We introduce a quantitative approach to the analysis of distributed systems which relies on a linear operator based network semantics. A typical problem in a distributed setting is how information propagates through a network, and a typical qualitative analysis is concerned with establishing whether some information will eventually be transmitted from one node to another node in the network. The quantitative approach we present allows us to obtain additional information such as an estimation of the probability that some data is transmitted within a given interval of time. We formalise situations like this using a probabilistic version of a process calculus which is the core of KLAIM, a language for distributed and mobile computing based on interactions through distributed tuple spaces. The analysis we present exploits techniques based on Probabilistic Abstract Interpretation and is characterised by compositional aspects which greatly simplify the inspection of the nodes interaction and the detection of the information propagation through a computer network.