
130 Cyber-Physical Systems: From Theory to Practice
Remark 5.3 The aforementioned basic characterizations of flow and synchronization processes
recover classical results for Markov chains [36]. At their essence, they follow simply from well-known
properties of reducible nonnegative matrices with unity eigenvalues. The results are interesting from
that standpoint that they inform the statistical analysis of several other models for stochastic network
dynamics, including infinite-server, queueing network, and voter models, in terms of a network graph.
The results also highlight the duality between flow and synchronization process asymptotics.
Remark ...