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Lecture 43

# JOUR 601 Lecture Notes - Lecture 43: Preferential Attachment, Graph Theory, Network Theory

by OC2033869

This

**preview**shows half of the first page. to view the full**3 pages of the document.**Network Dynamics

Traditionally, research in graph theory focuses its attention very much on studying graphs that

are static even though all real-world networks are dynamic in nature.

How they have evolved over time is a defining feature to their typology and overall makeup

Much of network theory is still focused on trying to explore the basics of static graphs as the

study of their dynamics results in the addition of a whole new setoff parameters in our models

that takes us on a whole new level of complexity

Growing a network means adding more nodes to it and adding more links to it that increase the

overall connectivity

In our random model, links were placed between nodes at random with some given probability

Growing the network here just meant growing the probability to have more links develop over

time

An interesting thing we find when we do this is that there are thresholds and phase transitions

during the network’s development

Thresholds mean by gradually increasing our link probability parameter some properties to the

network suddenly appear when we pass our critical value

Example:

Our first threshold is when our degree goes above 1 over the total number of nodes in the

network.

At this threshold, we start to get our first connection.

At degree 1, when every node has on average one connection, the network appears to be

connected. We see giant components emerging in the network. This means one dominant

cluster.

In this, we start to have cycles. This means that there are feedback loops in the network and a

threshold occurs when nodes have an average degree of log n.

At this point, everything starts to be connected meaning there is a path to other nodes in a

network. This is what we see in a random network.

Most real-world networks are not random as they are subject to some resource constraints

and they have preferential attachment giving them clusters that we don’t see in random graphs

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