1. Network Science: A Short Introduction i3 Workshop
Konstantinos Pelechrinis
Summer 2014
Figures are taken from:
M.E.J. Newman, “Networks: An Introduction”

2. Network models
We want to have formal processes which can give rise to networks with specific properties
E.g., degree distribution, transitivity, diameters etc.
These models and their features can help us understand how the properties of a network (network structure) arise
By growing networks according to a variety of different rules/models and comparing the results with real networks, we can get a feel for which growth processes are plausible and which can be ruled out
Random graphs represent the “simplest” model

3. Erdos-Renyi random network model
Simplest model
Basic idea: nodes are connected completely at random
Given: number of nodes n
Number of edge: m
In this case for each edge m we pick uniformly at random a pair of nodes
Probability that any two nodes in the network are connected: p
In this case, we go over all possible pairs of the n nodes and connect each one of them with a probability p
Both processes/models are equivalent

4. Properties of random networks
The random network model can generate networks with:
Short paths
Giant components
It cannot generate networks with:
High clustering
Skewed degree distribution

5. Small-world model Random graphs exhibit small paths but not clustering
If we consider an ordered network (lattice) exhibits high clustering but large paths
Why not combine both these models ?

6. Small-world models
The small-world model (Watts and Strogatz 1998) tries to do exactly this
We start with a circle model of n vertices in which every vertex has a degree of c
We go through each of the edges and with some probability p we rewire it
Remove this edge and pick two vertices uniformly at random and connect them with a new edge
Shortcut edge

7. Small-world models
The parameter p controls the interpolation between the circle model and the random graph
p=0  ordered situation/circle model
p=1  random graph
Intermediate values of p give networks somewhere in between
The crucial and interesting point is that small paths appear even for small values of p as we increase from p=0, while the high clustering remains until fairly large values of p
Hence, there is a regime for values of p where both small paths as well as high clustering exists!

8. Small-world models

9. Skewed degree distribution
Small world models
Small-world
regime
For c=6 and n=600
Cannot generate:
Skewed degree distribution

10. Preferential attachment
Both previous models cannot generate skewed degree distributions
How can we have networks where there are a few nodes with a large number of edges, while the majority of them has few edges only?
A simple growth process can provide insights!
Until now we have fixed topology models
Given number of nodes and edges from the beginning
In other words, nodes do not appear one-by-one in time
A growth process refers to the evolution of the network by the addition of nodes (and edges for these nodes)

11. Preferential attachment
Nodes prefer to attach to existing nodes that have high degree!
At every point of time a new node is created and this node generates b edges
Each of this edges is connected to the existing nodes randomly
NOT UNIFORMLY AT RANDOM BUT WITH A PROBABILITY PROPORTIONAL TO THE NUMBER OF EDGES AN EXISTING NODE ALREADY HAS!
Rich-gets-richer, cumulative advantage, Matthew effect etc.