  The paper examines the learning mechanism of adaptive agents over
weakly-connected graphs and reveals an interesting behavior on how information
flows through such topologies. The results clarify how asymmetries in the
exchange of data can mask local information at certain agents and make them
totally dependent on other agents. A leader-follower relationship develops with
the performance of some agents being fully determined by the performance of
other agents that are outside their domain of influence. This scenario can
arise, for example, due to intruder attacks by malicious agents or as the
result of failures by some critical links. The findings in this work help
explain why strong-connectivity of the network topology, adaptation of the
combination weights, and clustering of agents are important ingredients to
equalize the learning abilities of all agents against such disturbances. The
results also clarify how weak-connectivity can be helpful in reducing the
effect of outlier data on learning performance.
