The t$-$Party Evolving Network Model

In the t$-$party gender play no role, hence each newcomer is allowed to invite only one other participants to a dance. However, attractiveness plays a role: More attractive participants are more likely to be invited to a dance by a new participant. The party evolves following these rules:

Every participant corresponds to a node i and is assigned a time$-$independent attractiveness coefficient i$.$

At each time step a new node joins the t$-$party.

This new node then invites one already partying node to a dance, establishing a new link with it$.$

The new node chooses its dance partner with probability proportional to the potential partner's attractiveness. If there are t nodes already in the party, the probability that node i receives a dance invitation is i$=$ijj$=$it

where is the average attractiveness.

Derive the time evolution of the node degrees, telling us how many dances a node had.

Derive the degree distribution of nodes with attractiveness $.$

If half of the nodes have $=2,$ and the other half $=1,$ what is the degree distribution of the network after a sufficiently long time?