A natural gas distribution network connects consumers to a source of gas. It is managed by a network operator, whose task incurs various costs, some of which may not be attributable to a particular consumer. Assuming that the operator wishes to recover these costs by charging for its services, the problem is then to determine how much each consumer should pay. In other words, how should these costs be allocated among consumers. In this paper, we address this problem and propose cost allocation rules that depend on the network and the demands of the consumers. To that end, we adopt a normative approach and oppose two principles: (i) the connection principle and (ii) the uniformity principle. The Connection rule is proposed in accordance with (i), while the Uniform rule is developed in line with (ii). It appears that (i) and (ii) are incompatible. To make a trade-off between these two principles, we propose the family of Mixed rules, which compromise between the Connection rule and the Uniform rule by mean of convex combinations. For each cost allocation rule, an axiomatic characterization is provided. Then, we show that the Connection rule coincides with the multi-choice Shapley value of a specific multi-choice game derived from the network and the demands of the consumers. Moreover, the Connection rule is in the Core of this specific multi-choice game. Similarly, we show that the Uniform rule coincides with the multi-choice Equal division value and the Mixed rules coincide with the multi-choice Egalitarian Shapley values.