Sustainable Network Dynamics

We propose a dynamic graph-theoretic model for ecosystem management as a control over networked system composed of target nodes and unmarked nodes. The network is represented by a complete graph, in which all vertices are connected by a unique edge. Target nodes are attracted by the objective function issued from the external ecosystem management. They pull the network toward the objective position, which is either non-null or stationary. The management policy is considered successful if the graph remains connected in time, that is, target nodes attain the objective and unmarked nodes stay in the convex hull. At the time of the ecosystem network transfer, the model yields an Impossibility Theorem as well as a Sustainability Criterion to maintain full connectivity of the network. Our definition of sustainability can be easily linked to the general definition of sustainability as ecosystem integrity preservation. At last, we identify three management rules to ensure the maintenance of connectivity in time, given the property of the objective transposition function, the nature of connections and the utility-updating time-delays between the nodes.