On the Efficiency of P2P Service Trading through Social Networks

We consider a set U of users connected in a social network GS = (U,F ), where the social link set F ⊂ U × U defines the friends relations between users. If user i treats j as his friend, i connects to j through a directed link 〈i, j〉 ∈ F . Suppose each user can provide some services, and also want to consume some services provided by other users. We can define a demand graph GD = (U,D), where D ⊂ U × U is the service demand profile among users. For a service demand d = (k, l) ∈ D, k is the provider and l is the consumer of the service. User k charges user l a cost of h(k,l) for providing the service. We define the associated |U | × |U | demand matrix H = [H(k, l)] as H(k, l) = h(k,l) if (k, l) ∈ D, and H(k, l) = 0 if (k, l) / ∈ D. Assume there is no centralized bank and no common currency in the system. User k and l can trade their services synchronously if they need services from each other and h(k,l) = h(l,k). However, this synchronous trading is too restrictive. To relax the synchronous requirement, the trust between friends can be utilized to allow friends exchange services asynchronously. Specifically, we introduce pairwise credit between a pair of friends 〈i, j〉. User j can obtain a service d = (i, j) from i by borrowing h(i,j) amount of credits from i. Let bij be the amount of credit j owes to i. Automatically, we have bij = −bji. Initially, there is no credit balance between i and j, bij(0) = bji(0) = 0. After i serves j once, the balance becomes bij(1) = h(i,j). If the next time i obtains a service (j, i) from j, the balance becomes bij(2) = bij(1)− h(j,i) = h(i,j) − h(j,i). Let Cij be the maximum amount of credits j can borrow from i. Then the constraint on the credit balance is bij ≤ Cij , and bji ≤ Cji. Since bij = −bji, the credit balance constraints between i and j can be summarized as −Cji ≤ bij ≤ Cij