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

## NAME

`dms` - Grow a scale-free random graph with tunable exponent

## SYNOPSIS

`dms` N m n0 a

## DESCRIPTION

`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.

## PARAMETERS

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.

## OUTPUT

`dms` prints on STDOUT the edge list of the final graph.

## EXAMPLES

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)

## AUTHORS

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

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