Price model
* Introduction
- ER graph fails to explain the existence of fat tails in the degree
distribution of real networks. Thus, the natural question to ask is what
mechanisms are needed to produce heavy tails?
- Price was interested in citation networks, and found a presence of
approximate power-laws in the citations: most papers have very few
citations, while few papers have too many citations
- To explain this, he proposed a model in which a network grows with the
addition of new vertices. For example, in citation network, new papers
keep getting published and they cite the already published papers.
- New papers cite those papers with higher probability which are already
well cited.
- A similar mechanism was proposed by Simon for explaining heavy tails in
the wealth distributions. The basis of this mechansim was that people
gain money proportional to the wealth they have, and hence
rich-get-richer.
- Price coined the term `cumulative advantage' to this mechanism, although
now we usually call it `preferential attachment' coined by Barabasi and
Albert in 1999.
* Model description
- The price model consists of following steps:
1. Vertices are added one by one to the network, and every new vertex has
c out links (In the original model, c only the average number of links
the new vetex has, and the actual number is distributed around c.
However, for our aim, it doesn't matter much, and we will assume that
the number of out-links is exactly c).
2. Each new vertex connects to the c previous vertices (not necessarily
distinct) preferentially: the probability that it connects a given old
vertex v is proportional to the in-degree of v. Thus, the vertices
which already have higher in-degree will gain even more in-links.
- It is easy to see that the new vertex cannot connect a given old vertex
with probability exactly proportional to its in-degree; in that case the
vertices with zero in-degree will never accumulate any more links. Thus,
in the model, the probability is made proportional to (q + a), where q is
the in-degree of the old vertex, and a is constant.
* Degree distribution of the Price model
- Refer to the slides, or the Ch14 in the Newman's book (the old edition)