Anatol Rapoport

Connectivity of random nets

The weak connectivity γ of a random net is defined and computed by an approximation method as a function ofa, the axone density. It is shown that γ rises rapidly witha, attaining 0.8 of its asymptotic value (unity) fora=2, where the number of neurons in the net is arbitrarily large. The significance of this parameter is interpreted also in terms of the maximum expected spread of an epidemic under certain conditions.

Spread of information through a population with socio-structural bias: I. Assumption of transitivity

A previously derived iteration formula for a random net was applied to some data on the spread of information through a population. It was found that if the axon density (the only free parameter in the formula) is determined by the first pair of experimental values, the predicted spread is much more rapid than the observed one. If the successive values of the “apparent axon density” are calculated from the successive experimental values, it is noticed that this quantity at first suffers a sharp drop from an initial high value to its lowest value and then gradually “recovers”.

Spread of information through a population with socio-structural bias: II. Various models with partial transitivity

The assumption of transitivity treated in part I is modified in various ways to describe an information-diffusion process, in which a certain amount of randomness of contact does occur. In one model a parameter is introduced which is indicative of a tendency to go beyond ones immediate vicinity to spread the information as the vicinity becomes saturated with knowers. In another model the randomness appears in the assumption that new knowers are uniformly distributed among the knowers. Two of the equations thus derived, each with two free parameters are in good agreement with experimental results.

NETS WITH DISTANCE BIAS

A distance bias is imposed on the probability of direct connection between every pair of points in a random net. The probability that there exists a path from a given point in the net to another point is now a function of both the axone density and the distance between the points. A recursion formula is derived in terms of which this probability can be computed.

On the mathematical theory of rumor spread

The applicability of the theory of random nets to the theory of rumor spread is shown. In particular the “weak connectivity” of the net appears as the saturation fraction of “knowers” in a thoroughly mixed population through which a message diffuses where each knower tells the message to a finite average number of individuals. Further it is shown how the time course equation of rumor spread, where time is measured by the number of “removes” from the starters, can be translated into an ordinary continuous time course equation if the distribution of the telling intervals is known.

“Ignition” phenomena in random nets

The spread of excitation in a “random net” is investigated. It is shown that if the thresholds of individual neurons in the net are equal to unity, a positive steady state of excitation will be reached equal to γ, which previously had been computed as the weak connectivity of the net. If, however, the individual thresholds are greater than unity, either no positive steady state exists, or two such states depending on the magnitude of the axone density. In the latter case the smaller of the two steady states is unstable and hence resembles an “ignition point” of the net. If the initial stimulation (assumed instantaneous) exceeds the “ignition point,” the excitation of the net eventually assumes the greater steady state.

Cycle distributions in random nets.

Characteristics of random nets are derived from assumptions concerning the distribution of connections. The single aggregate of neurons with random connections without branching and two parallel chains with normal distribution of connections are considered. The cycle saturation is derived for each type of net, that is, the fraction of neurons which are members of cycles. It is shown that in the single aggregate with random connections, the cycle saturation varies inversely as the square root of the number of neurons; in the dense two-chain net it varies inversely as the square root of the neuron density and inversely on the square root of the standard deviation of the normal distribution.

Outline of a probabilistic approach to animal sociology: III

The probabilities of the emergence of the two kinds of social structure in a 3-bird flock (chain and cycle) are deduced under the assumption of certain biases acting on the social dynamics of the flock. In particular a bias against the reversal of peck order and a bias against encounters of individuals of disparate social rank are considered. Like-wise a distribution of an “inherent” fighting ability is considered which influences the outcomes of encounters. A functional relation is derived between the importance of this ability and the initial probability of a chain structure.

Spread of information through a population with socio-structural bias: III. Suggested experimental procedures

Experimental procedures are suggested to test the theory developed in previous papers which may have applications to predicting the spread of information by contact through a population. The experiment is designed to test the statistical properties of the “acquaintance net” of the population. Thus behavioral aspects are for the time being eliminated, and attention is focused exclusively in the properties of the potential communication net itself.

A statistical approach to the theory of the central nervous system

A “probabilistic” rather than a “deterministic” approach to the theory of neural nets is developed. Neural nets are characterized by certain parameters which give the probability distributions of different kinds of synaptic connections throughout the net. Given a “state” of the net (i.e., the distribution of firing neurons) at a given moment, an equation for the state at the next moment of quantized time is deduced. Certain very special cases involving constant distributions are solved. A necessary condition for a steady state is deduced in terms of an integral equation, in general non-linear.