Assessing performance tradeoffs in hierarchical organizations using a diffusive coupling model

Abstract

We study a continuous-time dynamical system of nodes diffusively coupled over a hierarchical network to examine the efficiency and performance tradeoffs that organizations, teams, and command and control units face while achieving coordination and sharing information across layers. Specifically, after defining a network structure that captures real-world features of hierarchical organizations, we use linear systems theory and perturbation theory to characterize the rate of convergence to a consensus state, and how effectively information can propagate through the network, depending on the breadth of the organization and the strength of inter-layer communication. Interestingly, our analytical insights highlight a fundamental performance tradeoff. Namely, networks that favor fast coordination will have decreased ability to share information that is generated in the lower layers of the organization and is to be passed up the hierarchy. Numerical results validate and extend our theoretical results.

Figures (10)

• Figure 5: Comparison between $\lambda_B$ and $\lambda_G$ computed using \ref{['eq:lambdaB_general']} and \ref{['eq:gh']}, respectively, for different breadths $M$. In the blue region $\lambda_B>\lambda_G$, in the orange region $\lambda_G>\lambda_B$.
• Figure 6: Convergence rate $r$ for the autonomous consensus problem, computed (a) analytically for $L=2$ and (b) numerically for $L=3$, for different values of $\alpha$ and $\beta$. In both scenarios $M=3$.
• Figure 7: Convergence rate $r$ of the bottom-up consensus problem, computed numerically (a) $L=2$ and b) $L=3$, for different values of $\alpha$ and $\beta$. In both scenarios $M=3$, $u=1$, and $\gamma_5=1$.
• Figure A.1: Convergence rate $r$ for the autonomous consensus problem, computed (a) analytically for $L=2$ and (b) numerically for $L=3$, for different values of $\alpha$ and $\beta$. In both scenarios $M=3$.
• Figure A.2: Convergence rate $r$ of the bottom-up consensus problem, computed numerically (a) $L=2$ and b) $L=3$, for different values of $\alpha$ and $\beta$. In both scenarios $M=3$, $u=1$, and $\gamma_5=1$.

Theorems & Definitions (17)

• Corollary 11
• Corollary 12
• Proposition 13
• Corollary 14
• Corollary 15
• Proposition 16
• Remark 3
• Example 5: Case $L=2$
• Proposition 17
• Proposition 18