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? In Jackson and Wolinsky (1996) – the case of :
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 !!!