A class of large-scale, multi-agent systems with decentralized information structures can be represented by a linear system with a generalized frequency variable. In this paper, we investigate fundamental properties of such systems, stability, and D-stability, exploiting the dynamical structure. Specifically, we first show that such system is stable if and only if the eigenvalues of the connectivity matrix lie in a region of the complex plane specified by the generalized frequency variable. The stability region is characterized in terms of polynomial inequalities, leading to an algebraic stability condition. We also show that the stability test can be reduced to a linear matrix inequality (LMI) feasibility problem involving generalized Lyapunov inequalities and that the LMI result can be extended for robust stability analysis of systems subject to uncertainties in the interconnection matrix. We then extend the result to D-stability analysis to meet practical requirements, and provide a unified treatment of D-stability conditions for ease of implementation. Finally, numerical examples illustrate utility of the stability conditions for the analysis of biological oscillators and for the design of cooperative stabilizers.