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Nonlinear Mapping Convergence and Application to Social Networks

Brian D. O. Anderson, Mengbin Ye

TL;DR

This paper discusses nonlinear discrete-time maps of the form $x (k+1)=F(x(k))$, focussing on equilibrium points of such maps, and makes an application to problems in social networks.

Abstract

This paper discusses discrete-time maps of the form $x(k + 1) = F(x(k))$, focussing on equilibrium points of such maps. Under some circumstances, Lefschetz fixed-point theory can be used to establish the existence of a single locally attractive equilibrium (which is sometimes globally attractive) when a general property of local attractivity is known for any equilibrium. Problems in social networks often involve such discrete-time systems, and we make an application to one such problem.

Nonlinear Mapping Convergence and Application to Social Networks

TL;DR

This paper discusses nonlinear discrete-time maps of the form , focussing on equilibrium points of such maps, and makes an application to problems in social networks.

Abstract

This paper discusses discrete-time maps of the form , focussing on equilibrium points of such maps. Under some circumstances, Lefschetz fixed-point theory can be used to establish the existence of a single locally attractive equilibrium (which is sometimes globally attractive) when a general property of local attractivity is known for any equilibrium. Problems in social networks often involve such discrete-time systems, and we make an application to one such problem.

Paper Structure

This paper contains 7 sections, 6 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Background on Lefschetz Fixed Point Theory
  3. Main Result
  4. Application to a Social Network Problem
  5. Existing Results
  6. Proof of a Unique Fixed Point Which Is Locally Exponentially Stable
  7. Concluding Remarks

Key Result

Theorem 1

The Lefschetz number of the identity map $\mathcal{I}_d: X\rightarrow X$ where $X$ is a compact oriented manifold or a compact triangulable space is $\chi(X)$, the Euler characteristic of $X$.

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • proof
  • Lemma 2: Corollary 7.6.2 in horn2012matrixbook
  • ...and 4 more