Abstract | We study network connection games where the nodes of a networ
k perform edge swaps
in order to improve their communication costs. For the model
proposed by [2], in which the selfish
cost of a node is the sum of all shortest path distances to the o
ther nodes, we use the probabilistic
method to provide a new, structural characterization of equ
ilibrium graphs. We show how to use this
characterization in order to prove upper bounds on the diame
ter of equilibrium graphs in terms of the
size of the largest
k
-vicinity (defined as the the set of vertices within distance
k
from a vertex), for
any
k
≥
1 and in terms of the number of edges, thus settling positivel
y a conjecture of [2] in the cases
of graphs of large
k
-vicinity size (including graphs of large maximum degree) a
nd of graphs which are
dense enough.
Next, we present a new swap-based network creation game, in w
hich selfish costs depend on the imme-
diate neighborhood of each node; in particular, the profit of
a node is defined as the sum of the degrees
of its neighbors. We prove that, in contrast to the previous m
odel, this network creation game admits
an exact potential, and also that any equilibrium graph cont
ains an induced star. The existence of the
potential function is exploited in order to show that an equi
librium can be reached in expected polyno-
mial time even in the case where nodes can only acquire limite
d knowledge concerning non-neighboring
nodes. |