Power and Core-Periphery Networks

1. Power and Core-Periphery Networks Dotan Persitz June 2011

2. Introduction • Why do core-periphery networks emerge? • Strategic network formation model with heterogeneous agents.

3. Core-periphery networks - definition • In a core-periphery network the nodes can be partitioned into two subsets: • Core – every two agents are connected. • Periphery – every two agents are disconnected. • No restriction on the links between the core and the periphery. • Various definitions in the literature. • Graph Theory: split graphs (Foldes & Hammer (1977)). • Sociology: Borgatti & Everett (1999). • Economics: Bramoulle & Kranton (2005) and Bramoulle (2007).

4. Core-periphery networks - definition

5. Real world core-periphery networks • Frequent (White et al. (1976)). • Geographical networks: • Highways, streets, airports and hardwired internet (Holme (2005)) • Social networks: • Scientific collaboration networks (Mullins et al. (1977), van der Leij & Goyal (2009), Moody (2004)). • Drug users (Curtis et. al. (1995)). • Industrial Networks: • Interlocking directorates (Mintz & Schwartz (1981), Davis et al. (2003)). • Research collaboration (Baker et al. (2008)).

6. Pharmaceutical-biotech alliances

7. Core-periphery networks – observation Mullins et al. (1977):

8. Possible story • Consider a set of agents that form a network. • Once in a while, one of the agents comes up with an innovative idea. • Other agents get the information regarding this new idea through the network, with a delay (increases with the distance from the source). • There are two types of agents - “superior” and “inferior”. • The probability that a “superior” agent will come up with a new idea is higher than the probability that an “inferior” agent will do so. • Also, “superior” agents are more able than “inferior” agents in exploiting new ideas. • In any other respect, the two types are identical. • It is more beneficial to be linked to a “superior” agent (directly or indirectly). • A “superior” agent benefits more than an “inferior” agent from any given path.

9. The Model • n individuals in the social network. • Two types of agents: • The type of an agent will be denoted by • agents of type a. • agents of type b.

10. The Model • The utility of agent i: • The intrinsic value function:

11. The intrinsic value function • The values are positive. • Interpretation: • Power-based preferences: . • Homophilic preferences: . • Heterophilic preferences: . • Jackson and Wolinsky (1996) preferences: .

12. The Model – solution concepts

13. Core Periphery networks – classification

14. Results • Characterization of the architecture under power-based preferences (stability and efficiency). • Fix the depreciation rate and the intrinsic value function and increase the linking costs. • AB-CP network is a core-periphery network in which: • All the core agents are of type a • All the periphery agents are of type b

15. Extremely low linking costs • The architecture does not reflect any heterogeneity.

16. Low linking costs • The type a agents acquire better position: • Very connected among themselves. • Serve as bridges for the type b agents.

17. Strong power-based preferences • Restrictions on the agents’ linking preferences: • Type a agents: . • Type b agents: . • In partially strong power-based preferences only the first holds. • Trade-off between decay and heterogeneity. • Decay gives an incentive to connect with “distant” agents. • Decay is a function of the decay factor and the distance. • Heterogeneity gives an incentive to connect with type a agents.

18. Medium linking costs

19. The Heterogeneous Connections Model

20. High linking costs

21. Q • What is Q? • Let g be AB-DisCP and let g’ be the AB-OGMinCP network. • The net benefits per payment from moving from g to g’ is: • In Jackson and Wolinsky (1996) – the case of : • The net benefits per payment from moving from the empty network to the star network is . • Q is interesting, but not a result of introducing heterogeneity.

22. Proposition 4 – proof (flavor) • AB-DisCP is pairwise stable by the linking costs range. • Efficiency: • Type a agents form a clique (positive externalities). • AB-OGMinCP is the best among the connected. • Type b agents which are not in the main component are isolates. • Either AB-DisCP or AB-OGMinCP is the best among the remaining candidates. • Q determines the efficient network. • Uniqueness when Q is negative: • Type a agents form a clique. • A general result: in a pairwise stable network each agent has non-negative utility. • Assume there is another pairwise stable network. Then, it must have higher total utility than AB-DisCP. Contradiction.

23. Extremely high linking costs • A-stars, AB-stars and the empty network dominate this extreme range of costs.

24. Issues • Homophily and Heterophily. • More complex architectures: • Semi periphery: • Weakening Assumption 1. • Introducing a third type. • Introducing simple linking costs heterogeneity. • Multiple peripheries: • Distinguishing the advantages of ‘superior agents’. • Stability concepts. • Pairwise Nash. • Calvo-Armengol & Ilkilic (2009). • “Switch”. • Needs some form of “farsighted” notion. • The formation process.

25. Conclusions • Core periphery structures are frequent in the “world”. • “Power-based” linking preferences are suggested as a possible explanation for the emergence of such architectures. • Such social preferences may deepen inequality by granting an additional positional advantage to the already exogenously privileged.

26. Go Canucks Go !!!