## Internet Mathematics

- Internet Math.
- Volume 1, Number 1 (2003), 91-113.

### The Average Distance in a Random Graph with Given Expected Degrees

Fan Chung and Linyuan Lu

#### Abstract

Random graph theory is used to
examine the "small-world phenomenon"---any two strangers are connected
through a short chain of mutual acquaintances.
We will show that
for certain families of
random graphs with given expected degrees, the average distance is
almost surely of order
$\log n / \log \tilde d$ where
$\tilde d$ is the weighted average of the sum of squares of the
expected degrees.
Of particular interest are
power law random graphs in which the number
of
vertices of degree *k* is proportional to $1/k^{\beta}$ for some fixed
exponent $\beta $. For the case of $\beta > 3$, we prove that
the average distance of
the power law graphs is almost surely of order
$\log n / \log \tilde d$. However, many Internet, social, and citation networks
are power law graphs with exponents in the range $2 < \beta < 3$ for which the power
law random graphs
have
average distance almost surely of order $\log \log n$,
but have diameter of order $\log n$
(provided having some mild constraints for the average distance and maximum degree).
In particular, these graphs
contain
a dense subgraph, that we call
the core, having $n^{c/\log \log n}
$ vertices. Almost all vertices are within distance
$\log \log n$ of the core although there are vertices at distance $\log n$ from the core.

#### Article information

**Source**

Internet Math., Volume 1, Number 1 (2003), 91-113.

**Dates**

First available in Project Euclid: 9 July 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.im/1057768561

**Mathematical Reviews number (MathSciNet)**

MR2076728

**Zentralblatt MATH identifier**

1065.05084

#### Citation

Chung, Fan; Lu, Linyuan. The Average Distance in a Random Graph with Given Expected Degrees. Internet Math. 1 (2003), no. 1, 91--113. https://projecteuclid.org/euclid.im/1057768561