Ordinal-time

Ordinal-time: Consider a complete time sequence of an evolving social network that includes each time a new network link is formed and each time an individual adopts a new behavior. For each k, consider the set of all individuals who were ever k-exposed at any time, and define po(k) to be the fraction of this set that became adopters before acquiring a (k + 1)st neighbor who is an adopter.

Figure 1 shows the shape of influence curves for the ordinal-time definitions using the Wikipedia data:

$p_{o}\left(k\right)$ is the fraction of cases in which a node that is k-exposed to a community C proceeds to join C before acquiring a (k + 1)st neighbor in C. That is, we define $p_{o}\left(k\right)$ as the ratio of two quantities, $p_{o}\left(k\right)=n_{o}\left(k\right)/d_{o}\left(k\right)$, where $d_{o}\left(k\right)$ is the number of triples (u;C; k) for which u was ever k-exposed to C, and $n_{o}\left(k\right)$ is the number of triples (u,C,k) for which u was k-exposed to C, and then joined C before acquiring a (k + 1)st neighbor in C.