The interacting array of fine magnetic particles has been studied with a hysteretic magnetic dipole model which simulates a dilute well-dispersed array of fine particles. In any state each dipole element is either parallel or antiparallel to the array axis. In the model the elements reverse according to vulnerability. The cubic array of assigned elements is extended in the calculations by adding hypothetical identical arrays to the original. This method of periodic extension eliminates the coherent demagnetizing field of finite aspect ratio. Calculations made using this model show that columnar breakdown is present to a large extent. This breakdown can be partially quenched by assigning a distribution of critical fields with large spread to the array elements. The average interaction field is observed to be a linear function of the magnetization in agreement with theory. The deviation of the interaction field is a function of the array magnetization in disagreement with the constant value predicted by the theory. The disagreement is attributed to the columnar breakdown. Support for this proposition is given by the observation that partial quenching of columnar breakdown yields a constant deviation over a range of magnetization up to approximately ±25 percent of saturation. The coercivity of the interacting array is greater than the value obtained in the absence of interaction. The increase in coercivity is attributed to the linear dependence of the interaction field expectation on magnetization. Two tests of this hypothesis yield agreement. First, when the expectation is nullified by an oppositely directed coherent demagnetizing field, the coercivity decreases to almost its noninteracting value. Second, the predicted increasing functional dependence of coercivity on saturation magnetization is observed both in these arrays and in independent experiments. The dependence of the deviation of the interaction field on the state of magnetization shows that even the - - modified Preisach functionI for these arrays is not stable over the entire range of magnetization. This conclusion is further justified by the observed distribution function that appears to grow from one state to another rather than simply shifting and maintaining its shape.