Link-centric preferential attachment

In mathematical modeling of social networks, link-centric preferential attachment^[1][2] is a node's propensity to re-establish links to nodes it has previously been in contact with in time-varying networks.^[3] This preferential attachment model relies on nodes keeping memory of previous neighbors up to the current time.^[1][4]

Background

In real social networks individuals exhibit a tendency to re-connect with past contacts (ex. family, friends, co-workers, etc.) rather than strangers. In 1970, Mark Granovetter examined this behaviour in the social networks of a group of workers and identified tie strength, a characteristic of social ties describing the frequency of contact between two individuals. From this comes the idea of strong and weak ties,^[5] where an individual's strong ties are those she has come into frequent contact with. Link-centric preferential attachment aims to explain the mechanism behind strong and weak ties as a stochastic reinforcement process for old ties in agent-based modeling where nodes have long-term memory.

Examples

In a simple model for this mechanism, a node's propensity to establish a new link can be characterized solely by ${\displaystyle n}$, the number of contacts it has had in the past. The probability for a node with n social ties to establish a new social tie could then be simply given by^[4]

${\displaystyle P(n)={c \over n+c}\,}$

where c is an offset constant. The probability for a node to re-connect with old ties is then

${\displaystyle 1-P(n)={n \over n+c}.}$

Figure 1. shows an example of this process: in the first step nodes A and C connect to node B, giving B a total of two social ties. With c = 1, in the next step B has a probability P(2) = 1/(2 + 1) = 1/3 to create a new tie with D, whereas the probability to reconnect with A or C is twice that at 2/3.

More complex models may take into account other variables, such as frequency of contact, contact and intercontact duration, as well as short term memory effects.^[1]

Effects on the spreading of contagions / weakness of strong ties

Understanding the evolution of a network's structure and how it can influence dynamical processes has become an important part of modeling the spreading of contagions.^[6][7] In models of social and biological contagion spreading on time-varying networks link-centric preferential attachment can alter the spread of the contagion to the entire population. Compared to the classic rumour spreading process where nodes are memory-less, link-centric preferential attachment can cause not only a slower spread of the contagion but also one less diffuse. In these models an infected node's chances of connecting to new contacts diminishes as their size of their social circle ${\displaystyle n}$ grows leading to a limiting effect on the growth of n. The result is strong ties with a node's early contacts and consequently the weakening of the diffusion of the contagion.^[1][4]

See also

• BA model
• Network science
• Interpersonal tie

References