Small-World Topology Can Significantly Improve the Performance of Noisy Consensus in a Complex Network

Theoretical Computer Science

2017

Self Cite

The size and number of maximum matchings in a network have found a large variety of applications in many fields. As a ubiquitous property of diverse real systems, power-law degree distribution was shown to have a profound influence on size of maximum matchings in scale-free networks, where the size of maximum matchings is small and a perfect matching often does not exist. In this paper, we study analytically the maximum matchings in two scale-free networks with identical degree sequence, and show that the first network has no perfect matchings, while the second one has many. For the first network, we determine explicitly the size of maximum matchings, and provide an exact recursive solution for the number of maximum matchings. For the second one, we design an orientation and prove that it is Pfaffian, based on which we derive a closed-form expression for the number of perfect matchings. Moreover, we demonstrate that the entropy for perfect matchings is equal to that corresponding to the extended Sierpiński graph with the same average degree as both studied scale-free networks. Our results indicate that power-law degree distribution alone is not sufficient to characterize the size and number of maximum matchings in scale-free networks.

Section: Structural Properties Of a Graphmentioning

confidence: 99%

Maximum matchings in scale-free networks with identical degree distribution

Theoretical Computer Science

2017

Self Cite

“…In [30], it is shown that in the small-world Farey network, the network coherence H FO scales logarithmically with the number of vertices N as H FO ∼ ln N . However, in the pseudofractal scale-free web, H FO is a constant, independent of N .…”

Section: Results Analysismentioning

confidence: 99%

“…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],…”

Section: Related Workmentioning

confidence: 99%

See 1 more Smart Citation

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

Patterson

2020

IEEE Trans. Cybern.

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The vast majority of real-world networks are scalefree, loopy, and sparse, with a power-law degree distribution and a constant average degree. In this paper, we study first-order consensus dynamics in binary scale-free networks, where vertices are subject to white noise. We focus on the coherence of networks characterized in terms of the H2-norm, which quantifies how closely agents track the consensus value. We first provide a lower bound of coherence of a network in terms of its average degree, which is independent of the network order. We then study the coherence of some sparse, scale-free real-world networks, which approaches a constant. We also study numerically the coherence of Barabási-Albert networks and high-dimensional random Apollonian networks, which also converges to a constant when the networks grow. Finally, based on the connection of coherence and the Kirchhoff index, we study analytically the coherence of two deterministically-growing sparse networks and obtain the exact expressions, which tend to small constants. Our results indicate that the effect of noise on the consensus dynamics in power-law networks is negligible. We argue that scale-free topology, together with loopy structure, is responsible for the strong robustness with respect to noisy consensus dynamics in power-law networks.

“…From Theorem 3.5 and Lemma 4.3, we obtain the exact expressions for p n+1,i (01), p n+1,i (02), p n+1,i (20), l n+1,i (01), and thus for P n+1,i (01), P n+1,i (02), P n+1,i (20), L n+1,i (01). We finally calculate S n,i (α) for an arbitrary vertex α.…”

mentioning

confidence: 98%

The number and degree distribution of spanning trees in the Tower of Hanoi graph

Theoretical Computer Science

et al. 2016

Self Cite

The number of spanning trees of a graph is an important invariant related to topological and dynamic properties of the graph, such as its reliability, communication aspects, synchronization, and so on. However, the practical enumeration of spanning trees and the study of their properties remain a challenge, particularly for large networks. In this paper, we study the number and degree distribution of the spanning trees in the Hanoi graph. We first establish recursion relations between the number of spanning trees and other spanning subgraphs of the Hanoi graph, from which we find an exact analytical expression for the number of spanning trees of the n-disc Hanoi graph. This result allows the calculation of the spanning tree entropy which is then compared with those for other graphs with the same average degree. Then, we introduce a vertex labeling which allows to find, for each vertex of the graph, its degree distribution among all possible spanning trees.