Networks - an Introduction

by M.E.J. Newman

Chapter 2: Technological Networks

I’m interested in bolstering my understanding of the internet as a packet-switching network, and how this compares to other networks (circuit-switching, and otherwise). What does network science have to teach us about network congestion?

The topic kicks off with a reminder that there is no cartographer approving new vertices or edges in the network, but rather the Border Gateway Protocol (BGP) implemented by the world’s routers. I like this quote, “There is no one whose job it is to maintain an official map of the network”. So how do we study the network’s structure if there’s not a map? I have the feeling traceroute is involved.

It appears most internet maps terminate at the router-level. Possible groupings of router IPs include subnets, domains, and autonomous systems.

Chapter 18: Dynamical Systems on Networks

The definition of dynamical systems given here suprised me - “any system whose state, as represented by a some set of quantitative variables, changes over time according to some given rules or equations”. For some reason I think I have been conflating the definition of dynamical systems with complex systems, which the Complex Systems Society defines as - “systems where the collective behavior of their parts entails emergence of properties that can hardly, if not at all, be inferred from properties of the parts.

One way to begin approaching the analysis of dynamical systems is to study steady states. We can solve our equation for that point, develop a taylor series to approximate a polynomial expression of the system, and then determine if the fixed point is a attracting or repelling. This is called linear stability analysis (LSA), and begins with identifying the set of equilibrium fixed points. You confirm these points by playing out the behavior of small shifts in intial position. Do these points attract towards a fixed point, repel from it, or just kind of chill (marginal)?