A two-dimensional small-world type network, subject to spatial prisoners'dilemma dynamics and containing an influential node defined as a special nodewith a finite density of directed random links to the other nodes in thenetwork, is numerically investigated. It is shown that the degree ofcooperation does not remain at a steady state level but displays a punctuatedequilibrium type behavior manifested by the existence of sudden breakdowns ofcooperation. The breakdown of cooperation is linked to an imitation of asuccessful selfish strategy of the influential node. It is also found thatwhile the breakdown of cooperation occurs suddenly, the recovery of it requireslonger time. This recovery time may, depending on the degree of steady statecooperation, either increase or decrease with an increasing number of longrange connections.
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