This brief considers the network topology optimization problem for event-triggered consensus of multi-agent systems (MASs) to reduce the triggered events and improve the convergence rate. For a MAS with an undirected graph as its communication network, edge swapping operations are applied to optimize its network topology, which keep the number of links incident to each agent unchanged. Based on the theory of graph spectra, a necessary condition is provided to determine an effective edge swapping operation which leads to less event triggered times and faster convergence rate. An iterative algorithm is developed to optimize the network topology for MASs under the first-order event-triggered consensus protocol. Finally, some numerical simulations are given to illustrate the effectiveness of the approach.
For a first-order leader-follower multi-agent system (MAS), its consensus convergence rate is determined by the algebraic connectivity (the smallest real part of nonzero eigenvalues) of the corresponding directed graph of its interaction topology. In this paper, effects of changing the weights of arcs among the followers on the algebraic connectivity are investigated for a leader-follower topology with a weighted strongly connected directed graph as the followers' interaction topology. If the weight of one arc decreases (increases), the algebraic connectivity increases (decreases) if and only if the entry of the Fiedler vector corresponding to its head is smaller than that of its tail. For arcs with a common head, the arc whose tail corresponds to the largest (smallest) entry of the Fiedler vector improves the algebraic connectivity most if the weight of one of these arcs decreases (increases). A necessary and sufficient condition for improving the algebraic connectivity is also proposed for decreasing (increasing) the weights of multiple arcs by the entries of the Fiedler vector corresponding to the vertices of the arcs and the amounts of the weight changes. Moreover, a method of choosing an optimal set of arcs that improve the algebraic connectivity most is proposed if the changing weights are given. Finally, several numerical experiments are given to illustrate the theoretical results.
Network similarity measures have proven essential in the field of network analysis. Also, topological indices have been used to quantify the topology of networks and have been well studied. In this paper, we employ a new topological index which we call the Ediz eccentric connectivity index. We use this quantity to define network similarity measures as well. First, we determine the extremal value of the Ediz eccentric connectivity index on some network classes. Second, we compare the network similarity measure based on the Ediz eccentric connectivity index with other well-known topological indices such as Wiener index, graph energy, Randić index, the largest eigenvalue, the largest Laplacian eigenvalue, and connectivity eccentric index. Numerical results underpin the usefulness of the chosen measures. They show that our new measure outperforms all others, except the one based on Wiener index. This means that the measure based on Wiener index is still the best, but the new one has certain advantage to some extent.
For a multiagent system (MAS), some redundant links can be removed without reducing its convergence rate. The deletion of redundant links is of benefit to an MAS in improving its lifetime, resources usage efficiency, and so on. In this article, the problem of determining redundant links of MASs is addressed. Based on the theory of graph spectra, necessary and sufficient conditions for determining a redundant link are given for an MAS under the first-order and the second-order consensus protocols. It is surprising to find that the convergence rate of the second-order protocol can even be improved by removing some redundant links, which can be determined by the proposed condition. Moreover, algorithms with running time $O(N^5)$ are designed to determine a maximal set of redundant links for MASs with $N$ agents. Numerical simulations are given to illustrate the effectiveness of the algorithms.
The unbalanced transportation problem is changed to the shortest path problem on network in this paper.A common iteration algorithm that can not only solve the balanced and unbalanced transportation problems,but also solve the balanced and unbalanced assignment is proposed by using the rule of Floyd algorithm.The algorithm is more convenient than closed circuit method of solving transportation problem and Hungary method of solving assignment problems for computers.
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