Complex Objects in the Polytopes of the Linear State-Space Process

TL;DR

This paper extends the DeGroot model to second order, allowing complex object resultants in the convex hull of initial states, and identifies conditions for convergence to these complex objects.

Contribution

It introduces a second order generalization of the DeGroot model that enables convergence to complex objects and identifies network and dampening conditions for such convergence.

Findings

Unique solutions exist for convergence to complex objects.

Dampening values can control the system's outcomes.

Strongly connected networks ensure control over convergence.

Abstract

A simple object (one point in $m$-dimensional space) is the resultant of the evolving matrix polynomial of walks in the irreducible aperiodic network structure of the first order DeGroot (weighted averaging) state-space process. This paper draws on a second order generalization the DeGroot model that allows complex object resultants, i.e, multiple points with distinct coordinates, in the convex hull of the initial state-space. It is shown that, holding network structure constant, a unique solution exists for the particular initial space that is a sufficient condition for the convergence of the process to a specified complex object. In addition, it is shown that, holding network structure constant, a solution exists for dampening values sufficient for the convergence of the process to a specified complex object. These dampening values, which modify the values of the walks in the network, control the system's outcomes, and any strongly connected typology is a sufficient condition of such control.