- dms(1)
- www.complex-networks.net
- dms(1)

`dms`

- Grow a scale-free random graph with tunable exponent

`dms`

`N` `m` `n0` *a*

`dms`

grows an undirected random scale-free graph with `N` nodes using
the modified linear preferential attachment model proposed by
Dorogovtsev, Mendes and Samukhin. The initial network is a clique of
`n0` nodes, and each new node creates `m` new edges. The resulting
graph will have a scale-free degree distribution, whose exponent
converges to `gamma=3.0 + a/m`

for large `N`.

`N`Number of nodes of the final graph.

`m`Number of edges created by each new node.

`n0`Number of nodes in the initial (seed) graph.

*a*This parameter sets the exponent of the degree distribution (

`gamma = 3.0 + a/m`

).*a*must be larger than`-m`.

`dms`

prints on STDOUT the edge list of the final graph.

Let us assume that we want to create a scale-free network with
`N=10000` nodes, with average degree equal to 8, whose degree
distribution has exponent

```
gamma = 2.5
```

Since `dms`

produces graphs with scale-free degree sequences with an
exponent `gamma = 3.0 + a/m`

, the command:

```
$ dms 10000 4 4 -2.0 > dms_10000_4_4_-2.0.txt
```

will produce the desired network. In fact, the average degree of the graph will be:

```
<k> = 2m = 8
```

and the exponent of the power-law degree distribution will be:

```
gamma = 3.0 + a/m = 3.0 -0.5 = 2.5
```

The following command:

```
$ dms 10000 3 5 0 > dms_10000_3_5_0.txt
```

creates a scale-free graph with `N=10000` nodes, where each new node
creates `m=3` new edges and the initial seed network is a ring of
`n0=5` nodes. The degree distribution of the final graph will have
exponent equal to `gamma = 3.0 + a/m = 3.0`

. In this case, `dms`

produces a Barabasi-Albert graph (see ba(1) for details). The edge
list of the graph is saved in the file `dms_10000_3_5_0.txt`

(thanks
to the redirection operator `>`

).

S. N. Dorogovtsev, J. F. F. Mendes, A. N. Samukhin. "Structure of Growing Networks with Preferential Linking". Phys. Rev. Lett. 85 (2000), 4633-4636.

V. Latora, V. Nicosia, G. Russo, "Complex Networks: Principles, Methods and Applications", Chapter 6, Cambridge University Press (2017)

V. Latora, V. Nicosia, G. Russo, "Complex Networks: Principles, Methods and Applications", Appendix 13, Cambridge University Press (2017)

(c) Vincenzo 'KatolaZ' Nicosia 2009-2017 `<v.nicosia@qmul.ac.uk>`

.

- www.complex-networks.net
- September 2017
- dms(1)