A friend of a friend is... a dense network

December 1, 2016, Santa Fe Institute
A new theoretical model shows that networks evolve very differently depending on how often friend-of-a-friend connections occur. Credit: Pixabay

It's a familiar request in the digital age: one of your friends on social media has a friend who wants to be your friend. Frequent linking among friends of friends can cause a rapid increase in social network connectivity.

A new theoretical model shows that networks evolve very differently depending on how often these "second neighbor" connections occur. The work could offer a better understanding of how dense networks form.

Networks—like those based on social media or internet connections—are often characterized by their degree, which is the number of links per member, or node. Previous models of networks have tended to focus on sparse networks in which the degree remains finite as a network grows.

By including friend-of-friend interactions in their model, Renaud Lambiotte (University of Namur, Belgium), Paul Krapivsky (Boston University), and Uttam Bhat and Sid Redner (both Santa Fe Institute) could control the link density of the network.

"It's an incredibly simple model that can produce both sparse and dense networks," says Redner, a Santa Fe Institute professor.

In their recent paper published in Physical Review Letters, the researchers constructed a general network evolution in which every new node links to one target node already in the network, as well as to each of the neighbors of the target (that is, friends of friends), with copying probability p. The likelihood of each of these "copying" steps turns out to be the crucial factor in how the network evolves.

If copying is unlikely, the network evolves into a sparse, skeleton-like framework. But when the copying probability is greater than 1/2, the network becomes dense, with the number of links growing faster than the network itself. This "densifying" behavior has been observed in real world data, such as research paper citation lists, internet router maps, and other networks.

The researchers also investigated multiple-node connections, such as triangles that consist of three mutually-linked nodes. They found that the triangle count grew faster than the for a copying probability greater than 2/3. In fact, they discovered an unlimited number of these growth transitions related to copying.

"It's kind of exotic, but cool, that such a generic model has all these transitions in it," Redner says.

If similar transitions are identified as real networks evolve—like those in —the model's copying mechanism could be an allegory for many real friend-of-friend interactions. The model may also offer a way to study the role of triangles and other so-called "cliques" as information or diseases spread in a population.

Explore further: Researchers develop algorithm to maximize friendship acceptance by strangers on social networks

More information: R. Lambiotte et al, Structural Transitions in Densifying Networks, Physical Review Letters (2016). DOI: 10.1103/PhysRevLett.117.218301

Related Stories

New paper focuses on degree centrality in networks

February 26, 2015

Social networks such as Facebook, LinkedIn and Twitter play an increasingly central role in our lives. Centrality is also an important concept in the theory of social networks. Centrality of an individual, called a "node" ...

Uncovering complex network structures in nature

December 10, 2014

The global spread of Ebola is due to the complex interactions between individuals, societies, and transportation and trade networks. Understanding and building appropriate statistical and mathematical models of these interactions ...

Recommended for you

Pattern formation—the paradoxical role of turbulence

February 19, 2018

The formation of self-organizing molecular patterns in cells is a critical component of many biological processes. Researchers from Ludwig-Maximilians-Universitaet (LMU) in Munich have proposed a new theory to explain how ...

Bringing a hidden superconducting state to light

February 16, 2018

A team of scientists has detected a hidden state of electronic order in a layered material containing lanthanum, barium, copper, and oxygen (LBCO). When cooled to a certain temperature and with certain concentrations of barium, ...

0 comments

Please sign in to add a comment. Registration is free, and takes less than a minute. Read more

Click here to reset your password.
Sign in to get notified via email when new comments are made.