The signaling system is a fundamental part of the cell, as it regulates essential functions including growth, differentiation, protein synthesis, and apoptosis. A malfunction in this subsystem can disrupt the cell significantly, and is believed to be involved in certain diseases, with cancer being a very important example. While the information available about intracellular signaling networks is constantly growing, and the network topology is actively being analyzed, the modeling of the dynamics of such a system faces difficulties due to the vast number of parameters, which can prove hard to estimate correctly. As the functioning of the signaling system depends on the parameters in a complex way, being able to make general statements based solely on the network topology could be especially appealing. We study a general kinetic model of the signaling system, giving results for the asymptotic behavior of the system in the case of a network with only activatory interactions. We also investigate the possible generalization of our results for the case of a more general model including inhibitory interactions too. We find that feedback cycles made up entirely of activatory interactions (which we call dynamically positive) are especially important, as their properties determine whether the system has a stable signal-off state, which is desirable in many situations to avoid autoactivation due to a noisy environment. To test our results, we investigate the network topology in the Signalink database, and find that the human signaling network indeed has only significantly few dynamically positive cycles, which agrees well with our theoretical arguments.