Power Power and and

Core-Periphery NetworksCore-Periphery Networks

Dotan PersitzDotan Persitz

June 2011June 2011

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Introduction

Why do core-periphery networks emerge?

Strategic network formation model with heterogeneous agents.

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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).

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Core-periphery networks - definition

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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)).

Pharmaceutical-biotech alliances

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Core-periphery networks – observation

Mullins et al. (1977):

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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.

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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.

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The Model

The utility of agent i:

The intrinsic value function:

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The intrinsic value function

The values are positive.

Interpretation:– Power-based preferences: .– Homophilic preferences: . – Heterophilic preferences: .– Jackson and Wolinsky (1996) preferences: .

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The Model – solution concepts

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Core Periphery networks – classification

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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

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Extremely low linking costs

The architecture does not reflect any heterogeneity.

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Low linking costs

The type a agents acquire better position:– Very connected among themselves.– Serve as bridges for the type b agents.

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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.

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Medium linking costs

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The Heterogeneous Connections Model

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High linking costs

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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.

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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.

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Extremely high linking costs

A-stars, AB-stars and the empty network dominate this extreme range of costs.

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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.

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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.

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Go Canucks Go !!!