A threshold function is a critical value or parameter in random structures, particularly in graph theory and probability, that determines a sudden change in the properties of the structure. When a certain threshold is crossed, typically related to the density of edges in a random graph, the structure transitions from one state to another, such as from being disconnected to connected or from lacking a particular substructure to possessing it. This concept is particularly important in understanding phase transitions and the behavior of random graphs and hypergraphs.
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