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\begin{document} 
\begin{center}
\vspace*{1.0cm}

{\LARGE{\bf Discrete Mathematics and Physics on the Planck-Scale
    exemplified by means of a Class of 'Cellular Network Modells' and
    their Dynamics}} 

\vskip 1.5cm

{\large {\bf Manfred Requardt }} 

\vskip 0.5 cm 

Institut f\"ur Theoretische Physik \\ 
Universit\"at G\"ottingen \\ 
Bunsenstrasse 9 \\ 
37073 G\"ottingen \quad Germany


\end{center}

\vspace{1 cm}

\begin{abstract}
Starting from the hypothesis that both physics, in 
particular space-time and the physical vacuum, and the corresponding
mathematics are discrete on the Planck scale we develop a certain
framework in form of a class of '{\it cellular networks}' consisting of cells
(nodes) interacting with each other via bonds according to a certain
{\it 'local law'} which governs their evolution. Both the internal states of
the cells and the strength/orientation of the bonds are assumed to be dynamical
variables. We introduce a couple of candidates of such local laws
which, we think, are capable of catalyzing the unfolding of the
network towards increasing complexity and pattern formation.
In section 3 the basis is laid for a version of '{\it discrete
analysis}' and {\it 'discrete topology/geometry'} which, starting from different, perhaps more physically
oriented principles, manages to make contact with the much more
abstract machinery of Connes et al. and may complement the latter
approach. In section 4 a, as far as we can see, promising concept of
'{\it topological dimension}' in form of a '{\it degree of
connectivity}' for graphs, networks and the like is developed. It is then indicated how
this '{\it dimension}', which for continuous structures or lattices being
embedded in a continuous background agrees with the ''usual''
notion of dimension (i.e. the respective embedding dimension) , may
vary dynamically as a result of a '{\it phase
transition like}' change of the '{\it connectivity}' in the network. A certain
(highly) speculative argument, along the lines of statistical mechanics, is
supplied in favor of the naturalness of dimension 4 of ordinary
(classical) space-time.  
\end{abstract} \newpage
\section{Introduction}
\noindent There exists a certain suspicion in parts of the scientific
community that nature may be "discrete" on the Planck scale. The
point of view held by the majority is however, at least as far as we
can see, that quantum theory as we know it holds sway down to
arbitrarily small scales as an allembracing general principle, being
applied to a sequence of increasingly fine grained effective field
theories all the way down up to, say, string field theory. But even
on that fundamental level one starts from strings moving in a
continuous background. It is then argued that "discreteness" enters
somehow through the backdoor via "quantisation".

The possibly most radical and heretical attempt, on the other
side, is it to try to generate both gravity and quantum theory as secondary and
derived concepts (in fact merely two aspects) of one and the same underlying 
more primordial theory instead of simply trying to quantise gravity,
which is the canonical point of view (see e.g. \cite{1}).

This strategy implies more or less directly that -- as gravity is
closely linked with the dynamics of (continuous) space-time -- the
hypothetical underlying more fundamental theory is supposed to live
on a substratum which does not support from the outset something like
continuous topological or geometrical structures. In our view these
continuous structures should emerge as derived concepts via some sort
of coarse graining over a relatively large number of "discrete" more
elementary building blocks.

This program still leaves us with a lot of possibilities. For various
reasons, which may become more plausible in the course of the
investigation, we personally favor what we would like to call a
"cellular network" as a realisation of this substratum, the precise
definitions being given below. Without going into any details at the
moment some of our personal motivations are briefly the following:\\
i) These systems are in a natural way discrete, the local state space
at each site being usually finite or at least countable.\\
ii) Systems like these or their (probably better known) close
relatives, the "cellular automata", are known to be capable of
socalled "complex behavior", "pattern generation" and
"selforganisation" in general while the underlying dynamical laws are
frequently strikingly simple (a wellknown example being e.g.
Conway's "game of life").\\
Remark: A beautiful introduction into this fascinating field is e.g.
\cite{2}. As a shorter review one may take the contribution of
Wolfram (l.c.). More recent material can be found in the proceedings
of the Santa Fee Institute, e.g. the article of Kauffman in \cite{3},
who investigates slightly different systems ("switching nets").\\
iii) Some people suspect (as also we do) that physics may be
reducible at its very bottom to some sort of "information processing
system" (cf. e.g. \cite{4,5}). Evidently cellular automata and the
like are optimally adapted to this purpose.\\
iv) In "ordinary" field theory phenomena evolving in space-time are
typically described by forming a fibre bundle over space-time (being
locally homeomorphic to a product). In our view a picture like this
can only be an approximate one. It conveys the impression that
space-time is kind of an arena or stage being fundamentally different
from the various fields and phenomena which evolve and interact in
it. In our view these localised attributes, being encoded in the various
field values, should rather be attributes of the -- in the
conventional picture hidden -- infinitesimal neighborhoods of
space-time points, more properly speaking, neighborhoods in a medium
in which space-time is immersed as a lower dimensional "submanifold".
To put it in a nutshell: We would prefer a medium in which what we
typically regard as irreducible space-time points have an internal
structure. To give a simple picture from an entirely different field:
take e.g. a classical gas, consider local pressure, temperature etc.
as collective coarse grained coordinates with respect to the
infinitesimal volume elements, regard then the microscopic degrees of
freedom of the particles in this small volume elements as the hidden
internal structure of the "points" given by the values of the above
collective coordinates (warning: this picture is of course not
completely correct as the correspondence between the values of
local pressure etc. and volume elements is usually not one-one).
It will turn out that a discrete structure as alluded to above is a
nice playground for modelling such features.\\[0.5cm]
Remark: Evidently there are close ties between what we have said in
iv) and certain foundational investigations in pure mathematics
concerning the problem of the 'continuum', a catchword being e.g.
"non-standard analysis".\vspace{0.5cm}

A lot more could be said as to the general physical motivations and a
lot more literature could be mentioned as e.g. the work of
Finkelstein and many others (see e.g. \cite{6,7}.  For further references
cf. also the papers of Dimakis and M\"uller-Hoissen (\cite{8}).  Most
similar in spirit is in our view however the approach of 't Hooft
(\cite{9}).

\section{The Concept of the "Cellular Network"}
While our primary interest is the analysis of various partly long
standing problems of current physics, which seem to beset physics
many orders away from the Planck regime, we nevertheless claim that
the understanding of the processes going on in the cellular network
at Planck level will provide us with strong clues concerning the
phenomena occurring in the "daylight" of
"middle-energy-quantum-physics". In fact, as Planck scale physics is
-- possibly for all times -- beyond the reach of experimental
confirmation, this sort of serious speculation has to be taken as a
substitution for experiments.

To mention some of these urgent problems of present day physics:\\
i) The unification of quantum theory and gravitation in general, and
in our more particular context: both the emergence of ''quantum
behavior'' and gravitation/space-time as two separate but related
aspects of the unfolding of the primordial network state, \\
ii) the origin of the universe, of space-time from "nothing" and
its very early period of existence,\\
iii) the mystery of the seeming vanishing of the 'cosmological
constant', which, in our view, is intimately related to the correct
understanding of the nature of vacuum fluctuations,\\
iv) the primordial nature of the "Higgs mechanism",\\
v) causality in quantum physics or, put differently, its strongly
translocal character,\\
vi) 'potential' versus 'actual' existence in the quantum world, the
ontological status of the wave function (e.g. of the universe))  and
the quantum mechanical measurement problem in general.

Most of these topics have been adressed in the thoughtful book of
S. Weinberg (\cite{weinberg}) and will be treated by us in much more
detail elsewhere in the near future. Therefore we refrain from making
more comments as to these fundamental questions at the moment apart
from the one remark that our approach will partly be based 
on the assumption that nature behaves or
can be imitated as a cellular network at its very bottom (\cite{10}).
It is however crucial for these investigations to have a sufficiently
highly developed form of ''discrete mathematics'' on graphs and
networks and the like at ones disposal. 
Therefore we will concentrate in the following mainly on establishing
the necessary (mostly mathematical) prerequisites on which the
subsequent physical investigations will be based.

This is the more so necessary because one of our central hypotheses
is that most of the hierarchical structure and fundamental building
blocks of modern physics come into being via a sequence of {\it unfolding
phase transitions} in this cellular medium. As far as we can see, the
study of phase transitions in cellular networks is not yet very far
developed, which is understandable given the extreme complexity of
the whole field. Therefore a good deal of work should be, to begin
with, devoted to an at least  qualitative understanding of this intricate
subject.

Furthermore discrete mathematics/physics of this kind is an
interesting topic as such irrespective of the applications mentioned
above which would justify a separate treatment of questions like the
following anyhow (cf. e.g. the very interesting paper by Mack,
\cite{mack}, where a complex of ideas is sketched to which we are
quite sympathetic) \\[0.5cm] 
{\bf 2.1 Definition(Cellular Automaton)}: A cellular automaton
consists typically of a fixed regular array of cells \{$C_i$\}
sitting on the nodes \{$n_i$\} of a regular lattice like, say, $\Ir^d$
for some d.  Each of the cells is characterized by its internal state
$s_i$ which can vary over a certain (typically finite) set $\cal S$
which is usually chosen to be the same for all lattice sites.

Evolution or dynamics take place in discrete steps $\tau$ and is
given by a certain specific 'local law' $ll$:
\begin{equation}s_i(t+\tau)=ll(\{s'_j(t)\}\;\;\underline{S}(t+\tau)=LL
(\underline{S}(t)) \end{equation}
where t denotes a certain "{\it clock time}" (not necessarily physical time), 
$\tau$ the elementary clock time interval, \{$s'_j$\} the
internal states of the nodes of a certain local neighborhood of the
cell $C_i$, {\it ll} a map:
\begin{equation}ll: {\cal S}^n \to \cal S 
\end{equation}
with n the number of neighbors occurring in (1), \underline{S}(t) the
global state at "time" t, {\it LL} the corresponding global map acting on
the total state space $X:=\{\underline{S}\}$. {\it LL} is called {\it
reversible} if it is a bijective map of X onto itself. \vspace{0.5cm}

Cellular automata of this type behave generically already very
complicated (see \cite{2}). But nevertheless we suspect they are
still not complicated enough in order to perform the specific type of
complex behavior we want them to do. For one, they are in our view
too regular and rigid for our purposes. For another, the occurring regular lattices
inherit quasi automatically such a physically important notion like
'{\it dimension}' from the underlying embedding space.

Our intuition is however exactly the other way round. We want to
generate something like dimension (among other topological notions)
via a dynamical process (of phase transition type) from a more
primordial underlying model which, at least initially, is lacking
such characteristic properties and features. 

There exist a couple of further, perhaps subjective, motivations
which will perhaps become more apparent in the following and which
result in the choice of the following primordial model
system:\\[0.5cm]
{\bf 2.2 Definition(Cellular Network)}: In the following we will
mainly deal with the class of systems defined below:\\
i) "Geometrically" they are {\it graphs}, i.e. they consist of nodes
\{$n_i$\} and bonds \{$\bi$\} where pictorially the bond $\bi$
connects the nodes $n_i$ and $n_k$ with $n_i\neq n_k$ implied (there
are graphs where this is not so), furthermore, to each pair of nodes
there exists at most one bond connecting them. In other words the
graph is 'simple' (schlicht). There is an intimate relationship
between the theory of graphs and the algebra of relations on sets. In
this latter context one would call a simple graph a set carrying a
homogeneous non-reflexive, (a)symmetric relation.

The graph is assumed to be {\it connected}, i.e. two arbitrary nodes
can be connected by a sequence of consecutive bonds, and {\it
regular}, that is it looks locally the same everywhere.
Mathematically this means that the number of bonds being incident
with a given node is the same over the graph ({\it 'degree' of a node}).
We call the nodes which can be reached from a given node by making
one step the {\it 1-order-neighborhood} ${\cal U}_1$ and by not more
than n steps ${\cal U}_n$.\\
ii) On the graph we implant a class of dynamics in the following way:\\[0.5cm]
{\bf 2.3 Definition(Dynamics)}: As for a cellular automaton each node
$n_i$ can be in a number of internal states $s_i\in \cal S$. Each
bond $\bi$ carries a corresponding bond state $J_{ik}\in\cal J$. Then
we assume:
\begin{equation} s_i(t+\tau)=ll_s(\{s'_k(t)\},\{J'_{kl}(t)\})
\end{equation}
\begin{equation} J_{ik}(t+\tau)=ll_J(\{s'_l(t)\},\{J'_{lm}(t)\})
\end{equation}
\begin{equation}
(\underline{S},\underline{J})(t+\tau)=LL((\underline{S},\underline{J}(
t))
\end{equation}
where $ll_s$, $ll_J$ are two mappings (being the same all over the
graph) from the state space of a local
neighborhood of a given fixed node or bond to $\cal S, J$, yielding the
updated values of $s_i$ and $\bi$.\\[0.5cm]
Remarks: i) The theory of graphs is developed in e.g. \cite{11,12}. As
to the connections to the algebra of relations see also
\cite{schmidt}. There are a lot of concepts in graph theory which are
useful in our context, some of which will be introduced below if
necessary. On the other hand we do not want to overburden this
introductory paper with to much technical machinery.  \\
ii) Synonyma for 'node' and 'bond' are e.g. 'site' and 'link' or
'vertex' and 'edge'.\\
iii) It may be possible under certain circumstances to replace or
rather emulate  a cellular network of the above kind by some sort of
extended cellular automaton (e.g. by replacing the bonds by
additional sites). The description will then however become quite
cumbersome and involved. \vspace{0.5cm}

What is the physical philosophy behind this picture? We assume the
primordial substratum from which the physical universe is expected to
emerge via a selforganisation process to be devoid of most of the
characteristics we are usually accustomed to attribute to something
like a manifold or a topological space. What we are prepared to admit
is some kind of "{\it pregeometry}" consisting in this model under
discussion of an irregular array of elementary grains and "direct
interactions" between them, more specifically, between the members of
the various local neighborhoods.

It is an essential ingredient of our approach that the strength of
these direct interactions is of a dynamical nature and allowed to
vary. In particular it can happen that two nodes or a whole cluster
of nodes start to interact very strongly in the course of the
evolution and that this type of {\it collective behavior} persists
for a long time or forever (becomes {\it locked in}) or, on the other
extreme, that the interaction between certain nodes becomes weak or
vanishes.

It is not an easy task to select from the almost infinity of possible
models an appropriate subclass which we think has the potential of
displaying some or possibly all of the complex features (typically on
length scales far away from the Planck regime) we are confronted with
in ''ordinary'' (middle energy - compared to the Planck scale - )
quantum physics, and being, on the other side, sufficiently
transparent on, say, its natural primordial scale. We studied in fact
a lot of alternatives (we do not mention) and want to present in the
following a typical representative of a certain class we are presently
favoring.

Our guiding principles have been roughly the following: Most of the
cellular automaton rules being in use today (cf. e.g. \cite{2}) are of
a pronouncedly dissipative flavor. It is even frequently argued that
some kinde of dissipation (or rather: shrinking of 'phase space') is a
necessary prerequisite in order to have {\it 'attractors'} and, as a
consequence, pattern generation. We are not entirely convinced that the
arguments along these lines are really conclusive (for a class of
reversible automata see e.g. the book of Toffoli and Margolus,
\cite{toffoli})

In any case, as we want our model system to generate ''quantum
behavior'' on a, however, much coarser scale and if being in a certain
specific {\it 'phase'}, we consider it to be essential to implant a certain
propensity for {\it 'undulation'} in the class of local laws under
discussion. Furthermore, it turns out to be extremely useful - in
order to tame the horribly large quantum fluctuations occurring on
Planck scale, when probing into space-time regions of larger extension
- to incorporate a tendency to screen destructive fluctuations. These
''boundary conditions'' (among other considerations) led us to the
following model system, which is only a simple representative of
possibly a whole class.

At each site $n_i$ there is sitting a one-dimensional discrete site
variable $s_i \in q\cdot\Ir$ with q, for the time being, a certain
elementary quantum. The bond variables $J_{ik}$ are, in this simple
case, assumed to be two-valued, i.e: $J_{ik}\in \{\pm 1\}$.\\[0.5cm]
Remarks: i)For the time being we let the site variables range over the
full $\Ir$ in order not to complicate the already sufficiently
complicated reasoning further. It is of course possible to impose
certain boundary conditions (e.g. switching to a subgroup of $\Ir$) if
it turns out to be sensible.\\
ii)In an extended model, which we will employ later on in order to
catalyze the {\it 'unfolding'} of the network together with the emergence of
space-time and gravitation, $J_{ik}$ can also take on the value $0$.\\
iii)In the next section on graphs we will give the graph an
{\it 'orientation'}, i.e. the bond $b_{ik}$ is assumed to point from $n_i$
to $n_k$ with $n_i$ initial node, $n_k$ terminal node, $b_{ki}$
denoting the same bond with reverse orientation (see Definition 3.1)
As a consequence, in order to be consistent, we assume:\\[0.5cm]
{\bf 2.4 Consequence}: 
\begin{equation}J_{ik}=-J_{ki}\end{equation}\vspace{0.5cm}

The physical idea behind this scheme is the following: If $J_{ik}$ is
positive an elementary quantum q is transported in the elementary
'clock-time step' $\tau$ from node (cell) $n_i$ to $n_k$. Then the
first half of the local law reads:
\begin{equation}s_i(t+\tau)-s_i(t)=-q\cdot\sum_k J_{ik}\end{equation}
which is sort of a master or continuity equation.

What remains to be specified is the backreaction of the node states
onto the bond states. We make the following choice:
\begin{equation}\mbox{If}\quad s_i(t)>s_k(t)\quad \mbox{then}\quad
  J_{ik}(t+\tau)=+1\end{equation}
\begin{equation}\mbox{and hence}\quad
  J_{ki}(t+\tau)=-J_{ik}=-1\end{equation}
For the borderline case $s_i(t)=s_k(t)$ we have roughly two options
$B_1,B_2$ depending on the admissible state space of $J_{ik}$,
i.e. $\{-1,+1\}$ or $\{-1,0,+1\}$. In the former case we decree:
\begin{equation}J_{ik}(t+\tau)=J_{ik}(t)\quad\mbox{if}\quad
  s_i(t)=s_k(t)\end{equation}
and in the latter case:
\begin{equation}J_{ik}(t+\tau)=0\quad\mbox{if}\quad
  s_i(t)=s_k(t)\end{equation}

Introducing the signum function sign with
\begin{equation}sign(x)=1,0,-1\quad \mbox{if}\quad x>0,=0,<0
\end{equation}
we then get:
\begin{equation}B_1)\; J_{ik}(t+\tau)=-sgn(s_k(t)-s_i(t))+(1-|sgn(s_k(t)-s_i(t))|)J_{ik}(t)\end{equation}
\begin{equation}\; B_2)J_{ik}(t+\tau)=-sgn(s_k(t)-s_i(t))\end{equation}

As already indicated above, in case we want to model the unfolding of
our ''network universe'' beginning with an extremely densly connected
initial state (a complete graph or simplex, say) with no genuine
physical neighborhood structure -- nodes are not experienced by each other
as near by or far away -- , we intensify the effect implemented in $B_2)$
and simulate what is called in catastrophy theory or in the realm of
self organisation a fold (in physics known as hystheresis):
\begin{equation}C)\,i)\,J_{ik}(t+\tau)=-sgn(s_k(t)-s_i(t))\quad\mbox{if}\end{equation}
\begin{equation} |s_k(t)-s_i(t)|\ge\lambda_1\;\mbox{and}\;J_{ik}(t)\neq
  0\;\mbox{or}\;
  |s_k(t)-s_i(t)|>\lambda_2\;\mbox{and}\;J_{ik}(t)=0\end{equation}
with $\lambda_2>\lambda_1>0$ two critical parameters, indicating the
hysteresis interval $I_{\lambda}=[\lambda_1,\lambda_2]$
\begin{equation}ii)\;J_{ik}(t+\tau)=0\quad\mbox{if}\quad|s_k(t)-s_i(t)|<\lambda_1
\end{equation}
{\bf 2.5 Class of Local Laws}: For our purposes an admissible class of
local laws is given by the representatives $A)$ plus $B_1,B_2$ or
$C$.\\[0.5cm]
Remarks:i)The reason why we do not choose the ''current'' $q\cdot
J_{ik}$ proportional to the ''voltage difference'' $(s_i-s_k)$ as
e.g. in Ohms's law is that we favor a network which is capable of
self-excitation and self-organisation rather than self-regulation
around a relatively uninteresting equilibrium state. The balance
between dissipation and amplification of spontaneous fluctuations has
however to be carefully chosen (''complexity at the edge of chaos'')\\
ii)Note that -- in contrast to e.g. euclidean lattice field theory -- for
the time being the socalled '{\it clock time}'$t$ is not standing on
the same footing as, say, potential coordinates in the network (e.g.
curves of bonds). We suppose anyhow that socalled '{\it physical
time}' will emerge as sort of a secondary collective variable in the
network, i.e. being different from the clock time (while being of
course functionally related to it).

In our view this is consistent with the spirit of relativity. What
Einstein was really teaching us is that there is a (dynamical)
interdependence between what we experience as space and time, not
that they are absolutely identical! In any case the assumption of an
overall clock time is at the moment only made just for convenience in
order to make the model system not too complicated. If our
understanding of the complex behavior of the network dynamics
increases, this assumption may be weakened in favor of a possibly
local or/and dynamical clock frequency. 
\vspace{0.5cm}

As can be seen from the definition of the cellular network it
separates quite naturally into two parts of a different mathematical
and physical nature. The first one comprises part i) of definition
2.2, the second one part ii) and definition 2.3. The first one is
more static and "geometric" in character, the latter one conveys a
more dynamical and topological flavor as we shall see in the
following.

We begin in section 3 with a representation of what may be called
discrete analysis on graphs and networks. This is followed in section
4 by making the first steps into an investigation of certain possible
dynamical processes in networks of
the defined type which have the character of  phase transitions or collective
behavior and may induce {\it
dimensional change}. Most importantly we develop a physically
appropriate concept of'{\it dimension}' for such irregular discrete
structures which may be of importance in a wider context.
\section{Discrete Analysis on Networks}
At first glance one would surmise that as an effect of discreteness
something like a network will lack sufficient structure for such a
discipline to exist, but this is not so. Quite the contrary, there
are intimate and subtle relations to various recent branches of pure
mathematics as e.g. '{\it cyclic (co)homology}', '{\it noncommutative
de Rham complexes}', '{\it noncommutative geometry}' in general and
the like (see e.g. \cite{13}-\cite{16}, as a beautiful and concise
survey we recommend also \cite{coque}).

The general aim of these recent developements is it to generate
something like a geometrical and differentiable structure within 
certain mathematical contexts which traditionally are not considered
to support such structures.  Particularly simple examples are
discrete sets of, say, points, e.g. lattices. In a series of papers
Dimakis and M\"uller-Hoissen have applied the general abstract
machinery to models like these, having a possible bearing to, say,
lattice field theory etc. (see e.g. \cite{8} and further references
there).

The fundamental object in these approaches is typically the socalled
'{\it universal differential algebra}' or '{\it differential
envelope}' which can be canonically constructed over any associative
algebra and which is considered to be a generalisation or surrogate
(depending on the point of view) of a differential structure in the
ordinary cases.

As this notion may already indicate, this scheme, paying tribute to its
universality and generality, is sometimes relatively far away from
the concrete physical models one is perhaps having in mind. In the
case of networks, for example, the inevitable starting point is the
'{\it maximally connected}' network or graph (also called a '{\it
complete graph}' or in algebraic topology a '{\it simplex}'), i.e. any
two nodes are directly connected by a bond.

As a consequence, the construction is lacking, at least initially,
something which is of tantamount importance in physical models, i.e.
a natural and physically motivated neighborhood structure. Typically
the interesting physical models are relatively lowly connected, which
implies that they usually exhibit a pronounced feeling of what is near
by or far away on the network.

One can of course pull this general structure down to the level of
the models one may have in mind by imposing '{\it relations}' between
various classes of '{\it differential forms}' employing a general
result that each differential calculus over an algebra is isomorphic
to the universal one modulo a certain 'differential ideal' but anyway, given a
concrete model this approach is relatively abstract and perhaps not
the most transparent and direct one. While being mathematically
correct we want nevertheless to make some reservations as to its
concrete meaning as far as specific models are concerned
(e.g. networks and graphs); see below. Furthermore, it stresses more
the global algebraic relations and perhaps not so much the inherent
topological/geometrical content of the given model theory. 

To put it differently, networks and graphs are only ''mildly
non-commutative'' or rather 'non-local'. On the other hand they convey
a lot more of extra structure (as most models do), which is not
automatically implemented in the general algebraic scheme but has to
be brought to light by scrutinizing the specific model class under discussion.

To put it in a nutshell: one can either go the way ''top down'',
starting from some branch of non-commutative geometry and realize in
the course of time that e.g. discrete sets or graphs may serve as
certain model systems for this abstract algebraic scheme, or one may
start from some concrete physical speculations about the supposed fine
structure of the physical vacuum and space-time as dynamical unfolding
networks and then make ones way ''bottom up'' observing that part of
the emerging mathematical structure  may be viewed as a variant of
non-commutative geometry.

We followed this latter route and think the two philosophies may
complement each other even if, coming from different directions, one
may sometimes end up at formally closely related concepts.

We begin with the introduction of some useful concepts borrowed from
algebraic topology and also known from graph theory (as to this we
recommend the beautiful book of Lefschetz, \cite{17}.

At first we have to give the graph an '{\it orientation}':\\[0.5cm]
{\bf 3.1 Definition(Orientation)}: With the notions defined in
definition 2.2 we say the bond $\bi$ points from node $n_i$ to node
$n_k$, the bond $b_{ki}$ from $n_k$ to $n_i$. We call $n_i$, $n_k$
initial and terminal node of $\bi$ respectively. We assume the up to
now formal relation:
\begin{equation} \bi=-b_{ki} \end{equation}
Remark: Note that orientation in the above (mathematical) sense is
different from what is understood in many applications as '{\it
directed bond}' in a network (as e.g. in typical "Kauffman nets",
\cite{3}). There a directed bond can typically "transport", say, a
message only in one given fixed direction. That is, nets of this type
behave, in physical terms, pronouncedly anisotropic locally. The
definition 3.1, on the other side, is rather implementing something
like the orientation of curves.\\[0.5cm]
{\bf 3.2 Definition(Chain Complexes)}: We introduce, to begin with,
the two vector spaces $C_0$, $C_1$ whose elements, {\it zero- and
one-chains} are defined by up to now formal expressions
\begin{equation} \underline{c}_0 :=\sum f_in_i\quad \underline{c}_1
:=\sum g_{ik}b_{ik} \end{equation} \\[0.5cm]
where the $f_i$'s and $g_{ik}$'s range over a certain given field or
ring, of in the simplest cases numbers (i.e.$\Ir$,$\Rl$,$\Cx$), the
$n_i$'s and $b_{ik}$'s serve as generators of a free abelian
group.\\[0.5cm]
Remarks: i) Evidently one could in a next step choose much more general
spaces like, say, groups or manifolds.\\
ii) Furthermore, for the time being, the $f_i$'s and $g_{ik}$'s
should not be confused with the $s_i$'s and $J_{ik}$'s introduced in
section 2. The $f_i$'s and $g_{ik}$'s are e.g. allowed to vanish
outside a certain given cluster of nodes in various calculations or,
put differently, it may be convenient to deal only with certain
subgraphs.\\
iii) The spaces $C_0$, $C_1$ are in fact only the first two members
of a whole sequence of spaces.\\[0.5cm]
{\bf 3.3 Definition (Boundary)}: we now define a {\it boundary
operator} by
\begin{equation} \delta \bi:=n_k-n_i \end{equation}
which by linearity induces a linear map from $C_1$ to $C_0$:
\begin{equation} \delta: C_1\ni \sum g_{ik}\bi\to \sum
g_{ik}(n_k-n_i)\in C_0 \end{equation}
The kernel, $Z_1$ of this map, the 1-chains without '{\it boundary}',
consist of the '{\it 1-cycles}'. A typical example is a '{\it loop}',
i.e. a sequence of bonds, $\sum_{\nu}b_{i_{\nu}k_{\nu}}$ s.t.
$k_{\nu}=i_{\nu+1}$ and $k_n=i_1$. (However not every cycle is a
loop!).\\[0.5cm]
{\bf 3.4 Definition(Coboundary)}: Analogously we can define the coboundary
operator as a map from $C_0$ to $C_1$:
\begin{equation} dn_i:=\sum_k b_{ki} \end{equation}
where the sum extends over all bonds having $n_i$ as terminal node,
and by linearity:
\begin{equation} d: \sum_i f_in_i \to \sum_i f_i\,(\sum_k b_{ki})
\end{equation}
Remarks:i)In algebraic topology 'cotheory' is frequently defined on
'dual spaces'. At the moment we do not make this distinction.\\
ii)To avoid possible formal complications, we always assume the
'degree' of the nodes to be uniformly bounded away from
infinity. These matters could however be more appropriately dealt with
after the introduction of suitable metrics, norms and related
topological concepts.\vspace{0.5cm}

We will now show that these two operations, well known in algebraic
topology, can be fruitfully employed to create something like a
discrete calculus. Evidently, the 0-chains can as well be considered as
functions over the set of nodes; in this case we abbreviate them by
f,g etc. (if necessary, chosen from  a certain subclass of 0-chains ${\cal A} \subset  
C_0$, e.g. of '{\it finite support}', $L^1$, $L^2\ldots$). $\cal A$ is
trivially a module over itself (pointwise multiplication) freely
generated by the nodes $\{n_i\}$ which can be identified with the
'{\it elementary functions}' $e_i:=1\cdot n_i$.

With $b_{ik}=-b_{ki}$ we can write the action of $d$ on $f$
differently, thus making more transparent its slightly hidden
meaning:\\
with $b_{ki}=1/2(b_{ki}-b_{ik})$ we get
\begin{equation}\sum_i f_i \sum_k b_{ki}=1/2\sum_{ik}
  (f_k-f_i)\,b_{ik}\end{equation}
i.e:\\[0.5cm]
{\bf 3.5 Observation}: \begin{equation} df=d(\sum_i
f_in_i)=1/2\cdot\sum_{ik}(f_k-f_i)\,\bi \end{equation} \vspace{0.5cm}

We have still to show to what extent the operation d defined above
has the properties we are expecting from an (exterior) derivation.
The really crucial property in the continuum case is the (graded)
Leibniz rule. This is in fact a subtle and interesting point. To see
this we make a short aside about how discrete differentiation is
usually expected to work.

Take the following definition:\\[0.5cm]
{\bf 3.6 Definition (Partial Forward Derivative and Partial
Differential at Node (i))}:
\begin{equation} \nabla_{ik}f(i):=f(k)-f(i) \end{equation}
where $n_i,n_k$ are '{\it nearest-neighbor-nodes}', i.e. being
connected by a bond $\bi$.\\[0.5cm]
{\bf 3.7 Observation}: 
\begin{eqnarray}
\nabla_{ik}(f\cdot g)(i) & = & (f\cdot g)(k)-(f\cdot g)(i)
\nonumber\\
                         & = & \nabla_{ik}f(i)\cdot
g(i)+f(k)\cdot\nabla_{ik}g(i)\\
                         & = &
\nabla_{ik}f(i)g(i)+f(i)\nabla_{ik}g(i)+\nabla_{ik}f(i)\nabla_{ik}g(i)
\end{eqnarray}
In other words the "derivation" $\nabla$ does {\bf not} obey the ordinary(!)
Leibniz rule. In fact, application of $\nabla$ to, say, higher powers
of f becomes increasingly cumbersome (nevertheless there is a certain
systematic in it). One gets for example:
\begin{equation} \nabla_{ik}f^{(n)}(i)=\nabla_{ik}f(i)\cdot
\{f^{(n-1)}(k)+f^{(n-2)}(k)f(i)+\ldots+f(k)f^{(n-2)}(i)+f^{(n-1)}(i)\}
\end{equation} 

Due to the discreteness of the formalism and, as a consequence, the
inevitable bilocality of the derivative there is no chance to get
something as a true Leibniz rule on this level. (That this is
impossible has also been stressed from a different point of view in
e.g. example 2.1.1 of \cite{14}).\\[0.5cm]
Remark: We will come back to the non-Leibnizean character of $\nabla$
below when establishing a {\it duality} between $d$ and $\nabla$. It
is in fact a rather interesting relation even from a purely algebraic
point of view, as it is a structural relation known in algebraic
topology as {\it 'Kuntz algebra'} (cf. \cite{coque}). Before however
doing that we will further clarify the role of $d$.\vspace{0.5cm} 

In some sense it is considered to be one of the merits of the
abstract algebraic framework (mentioned at the beginning of this
section) that a graded Leibniz rule holds in that generalized case
almost automatically.
The concrete network model under investigation offers a good
opportunity to test the practical usefulness of concepts like these.

To write down something like a Leibniz rule an important structural
element is still missing, i.e. the multiplication of node functions
from, say, some $\cal A$ with the members of $C_1$, in other words a
'{\it module structure}' over $\cal A$. One could try to make the
following definition:
\begin{equation} f\cdot\bi:=f(i)\cdot\bi\quad \bi\cdot
f:=f(k)\cdot\bi \end{equation}
and extend this by linearity.

Unfortunately this "definition" does not respect the relation
$\bi=-b_{ki}$. We have in fact:
\begin{equation} f(i)\bi=f\cdot\bi=-f\cdot b_{ki}=-f(k)b_{ki}=f(k)\bi
\end{equation}
which is wrong in general for non-constant $f$!

Evidently the problem arises from our geometrical intuition which
results in the natural condition $\bi=-b_{ki}$, a relation we however
want to stick to. On the other side we can extend or embed our
formalism algebraically in a way which looses the immediate contact with
geometrical evidence but grants us with some additional mathematical
structure. This is in fact common mathematical practice and a way to
visualize e.g. the {\it 'universal differential algebra'} in {\it
  'non-commutatine geometry'} (see e.g. \cite{coque}). We want however
to complement this more algebraic extension scheme by a, as we think,
more geometric one below. 

We can define another relation between nodes, calling two nodes
related if they are connected by a bond with a built-in direction 
from the one to the other. We write this in form of a {\it tensor
product} structure. In the general tensor product $C_0\otimes C_0$ we
consider only the subspace $\C$ spanned by the elements $\n$ with
$n_i,n_k$ connected by a bond (i.e. $i\neq k$!) and consider $\n$ to be unrelated to
$n_k\otimes n_i$, i.e. they are considered to be linearly independent
basis elements.\\[0.5cm]
{\bf 3.8 Observation}: There exists an isomorphic embedding of $C_1$ onto
the subspace generated by the antisymmetric elements in $\C$, i.e:
\begin{equation} \bi\to \nn=:\na\quad (\mbox{with}\; i\neq k) \end{equation}
generate an isomorphism by linearity between $C_1$ and the
corresponding subspace $C_0\wedge C_0 \subset \C$.\\[0.5cm]
Proof: Both $\bi$ and $\na$ are linearly independent in there
respective vector spaces. \vspace{0.5cm}

In contrast to $C_1$ the larger $\C$ now supports a non-trivial and natural
bimodule structure:\\[0.5cm]
{\bf 3.9 Observation/Definition (Bimodule)}: We can now define
\begin{eqnarray} f\cdot (\n) & := & f(i)(\n)\\
          (\n)\cdot f        & := & f(k)(\n)
\end{eqnarray}
and extend this by linearity to the whole $\C$, making it into a
bimodule over some ${\cal A} \subseteq C_0$.\\[0.5cm]
Remarks:i) Equivalently one could replace $\n$ by $e_i\otimes e_k$, the
corresponding elementary functions. If one now identifies
$e_i\otimes e_k$ with the abstract symbols $e_{ik}:=e_ide_k$ employed
in \cite{8}, one may establish a link to perhaps more abstract but
related approaches.\\
ii) Another case in point is our definition $dn_i=\sum_k b_{ki}$ and
the representation $da=\idty\otimes a-a\otimes\idty$ which is employed
within the context of the universal differential algebra. With
$\idty=\sum_i 1\cdot n_i$ the close relation becomes apparent.
\\[0.5cm]
{\bf 3.10 Lemma}: As a module over $\cal A$,\, $\C$ is generated by
$C_0\wedge C_0$.\\[0.5cm]
Proof: It suffices to show that every $\n$ can be generated this way.
\begin{equation} n_i\cdot\nn=\n \end{equation}
as $n_i\cdot n_k=0$ for $i\neq k$. \vspace{0.5cm}

With the $\bi$ so embedded in a larger space and identified with
\begin{equation}
\nn=\na 
\end{equation}
we are in the position to derive a graded Leibniz
rule on the module (algebra) $\A$. Due to linearity and the structure
of the respective spaces it suffices to show this for products of
elementary functions $e_i=n_i$. The same relation could of course be
directly verified in a slightly more elegant way by regrouping 
\begin{equation}d(f\cdot g)=\sum_i f_ig_idn_i=\sum_i f_ig_i(\sum_k \nn)
\end{equation}
appropriately, employing {\bf 3.9 Observation}.
We in fact have:\\[0.5cm]
($i\neq k$ not nearest neighbors):\begin{equation}d(n_i\cdot
n_k)=0,\,dn_i\cdot n_k=n_i\cdot dn_k=0 \end{equation}
($i\neq k$ nearest neighbors):\begin{equation}d(n_i\cdot
n_k)=d(0)=0\;\;\mbox{and} \end{equation}
\begin{eqnarray}dn_i\cdot n_k+n_i\cdot dn_k & = &
(\sum_{k'}b_{ik'})\cdot n_k+n_i\cdot(\sum_{i'}b_{ki'})\\ & = &
\bi\cdot n_k+n_i\cdot b_{ki}\\
& = & \{\nn n_k+n_i(n_k\otimes n_i-n_i\otimes
n_k)\}\\ & = & \{n_i\otimes n_k-n_i\otimes n_k\}=0 \end{eqnarray} 
$(i=k)$:
\begin{equation} d(n_i^2)=d(n_i)=\sum_k\bi\;\;\mbox{and} \end{equation}
\begin{eqnarray} dn_i\cdot n_i+n_i\cdot dn_i & = &
(\sum_k\bi)\,n_i+n_i(\sum_k\bi)\\ & = & \sum_k\nn=\sum_k\bi=dn_i
\end{eqnarray}
{\bf 3.11 Conclusion}: As a map from the bimodule ${\cal A} \subseteq
C_0$ to the  bimodule $\C$ generated by the elements $\bi$
over $\A$ the map d fulfills the Leibniz rule, i.e:
\begin{equation} d(f\cdot g)=df\cdot g+f\cdot dg \end{equation}

>From the above we see also that functions, i.e. elements from $\A$
and bonds or differentials of functions do no longer commute (more
specifically, the two possible ways of imposing a module structure
could be considered this way). We have for example:\\[0.5cm]
{\bf 3.12 Commutation Relations}:\\
($i\neq k$ not nearest neighbors): \begin{equation} n_i\cdot
dn_k=dn_k\cdot n_i=0 \end{equation}
($i\neq k$ nearest neighbors). \begin{eqnarray} n_i\cdot
dn_k+dn_k\cdot n_i & = & 1/2\sum_{i'}\{n_i(n_k\otimes
n_{i'}-n_{i'}\otimes n_k)\\ & + & (n_k\otimes
n_{i'}-n_{i'}\otimes n_k)\,n_i\}\\ & = & -1/2\nn=-\bi \end{eqnarray}
($i=k$): \begin{equation} n_i\cdot dn_i+dn_i\cdot n_i=\sum_k\bi=dn_i
\end{equation}

Another important relation we want to mention is the followng:
$\delta\,d\,f$ is a map from $C_0\to C_0$ and reads in detail:\\[0.5cm]
{\bf 3.13 Observation (Laplacian)}: \begin{equation}
\delta\,d\,f=-\sum_i(\sum_k f(k)-n\cdot f(i))\,n_i=:-\Delta\,f \end{equation}
with n the number of nearest neighbors of $n_i$.\\[0.5cm]
Proof: \begin{eqnarray} \delta\,d\,f & = &
1/2\sum_{ik}(f(k)-f(i))(n_k-n_i)\\ & = &
1/2\sum_{ik}(f(k)\,n_k+f(i)\,n_i-f(i)\,n_k-f(k)\,n_i)\\ & = &
-\sum_i(\sum_kf(k)-n\cdot f(i))\,n_i \end{eqnarray}

Having now established the first steps in setting up this particular
version of discrete calculus one could proceed in various directions.
First, one can develop a discrete Lagrangian variational calculus,
derive Euler-Lagrange-equations and Noetherian theorems and the like
and compare this approach with other existing schemes in discrete
mathematics.  

Second, one can continue the above line of reasoning and proceed to
more sophisticated geometrical concepts and set them into relation to
existing work, e.g. \cite{8}. For the time being we would like to
follow this latter route.

The philosophy underlying non-commutative geometry is that
e.g. individual points of, say, a manifold have to be dispensed with
and replaced by some equivalent of the algebra of functions over the
manifold. On the other side networks are, as was already mentioned
above, only mildly non-commutative and carry still a pronounced local
structure (even the notion of points make still sense),
which it may be advisable to implement in the discrete calculus.

As a consequence we will develop, as in the case of ordinary
manifolds, differentials and partial derivatives in
parallel as dual concepts. This may be, in our opinion, the main
difference of our approach to other existing work in the field.

A characteristic feature of network calculus is its
non-locality. According to 3.6 Definition the partial derivatives
$\nik$ act locally, i.e. one can consider them as acting at node
$n_i$. On the other side, this is not so for the $\bi$; it leads to
inconsistencies if one tries to relate them somehow to a definite
node. We are however free to introduce the dual concept with respect
to the $\nik$ and define: \\[0.5cm]
{\bf 3.14 Definition ((Co)Tangential Space)}:\\i) We call the space
spanned by the $\nabla_{ik}$ at node $n_i$ the tangential space
$T_i$.\\ ii) Correspondingly we introduce  the space spanned by the
$d_{ik}$ at node $n_i$ and call it
the cotangential space $T_i^{\ast}$ with the $d_{ik}$ acting as linear
forms over $T_i$ via:
\begin{equation} <d_{ik}|\nabla_{ij}>=\delta_{kj} \end{equation}\\[0.5cm]
{\bf 3.15 Definition/Observation}: Higher tensor products of differential forms
at a node $n_i$ can be defined as {\it multilinear forms}:
\begin{equation} <d_{ik_1}\otimes\cdots\otimes
d_{ik_n}|(\nabla_{il_1},\cdots,\nabla_{il_n})>:=\delta_{k_1l_1}
\times\cdots{\times} \delta_{k_nl_n} \end{equation}
and linear extension.\vspace{0.5cm}

In a next step we extend these concepts to functions $f\in
C_0$ and {\it 'differential operators'} or {\it 'vector fields'} $\sum a_{ik}\nik$
and make the following interpretation:\\[0.5cm]
{\bf 3.16 Interpretation}: We identify $\bi$ with
$d_{ik}-d_{ki}$\vspace{0.5cm}

We have to check whether this is a natural(!) identification.\\[0.5cm]
{\bf 3.17 Observation}: Vector fields $v:=\sum a_{ik}\nik$ are assumed
to act on functions $f=\sum f_in_i$ in the following manner:
\begin{equation}v(f):=\sum a_{ik}(f_k-f_i)n_i\end{equation}
i.e. they map $C_0\to C_0$.\\[0.5cm]
{\bf 3.18 Corollary}: Note that this implies:
\begin{equation}\nik n_k=n_i\quad \nik n_i=-n_i\end{equation}
\begin{equation}\nki n_k=-n_k\quad \nki n_i=n_k\end{equation}\\[0.5cm]
{\bf 3.19 Observation}: {\it 'Differential forms'} $\omega =\sum
g_{ik}d_{ik}$ act on vector fields $v=\sum a_{ik}\nik$ according to:
\begin{equation}<\omega|v>=\sum
  g_{ik}a_{ik}n_i\end{equation}\vspace{0.5cm}

With these definitions we can calculate $<df|v>$ with
\begin{equation}df=1/2\sum (f_k-f_i)\bi=\sum
  (f_k-f_i)d_{ik}\end{equation}
according to 3.16 Interpretation. Hence:
\begin{equation}<df|v>=<\sum (f_k-f_i)d_{ik}|\sum a_{ik}\nik>=\sum
  (f_k-f_i)a_{ik}n_i\end{equation}
which equals:
\begin{equation}(\sum a_{ik}\nik)(\sum
  f_in_i)=v(f)\end{equation}\\[0.5cm]
{\bf 3.20 Consequence}: Our geometric interpretation of the algebraic
objects reproduces the relation:\\
i) \begin{equation}<df|v>=v(f)\end{equation}
known to hold in ordinary differential geometry, as is the case for
the following relations, and shows that the definitions made above
seem to be natural.\\
Furthermore vector and covector fields are left modules under the
action of ${\cal A} \subseteq C_0$:\\
ii) \begin{equation}(\sum f_in_i)(\sum a_{ik}\nik):=\sum f_ia_{ik}\nik
\end{equation}
iii)\begin{equation}(\sum f_in_i)(\sum g_{ik}d_{ik}):=\sum
f_ig_{ik}d_{ik}\end{equation}
iv) As was the case with the $\n$ the $d_{ik}$ generate also in a
natural way a right module over ${\cal A}$.\vspace{0.5cm}

As was already mentioned above, the partial derivatives $\nik$ do not
obey the ordinary (graded) Leibniz rule in contrast to the operator
$d$. In the latter case this is however only effected by embedding the
''natural'' geometric objects in a bigger slightly more abstract
space. The structural relation the $\nik$ actually are obeying (see 3.7
Definition) is known from what is called by algebraic topologists a
{\it 'Kuntz algebra'} (cf. \cite{coque}); which however does occur
there in
a different context. With $q:=\nik$ we have:\\[0.5cm]
{\bf 3.21 Observation (Kuntz algebra)}:
\begin{equation}q(f\cdot g)=q(f)\cdot g+f\cdot q(g)+q(f)\cdot
  q(g)\end{equation}
and analogously for vector fields $\sum a_{ik}\nik$.\\
With $u:=1+q$ we furthermore get:
\begin{equation}u(f\cdot g)=u(f)\cdot u(g) \end{equation}
and
\begin{equation}q(f\cdot g)=q(f)\cdot g+u(f)\cdot q(g)\end{equation}
i.e. a {\it 'twisted derivation'} with $u$ an endomorphism from $\A$
to $\A$.\vspace{0.5cm}

As with $\nik$ the product rule for higher products can be inferred
inductively:
\begin{equation}q(f_1\cdots f_n)=\sum_i f_1\cdots q(f_i)\cdots
  f_n+\sum_{ij} f_1\cdots q(f_i)\cdots q(f_j)\cdots f_n+\ldots
  q(f_1)\cdots q(f_n) \end{equation}\vspace{0.5cm}

It is an interesting task to compare our framework sketched above with
the perhaps more algebraic approaches which can be found in the
mentioned literature or more recently in \cite{baehr}. It should
however be stressed that the other approaches are typically concerned
with energy scales (e.g. standard model) different from our more
primordial level. So we prefer to do this elsewhere apart from making
some remarks of perhaps more principal interest concerning the
appropriate geometric interpretation or meaning of the various
abstract and algebraic symbols; among other things about the relation
between the socalled universal differential algebra and the highly
{\it 'reduced calculi'} occurring typically in physical applications.

In contrast to the universal differential algebra mentioned above,
where every two nodes are connected by a bond, this is not so for our
'{\it reduced}' calculus. As a consequence certain operations are
straightforward to define in the former approach. However, descending
afterwards  to the lower-connected more realistic models is tedious
in general and not always particularly transparent. That is, this
method does not always really save calculational efforts (for a
discussion of certain simple examples see \cite{8}). It even may lead
to (in our view) unnatural results as we want to show below.

By the way, the mathematical "triviality " of the differential envelope is
reflected by the trivialty of the corresponding '{\it (co)homology
groups}' of the maximally connected graph (simplex). This trivialty
is then broken by deleting graphs in the reduction process.

To mention a typical situation: Take e.g. the subgraphs of a graph G
consisting of, say, four nodes $n_i,n_k,n_l,n_m$
and all the bonds between them which occur in G. In the case G being a simplex (i.e. non-reduced case) all these
subgraphs are geometrically/topologically equivalent. An important
consequence of this is that the four nodes can be connected by a
{\it 'path'}, i.e. a sequence of consecutive bonds, each being passed
only once, the effect being that one can naturally define a 
multiplication in this scheme via '{\it concatenation}', e.g:
\begin{equation} (\n)\cdot(n_l\otimes
n_m)=n_i\otimes n_k\cdot n_l\otimes n_m \end{equation}
with $n_k\cdot n_l=\delta_{kl}\cdot n_k$. Correspondingly, all the
higher differentials can be generated by the 1-forms via
concatenation. The reason why it is sufficient to concatenate only at
the extreme left and right of a '{\it word}' stems exactly from the
simplex-character of each subgraph.

In typical reduced cases all this is no longer the case; the
combinatorical topology becomes non-trivial. To give an example: Take
as G a regular graph with the degree of the nodes (number of incident
bonds) $n=3$. In the analogous case of 4-node-subgraphs there exists
now a kind of subgraph the nodes of which cannot be concatenated in
the above way. Take e.g. the subgraph with bonds existing only
between, say, $n_1\,n_4$, $n_2\,n_4$, $n_3\,n_4$. I.e., one has the
1-forms $b_{14},\;b_{24},\;b_{34}$ or:
\begin{equation} n_1\otimes n_4,\;n_2\otimes n_4,\;n_3\otimes n_4
\end{equation}
but there is no obvious way to generate the corresponding reduced
subgraph by concatenating them sequentially(!) without passing certain
bonds twice.

We have developed above two pictures, one of a more algebraic flavor,
the other of a perhaps more geometric content. On the one side we
identified $C_1=\{\sum g_{ik}b_{ik}\}$ with the antisymmetric part of
a certain subspace $\C$ of the tensor product over $C_0$, see 3.8
Observation. $\C$ was built from the elements $\n$ being connected by
a bond; in physical terms: sites, coupled dynamically, and $\bi =\nn$.

Along these lines one can build ''higher elements'' making the
following definition:\\[0.5cm]
{\bf 3.22 Definition}: The space 
\begin{equation}\Omega_{k-1}:=C_0\hten\cdots\hten
C_0\;\mbox{(k-times)}\end{equation} 
is built from elements 
\begin{equation}n_{i_0}\otimes\cdots\otimes
n_{i_k}=:n_{i_1\ldots i_k}\end{equation}
with each pair of consecutive nodes
$n_{i_l},n_{i_{l+1}}$ being connected by a bond. In graph language
these elements may be associated with {\it bond sequences}.\\
If a pair of consecutive nodes occurs only once in the sequence (in
other words: a bond) the bond sequence is called a {\it path}. If,
furthermore, initial and terminal node coincide the bond sequence is
called a {\it closed path} or {\it loop}. If in a closed path none of
the inner nodes occurs twice it is called in graph theory a {\it
  circuit}.\\[0.5cm]
Remarks: i) In the following it is always understood that $\otimes$
occurs always between nodes which are connected by a bond.\\
ii) Note that sequences like e.g. $n_i\otimes n_k\otimes n_i$ are
admitted.\vspace{0.5cm}

Multiplication can now naturally be defined via {\it
  'concatenation'}:\\[0.5cm]
{\bf 3.23 Definition (Concatenation)}:
\begin{equation}(n_{i_0}\otimes\cdots\otimes
  n_{i_k})\cdot(n_{j_0}\otimes\cdots\otimes
  n_{j_l})=n_{i_0}\otimes\cdots\otimes n_{i_k}\cdot
  n_{j_0}\otimes\cdots\otimes n_{j_l}\end{equation}
with the rhs an element of $\Omega_{k+l}$. (Note that $n_{i_k}\cdot
n_{j_0}$ means multiplication in $C_0$; i.e. this yields either $0$ or
$n_{i_k}=n_{j_0}$).\\[0.5cm]
Remarks:i) Hence, multiplication is basically the pasting together of
bond sequences, i.e. a geometric process.\\
ii) The above shows that the ring which can be formed this way has
typically a
lot of {\it 'zero divisors'} apart from the special situation where
$G$ is a complete graph (simplex).\\[0.5cm]
{\bf 3.24 Visualisation}: Following this philosophy the network
or graph $G$ we started from can be identified algebraically with the
ring or bimodule with the above multiplication (concatenation) and
non-zero elements all the admissible bond sequences defined
above.\\[0.5cm]
Remark: Note however that an object like $n_0\otimes\cdots\otimes
n_k\in \Omega_k$ should not be confused with $n_0\otimes
n_1+n_1\otimes n_2+\cdots n_{k-1}\otimes n_k\in \Omega_1$. Both
elements can be associated with a bond sequence or path leading from
$n_0$ to $n_k$, but they are of an entirely different algebraic (and
geometric(!)) character. In the former case the pieces of the bond
sequence are concatenated multiplicatively in the latter case they are
composed additively.\vspace{0.5cm}

On the other side we can concatenate the {\it dual forms} $d_{ik}$
non-locally (a process which is not employed in the ordinary continuum
differential geometry, which deals typically with local fields, but
which could also be performed in the latter context, leading perhaps
to new and interesting structures; cf. e.g. \cite{coque} where
non-commutative calculus is introduced on the commutative algebra of
smooth functions over a manifold).

For example one can form the space $T_i^{\ast}\hten T_k^{\ast}$, its
elements acting bilinearly on pairs of vectors $v_i,v_k$ with $v_i\in
T_i,v_k\in T_k$.\\[0.5cm]
{\bf 3.25 Conclusion}: From the above we can infer that, whereas $\n$
should not be viewed as an element of $T_i^{\ast}$ (they are different
geometric objects), the whole algebraic structure built over the set
$\{\n\}$ via concatenation is isomorphic to the corresponding
structure built over the set $\{d_{ik}\}$ with e.g:
\begin{equation}n_{i_0}\otimes\cdots\otimes n_{i_k}\to
  d_{i_0i_1}\otimes\cdots\otimes d_{i_{k-1}i_k}\end{equation}

The above observations give us the chance to generate higher
dimensional geometric objects which may be considered as equivalents
of the building blocks of piecewise affine or triangulated smooth
manifolds. Following our general philosophy of creating
geometric/topological concepts for {\it 'non-standard'} spaces (cf. in
particular the next section), one can try to catch the abstract
essence of a notion like surface or volume in developing the following
scheme which we however only sketch, for the time being, with the help
of an example.

The map $d$ mapped a node $n_i$ onto the sum over bonds $\sum b_{ki}$
with endpoint $n_i$, where the oriented $b_{ki}$ was the
antisysmmetric combination $d_{ki}-d_{ik}$ or $n_{ki}-n_{ik}$. In the
same sense one can proceed geometrically if the graph has an
appropriate structure:\\[0.5cm]
{\bf 3.26 Definition}: A triangle in a graph is a triple of nodes
$n_1,n_2,n_3$ with $n_1\,n_2,n_2\,n_3,n_3\,n_1$ connected by bonds.\\
In case $n_1,n_2,n_3$ form a triangle it should be
algebraically/geometrically realized (as is the case with the bond
$b_{ik}$) as a geometric object being a totally antisymmetric
combination of elements of $\Omega_2$ in the following sense:
\begin{equation}s_{123}:=\sum_{per}
  \sigma(per)n_{i_1i_2i_3}\end{equation}
with $\sigma(per)$ the signum of the corresponding permutation of
$\{1,2,3\}$. \\[0.5cm]
{\bf 3.27 Observation}: With this definition the triangle has exactly
two possible orientations , i.e:
\begin{equation}s_{123}=-s_{321}\end{equation}
which corresponds with the two possible orientations of the path
around the triangle.\\[0.5cm]
Remarks: i) It is of course possible that $n_{12},n_{23}$ and therefore
$n_{123}$ can be built while $n_1\,n_3$ are not connected s.t. they do
not form a triangle. A fortiori a graph need not have triangles.\\
ii) Following similar lines one can build also higher geometric
objects.\vspace{0.5cm}

{\bf 3.28 Extension of d}: The geometric idea behind the attempt to
extend in a next step the range of the map $d$ is that it should
relate a bond (edge) with an appropriate combination of the triangles
being incident with it, i.e:
\begin{equation}d(b_{12})=\sum_i s_{12i}\end{equation}

This will cause certain algebraic problems of a general nature for
arbitrary, in particular non-regular, graphs (see the above remark) as
typical relations known to hold e.g. in {\it 'simplicial cohomology'}
like $d\circ d=0$ will not hold automatically. The necessary
preconditions could however be analyzed but we shall not make the
attempt to do this here due to lack of space. Instead of that we will
close this section by addressing another topic which is in fact
closely related to these problems and will make contact with a
different approach which starts from the universal differential
envelope, i.e., given the class of nodes, from the, in our language,
complete graph (simplex); see e.g. \cite{8}. 

In that extremely regular situation matters are rather smooth and can
directly be taken over from the general case of the universal
differential algebra $\Omega(\A)$ defined over an arbitrary algebra
$\A$. In our notation one defines e.g:
\begin{equation}d_u(n_{1\ldots
    k})=\sum_i(\sum_{\nu=0}^{k+1}(-1)^{\nu}n_{1\ldots i\ldots
    k})\end{equation}
with $\nu$ denoting the place of the insertion of node $n_i$,
beginning with $\nu=0$, i.e. before $n_1$ and, as always, consecutive
nodes being different understood. One shows immediately that
e.g. $d_u\circ d_u=0$ on $\Omega$.\\[0.5cm]
Remarks:i) Note that each term on the rhs is well defined since all
the nodes are connected with each other.\\
ii) With $d_u$ we denote the {\it universal} derivation.\vspace{0.5cm}

We remarked at the beginning of this section that it is a general
result that each differential calculus (more specifically: a certain
differential algebra) over a given algebra $\A$ is isomorphic to the
universal one divided by a certain {\it 'differential ideal'}. This
was exploited in \cite{8} to construct a differential calculus on
certain simple examples of {\it 'reduced'} (smaller) differential
algebras.

In our specific context of networks and graphs we may translate this
general result in the following way: Let $\Omega^u=\sum \Omega^u_k$ be
the universal differential algebra with $\Omega^u_k$ consisting of the
$(k+1)$-fold tensor products of arbitrary $(k+1)$-tuples of nodes. We
can define a projector $\Pi$ which projects $\Omega^u$ onto
$\Omega=\sum \Omega_k$ with $\Omega_k$ consisting of the $(k+1)$-fold
{\it admissible} tensor products (bond sequences) of connected nodes
in our actual network. We have:
\begin{equation}\Pi(n_0\otimes\cdots\otimes n_k)=0\end{equation}
if $(n_0,\ldots ,n_k)$ is not admissible
\begin{equation}\Pi(n_0\otimes\cdots\otimes
  n_k)=n_0\otimes\cdots\otimes n_k\end{equation}
if $(n_0,\ldots ,n_k)$ is admissible.\\[0.5cm]
{\bf 3.29 Consequence}: We have 
\begin{equation}\Pi=\Pi^2\; , \;
  \Omega^u=\Pi\Omega^u+(\idty-\Pi)\Omega^u\end{equation}
with $\Pi\Omega^u=\Omega$.\\
We can define
\begin{equation}d:=\Pi\circ d_u\circ\Pi\end{equation}
which leaves $\Omega$ invariant but in general $d\circ d\neq 0$ in
contrast to $d_u\circ d_u=0$.\vspace{0.5cm}

The reason is the following: We can make $Ker(\Pi)$ into a two-sided
ideal $I$ consisting of the elements $n_{0\ldots k}$ having at least
one pair of consecutive nodes {\it not} being connected by a
bond. This ideal $I$ is {\it not} invariant under the action of $d_u$!
A closer analysis shows that $d_u(n_{0\ldots k})\notin I$ if there
occur {\it 'insertions'} between non-connected neighbors
s.t. non-admissible elements become admissible,
i.e. connected.\\[0.5cm]
{\bf 3.30 Observation}: In general there exist elements $n_{0\ldots
  k}\in Ker(\Pi)$ s.t. $\Pi(d_u(n_{0\ldots k}))\neq 0$, in other
words, $I$ is in general not left invariant by $d_u$.\\This is the
reason for e.g. the non-vanishing of $d\circ d$.\\
If one wants to make $\Omega$ a real differential algebra one has to
enlarge $I$!\\[0.5cm]
{\bf 3.31 Consequence}: The ideal $I'=I+d_u\circ I$ is invariant under
$d_u$ and $d$ defines a differential algebra on the smaller algebra
$\Omega^u/I'\subset \Omega$ with $\Omega=\Omega^u/Ker(\Pi)$.\\[0.5cm]
(That $I'$ is an ideal left invariant by $d_u$ is easy to prove with
the help of the property $d_u\circ d_u=0$).\vspace{0.5cm}

So much so good, but in our view there exists a certain
problem. $\Omega^u/I'$ is the algebra one automatically arrives at if
one defines the homomorphism $\Phi$ from $\Omega^u$ to the reduced
differential algebra in the following canonical way:
\begin{equation}\Phi:\; n_i\to n_i\quad d_un_i\to dn_i\end{equation}
i.e. under the premise that $d$ defines already another differential
algebra. It is in this sense the general result mentioned above has to
be understood.

On the other hand this may lead to a host of, at least in our view,
unnatural relations in concrete examples as e.g. our network which
may already carry a certain physically motivated interpretation going
beyond being a mere example of an abstract differential algebra. Note
e.g. that in our algebra $\Omega$ an element like $n_{123}$ is
admissible (i.e. non-zero) if $n_1,n_2$ and $n_2,n_3$ are
connected. $n_{123}$ may however arise from a differentiation process
(i.e. from an insertion) like $d_u(n_{13})$ with $n_1,n_3$ not(!)
connected.

This is exactly the situation discussed above:
\begin{equation}n_{13}\in I \quad\mbox{but}\quad d_u(n_{13})\notin
  I\end{equation}
Dividing now by $I'$ maps $d_u(n_{13})$ onto zero whereas there may be
little physical/geometric reason for $n_{123}$ or a certain
combination of such admissible elements being zero in our
network.\\[0.5cm]
{\bf 3.32 Conclusion}: Given a concrete physical network $\Omega$ one
has basically two choices. Either one makes it into a full-fledged
differential algebra by imposing further relations which may however
be unnatural from a physical point of view and very cumbersome for
complicated networks. This was the strategy
e.g. followed in \cite{8}.\\
Or one considers $\Omega$ as the fundamental object and each
admissible element in it being non-zero. As a consequence the
corresponding algebraic/differential structure on $\Omega$ may be less
smooth at first glance but on the other side more natural (the {\it
  'graded Leibniz-rule'} or $d\circ d=0$ may only hold up to a certain
order).\\
At the moment we refrain from making a general judgement whereas we
would probably prefer the latter choice. 

\section{Intrinsic Dimension in Networks, Graphs and other Discrete Systems}
There exist a variety of concepts in modern mathematics which
generalize the notion of '{\it dimension}' one is accustomed to in
e.g. differential topology or linear algebra. In fact, '{\it
topological dimension}' is a notion which seems to be even closer to
the underlying intuition (cf. e.g. \cite{18}).

Apart from the purely mathematical concept there is also a physical
aspect of something like dimension which has e.g. pronounced effects
on the behavior of, say, many-body-systems, especially their
microscopic dynamics and, most notably, their possible '{\it phase
transitions}'.

But even in the case of e.g. lattice systems they are usually
considered as embedded in an underlying continuous background space
(typically euclidean) which supplies the concept of ordinary
dimension so that the {\it 'intrinsic dimension'} of the discrete array itself
does usually not openly enter the considerations.

Anyway, it is worthwhile even in this relatively transparent
situations to have a closer look on where attributes of something
like dimension really come into the physical play. Properties of
models of, say, statistical mechanics  are almost solely derived from
the structure of the microscopic interactions of their constituents.
This is more or less the only place where dimensional aspects enter
the calculations.

Naive reasoning might suggest that it is the number of nearest
neighbors (in e.g. lattice systems) which reflects in an obvious way
the dimension of the underlying space and influences via that way the
dynamics of the system. However, this surmise, as we will show in the
following, does not reflect the crucial point which is considerably
more subtle.

This holds the more so for systems which cannot be considered as being
embedded in a smooth regular background and hence do not get their
dimension from the embedding space. A case in point is our primordial
network in which Planck-scale-physics is assumed to take
place. In our approach it is in fact exactly the other way round:
Smooth space-time is assumed to emerge via a {\it phase transition} or a
certain {\it cooperative behavior} and
after some '{\it coarse graining}' from this more fundamental
structure.\\[0.5cm]
{\bf 4.1 Problem}: Formulate an intrinsic notion of dimension for
model theories without making recourse to the dimension of some
embedding space.\vspace{0.5cm}

In a first step we will show that graphs and networks as introduced
in the preceding sections have a natural metric structure. We have
already introduced a certain neighborhood structure in a graph with
the help of the minimal number of consecutive bonds connecting two
given nodes.

In a connected graph any two nodes can be connected by a sequence of
bonds. Without loss of generality one can restrict oneself to '{\it
paths}'. One can then define the length of a path (or sequence of
bonds) by the number l of consecutive bonds making up the
path.\\[0.5cm] 
{\bf 4.2 Observation/Definition}: Among the paths connecting two
arbitrary nodes there exists at least one with minimal length which
we denote by $d(n_i,n_k)$. This d has the properties of a '{\it
metric}', i.e:
\begin{eqnarray} d(n_i,n_i) & = & 0\\ d(n_i,n_k) & = &
d(n_k,n_i)  \\d(n_i,n_l) & \leq & d(n_i,n_k)+d(n_k,n_l) \end{eqnarray}
(The proof is more or less evident).\\[0.5cm]
{\bf 4.3 Corollary}: With the help of the metric one gets a natural
neighborhood structure around any given node, where ${\cal U}_m(n_i)$
comprises all the nodes with $d(n_i,n_k)\leq m$, $\partial{\cal U}_m(n_i)$
the nodes with $d(n_i,n_k)=m$. \vspace{0.5cm}

With the help of the above neighborhood structure we can now develop
the concept of an intrinsic dimension on graphs and networks. To this
end one has at first to realize what property really matters
physically (e.g. dynamically) independently of the model or embedding
space. \\[0.5cm]
{\bf 4.4 Observation}: The crucial and characteristic property of,
say, a graph or network which may be associated with something like
dimension is the number of '{\it new nodes}' in ${\cal U}_{m+1}$ compared
to ${\cal U}_m$ for m sufficiently large or $m\to \infty$. The deeper
meaning of this quantity is that it measures the kind of '{\it
wiring}' or '{\it connectivity}' in the network and is therefore a
'{\it topological invariant}'.\\[0.5cm]
Remark: In the light of what we have learned in the preceding section
it is tempting to relate the number of bonds branching off a node,
i.e. the number of nearest neighbors or order of a node, to something
like dimension. On the other side there exist quite a few different
lattices with a variety of number of nearest neighbors  in, say, two- or three-
dimensional euclidean space. What however really matters in physics
is the embedding dimension of the lattice (e.g. with respect to phase
transitions) and only to a much lesser extent the number of nearest
neighbors. In contrast to the latter property dimension reflects the degree of
connectivity and type of wiring in the network. \vspace{0.5cm}

In many cases one expects the number of nodes in ${\cal U}_m$ to grow like
some power D of m for increasing m. By the same token one expects the
number of new nodes after an additional step to increase proportional
to $m^{D-1}$. With $|\,\cdot\,|$ denoting number of nodes
we hence have:
\begin{equation} |{\cal U}_{m+1}|-|{\cal U}_m|=|\partial {\cal U}_{m+1}|=f(m)
\end{equation}
with
\begin{equation} f(m)\sim m^{D-1} \end{equation}
for m large.\\[0.5cm]
{\bf 4.5 Definition}: The intrinsic dimension D of a regular
(infinite) graph is given by 
\begin{equation} D-1:=\lim_{m\to \infty}(\ln f(m)/\ln m)\; \mbox{or}
\end{equation}
\begin{equation} D:=\lim_{m\to \infty}(\ln |{\cal U}_m|/\ln m)
\end{equation}

That this definition is reasonable can be seen by applying it to
ordinary cases like regular translation invariant lattices. It is
however not evident that such a definition makes sense for arbitrary
graphs, in other words, a (unique) limit point may not always
exist. It would be tempting to characterize the conditions which
entail that such a limit exists. We, however, plan to do this elsewhere.
\\[0.5cm]
{\bf 4.6 Observation} For regular lattices D coincides with the
dimension of the euclidean embedding space $D_E$.\\[0.5cm]
Proof: It is instructive to draw a picture of the consecutive series
of neighborhoods of a fixed node for e.g. a 2-dimensional Bravais
lattice. It is obvious and can also be proved that for m sufficiently
large the number of nodes in $\U_m$ goes like a power of m with the
exponent being the embedding dimension $D_E$ as the euclidean volume
of $\U_m$ grows with the same power.\\[0.5cm]
Remarks:i) For $\U_m$ too small the number of nodes may deviate from
an exact power law which in general becomes only correct for
sufficiently large m.
\\ii) The number of nearest neighbors, on the other side, does not(!)
influence the exponent but rather enters in the prefactor. In other
words, it influences $|\U_m|$ for m small but drops out
asymptotically by taking the logarithm. For an ordinary Bravais
lattice with $N_C$ the number of nodes in a unit cell one has
asymptotically:
\begin{equation} |\U_m|\sim N_C\cdot m^{D_E} \quad\mbox{and hence:}
\end{equation}
\begin{equation} D=\lim_{m\to\infty}(\ln(N_C\cdot m^{D_E})/\ln
m)=D_E+\lim_{m\to\infty}(N_C/\ln m)=D_E 
\end{equation}
independently of $N_C$.\vspace{0.5cm}

Before we proceed a remark should be in order concerning related ideas
on a concept like dimension occurring in however completely different
fields of modern physics:

When we started to work out our own concept we scanned in vain the
literature on e.g. graphs accessible to us and consulted various
experts working in that field. From this we got the impression that
such ideas have not been pursued in that context. (It is however
apparent that there exist conceptual relations to the geometry of {\it
  'fractals'}.)

Quite some time after we developed the above introduced concept we
were kindly informed by Th. Filk that such a concept had been employed
in a however quite different context by e.g. A.A. Migdal et al and by
himself (see e.g. \cite{filk} and \cite{migdal}). As a consequence one
should say that, while a concept like this may perhaps not be
widely known for discrete structures like ours, it does, on the other
side, not seem to be entirely new. We hope to come back to possible
relations between these various highly interesting approaches
elsewhere (see our remarks before 4.6 Observation). 
 
Matters become much more interesting and subtle if one studies more
general graphs than simple lattices. Note that there exists a general
theorem showing that practically every graph can be embedded in
$\Rl^3$ and still quite a few in $\Rl^2$ ('{\it planar graphs}'). 

The point is however that this embedding is in general not invariant
with respect to the euclidean metric. But something like an apriori
given euclidean length is unphysical for the models we are after anyhow. 
This result has the advantage that one can visualize many graphs
already in, say, $\Rl^2$ whereas their intrinsic dimension may be much
larger.

An extreme example is a '{\it tree graph}', i.e. a graph without
'{\it loops}'. In the following we study an infinite, regular tree
graph with node degree 3, i.e. 3 bonds branching off each node. The
absence of loops means that the '{\it connectivity}' is extremely low
which results in an exceptionally high '{\it dimension}' as we will
see.

Starting from an arbitrary node we can construct the neighborhoods
$\U_m$ and count the number of nodes in $\U_m$ or $\partial\U_m$.
$\U_1$ contains 3 nodes which are linked with the reference node
$n_0$. There are 2 other bonds branching off each of these nodes.
Hence in $\partial\U_2=\U_2\backslash\U_1$ we have $3\cdot2$ nodes
and by induction:
\begin{equation} |\partial\U_{m+1}|=3\cdot2^m \end{equation}
which implies
\begin{equation} D-1:=\lim_{m\to\infty}(\ln|\partial\U_{m+1}|/\ln m)=
\lim_{m\to\infty}(m\cdot ln 2/\ln m+3/\ln m)=\infty \end{equation}
Hence we have: \\[0.5cm]
{\bf 4.7 Observation(Tree)}: The intrinsic dimension of an infinite
tree is $\infty$ and the number of new nodes grows exponentially like
some $n(n-1)^m$ with m (n being the node degree).\\[0.5cm]
Remark: $D=\infty$ is mainly a result of the absence of loops(!), in
other words: there is exactly one path, connecting any two nodes. 
This is usually not so in other graphs, e.g. lattices, where the
number of new nodes grows at a much slower pace (whereas the number
of nearest neighbors can nevertheless be large). This is due to the
existence of many loops s.t. many of the nodes which can be reached
from, say, a node of $\partial\U_m$ by one step are already
contained in $\U_m$ itself. \vspace{0.5cm}

We have seen that for, say, lattices the number of new nodes grows
like some fixed power of m while for, say, trees m occurs in the
exponent. The borderline can be found as follows:\\[0.5cm]
{\bf 4.8 Observation}: If for $m\to\infty$ the average number of new
nodes per node contained in $\partial\U_m$, i.e:
\begin{equation} |\U_{m+1}|/|\U_m|\geq 1+\varepsilon \end{equation}
for some $\varepsilon\geq 0$ then we have exponential growth, in other
words, the intrinsic dimension is $\infty$. \\[0.5cm]
Proof: If the above estimate holds for all $m\geq m_0$ we have by
induction:
\begin{equation} |\U_m|\geq |\U_{m_0}|\cdot (1+\varepsilon)^{m-m_0}
\end{equation} 

Potential applications of this concept of intrinsic dimension are
manifold. Our main goal is it to develop a theory which explains how
our classical space-time and what we call the '{\it physical vacuum}'
has emerged from a more primordial and discrete background via some
sort of phase transition.

In this context we can also ask in what sense space-time dimension 4
is exceptional, i.e. whether it is merely an accident or whether
there is a cogent reason for it.

As the plan of this paper was mainly to introduce and develop the necessary
conceptual tools and to pave the ground, the bulk of the investigation
in this particular direction shall be presented elsewhere as it is
part of a detailed analysis of the (statitical) dynamics on networks
as introduced above, their possible phase transitions,
selforganisation, emergence of patterns and the like.

In the following we will only
supply a speculative and heuristic argument in favor of space-dimension
3. We emphasized in section 2 that also the bond states, modelling the
strength of local interactions between neighboring nodes, are in our
model theory dynamical variables. In extreme cases these couplings
may completely die out and/or become {\it 'locked in'} between certain
nodes, depending on the kind of model.

It may now happen that in the course of evolution a local island of
'higher order' (or several of them) emerges via a spontaneous
fluctuation in a, on large scales, unordered and erratically
fluctuating network in which couplings between nodes are switched on
and off more or less randomly.

One important effect of the scenario we have in mind (among others) is
that there may occur now a pronounced near order in this island,
accompanied by an increase in correlation length and an effective
screening of the dangerously large {\it 'quantum fluctuations'} on
Planck scale, while the global state outside remains more or less
structureless. We assume that this will be effected by a reduction of
intrinsic dimension within this island which may become substantially lower
than outside, say, finite as compared to (nearly) infinity.

If this '{\it nucleation center}' is both sufficiently large and its
local state '{\it dynamically favorable}' in a sense to be specified
(note that a concept like '{\it entropy}' or something like that
would be of use here) it may start to unfold  and trigger a global phase
transition.

As a result of this phase transition a relatively smooth and stable
submanifold on a certain coarse-grained scale (alluding to the
language of synergetics we would like to call it an '{\it order parameter
manifold}') may come into being which displays certain properties we
would attribute to space-time.

Under these premises we could now ask what is the probability for
such a specific and sufficiently large spontaneous fluctuation? As we
are at the moment talking about heuristics and qualitative behavior
we make the following thumb-rule-like assumptions:\\[0.5cm]
i) In the primordial network '{\it correlation lengths}' are
supposed to be extremely short (more or less nearest neighbor), i.e.
the strengths of the couplings are fluctuating more or less
independently.\\
ii) A large fluctuation of the above type implies in our picture that
a substantial fraction of the couplings in the island passes a certain
threshold (cf. the models of section 2) i.e. become sufficiently weak/dead  and/or cooperative. The probability per
individual bond for this to happen be p. Let L be the diameter of the
nucleation center, $const\cdot L^d$ the number of nodes or bonds in
this island for some d. The probability for such a fluctuation is
then roughly (cf. i)):
\begin{equation} W_d=const\cdot p^{(L^d)} \end{equation}
iii) We know from our experience with phase transitions that there
are favorable dimensions, i.e. the nucleation centers may fade away
if either they themselves are too small or the dimension of the
system is too small. Apart from certain non-generic models $d=3$ is
typically the threshold dimension.\\
iv) On the other side we can compare the relative probabilities for
the occurrence of sufficiently large spontaneous fluctuations for
various d's. One has:
\begin{equation} W_{d+1}/W_d\sim p^{(L^{d+1})}/p^{(L^d)}=p^{L^d(L-1)} 
\end{equation}
Take e.g. $d=3,\,L=10,\,p=1/2$ one gets:
\begin{equation} W_4/W_3\sim 2^{-(9\cdot10^3)} \end{equation}
In other words, provided that this crude estimate has a grain of
truth in it, one may at least get a certain clue that space-dimension
3 is both the threshold dimension and, among the class of in principle
allowed dimensions (i.e. $d\geq3$) the one with the dominant
probability. 
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