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The magnetic structure and magnetization process in a permalloy wire-based honeycomb network have been investigated by means of magnetic-force microscopy MFM and magnetoresistance measurement. The MFM measurements show the remanent magnetic structures to be governed by magnetic interaction similar to the ice rule, which provides a direct analogy between the present honeycomb network and an Ising system on a kagome lattice. The magnetoresistance measurements reveal that this interaction also dominates the magnetization processes in the network. By decreasing the exchange energy at the vertices of the network, the ice-rule type of interaction causes a transition of the magnetization process in the network.