(pdf of full article)

A Hierarchical Model of Information Transmission

One strength of the function-output approach to information is its ability to describe information on a number of different levels: the electronic pulse in a wire representing a bit, sound waves representing linguistic phonemes, or thoughts being represented through written language or printed image. Consider receiving a speech sound x generated by a speaker. We represent this action by

$$\strut\displaystyle {\mathit phoneme}(x).$$

The values returned by the phoneme function are the information produced by the hearer.

We will first examine inverse functions. The inverse of function f(x) is denoted as $f^{-1}(x) = \frac{1}{f(x)}$. Thus, f^-1(f(x)) = xfor all x and for all f that are reversible deterministic functions. For positive numbers, the mathematical square function has as its inverse the square root function. If f represents here the square function, f(4)=16 and f^-1(16) = 4. Thus, f^-1(f(4))=4.For notational consistency below, we assume that a function f(x) acts to decode the encoded message x. The inverse function f^-1(x) acts to encode x.

By combining hierarchically arranged functions together, with one ``feeding" the other in a hierarchical manner, one can capture common principles when describing complex information systems. These functions now may be ``stacked" so that one function provides input to another in order of increasing complexity. Given that f^-1(f(x)) = f(f^-1(x))= x and g^-1(g(x))=g(g^-1(x)) =x for all x, and assuming that f and g are reversible, it will always be the case that

$$\strut\displaystyle f(g(g^{-1}(f^{-1}(x)))) = x.$$

Note that the assumption of reversibility is an assumption that no ``errors" or ``loss" occurs in processing; in most systems, some loss occurs. These losses are often acceptable in situations where processes are not very sensitive to variances in their inputs. For example, slight changes in light intensity or in color usually go unnoticed in human vision, and slight variations in the type before the reader will not cause the content to be lost.

Through the combination of functions and their inverses, a variety of processes may be modeled, including high level communication behaviors such as how language is sent and received through a sonic medium. The information in the language is transmitted from sender to receiver through the encoding of the thought in a sonic form. We thus have two forms of representation: phrase and sound. This process may be represented by

$$\strut\displaystyle {\mathit phrase}({\mathit phoneme}({\mathit phoneme}^{-1}({\mathit phrase}^{-1}(x))))$$

where ${\mathit phrase}(x)$ is the phrase received by the hearer. If I desire to transmit a word to you, it is assigned to x and the ${\mathit phrase}^{-1}$ function encodes the word, producing the input to the ${\mathit phoneme}^{-1}$ function. This function places the coded sound into the atmosphere, where it is picked up and decoded by the ${\mathit phoneme}$ function, which decodes the message into a form acceptable to the ${\mathit phrase}$ function, which decodes its input, producing the original word if there was no interference with the process.

  

Figure 2: Encoding and decoding processes in communication.

  

Figure 3: Hierarchical model of human communications. The ``U" shaped arrow represents the passage of something being transmitted, being encoded and decoded as in Figure 2.

When one desires to add additional functionality to this hierarchical model, it becomes necessary to add both encoding and decoding functions for an added layer, e.g. ${\mathit knowledge}^{-1}$ and ${\mathit knowledge}(x)$. We may add a ``knowledge" layer as follows:

$$\strut\displaystyle {\mathit knowledge} \left({\mathit phrase... ...{\mathit knowledge}^{-1}(x)\right)\right)\right)\right)\right).$$

This assumes functions with lossless operation, and is illustrated in Figure 2.

Each function in Figure 3 can be thought of as a black box that accepts communications from above (on the left side) and ``processes" the input. The output (from the bottom of these devices) indirectly feeds into the inverse of the function (flowing upwards on the same level on the right side) but goes directly into another function below it (on the left.) Lower level functions are necessary but not sufficient if communication is to occur; additional functions must be added until the function at the bottom can provide a physical linkage between the bottom layer on one hierarchy and the corresponding functional layer on another hierarchy.

One of the advantages of this form of model is that it allows for personal processes to operate differently from one individual to another, as long as they are largely reversible within the individual. This allows one to incorporate personal factors, cultural biases, etc. into higher level cognitive processes near the top of the hierarchies.

Information is always and only transmitted through a series of physical processes. In some cases, it may be helpful although not necessary to view this set of processes in a hierarchical manner because of the inherent hierarchical nature of communication and representation systems. In representational systems, one representation may represent another representation, such as when these printed words represent English language terms representing some ideas in the author's brain. The movement of a neural signal from a sensory organ to various parts of the brain is evidence of both a simple connection of processes and a hierarchical view of communication.

A given level has below (or above) it one or more hierarchies, referred to here as ``legs," stretching out and touching the ends of other legs. For example, a given human mind may communicate through a number of different physical processes, such as through speech, gesture, or the written word. The same knowledge process can be working in the brain, but the transmitted message is traveling to its destination through one or more of several different physical methods. More complex networks of information transmission structures, such as those used in information diffusion [Cha86,LB75,Wil92] also may be modeled using ``legs." These networks may be studied by examining the operation and transmission provided by "legs" taken as a whole, or by examining how the transmitting characteristics of each individual process or type of process within a leg.

A communication hierarchy consists of a number of layers. Our hierarchy is modeled after the telecommunications hierarchies that have been proposed by standards organizations (e.g. the ISO) and companies (e.g. IBM and SNA). What should constitute a layer, and what set of processes should be agglomerated into a single layer or process? Each layer is defined by the interface with the layer above or below as well as by the process producing the layers above, or below, or producing itself. Yet, any process can be continually broken down until the smallest possible physical process is reached. For the study of macroscopic environments, the arbitrary choice of the grouping of layers is required. They may be most beneficially selected based upon naturalness considerations, processes and layers being described so that the processes are easily understood.

Treating a specific range of possible levels as a single unit is consistent with information being viewed as a ``thing" [Buc91]. Lumping together all levels below a certain point for several hierarchies results in a channel that can be treated as a unit. This channel is sometimes referred to as ``information as thing."

The defining limits of a function are somewhat arbitrary and most hierarchies can usually be decomposed further. At the bottom of each hierarchy is a layer that contains the physical mechanism that allows communication to occur. We refer to this bottom layer as the ``physical layer," no matter what size the function. The physical layer, as with all the other layers, can be defined such that it is very large, so that one layer performs a particular human sized task, or it may be very small, with several different smaller layers performing a particular task.

The value of characteristics of one level may be passed on to other levels or the values may be dropped. Loss of a value may be irrecoverable; the information will be permanently lost if the characteristic is independently valued and can't be inferred from other characteristics. For example, a lost value for the author's gender variable can be easily recovered from the author's name or knowledge about the presence of a beard, while the loss of the value for the author's first name would be a permanent loss, not being easily deduced from other characteristics, such as gender or the presence of a beard.

Most definitions and measures of information address the transmission of these characteristics' values from one level to the corresponding level at the destination. The use of the hierarchical model allows one to focus on the level in the hierarchy that is of greatest interest, rather than getting into a debate about whether information is of one nature or another, whether it is located at one level in the hierarchy or another. Information is produced and may be studied at your level of interest and the processes at your level are worthy of discussion, as are the information producing processes at other levels in the hierarchy.