A process for preparing high molecular weight, branched, acyclic polyalkyleneamines comprising transaminating a reaction mixture that includes at least a first polyalkyleneamine component that contains at least two non-tertiary amine groups separated from one another by a ternary or higher carbon atom and a second polyalkyleneamine component having the formula wherein x, y, and z are the same or different and are integers of from 1 to 10; a, b, c, d, e, and f are the same or different and are H or hydrocarbyl of from 1 to 10 carbon atoms; A, B, C, D, E, are the same or different and are H or hydrocarbyl of from 1 to 10 carbon atoms; provided that at least two of the amine groups are primary or secondary.