Rearranging trees for robust consensus

In this paper, we use the H2 norm associated with a communication graph to characterize the robustness of consensus to noise. In particular, we restrict our attention to trees and by systematic attention to the effect of local changes in topology, we derive a partial ordering for undirected trees according to the H2 norm. Our approach for undirected trees provides a constructive method for deriving an ordering for directed trees. Further, our approach suggests a decentralized manner in which trees can be rearranged in order to improve their robustness.Comment: Submitted to CDC 201

Superdiffusion in a class of networks with marginal long-range connections

A class of cubic networks composed of a regular one-dimensional lattice and a set of long-range links is introduced. Networks parametrized by a positive integer k are constructed by starting from a one-dimensional lattice and iteratively connecting each site of degree 2 with a $k$th neighboring site of degree 2. Specifying the way pairs of sites to be connected are selected, various random and regular networks are defined, all of which have a power-law edge-length distribution of the form $P_>(l)\sim l^{-s}$ with the marginal exponent s=1. In all these networks, lengths of shortest paths grow as a power of the distance and random walk is super-diffusive. Applying a renormalization group method, the corresponding shortest-path dimensions and random-walk dimensions are calculated exactly for k=1 networks and for k=2 regular networks; in other cases, they are estimated by numerical methods. Although, s=1 holds for all representatives of this class, the above quantities are found to depend on the details of the structure of networks controlled by k and other parameters.Comment: 10 pages, 9 figure