Revisiting Finite-Time Distributed Algorithms via Successive Nulling of Eigenvalues

Finite-Time Weighted Average Consensus and Generalized Consensus Over a Subset

“…These results are formulated using the novel framework of generalized consensus, meaning that consensus is reached on the network except for some prescribed subset. This framework contrasts our results with other existing algorithms of distributed average consensus, such as [19], where disagreement is prohibited and the final consensus state does not extend straightforwardly to other values.…”

“…The following theorem is based on the method of successive nulling of eigenvalues [19]. Theorem 1: Suppose that W is a linear operator on G with K distinct eigenvalues λ 1 , · · · , λ K , and multiplicities m 1 , · · · , m K .…”

“…Recently, by introducing a novel approach termed as successive nulling of eigenvalues, the authors in [19] provide a necessary and sufficient condition for reaching average consensus in finite time with linear protocols. They design parametric distributed iterations which are a function of a linear operator W on an arbitrary undirected graph, and show that average consensus can be reached in K steps with K being the number of distinct eigenvalues of the operator W if and only if W has at least one simple eigenvalue having the eigenvector of all constants.…”

“…In this letter, we first relax the condition in [19] by showing that weighted average consensus can be reached in K steps if and only if the operator W has at least one simple eigenvalue whose eigenvector has non-zero entries for all variables. When there are zero entries in the eigenvector associated to the simple eigenvalue, we further prove that finite-time weighted average consensus is reached only by a subset S o of nodes corresponding to the non-zero entries.…”

Interpolation of graph signals using shift-invariant graph filters

“…By default, we will assume that L = D + 1 and set the coefficients h * that solve (11) 2 http://www.seas.upenn.edu/ ∼ ssegarra equal to the unit vector that spans the one-dimensional kernel of CΨ. Note also that for the case where all the eigenvalues are distinct, we need L > N − K. An alternative way to design the filter coefficients that annihilate a specific set of frequencies is to use the "successive nulling of eigenvalues" approach in [8], which relies on a slightly different definition of a graph filter. Once the coefficients of the filter are designed, the next step is to find the optimum seeding signal.…”