric data structure to conduct feature selection. To demonstrate the generality of our framework, we employ methods that are well known in the literature to model the local geometric data structure and develop three specific GLSPFS-based feature selection algorithms. Also, we develop an efficient optimization algorithm with proven global convergence to solve the resulting feature selection problem. A comprehensive experimental study is then conducted in order to compare our feature selection algorithms with many state-of-the-art ones in supervised, unsupervised, and semisupervised learning scenarios. The result indicates that: 1) our framework consistently achieves statistically significant improvement in selection performance when compared with the currently used algorithms; 2) in supervised and semisupervised learning scenarios, preserving global pairwise similarity is more important than preserving local geometric data structure; 3) in the unsupervised scenario, preserving local geometric data structure becomes clearly more important; and 4) the best feature selection performance is always obtained when the two factors are appropriately integrated. In summary, this paper not only validates the advantages of the proposed GLSPFS framework but also gains more insight into the information to be preserved in different feature selection tasks. [ABSTRACT FROM AUTHOR]

We consider the distributed integer-weight-balancing problem in networks of nodes that are interconnected via directed edges, each able to admit a positive integer weight (or flow) within a certain interval, captured by lower and upper limits. We propose and analyze distributed iterative algorithms for obtaining admissible and balanced integer weights, i.e., integer weights within the given interval constraints, such that, for each node, the sum of the weights of the incoming edges equals the sum of the weights of the outgoing edges. The proposed algorithms assume that communication among pairs of nodes that are interconnected is bidirectional and are shown to lead to a set of admissible and balanced integer weights after a finite number of time steps (as long as the necessary and sufficient integer circulation conditions are satisfied on the given digraph). [ABSTRACT FROM AUTHOR]