1 research outputs found
Randomized Consensus with Attractive and Repulsive Links
We study convergence properties of a randomized consensus algorithm over a
graph with both attractive and repulsive links. At each time instant, a node is
randomly selected to interact with a random neighbor. Depending on if the link
between the two nodes belongs to a given subgraph of attractive or repulsive
links, the node update follows a standard attractive weighted average or a
repulsive weighted average, respectively. The repulsive update has the opposite
sign of the standard consensus update. In this way, it counteracts the
consensus formation and can be seen as a model of link faults or malicious
attacks in a communication network, or the impact of trust and antagonism in a
social network. Various probabilistic convergence and divergence conditions are
established. A threshold condition for the strength of the repulsive action is
given for convergence in expectation: when the repulsive weight crosses this
threshold value, the algorithm transits from convergence to divergence. An
explicit value of the threshold is derived for classes of attractive and
repulsive graphs. The results show that a single repulsive link can sometimes
drastically change the behavior of the consensus algorithm. They also
explicitly show how the robustness of the consensus algorithm depends on the
size and other properties of the graphs