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The phrase "a small network" is correct and usable in written English.
You can use it when referring to a limited or restricted group of interconnected individuals, devices, or systems.
Example: "In our research, we focused on a small network of local businesses to understand their impact on the community."
Alternatives: "a limited network" or "a minor network."
Exact(58)
Layla maintains a small network of tipsters whom she pays for information.
On a small network, you get a little pop in the numbers and everybody's thrilled".
Shaab Samadai: home to a small network of caves and a number of spinner dolphins.
The Excellence School is part of Uncommon Schools, a small network of charter schools.
France Inc continues to be influenced by a small network of politically connected power brokers.
But Mr. Baine said that a small network is a low priority for cable operators.
"Being a small network infrastructure company, that's a very hard business," Mr. Kuittinen said.
For a small network, that can help level the playing field.
Not only that, but I have a small network of people I can discuss the book with.
A few years ago, Mr. Kardon and a small network of lawyers across the country tried a different tack.
A small network of community legal centres exists to empower local communities to protect themselves and their environment.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com