Transcript by Jackie Crescimanno

Strategic Network Formation With
Structural Holes
By Jon Kleinberg, Siddharth Suri, Eva Tardos,
Tom Wexler
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
Structural Holes
The paper looks at
what would
happen to a social
network graph if all
the nodes were
incentivized to
become 'bridging'
nodes
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
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
What kinds of graphs does this
create?


Does equilibria exist
for any number of
nodes?
Can we always reach
equilibria using best
response updates?
Experiments: The possibility of
cycling
a = .9
β(r) = 2a/r
cxy = 1
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?
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
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
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?
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
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
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)
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?
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
Questions?