A discrete computer network model with expanding dimensions

TL;DR

This paper introduces a novel discrete network model with expanding dimensions, analyzing stability properties, including a new concept called devil networks, relevant for data transmission in computer networks.

Contribution

It formulates a rigorous model for discrete linear time-varying networks with expanding dimensions and establishes stability criteria, including for unstable devil networks.

Findings

Network stability is significantly affected by structure and connection weights.

New stability and instability criteria are established for general network classes.

A novel concept of devil networks is introduced and analyzed.

Abstract

Complex networks with expanding dimensions are studied, where the networks may be directed and weighted, and network nodes are varying in discrete time in the sense that some new nodes may be added and some old nodes may be removed from time to time. A model of such networks in computer data transmission is discussed. Each node on the network has fixed dimensionality, while the dimension of the whole network is defined by the total number of nodes. Based on the spectacular properties of data transmission on computer networks, some new concepts of stable and unstable networks differing from the classical Lyapunov stability are defined. In particular, a special unstable network model, called devil network, is introduced and discussed. It is further found that a variety of structures and connection weights affects the network stability substantially. Several criteria on stability, instability, and devil network are established for a rather general class of networks, where some conditions are actually necessary and sufficient. Mathematically, this paper makes a first attempt to rigorously formulate a fundamental issue of modeling discrete linear time-varying systems with expanding dimensions and study their basic stability property.