And very briefly speaking, Before going to mass material preferential

attachments says that when new nodes added to a graph, they are more likely to be

connected to those with a lot of connections already.

So the rich gets richer more connected gets even more connected.

Conceptually this is said the self organizing growth mechanism of the graph

leads to paralogical distribution And mathematically, just, turns out that

sampling, by density leads to a difference equation.

We'll see that in advanced material whose solution gives you equilibrium that

satisfies a power law. In contrast,

Constraint optimization generates power law distribution in very different ways,

so that the graph topology is really designed with some objective functions,

and some constraints in mind. And yet, resulting topology can also show

power law. Conceptually it said that power law is the

natural outcome of constraint optimization.

With objective driven by economics, for example in constraints driven by

technology, mathematically says that either entropy maximization, or what's

called iso-elastic utility maximization and the linear constraints.

We're going to see these in the advanced material part of the reading for this

lecture. Either of them, among many others would

give rise to a power law. So there lies, the main differences

between preferred attachment, and constraint optimization.

So the question is which one to use? Well, it depends on which one gives you

the predictive power that you need on other properties.

Not the power law distribution anymore, because they both generate that.

Other properties of the graph, such as robustness to a target attack of highly

connected nodes. Would that break the network?

Would that break just the edge? Whichever, generate a model can also have

a predict on other properties that you care about, should be the one that you

use. And there lies an interesting story on the

pitfall of generated models. In 2000 There were so much talk about the

internet having an Achilles heel. It turns out that if you look at empirical

data, you go talk to At and t or Cisco-Juniper you'll realize that, that's

just now supported by factual data. It highlights the importance of

fortification of any self consistent theory through empirical data of the field

that you are talking about. It also highlights the risk of over

generalizing something called a network science, which is a fuzzy and mostly, a

meaning less term, unless you provide some concrete meaning to it By overgeneralizing

it to some universally true property it actually loses domain-specific

functionality models. And thereby often loses predictive power

and even lead to confusing misleading conclusions.

Such as the non-existence of this Achilles heels So we have seen generative models.

Reverse engineering of network apology, small whirls and scale free, and later

we'll also look at reverse engineering of network protocols.

We have also seen. The interplay between topology, say a

graph and the corresponding matrices, versus functionalities.

Whether there's navigation or search. Routing or robustness to a specific kind

of attacks. And, the interplay between these two

dimensions of what we call a network whether socio-economic or tech network

will continue to show up in the rest of the course.

As we conclude this point. The midpoint of this twenty questions

about our net (for) life.