People communicate with those who have the same background or share a common
interest by using a social networking service (SNS). News or messages propagate
through inhomogeneous connections in an SNS by sharing or facilitating
additional comments. Such human activity is known to lead to endogenous
bursting in the rate of message occurrences. We analyze a multi-dimensional
self-exciting process to reveal dependence of the bursting activity on the
topology of connections and the distribution of interaction strength on the
connections. We determine the critical conditions for the cases where
interaction strength is regulated at either the point of input or output for
each person. In the input regulation condition, the network may exhibit
bursting with infinitesimal interaction strength, if the dispersion of the
degrees diverges as in the scale-free networks. In contrast, in the output
regulation condition, the critical value of interaction strength, represented
by the average number of events added by a single event, is a constant
1β1/2ββ0.3, independent of the degree dispersion. Thus, the
stability in human activity crucially depends on not only the topology of
connections but also the manner in which interactions are distributed among the
connections.Comment: 8 pages, 8 figure