Absorbing phase transition in a queueing model of coupled adaptive agents

Abstract

What decides whether people do things together or separately? Many activities cannot be carried out alone, and an individual must rank them against the private tasks competing for the same time. We address this within the priority-queue description of human activity by letting each agent choose the priority of a shared task rather than drawing it from a fixed distribution: the value of the joint activity, discounted by the estimated risk that the partner will not take part. Participation becomes strategic, and the model acquires a phase transition. A coupled phase, in which joint activity is sustained, is separated from an absorbing solitary phase by a saddle-node bifurcation that we obtain in closed form. The transition is discontinuous and the solitary phase is absorbing, so collapse is irreversible unless an agent persists unilaterally for of order one memory time, a cost we also compute. The heavy-tailed interevent statistics that motivate queueing models of human dynamics survive only in a narrow window at the transition, and there the exponent is fixed by the fraction of time spent coupled rather than by queue length: the universality classes of the non-strategic model do not survive the introduction of choice. On a network, attention divides as $1/(k+a)$ and fixes a critical degree beyond which no coupled state exists, so the solitary phase percolates according to the Molloy--Reed criterion with the second moment truncated at that degree --- formally an attack on hubs, with no attacker. For group activities the critical degree falls as the $(m-1)$th root, implying a maximum group size. The transition organises two quantities already measured in communication records: a finite capacity for keeping ties active, and the decay of ties whose rhythm is interrupted.