1. Section 8.2: Shortest path and small world effect
By: Ralucca Gera, NPS
Most pictures are from Newman’s textbook

2. The small world effect
The typical network average distances between vertices are surprisingly small in real life networks: coined as small world effect
Recall Stanley Milgram’s letter-passing experiment had an average of 6 hops (in the inferred network).
In math terms the small world effect is a hypothesis that the mean distance 𝑙 is “small”
Recall from Chapter 7 that 𝑙= 𝑖, 𝑗 𝑑(𝑖,𝑗) 𝑛 2 , setting 𝑑 𝑖,𝑗 =0, if there is no 𝑖𝑗 path (𝑖 and 𝑗 belong to different components)

3. Stanley Milgram’s experiment

4. Funneling was observed by Milgram in his experiment:
The small world effect
Funneling was observed by Milgram in his experiment:
Most of the shortest paths to a sink vertex i go through one of its neighbors, so there is this funneling towards the destination

5. The small world effect
In math terms the small world effect is a hypothesis that the mean distance 𝑙 is “small”
Typically networks have been found to have mean distance less than 20 – or in many cases less than 10 – even though the networks themselves have millions of nodes
This has implications such as
rumor spread in a social networks,
response time in the Internet
disease spreading in social networks
What is 𝑙 for your networks?
Thus it is no surprise that real networks have a small 𝑙

6. Statistics for real networks
𝑙 is the average distance

7. The small world effect
Mathematical models for networks try to mimic: small average path length 𝑙 and high clustering
Observed:
𝑙 increases slowly with the number 𝑛 of vertices in the network: 𝑙 ~ log 𝑛 log <𝑘>
The diameter of a network is relatively small as well: 𝑑𝑖𝑎𝑚 ~ log 𝑛 (in scale free 𝑑𝑖𝑎𝑚 ~ log ( log 𝑛 )
High average clustering coefficient
Average degree
Reuven Cohen and Shlomo Havlin
Phys. Rev. Lett. 90, – Published 4 February 2003

8. How to construct Small-words?
This is a model introduced by Watts-Strogatz: Networks that share properties of both regular and random graphs (Watts and his advisor Strogatz)
Regular/lattice
Small world
Random graphs
clustering coefficient
High
Low
average path length
p = probability of rewiring edges of the lattice
Source: Watts, DJ; Strogatz, S H Collective dynamics of 'small-world' networks, NATURE 393(668).

9. Example small word
Avg path
Avg clust
From Ernesto Estrada

10. Example small word
Avg path
Avg clust
From Ernesto Estrada

11. Example small word
Avg path
Avg clust
From Ernesto Estrada