In the future, analysis of social networks will conceivably move from graphs to hypergraphs. However, theory has not yet caught up with this type of data organizational structure. By introducing and analyzing a general model of preferential attachment hypergraphs, this paper makes a step towards narrowing this gap. We consider a random preferential attachment model H(p, Y) for network evolution that allows arrivals of both nodes and hyperedges of random size. At each time step t, two possible events may occur: (1) [vertex arrival event:] with probability p > 0 a new vertex arrives and a new hyperedge of size Yt, containing the new vertex and Yt − 1 existing vertices, is added to the hypergraph; or (2) [hyperedge arrival event:] with probability 1 − p, a new hyperedge of size Yt, containing Yt existing vertices, is added to the hypergraph. In both cases, the involved existing vertices are chosen independently at random according to the preferential attachment rule, i.e., with probability proportional to their degree, where the degree of a vertex is the number of edges containing it. Assuming general restrictions on the distribution of Yt, we prove that the H(p, Y) model generates power law networks, i.e., the expected fraction of nodes with degree k is proportional to k−1−⌈, where [EQUATION]. This extends the special case of preferential attachment graphs, where Yt = 2 for every t, yielding ⌈ = 2/(2 − p). Therefore, our results show that the exponent of the degree distribution is sensitive to whether one considers the structure of a social network to be a hypergraph or a graph. We discuss, and provide examples for, the implications of these considerations.

Consider a random preferential attachment model G(p) for network evolution that allows both node and edge arrivals. Starting with an arbitrary nonempty graph G0, at each time step, there are two possible events: with probability p > 0 a new node arrives and a new edge is added between the new node and an existing node, and with probability 1 - p a new edge is added between two existing nodes. In both cases, the involved existing nodes are chosen at random according to preferential attachment, i.e., with probability proportional to their degree. G(p) is known to generate power law networks, i.e., the fraction of nodes with degree k is proportional to k-β. Here β=(4-p)/(2-p) is in the range (2,3].

In this paper we study distributed agent matching in environments characterized by uncertain signals, costly exploration, and the presence of an information broker. Each agent receives information about the potential value of matching with others. This information signal may, however be noisy, and the agent incurs some cost in receiving it. If all candidate agents agree to the matching the team is formed and each agent receives the true unknown utility of the matching, and leaves the market. We consider the effect of the presence of information brokers, or experts, on the outcomes of such matching processes. Experts can, upon payment of a fee, perform the service of disambiguating noisy signals and revealing the true value of a match to any agent. We analyze equilibrium behavior given the fee set by a monopolist expert and use this analysis to derive the revenue maximizing strategy for the expert as the first mover in a Stackelberg game. Surprisingly, we find that better information can hurt: the presence of the expert, even if the use of its services is optional, can degrade both individual agents' utilities and overall social welfare. While in one-sided search the presence of the expert can only help, in two-sided (and general k-sided) search the externality imposed by the fact that others are consulting the expert can lead to a situation where the equilibrium outcome is that everyone consults the expert, even though all agents would be better off if the expert were not present. As an antidote, we show how market designers can enhance welfare by taxing use of expert services.