- f3m(1)
- www.complex-networks.net
- f3m(1)

`f3m`

- Count all the 3-node subgraphs of a directed graph

`f3m`

`graph_in` [`num_random`]

`f3m`

performs a motif analysis on `graph_in`, i.e., it counts all the
3-node subgraphs and computes the z-score of that count with respect
to the corresponding configuration model ensemble.

`graph_in`input graph (edge list). It must be an existing file.

`num_random`The number of random graphs to sample from the configuration model for the computation of the z-score of the motifs.

`f3m`

prints on the standard output a table with 13 rows, one for each
of the 13 possible 3-node motifs. Each line is in the format:

```
motif_number count mean_rnd std_rnd z-score
```

where `motif_number`

is a number between 1 and 13 that identifies the
motif (see MOTIF NUMBERS below), `count`

is the number of
subgraphs ot type `motif_number`

found in `graph_in`, `mean_rnd`

is
the average number of subgraphs of type `motif_number`

in the
corresponding configuration model ensemble, and `std_rnd`

is the
associated standard deviation. Finally, `z-score`

is the quantity:

```
(count - mean_rnd) / std_rnd
```

The program also prints a progress bar on STDERR.

We report below the correspondence between the 13 possible 3-node
subgraphs and the corresponding `motif_number`

. In the diagrams,
'O--->O' indicates a single edge form the left node to the right node,
while 'O`==`O' indicates a double (bi-directional) edge between the
two nodes:

```
(1) O<---O--->O
(2) O--->O--->O
(3) O<==>O--->O
(4) O--->O<---O
(5) O--->O--->O
\ ^
\_______|
(6) O<==>O--->O
\ ^
\_______|
(7) O<==>O<---O
(8) O<==>O<==>O
(9) O<---O<---O
\ ^
\_______|
(10) O<==>O<---O
\ ^
\_______|
(11) O--->O<==>O
\ ^
\_______|
(12) O<==>O<==>O
\ ^
\_______|
(13) O<==>O<==>O
^\ ^/
\\_____//
\_____/
```

To perform a motif analysis on the E.coli transcription regulation graph, using 1000 randomised networks, we run the command:

```
$ f3m e_coli.net 1000
1 4760 4400.11 137.679 +2.614
2 162 188.78 8.022 -3.338
3 0 0.89 3.903 -0.228
4 226 238.32 7.657 -1.609
5 40 6.54 2.836 +11.800
6 0 0.01 0.077 -0.078
7 0 0.12 0.642 -0.192
8 0 0.00 0.032 -0.032
9 0 0.01 0.109 -0.110
10 0 0.00 0.000 +0.000
11 0 0.00 0.032 -0.032
12 0 0.00 0.000 +0.000
13 0 0.00 0.000 +0.000
$
```

Notice that the motif `5`

(the so-called "feed-forward loop") has a
z-score equal to 11.8, meaning that it is highly overrepresented in
the E.coli graph with respect to the corresponding configuration model
ensemble. Conversely, the motif `2`

(three-node chain) is
underrepresented, as made evident by value of the z-score (-3.338).

R. Milo et al. "Network Motifs: Simple Building Blocks of Complex Networks". Science 298 (2002), 824-827.

R. Milo et al. "Superfamilies of evolved and designed networks." Science 303 (2004), 1538-1542

V. Latora, V. Nicosia, G. Russo, "Complex Networks: Principles, Methods and Applications", Chapter 8, Cambridge University Press (2017)

V. Latora, V. Nicosia, G. Russo, "Complex Networks: Principles, Methods and Applications", Appendix 16, Cambridge University Press (2017)

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

.

- www.complex-networks.net
- September 2017
- f3m(1)