H. G. Landau

On dominance relations and the structure of animal societies: I. Effect of inherent characteristics

Societies are considered in which a non-transitive dominance relation exists between every pair of members, such as the peck-right in a flock of hens. A one-dimensional measure of the structure of such a society,h, is defined, withh=0 for equality andh=1 for the hierarchy. It is assumed that each member of the society is characterized by an ability vector whose components depend on individual characteristics such as size, concentration of sex hormone, etc., but not on social factors such as social rank. The distribution of abilities among members of the society is assumed to be given by a distribution function which is the same for all members, and the probability that one member dominates another is given by a function of the ability vectors of the two.

On dominance relations and the structure of animal societies: III The condition for a score structure

The necessary and sufficient condition is given forn integers to be the score structure of a society with a dominance relation.

On dominance relations and the structure of animal societies: II. Some effects of possible social factors

In a previous paper (Landau, 1951) it was shown that a society with a dominance relation would rarely tend to be close to the hierarchy in structure if dominance is determined solely by the inherent characteristics of the members. Here we consider the effects of other factors, due to social rank or to the outcome of previous encounters which affected dominance.

Contribution to the Mathematical Theory of Contagion and Spread of Information: I. Spread through a Thoroughly Mixed Population

An equation is derived from the spread of a “state” by contact through a thoroughly mixed population, in which the probability of transmission depends both on the over-all duration of the process and on the time an individual has been in the “state.” Cases in which this probability is a function of only one or the other of the two “times” are worked out. It is shown that in the case of dependence on “private time” alone the asymptotic value of the fraction of the population effected is the same as that derived by the random net approach.

On some problems of random nets

The probability problems connected with random nets are restated as probabilities of drawings from an urn containing black and white balls. A partial difference equation is obtained and its solution is given. For large nets a series expression is obtained for the connectivity γ, and this is shown to be equivalent to the transcendental equation obtained by R. Solomonoff and A. Rapport (1951).

Development of structure in a society with a dominance relation when new members are added successively.

A society with a dominance relation is considered to be built up by starting with a small society and adding new members in succession. As each member is added he engages in contests with each of the older members to determine the dominance relation between them. The probability that the older member dominates is considered to depend on the size of the society and linearly on the older members score. A recurrence relation for the hierarchy index is derived. The approach of the society to a hierarchical structure is considered for various special cases of this probability. Reasonable assumptions concerning this dominance probability are shown to lead to structures close to the hierarchy. If the new member dominates all the older ones below a certain rank, and is dominated by all those above this rank, then the hierarchy will persist if it is the initial structure, or the structure will tend to hierarchy as the size increases, if it is not the initial structure.

Note on the effect of imitation in social behavior

The discussion given by N. Rashevsky (1949) on the effect of imitation in the mathematical biology of social behavior is generalized by assuming the distributions involved to be normal rather than Laplace distributions, and also by showing how most of the results can be derived without assuming any specific form for the distributions. In particular, it is demonstrated that it is possible, in a sufficiently large population, to have a stable behavior pattern which is quite independent of the desires of the population or of their inherent pattern of response.

Models of social structure

The mathematical treatment of three models of possible development of a society with a dominance relationship is discussed. The conclusion is reached that social factors as well as inherent characteristics need to be introduced to account for near-hierarchical structures. This is not a surprising conclusion; however, deriving it from mathematical considerations should be of interest to the mathematical sociologist since it puts the problem into theoretical perspective. Also these considerations may suggest quantitative observations or experimental tests and given indications as to their analysis.

The distribution of completion times for random communication in a task-oriented group

The task-oriented groups considered here consist of a number of individuals, each having initially one piece of information which must be transmitted to all the others to complete the task. Interest is centered on the communication net which restricts the possible channels for messages. At every sending time each individual sends all the information he has acquired to one other individual; the major assumption here is that this recipient is chosen at random from the possibilities given by the communication net. The information state is defined as a matrix which shows where the initial information has spread. These matrices can be considered as the states of a Markov chain, and in this way the distribution of completion times for the task is obtained. Some special cases are worked out and generalizations are indicated. A proof is given of the formula for the shortest possible completion time in any net when a fixed number of messages is sent by each individual at each sending time.

A problem in radiobiology: Diffusion and recombination of ions

A perturbation treatment is given for the cylindrically symmetrical distribution of ions or their products resulting from the passage of a single ionizing particle, and taking into account ion diffusion and recombination. The results apply to cases in which diffusion is more important than recombination.