Kemeny’s constant for a graph with bridges

Breen, Jane; Crisostomi, Emanuele; Kim, Sooyeong

doi:10.1016/j.dam.2022.08.004

Cited by 2 publications

(3 citation statements)

References 16 publications

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“…Similarly, our process motif interpretation could be usefully applied to the use of effective graph resistance to study the robustness of complex networks, which has known applications in mitigating cascading failures in electrical networks ( 71 ). Finally, we note that most work on Kemeny’s constant has considered undirected networks ( 68 , 69 , 71 ), and as such, our directed process motif interpretation may contribute to studies on directed networks ( 72 , 73 ) here.…”

Section: Further Analysis and Discussionmentioning

confidence: 81%

“…10 is directly related to Kemeny’s constant

( 67 , 68 ), as

. Kemeny’s constant captures the (weighted) average travel times of random walks between two nodes in the network and can also be interpreted in terms of how well connected the nodes are, and as an indication of clustering ( 69 ). Kemeny’s constant also directly maps to the “effective graph resistance” or Kirchhoff index for regular graphs ( 70 , 71 ), which is interpreted as the difficulty of transport in the network, and so, we see that

also directly maps to this quantity in this case.…”

Section: Further Analysis and Discussionmentioning

confidence: 99%

“…Moreover, the insights our formalism brings in terms of the relative abundance of convergent process motifs are thus directly applicable for Kemeny’s constant and effective graph resistance and will find further utility in those studies. This is particularly important for Kemeny’s constant, for example, where gaining intuition into the measure has been an ongoing concern ( 67 , 69 ). Similarly, our process motif interpretation could be usefully applied to the use of effective graph resistance to study the robustness of complex networks, which has known applications in mitigating cascading failures in electrical networks ( 71 ).…”

Section: Further Analysis and Discussionmentioning

confidence: 99%

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Analytic relationship of relative synchronizability to network structure and motifs

Lizier,

Bauer,

Atay

et al. 2023

Proc. Natl. Acad. Sci. U.S.A.

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Synchronization phenomena on networks have attracted much attention in studies of neural, social, economic, and biological systems, yet we still lack a systematic understanding of how relative synchronizability relates to underlying network structure. Indeed, this question is of central importance to the key theme of how dynamics on networks relate to their structure more generally. We present an analytic technique to directly measure the relative synchronizability of noise-driven time-series processes on networks, in terms of the directed network structure. We consider both discrete-time autoregressive processes and continuous-time Ornstein–Uhlenbeck dynamics on networks, which can represent linearizations of nonlinear systems. Our technique builds on computation of the network covariance matrix in the space orthogonal to the synchronized state, enabling it to be more general than previous work in not requiring either symmetric (undirected) or diagonalizable connectivity matrices and allowing arbitrary self-link weights. More importantly, our approach quantifies the relative synchronization specifically in terms of the contribution of process motif (walk) structures. We demonstrate that in general the relative abundance of process motifs with convergent directed walks (including feedback and feedforward loops) hinders synchronizability. We also reveal subtle differences between the motifs involved for discrete or continuous-time dynamics. Our insights analytically explain several known general results regarding synchronizability of networks, including that small-world and regular networks are less synchronizable than random networks.

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