Effect Of Adding Edges To Consensus Networks With Directed Acyclic Graphs

Directed acyclic graphs?DAGs?are a particular class of network that does not contain directed cycles,which are often encountered in natural,industrial and social networks.In recent year,the synchronization/consensus issues have attracted more and more attentions from biologists,computer scientists and system science scholars.But till now,the role of reverse edges on the directed acyclic graphs is still far from fully explored.To fulfill such a task,this work investigates the effect of additional reverse edges to a directed acyclic graph that may result in directed cycles.Significantly,it is shown that the effect on consensus performance of the whole network is determined solely by the subnetwork containing the additional edges.More precisely,by adding reverse edge?s?,the decreased amount is merely determined by the surpassing range of the reverse edge?s?,which is independent of the network size,topology or the location of the added edge.Numerical simulations on chain-and grid-networks have been carried out to show the feasibility of the results.In this thesis,we firstly give the results of the chain and grid network,and then put the results to a more general situation,the directed acyclic graph.The main results we get are below:1.)In the chain topology,the impact of an added reverse edge on the performance of the chain's synchronization is only related to the range of the reverse edge,that's,have nothing to do with the location of the reverse edge and the scale of the chain network.And the consensus rate decreases with the range of reverse edge increases.2.)In the grid network,except the similar result with the chain result on the diagonal of grid,there are some other unique results of the grid network.the symmetric distribution of the ?₂?L?,the core inheritance phenomenon and reverse edge with the same shape has the same impact on the synchronization performance of the network are what we find in this work.3.)In the tree topology,we get the mathematically same result of the chain when we investigate how the reverse edge?B?A? with a original path?B?A?impact the synchronization of the tree topology.and we found the reverse edge that has no impact on the synchronization performance of the network.In the weighted tree topology,we research the relationship between the weighted reverse edge with different land point in-degree and the synchronization performance of the tree topology.4.)In a more general situation,we investigated the network shaped like a umbrella and get the result that the consensus rate ?₂?L? of a network with a reverse edge is a binary function of the range and land point in-degree of the added reverse edge.And also,this work analyze the situation of DAGs with different sub-nets with some calculation and simulation results.5.)We proposed a polynomial method and proved the conclusions,and we revealed the mathematical core of how the reverse edge impact the convergence performance of the directed acyclic graphs.