Strategic Network Formation With Structural Holes
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Strategic Network Formation With Structural Holes
By Jon Kleinberg, Siddharth Suri, Eva Tardos, Tom Wexler
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Structural Holes
Structural holes theory suggests that node A is in a stronger position than the other nodes, because it can control the flow of information between the three otherwise independent groups of nodes
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Structural Holes
The paper looks at what would happen to a social network graph if all the nodes were incentivized to become 'bridging' nodes
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The Model
The payoff for a node u is
a |N(u)| + Σv,w N∈ (u) (β(rvw ) ) −Σv L(u)∈ (cuv ), where
a = the static benefit associated with having a link with another node
N(u) = the number of nodes connected to u
β = any decreasing function
rvw = the number of length 2 paths between v and w, if v and w are not connected, 0 otherwise
L(u) = the number of nodes u has bought a link to
cuv = the cost associated with the u,v edge
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Computing a node's best move
Can be done in polynomial time
Proof in the paper, via a reduction to the largest weight ideal problem, which can be reduced to the minimum cut of a network
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What kinds of graphs does this create?
Does equilibria exist for any number of nodes?
Can we always reach equilibria using best response updates?
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Experiments: The possibility of cycling
a = .9β(r) = 2a/rcxy = 1
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The Cost Matrix: Uniform
We first look at what would happen if the 'cost' of maintaining an edge was constant (in this case, cuv = 1 for every edge), and will try to answer the following questions:
Does there always exist some equilibrium, for a graph of n nodes?
If so, is it always reachable by round robin best response updates?
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Does equilibrium exist: Uniform Metric
Let Gn,k be a multipartite graph of n nodes, where the nodes are split up into n/k roughly equal sized groups, and every node in the ith group buys connections to every node in the jth group, for all j<i
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Does equilibrium exist: Uniform Metric
Can we chose k such that Gn,k is at equilibrium?
Yes - we do this by defining a benefit function B(n,k) = k(a-1) + Ck,2 β(n-k), and picking k' such that B(n, k')>0 and B(n, k'-1)<=0
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Can we always reach equilibrium: Uniform Metric
We have shown that for any n, there is always a k, such that Gn,k is in equilibrium.
Will our algorithm for computing best response dynamics reach an equilibrium?
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Do other Equilibria exist: Uniform
Yes! After running several
experiments, all equilibria were found to be dense, Ω(n^2) edges
The paper then proves that all equilibria are dense, assuming rβ(r) >0
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The Cost Matrix: Hierarchical
Useful for situations like the dynamics of a large company's social network
Here, we let the cost cuv, be the unique simple path between nodes u,v in the tree
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Does equilibrium exist: Hierarchical Metric
This is still an open question, for arbitrarily large n
However, running experiments suggest that when equilibrium does exist, it occurs with a small group of people with links to everyone, a few people with a significant number of links, and most with very few links
Average degree being O(√n)
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Conclusions
In both hierarchical and uniform metrics, we end up with a network divided into social classes, where a small number of nodes maintain O(n) links, and most nodes have much less
Even starting from an empty graph, the bridging incentive causes a break in the symmetry, but what happens under different bridging conditions?
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Other Research Sanjeev Goyal and Fernando Vega-Redondo's
'Structural holes in social networks' uses a model where a node u receives benefits from residing on arbitrarily long paths between two other nodes, w and v. Here, star networks turn out to be the most robust equilibrium, for a wide range of parameters
Vincent Buskens and Arnout van de Rijt's 'Dynamics of networks if everyone strives for structural holes' looks at only benefits from length 2 paths, but uses a stricter form of equilibrium, which they call unilateral stability
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Questions?