The Role of Abstraction in Network Models

D. C. Mikulecky

Fellow, ISCE

Professor of Physiology

Virginia Commonwealth University

June 2001

Abstraction is often seen as something to be avoided and having negative attributes.  This is a strange notion since the very act of using language is a form of abstraction.  The way to see this is to look at a number of ideas best explained by Rosen’s modeling relation and its relationship to semiotics.  This short note will assume you have done that.

In a network, there are a number of interrelated features that help make the whole more than the sum of its parts.  In its most abstract form, we can write expressions for what also can be pictorialized quite easily.   The picture of a network looks like this:

Yet it has another, more compact and equivalent representation using some ideas from mathematics.

Simple defintion: A bunch of things connected to one another in some pattern.

Formal Definition:  Define a "network" in terms of a set , V, of n vertices, vi ={v1,v2, …, , , vn} (These are the “dots” in the picture above) and k edges, ej, E = {e[1],e[2],...,e[k]} (These are the lines in the picture above).  These sets can be caused to "grow" or "shrink" by adding (algebraic) others like them as direct sums. Given the Cartesian product   V x V  leading to the set of ordered vertices, {v[m],v[n]}  we can then define a specific network in terms of

a relation in this set which ties together members of E with members of {v[m],v[n]}.  For each E[n] in this subset, their are pair of vertices (v[m],v[n]) which give the edge direction from v[m] to v[n].  This relation defines a network topology which can be represented as an p x q  incidence matrix A having as  its  elements the numbers  1, 0, -1 .  Each row in A corresponds to a particular edge in E[n] and each column to a particular vertex in V.  These numbers are assigned in the

following manner: Letters in brackets are subscripts.  a[rs] = 1 if the edge is incident on v[m];  a[rs] =-1 if it is incident on v[n], and a[rs] = 0 if it is not  incident on either.  Here r runs over 1,2,...,p and s over 1,2,...,q.

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