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This study advocates a mathematical framework of ''transport relations'' on anetwork. They single out a subset of ''traffic states'' described by time,duration, position and other traffic attributes (called ''monads'' for short).Duration evolutions are non-negative, decreasing toward zero for incomingdurations, increasing from zero for outgoing durations, allowing the detectionof ''junction states'' defined as traffic states with ''zero duration''. A''junction relation'' (crossroads, synapses, clearing houses, etc.) Is a subsetof the transport relation made of junction states? The objective is toconstruct a ''transport regulator'' associating with traffic states a set of''celerities'' that mobiles circulating in the network can use as velocities.In other word, a network is regarded as a ''provider of velocity information''to the mobiles for travelling from one departure state to an arrival stateacross a junction relation (a kind of geodesic problem). This investigationassumes that a system governs the evolution of monads in function of time,duration and position using celerities as controls and provides the transportregulator, a feedback from transport states to celerities. The proposedmathematical framework can acclimate road or aerial networks, endocrine(hormonal) or synaptic (neurotransmitters) networks, financial or economicnetworks, which motivated this investigation. This framework could probablyaccommodate computer and even social networks. This investigation is restrictedto junctions between two routes.

Transports Regulators of Networks with Junctions Detected by Durations Functions