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Fastest mixing reversible Markov chain on friendship graph: Trade-off between transition probabilities among friends and convergence rate

Saber Jafarizadeh

TL;DR

This work analyzes the Fastest Mixing Reversible Markov Chain (FMRMC) on a friendship graph under multi-objective optimization with fixed edges, using the Pareto frontier to trade off convergence rate against edge-modification costs. By transforming to a symmetric-Laplacian formulation and employing semidefinite programming, the authors derive explicit optimal weights and SLEM expressions across different blade counts $m$, revealing that for $m\ge3$ the Pareto frontier collapses to the star topology (minimum spanning tree), with the optimal SLEM independent of fix-edge probabilities and a non-impact region for within-friend transitions. For smaller $m$ (notably $m=2$ and $m=1$), the frontier can form lines or piecewise regions in the feasible space, with closed-form results contingent on the equilibrium distribution $\\\\pi$. Overall, the results highlight a fundamental trade-off structure in information diffusion on friendship graphs and identify conditions under which the star topology optimally accelerates convergence, informing network design under cost and performance constraints.

Abstract

A long-standing goal of social network research has been to alter the properties of network to achieve the desired outcome. In doing so, DeGroot's consensus model has served as the popular choice for modeling the information diffusion and opinion formation in social networks. Achieving a trade-off between the cost associated with modifications made to the network and the speed of convergence to the desired state has shown to be a critical factor. This has been treated as the Fastest Mixing Markov Chain (FMMC) problem over a graph with given transition probabilities over a subset of edges. Addressing this multi-objective optimization problem over the friendship graph, this paper has provided the corresponding Pareto optimal points or the Pareto frontier. In the case of friendship graph with at least three blades, it is shown that the Pareto frontier is reduced to a global minimum point which is same as the optimal point corresponding to the minimum spanning tree of the friendship graph, i.e., the star topology. Furthermore, a lower limit for transition probabilities among friends has been provided, where values higher than this limit do not have any impact on the convergence rate.

Fastest mixing reversible Markov chain on friendship graph: Trade-off between transition probabilities among friends and convergence rate

TL;DR

This work analyzes the Fastest Mixing Reversible Markov Chain (FMRMC) on a friendship graph under multi-objective optimization with fixed edges, using the Pareto frontier to trade off convergence rate against edge-modification costs. By transforming to a symmetric-Laplacian formulation and employing semidefinite programming, the authors derive explicit optimal weights and SLEM expressions across different blade counts , revealing that for the Pareto frontier collapses to the star topology (minimum spanning tree), with the optimal SLEM independent of fix-edge probabilities and a non-impact region for within-friend transitions. For smaller (notably and ), the frontier can form lines or piecewise regions in the feasible space, with closed-form results contingent on the equilibrium distribution . Overall, the results highlight a fundamental trade-off structure in information diffusion on friendship graphs and identify conditions under which the star topology optimally accelerates convergence, informing network design under cost and performance constraints.

Abstract

A long-standing goal of social network research has been to alter the properties of network to achieve the desired outcome. In doing so, DeGroot's consensus model has served as the popular choice for modeling the information diffusion and opinion formation in social networks. Achieving a trade-off between the cost associated with modifications made to the network and the speed of convergence to the desired state has shown to be a critical factor. This has been treated as the Fastest Mixing Markov Chain (FMMC) problem over a graph with given transition probabilities over a subset of edges. Addressing this multi-objective optimization problem over the friendship graph, this paper has provided the corresponding Pareto optimal points or the Pareto frontier. In the case of friendship graph with at least three blades, it is shown that the Pareto frontier is reduced to a global minimum point which is same as the optimal point corresponding to the minimum spanning tree of the friendship graph, i.e., the star topology. Furthermore, a lower limit for transition probabilities among friends has been provided, where values higher than this limit do not have any impact on the convergence rate.
Paper Structure (22 sections, 4 theorems, 67 equations, 3 figures)

This paper contains 22 sections, 4 theorems, 67 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Friendship Graph
  4. Reversible Markov Chain on Friendship Graph
  5. FMRMC in terms of Symmetric Laplacian
  6. Main Results
  7. Case of $m=2$
  8. Case of $m=1$
  9. Equilibrium distributions with $\pi_{3}^{2} > \pi_{1} \pi_{2}$
  10. Equilibrium distributions with $\pi_{3}^{2} < \pi_{1} \pi_{2}$
  11. Case of $\pi_{1} = \pi_{2}$
  12. Proof of Main Results
  13. Interlacing
  14. Case of $m=2$
  15. Equilibrium Distributions Satisfying (\ref{['eq:Eq201810142587']})
  16. ...and 7 more sections

Key Result

Lemma 1

Values of $q_{2i-1,2i}$ greater than the limits provided in (eq:corner-constraints-star-interior) and (eq:corner-constraints-star-non-interior) result in optimal value of $SLEM$ greater than the value of $SLEM$ obtained for values of $q_{2i-1,2i}$ that satisfy (eq:corner-constraints-star-interior) a

Figures (3)

  • Figure 1: Friendship graph with (a) $m=5$, (b) $m=2$ and (c) $m=1$ blades.
  • Figure 2: Pareto frontier of (a) Example \ref{['example:1']} and (b) Example \ref{['example:2']}.
  • Figure 3: Pareto frontier of friendship graph with $m=1$ blade for (a) $\pi_{3}^{2} \geq \pi_{1} \pi_{2}$ and (b) $\pi_{3}^{2} < \pi_{1} \pi_{2}$ (Example \ref{['example:3']}).

Theorems & Definitions (14)

  • Lemma 1
  • Remark 1
  • Theorem 1
  • Proof 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Example 1
  • Example 2
  • Example 3
  • ...and 4 more