Dynamics of Networks If Everyone Strives for

Structural Holes

Vincent Buskens (Utrecht University)Arnout van de Rijt (Cornell University)

Sociology Seminar

Zürich, June 6, 2006

Importance of Dynamic Networks

• If actors benefit from certain networks and are aware of those benefits, why wouldn’t they attempt to build them? Examples:

• Introducing one’s friends to one another (balance / trust)

• Buying from dependent suppliers and vice versa (Network Exchange Theory)

• Maintaining many weak ties when searching for a job (access to information)

• ‘Networking’ in the management world (brokerage / control)

• If all actors build networks with a certain aim, which networks will emerge? And, as a result, which benefits will the actors eventually receive?

A Specific Type of Network Dynamics

• Actors minimize their “network constraint” as defined by Burt (1992)

• Utility is a strictly decreasing function of network constraint

• In other words, actors optimize structural holes in their network

What is a “Structural Hole”?

• Due to Ronald Burt (1992: p. 2): disconnections between neighbors and hence “entrepreneurial opportunities for information access, timing, referrals, and control.”

The Infamous Robert and James

Is Burt Right?

• Jobs are found faster through ties that connect otherwise disconnected groups.• The jobs found are more desirable• Salaries are higher for managers with higher structural hole scores• And structural holes are positively correlated with income, positive performance evaluations, peer reputations, promotions, and good ideas.

Evidence:Bian 1994; Talmud 1994; Burt 1995, 1997, 1998, 2000, 2001, 2004; Granovetter 1995 [1974]; Yasuda 1996; Gabbay 1997; Jang 1997; Podolny and Baron 1997; Leenders & Gabbay 1999; Lin 1999; Burt, Hogarth, and Michaud 2000; Mehra, Kilduff, and Brass 2000; Lin, Cook, & Burt 2001; Mizruchi and Sterns 2001; Burt et al. 2002

How to Measure Benefits from Structural Holes?

• Use the ‘network constraint’ measure (Burt 1992: 54):

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Does ANYONE Care About Structural

Holes?

What if EVERYONE

Pursued Holes?

Our Approach

• In accordance with recently emerging economics literature on dynamic networks

• “Two-sided link formation” model• Ties are added if both actors

agree on adding a tie• Ties can be removed without

permission

• Specify stable and efficient networks

Stability and Efficiency Concepts

• Pairwise stability: No one wants to delete a link and no pair wants to add a link

• Strong pair-wise stability: No one wants to delete a set of links and no pair wants to add a link (also pair-wise Nash equilibrium)

• Unilateral stability: No one can profitably and with consent reconfigure his links

• Pareto efficiency: There is no network in which no actor is worse off and at least one actor is strictly better off

Definitions Bipartite Networks

• Bipartite networks: actors can be divided into two (non-empty) groups and there are no ties within the two groups

• Complete bipartite networks: bipartite networks in which all ties between the two groups exist

• Balanced complete bipartite networks: complete bipartite networks for which the number of actors in the two groups are as equal as possible

Analytic Results

• Shortest path length in pair-wise stable networks is smaller than or equal to 2

• Adding a tie without creating closed triads is always beneficial

• Pair-wise stable networks are connected

• Incomplete bipartite networks are not pair-wise stable

• Complete bipartite networks are pair-wise stable except for “stars” with more than 4 actors

• Complete bipartite networks are Pareto-efficient

Further Results

• A complete bipartite network is unilaterally stable if and only if it is balanced

• Checking all networks with ten or less actors

• Limited number of stable networks

Number of Stable Networks

# actors # networks Pairwise Unilaterally

2 2 1 1

3 4 1 1

4 11 2 1

5 34 2 2

6 156 4 1

7 1044 3 1

8 12346 10 2

9 274668 9 1

10 > 12·106 14 2

Other Unilaterally Stable Networks

The Simulation

• Start from a “random” network (size 2-25)

• At each point in time, one randomly chosen actor may propose or delete one link (pair-wise stability)

• For addition, consent is needed• Actors are myopic, i.e., they will

change the link that provides them with the largest utility gain given all other existing links

• No “noise”• Run until convergence for a set of

networks

Emergence of Bipartite Networks

• Number of actors 8• Convergences to a complete

bipartite network in 85-90% of the cases

• Number of actors is odd• 75-85% to the balanced

complete bipartite network• Number of actors is even

• 50-68% to the balanced complete bipartite network

• Around 30% to the least unbalanced complete bipartite network

• Density starting network correlates with density final network

Sometimes a Social Trap…

• Some pair-wise stable networks are surprisingly inefficient

• Example: The Octahedron

(strongly Pareto dominated by, e.g., the 3-3-complete bipartite network)

…but usually a balanced complete bipartite network will emerge!

What is So Special about Balanced Complete Bipartite Networks?

• Densest one can get without forming a triangle

• No one really is a broker (if everyone wants to be in the center, there is no center)

• They are the inverse of two complete segregated clusters

Alternative analyses

• Similar results if utility of structural holes is interpreted as relative utility compared to others

• Balanced complete bipartite networks are even stronger attractors if noise is added

• If utility is simplified to two components

• Benefits of direct relations• Relatively high costs of closed

triadsStill balanced complete bipartite networks are the dominant emerging structure

First experimental test

• Subjects are in groups of six in the laboratory

• Their utility function has three components• Linear benefits of ties• Quadratic costs of ties (inducing a

capacity constraint)• Costs of closed triads

• They get 5 minutes to configure their network (10 times 30 seconds)

• After each 30 seconds subjects get paid according to the structure

• A network is considered stable if it is the same at the end of three consecutive periods of 30 seconds

Universiteit Utrecht

.

Flavor of the results

• Stable networks are in a variety of conditions predominantly pair-wise stable networks

• In the condition that resembles best Burt constraint (low capacity constraint and high costs for closed triads), all emerged stable networks (14) were balanced complete bipartite networks

• In general, unilateral stability was not necessarily a good “stable network selection criterion”