Robustness of First- and Second-Order Consensus Algorithms for a Noisy Scale-Free Small-World Koch Network

Scale-Free Loopy Structure is Resistant to Noise in Consensus Dynamics in Complex Networks

“…The reason for the distinction of their coherence lies in, at least partially, the scale-free property of the pseudofractal scale-free web that is absent in the Farey network. On the other hand, the coherence H FO for the Koch network [32] is a logarithmic function of N , despite its scale-free structure. The reason the coherence for the Koch network is not a constant is because it has only triangles, lacking cycles of different length.…”

“…The first-order network coherence H FO and its scaling behavior in different networks have been extensively studied. [30], and Koch graph [32], H FO ∼ ln N ; in the complete graph [27],…”

“…It is then ideal that the deviation for each agent from the average value is small, which is referred to as network coherence [21][22][23][24]. Due to its practical significance, consensus problems under environmental disturbances have attracted considerable attention [25][26][27][28][29][30][31][32]. In this paper, we focus on a consensus protocol with Gaussian white noises added on the first-order derivatives of the vertex states, which is the most often discussed model in related literature.…”

Submodular Optimization for Consensus Networks With Noise-Corrupted Leaders

“…The optimal leader set of a given size k can be found through an exhaustive search of all possible subsets of nodes of size k. This approach, however, is not computationally tractable for anything other than small networks or small values of k. Therefore, much research has been done into deriving efficient approximation algorithms and corresponding bounds for leader selection. Several works have investigated the leader selection problem using convergence rate as a performance measure [5], [6], [7], [8], [9], [10]. Most notably, the works [11], [8] propose relaxations of the leader *This work was supported in part by NSF grants CNS-1527287 and CNS-1553340.…”

“…Further, analytic solutions have been developed for undirected paths and cycles for an arbitrary number of leaders [15], [16], [17], and for regular trees when the number of leaders is restricted to two [18]. A closed-form expression for coherence in Koch networks with a single noise-free leader was derived in [10], and analysis of the asymptotic scaling of coherence in 1D and 2D directed lattice graphs when leaders are on the boundary was given in [19]. Of particular note is the work of Clark et al [14], which showed that the leader selection problem with noise-free leaders can be expressed as an optimization problem over a submodular set function.…”