7 research outputs found
Stabilization Bounds for Influence Propagation from a Random Initial State
We study the stabilization time of two common types of influence propagation.
In majority processes, nodes in a graph want to switch to the most frequent
state in their neighborhood, while in minority processes, nodes want to switch
to the least frequent state in their neighborhood. We consider the sequential
model of these processes, and assume that every node starts out from a uniform
random state.
We first show that if nodes change their state for any small improvement in
the process, then stabilization can last for up to steps in both
cases. Furthermore, we also study the proportional switching case, when nodes
only decide to change their state if they are in conflict with a
fraction of their neighbors, for some parameter . In this case, we show that if , then there
is a construction where stabilization can indeed last for
steps for some constant . On the other hand, if ,
we prove that the stabilization time of the processes is upper-bounded by
Stabilization Time in Weighted Minority Processes
A minority process in a weighted graph is a dynamically changing coloring.
Each node repeatedly changes its color in order to minimize the sum of weighted
conflicts with its neighbors. We study the number of steps until such a process
stabilizes. Our main contribution is an exponential lower bound on
stabilization time. We first present a construction showing this bound in the
adversarial sequential model, and then we show how to extend the construction
to establish the same bound in the benevolent sequential model, as well as in
any reasonable concurrent model. Furthermore, we show that the stabilization
time of our construction remains exponential even for very strict switching
conditions, namely, if a node only changes color when almost all (i.e., any
specific fraction) of its neighbors have the same color. Our lower bound works
in a wide range of settings, both for node-weighted and edge-weighted graphs,
or if we restrict minority processes to the class of sparse graphs
A General Stabilization Bound for Influence Propagation in Graphs
We study the stabilization time of a wide class of processes on graphs, in
which each node can only switch its state if it is motivated to do so by at
least a fraction of its neighbors, for some . Two examples of such processes are well-studied dynamically changing
colorings in graphs: in majority processes, nodes switch to the most frequent
color in their neighborhood, while in minority processes, nodes switch to the
least frequent color in their neighborhood. We describe a non-elementary
function , and we show that in the sequential model, the worst-case
stabilization time of these processes can completely be characterized by
. More precisely, we prove that for any ,
is an upper bound on the stabilization time of
any proportional majority/minority process, and we also show that there are
graph constructions where stabilization indeed takes
steps