The concept of fairness index for self-authority servers in a large-scale network is introduced in this paper. The index quantifies the relative contributions of the servers to network routing, and can be used in network administration processes, such as negotiation of Multi-Lateral Peering Agreements. The fairness index concept leads naturally to the idea of an absolutely fair solution, which is a study focus in this paper. Although, an absolutely fair solution may not be an ideal operating point due to efficiency considerations, it serves as a reference point for comparing contribution from various servers in a network. Uniqueness and existence properties of absolutely fair solutions are examined in general as well as for certain specially structured networks of interest. Via the concept of a pricing duality, the connection of absolutely fair solutions to the von Neumann economic model is established. For implementation considerations, a distributed, low-data-rate control algorithm that converges to pre-defined fairness index targets is introduced and analyzed. A heuristic extension is studied to provide a practical approach for realistic situations.

This paper gives three versions of the small gain theorem with restrictions for uncertain time-varying nonlinear systems. The result can be viewed as an extension of the small gain theorem with restrictions for time-invariant nonlinear systems or the small gain theorem without restrictions for time-varying nonlinear systems. The result can be applied to study the stabilization problem or the output regulation problem of uncertain nonlinear systems.

In the paper a theoretical analysis is given for the smallest ball that covers a finite number of points $p_1, p_2, \cdots, p_N \in \Bbb R^n$. Several fundamental properties of the smallest enclosing ball are described and proved. Particularly, it is proved that the $k$-circumscribing enclosing ball with smallest $k$ is the smallest enclosing ball, which dramatically reduces a possible large number of computations in the higher dimensional case. General formulas are deduced for calculating circumscribing balls. The difficulty of the closed-form description is discussed. Finally, as an application, the problem of finding a common quadratic Lyapunov function for a set of stable matrices is considered.