###### Abstract

As in an earlier paper we start from the hypothesis that physics on the Planck scale should be described by means of concepts taken from discrete mathematics. This goal is realized by developing a scheme being based on the dynamical evolution of a particular class of cellular networks being capable of performing an unfolding phase transition from a (presumed) chaotic initial phase towards a new phase which acts as an attractor in total phase space and which carries a fine or super structure which is identified as the discrete substratum underlying ordinary continuous space-time (or rather, the physical vacuum). Among other things we analyze the internal structure of certain particular subclusters of nodes/bonds (maximal connected subsimplices, ) which are the fundamental building blocks of this new phase and which are conjectured to correspond to the physical points of ordinary space-time. Their mutual entanglement generates a certain near- and far-order, viz. a causal structure within the network which is again set into relation with the topological/metrical and causal/geometrical structure of continuous space-time. The mathematical techniques to be employed consist mainly of a blend of a fair amount of stochastic mathematics with several relatively advanced topics of discrete mathematics like the theory of random graphs or combinatorial graph theory. Our working philosophy is it to create a scenario in which it becomes possible to identify both gravity and quantum theory as the two dominant but derived(!) aspects of an underlying discrete and more primordial theory (dynamical cellular network) on a much coarser level of resolution, viz. continuous space-time.

Emergence of Space-Time on the Planck Scale described as an Unfolding Phase Transition within the Scheme of Dynamical Cellular Networks and Random Graphs

Manfred Requardt

Institut für Theoretische Physik

Universität Göttingen

Bunsenstrasse 9

37073 Göttingen Germany

## 1 Introduction

In a previous paper ([1]), starting from the hypothesis that both physics and mathematics are discrete on the Planck scale, we developed a certain framework in form of a class of ’cellular network models’ consisting of cells (nodes) interacting with each other via bonds according to a certain ’local law’ which governs their evolution.

Our personal philosophy in this endeavor is the following: If one wants to go beyond the metric continuum (a possibility which was already contemplated by B.Riemann in [2]), an entirely different and in some respects perhaps even new arsenal of physical/mathematical concepts and calculational tools is called for respectively has to be invented. (Ideas in a spirit similar to ours can e.g. be found in the work of R.Sorkin et al; [3] or C.Isham; [6]. A slightly different line of thought is persued in e.g. [5]; see further remarks at the end of section 2). Among other things one has to be extremely careful not to reimport notions and ideas through the backdoor which carry openly or implicitly a meaning or connotation having its origin in continuum concepts and, more generally, to obey Occam’s principle, i.e. stick to as few as possible clear and natural hypotheses.

A case in point is the concept of dimension which appears to be, at least traditionally, a typical continuum concept. As ordinary physics can successfully be described by employing a framework which has as one of its cornerstones the postulate of space-time as a metric continuum, a serious candidate for an underlying discrete and more primordial substratum shall have the capability to generate a concept like dimension as an ’emergent’ collective quality without(!) having it already among its elementary buildingblocks in one or the other disguise.

We showed in [1] that our class of discrete model theories do
have indeed a rich enough structure to accomplish, among other
things, this goal. More specifically, scrutinizing what are in fact
the crucial ingredients of something like dimension from the point of
view of physics, in particular dynamics and interaction in physical
systems, we managed to develop a notion of ’generalized
dimension’ also in discrete and quite irregular systems and which
mirrors both a characteristic and ’intrinsic’ property of the
discrete network models under discussion by measuring its ’connectivity’ and, on the other side, does not resort to some sort
of underlying dimension of a possible ’embedding space’. In this
sense it is in exactly the same way an ’intrinsic’ concept as,
say, the concept of ’intrinsic curvature’ for
manifolds. Furthermore it is indeed a true generalisation as it
coincides in the regular and ordinary situations with the usual
definition of dimension.

Remark: One can introduce of course other concepts over discrete
structures which carry a certain dimensional flavor but which have
their basis not so much in physics and dynamics but rather in the
algebraic topology of, say, ’simplicial complexes’
(cf. e.g. [4]). To some extent such ideas were also persued in
[1] but we have the (subjective) impression that the ’connectivity-dimension’ of the network is the really crucial
property from the physical point of view (in particular when something
like ’unfolding phase transitions’ and other large scale phenomena become
relevant). On the other hand these different concepts may neatly
complement each other in encoding appropriately various facets of
’complex systems’.

In [1], besides introducing our network concept, possible
dynamics on it and defining the concept of connectivity dimension, our
main concern was the development of a kind of ’discrete
analysis’ and the clarification of its relation to various other modern
approaches which follow different lines of reasoning but seem,
nevertheless, to be inspired by the same philosophy, i.e. the complex
of ideas and methods called ’non-commutative geometry’. On the
other side, the possible dynamical processes in our network models,
which are expected to lead, among other things, to the emergence of
something like ’space-time’ and continuum physics in general
were only briefly sketched.

The main purpose of this paper is a more detailed investigation of
this ’unfolding’ of ’phase transition type’ and the
development of the necessary mathematical, mostly
topological/geometric, tools and concepts which will allow us to cast
vague and qualitative reasoning into a more rigorous and precise
framework.

Remark: As to references to the field in general cf. e.g. [1] and
further references given in the papers mentioned there. A deserving
bibliography of papers dealing with more or less non-orthodox ideas
about space-time and related concepts is [7]. Various
further mathematical sources we have found useful for our approach
will be cited below.

A further clarifying remark (a kind of ”petitio principii”)
concerning the (mathematical) rigor of the type of reasoning we are
going to present in the following seems to be in order. In some sense
one should regard it rather as a blending of some ”educated
speculation” with various pieces of more rigorous mathematics. The
reasons for this are, as we think, almost inescapable. For one, we
have to set up and fix both the complete abstract framework and
conceptual tools and then to infer practical, observable consequences
for ordinary space-time physics more or less at the same time in order
to motivate our primordial assumptions. This is already difficult
enough as we have to close a gap between Planck-scale physics and,
say, ordinary quantum physics which extends over many orders of
magnitude. On the other side we consider this to be of tantamount
importance since it is, given the complete absence of reliable
experimental data, the only selection principle allowing to judge the
scientific value of the various assumptions in our model building
process.

For another, many of the mathematical disciplines to be employed in the following are either almost in their infancies (like e.g. the correct statistical treatment of extremely complex and densely entangled dynamical systems, consisting of an extraordinarily large number of degrees of freedom, typical cases in point being our cellular networks), or do not belong to the usual armory of mathematical physics (as e.g. various branches of discrete mathematics) and have to be developed to a certain extent almost from scratch, at least as applications to our field are concerned.

Therefore our general strategy will be as follows: we will prefer to develop in this paper what we think is the generic scenario in broad outline up to the point that we can make transparent how several (partly long standing and puzzling) problems of ordinary space-time physics might be successfully approached within our much wider framework with, among other things, its deeper conception of the nature and role of space-time and the corrresponding fine structure of the physical vacuum.

There remain quite a number of technical details to be filled in, sometimes even veritable subtheories, which may have some interest of their own. This shall in part be postponed to subsequent work in case it would blow up the paper to much and after the general scheme has proved its value. For the time being we will content ourselves with inserting some preliminary remarks at appropriate places in the following sections which are to serve as hints as to where more detailed investigations seem to be worthwhile. In our view the central section of this paper (both with respect to applied mathematical techniques and physical content) is section four.

Concluding the introduction we would like to add the remark that after having nearly completed this paper we stumbled by chance over an exceptionally beautiful (but, as is probably the case with the whole discipline, perhaps not so widely known) book by Bollobas ([23]; unfortunately it is even out of print at the moment) about ’random graphs’, a field originally founded by Erdoes and Renyi in the late fifties. By brousing through the book we realized that the results, being presented there, may confirm, if appropriately translated, to a large extent the soundness of several of our speculations about the assumed generic behavior of our unfolding network. In section 4 of our paper we briefly mention some results from this book; we plan however to embark on a closer inspection of the relations between these two fields in subsequent work (note that the theory of random graphs was founded to tackle some deep problems in graph theory proper, not so much for its probabilistic aspects or even possible applications in physics).

## 2 The Class of Dynamical Cellular Networks

The idea to base ’discrete dynamics’ on the Planck scale on something like a cellular network was partly inspired by the role which its close relatives, the cellular automata, are playing in the science of complexity (see the references given in e.g. [1]) and the general philosophy to achieve complex, emergent behavior and pattern generation by means of surprisingly simple looking microscopic ’local laws’.

On the other side, a cellular automaton proper appeared to be a far too regular and rigid array in our view, in particular in the context of possible formation of something like space-time as an ordered superstructure being embedded in or floating above a less organized and probably almost chaotic substratum. More specifically, the organisation of the network as such should be pronouncedly dynamical, most notably its kind of wiring. Furthermore, we maintain that really fundamental dynamical laws must have a quite specific structure, the basic ingredient being the mutual dynamical coupling of more or less two distinct classes of entities, in this context the one being of a local character (nodes or sites), the other of a pronouncedly ”geometrical” quality (bonds or links); the state of the one class being governed by the previous state of the other class. Typical cases in point are e.g. general relativity and gauge theory. In contrast to our network models the bonds in a cellular automaton are typically both rigid and regular and not(!) dynamical so that a ’backreaction’ of the node states (”matterfields”) on the bond states (”geometry/curvature”) is supressed.

To achieve this we assumed our system to be made up of ’cells’ and elementary interactions mediating among these cells, both of which are taken to be dynamical variables. This qualitative picture is then condensed into the concept of a ’network’ consisting of ’nodes’ and ’bonds’ (synonyma being sites, vertices or links, edges respectively) with each bond connecting two different nodes.

At each node ( running through a certain index set) the
corresponding cell can be in a specific internal state ( typically being some countable or finite set not
further specified at the moment). Correspondingly the bonds
, the bond between node and node ,
carry a dynamical ’valuation’ . A
dynamical ’local (global) law’ , () is then introduced
via the general evolution equation given below.

Remark: For the time being we employ a synchroneous ’clock
time’ , proceeding in discrete elementary steps ,
i.e. . That means, the whole system is updated in
discrete steps. This clock time should not(!) be confused with
socalled ’physical time’ which is expected to arise as an
emergent collective concept on a possibly much coarser scale. In any
case we conjecture that ’local’ physical time will turn out to be some
kind of ’order parameter field’ coming into existence via the
dynamical interaction of many nodes/bonds within the local grains
under discussion (see below). The choice of an external overall clock
time is made mainly for technical convenience in order to keep already
quite complicated matters reasonably simple and can be modified if
necessary.

To sum up, we make the following definitions:

2.1 Definition (Cellular Network): In the following we will deal with the class of systems defined below:

i) ”Geometrically” they are graphs, i.e. they consist of nodes
{} and bonds {} where pictorially the bond
connects the nodes and with 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 connected, i.e. two arbitrary nodes
can be connected by a sequence of consecutive bonds, and 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 (’degree’ of a node).
We call the nodes which can be reached from a given node by making
one step the 1-order-neighborhood and by not more
than n steps .

ii) On the graph we implant a class of dynamics in the following way:

2.2 Definition(Dynamics): As for a cellular automaton each node
can be in a number of internal states . Each
bond carries a corresponding bond state . Then
we assume:

(1) |

(2) |

(3) |

where , 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 , yielding the
updated values of and . and denote the
internal states of the nodes and bonds of this neighborhood under
discussion, the corresponding global
extended states and law.

Remark: The theory of graphs is developed in e.g. [8, 9]. As
to the connections to the algebra of relations see also
[10]. There exist a lot of further concepts in ’graph theory’ and ’discrete mathematics’ in general which are
useful in our context, some of which (together with some further
relevant references )will be introduced below, especially in section 4.

2.3 Lemma: The kind of discrete analysis we developed in
[1] made it necessary to give the bonds an ’orientation’
(which is extremely natural anyhow). Consistency then requires an
analogous relation to hold for the interactions, i.e:

(4) |

It is now our aim to attempt to motivate a kind of unfolding and pattern creation in such a network, which, starting from a certain initial state (or better: phase), is supposed to lead in the coarse of a dynamical process – among other things – to the emergence of something we are experiencing on a coarser scale of resolution as ’space-time’. In order to achieve this we have in a first step to model a kind of concrete ’critical’ dynamics which, on the one hand, fits in the general scheme of Definition 2.2. On the other hand it turns out to be a subtle task to appropriately implement the crucial ingredients which are to catalyze the unfolding process.

We presented several types of local laws in [1]. We in fact studied quite a few other possible dynamical laws, the corresponding scenarios and mappings between the mathematical entities occurring in these scenarios and the concepts carrying a certain physical meaning on a possibly much coarser scale of resolution like e.g. space-time as such, matter, (quantum)fields, gravitation etc. The main difficulty was to implement the unfolding process of our ’network universe’ in a natural way and understand as what kind of ’order parameter manifold’ space-time was going to emerge in this evolving and quite chaotic background. In this selection or identification process an important role was played by ’Occam’s principle’, i.e. to infer seemingly complicated phenomena from as simple as possible and, as we think, uncontorted microscopic laws. A crucial ingredient is the possible switching-on or -off of bonds , more specifically, of the corresponding elementary interactions in a ’hystheresis like’ manner (in catastrophy theory called a ’fold’).

These criteria can probably be fulfilled by a whole class of dynamical network laws, or expressed more carefully: there may exist a whole class of possible candidates, the behavior of which could be checked by simulation on a computer, from which we are presenting in the following the probably most simple ones. At the moment implementations of various possible network models on a computer are under the way. These investigations are quite time consuming for various reasons (complex behavior is difficult to study anyhow; cf. e.g. reference [17]), viz. it may turn out that the simple laws we are going to describe below have to be replaced in the end by another law from the same class in case they e.g. do not behave chaotically enough. These numerical investigations will be presented elsewhere.

The naive picture encoded in the following law is that a, for the time being, not further specified
substance consisting of elementary quanta is transported through
the network according to the following law:

2.4 Definition of the ’Unfolding’ Local Law: The local state
space at each site is assumed to be either

(5) |

The bonds can carry the ’valuation’

(6) |

Remark: As to the possible allowed range of the variables some
additional remarks are in order which will be given below.

Then the first half of the dynamical local law reads in the case
:

A)

(7) |

the sum extending over all the bonds being incident with the node
.

The second half, describing the backreaction of the node states on the
bond states, is slightly more complicated:

B)

(8) |

(9) |

with

(10) |

denoting the ’hystheresis interval’.

(11) |

To say it in words: Below bonds are temporarily
annihilated, above they are turned on again, and in between
the bonds may only switch between and , which is essentially a
switch in orientation or local direction of transport which can be
seen from the first half of the law.

C) The roles of are interchanged,
i.e. bonds are switched off for and
switched on again for .

Remark: Both possible laws are presently carefully studied on the
computer. As the simulations will take some time, the results will be
published elsewhere. What one can however already say is that one of
the many fascinating observations is the ability of systems like these
to self-organize themselves and to find a variety of different
attractors, i.e. to typically occupy a relatively small region of the
potentially accessible huge phase space. Furthermore type C)
leads to a totally different behavior as compared with type B).

is probably appropriate if one does not want to bother about boundary conditions and should be considered as the idealisation of the scenario where typical local fluctuations of , i.e.

(12) |

the average taken over some suitable time sequence and/or suitably large spacial array of nodes, are sufficiently far away from the possible internal boundaries of the concrete system with a finite local state space, e.g.

(13) |

One could hence regard the former system as a certain limit for and shifting at the same time the reference point from, say, to zero.

On the other side, for finite one can introduce boundary conditions like the following ones, arguing that they do not change the qualitative behavior provided is much larger than the generic local fluctuations about some average value with

(14) |

2.4 A’):

(15) |

i.e. there is no transport if it would transcend the maximal/minimal
’charge’ of the node, or .

A”): Perhaps more natural are ’periodic boundary
conditions’, i.e. the above equation has to be understood modulo .

We want to conclude this section with a couple of additional remarks
as to this particular type of local law:

Remarks: i) The above dynamical law shows that under certain
circumstances ’elementary interactions’ may become zero
for some lapse of time. It may even be possible that a substantial
fraction becomes ’locked in’ at the value zero for a rather long
time. This will then have some practical consequences for the
(partial) representation of our network as a graph. On the one side
one may consider the graph, in particular its wiring, to be given as a
fixed static underlying substratum, i.e. its bonds ”being there” as
elements of the graph even if a substantial fraction of the
corresponding elementary interactions is zero temporarily or
even locked in at zero.

On the other side, if one wants to deal with e.g. phase transition
like ’topological/geometrical’ changes of the whole wiring of
the network it may be advantageous to change the point of view a
little bit and regard not only the bond valuations as
dynamical variables but rather the underlying graph as such, more specifically,its bonds , i.e. allow them to be annihilated or
created. As a consequence the underlying graph will change its shape
in the course of clock time which may perhaps be a more fruitful mode
of representation. This latter point of view will be adopted in the
following where it seems to be appropriate.

ii) There is a certain (faint) resemblance to non-linear electrical
networks. One can in fact, if one likes so, consider the -field as
a charge distribution (or potential-field) with as kind of
voltage difference between neighboring nodes. Part A) of the dynamical
law reflects then the usual charge conservation. The interpretation of
B) is more complicated. One could e.g. regard it as a non-linear
dependence of the resistances of links on the applied voltages. In any
case, we are confident that it be possible to realize or emulate our
system as some kind of non-linear network, thus producing possibly
effects which resemble the unfolding of our universe in the
laboratory!

iii) We would like to emphasize that our dynamical law is a genuine
cellular network law and not(!) some Lagrangian field theory in
disguise (which is frequently the case in other approaches). In our
view it is not at all selfevident that it be advisable to model
fundamental laws on the Planck scale according to one or the other
kind of an (at best) ’effective quantum field theory’ which
lives many scales above the Planck regime.

On the other side, we are quite confident that these known types of
field theories will emerge as effective theories after some
appropriate kind of
’renormalisation transformation’ on a much coarser level,
describing the interaction of patterns (e.g. fields) which are
themselves rather ’collective excitations’. In this sense our
approach is very much in the spirit of the philosophy of t’Hooft
(cf. [12]).

iv) Our personal philosophy is ”iconoclastic” in the sense described
in the beautiful reviews about quantum gravity by Isham (see
e.g. [11]) in sofar as we want to show that and how both quantum
(field) theory and gravitation emerge as two different but related
aspects of one and the same underlying and more primordial theory of
the kind introduced above. The first steps of this endeavor will be
undertaken in this paper, which is mainly concerned with the analysis
of the kind of embedding of a certain ’order parameter manifold’
(space-time) in the background space (the cellular
network). Most importantly we are concerned with the internal
structure of certain subclusters of nodes/bonds which are on the
”continuous” (coarse grained) level ”experienced” as physical
points and their mutual causal relations.

In this context a lot of topological analysis has to be carried out which is performed in a similar spirit as the work of Sorkin et al. ([3]) or Isham ([6]) at least as far as the general philosophy is concerned. The technical tools and concepts being employed may however be different. In our case it is mainly a blend of various fields of ’discrete mathematics’ as e.g. graph theory, finite geometry, calculus of relations and the like with arguments boroughed from statistical physics/mathematics and the science of complexity. Some of the concepts developed below may even be new but this is difficult to decide from the widely scattered literature, a substantial fraction of which may have escaped our notice up to now.

## 3 The Primordial (Chaotic) Network Phase

From the structure of the dynamical law we have elaborated in section
2 we can infer that our dynamical network, which we henceforth denote
by (’quantum space’; at the moment this should however only
be considered as a metaphor), is at each fixed clock time a
certain (dynamic) graph if we do not take into account the
details of the status of the fields of node and bond states but
concentrate solely on its purely geometric content. The dynamics of
consists of the possible switching off or on of some bonds in
the time step from to .

3.1 Definition: The full dynamical network (with its
distribution of bond and node states being included) we call or
. If we want to concentrate on its purely geometric content we
view it as a graph or .

Our dynamical law is up to now ’deterministic’. Therefore we
should specify the kind of ’initial state’ (or rather a generic
group of initial states) from which our network is supposed to
evolve. As was the case with the class of admissible dynamical laws,
we experimented with quite a few types of possible initial states,
looking to what kind of scenarios they would probably lead. It is our
impression that both the most natural (having Occam’s principle in
mind) and most promising assumption is to start from a maximally
connected graph, a socalled ’complete graph’ or ’simplex’,
i.e. each two nodes are connected by a bond
or (see below) from a class of graphs which are ’almost
complete’.

3.2 Assumption: We assume that our ’initial phase
consists generically (for reasons explained below) of ’almost
complete graphs’, i.e. the ’average number’ of active bonds
() is almost maximal (see the following
remarks).

Some remarks are here in order: One knows already from the analysis of cellular automata that these systems behave typically in an extremely complicated manner and that for exactly the same reasons as in e.g. statistical mechanics it makes frequently only sense to study ’generic’ properties and behavior, mostly with the help of statistical means (cf. e.g. the references on cellular automata in [1], in particular the paper by Wolfram; [13]).

In other words, it seems more appropriate to speak of different ’phases’ instead of individual states. Our dynamical law shows that,
typically, a certain percentage of bonds are switched on or off at
each clock time step or during a certain interval. That means, it is
more natural to assume that was ”initially” or better:
remained for possibly quite some time in a phase with two arbitrary
nodes being connected on average with a probability ’near
one’. If we assume that the number of nodes in our network is a very
large but finite number , a simplex has
bonds. This means:

3.3 Assumption (Statistical Version): We assume that our network
evolves from an initial phase with the average number of
bonds (taken e.g. over a suitable clock time interval) being
”near” .

Henceforth our arguments will frequently (for obvious reasons) carry a
markedly statistical flavor. Therefore we will introduce a couple of
corresponding concepts and notations.

3.4 Definition: i) We denote the set of vertices (nodes) and
edges (bonds) by , respectively (we conform here to the usual
notations of graph theory; note that in our case - simple graphs -
can be viewed as a certain subset of and represents a
homogeneous, non-reflexive, symmetric relation), their cardinalities
by . The degree of a node , i.e. the number of bonds being
incident with it, is . The graph is called regular
if is constant over the whole graph.

ii) As in our case or are time dependent (if we adopt
the point of view developed in Remark i) at the end of the previous
section, i.e. consider the underlying graph as such as a dynamical
object), it makes sense to build the corresponding statistical
concepts:

(16) |

are the respective averages of , with the
”spacial” average over the graph at fixed clock time ,
some temporal average (e.g. at a fixed node),
a space-time average. The time average is assumed to be
taken over an appropriate time interval, the length of which depends
on the specific context under discussion (e.g. the particular ’phase’, various correlation lenghts etc.).

Our philosophy concerning the possible time dependence is, for the
time being, that a bond is contributing in, say, or
at clock time if .

Evidently there exist certain relations between these notions,
e.g:

3.5 Observation: With

(17) |

we have

(18) |

and

(19) |

Remark: In [1] we gave each bond an orientation
s.t. . We are however counting in Observation 3.5
these two differently oriented bonds as one and the same bond (as
usual), hence the factor . In any case, this does not make a big
difference as one can always associate an ’undirected’ graph
with a ’bidirected’ graph.

3.6 Conclusion: With the help of what we have said above,
Assumption 3.3 can now be framed this way:

and/or remain in a small neighborhood of
in the ’initial phase’ for a possibly long
time, where ”small” has to be understood pragmatically, depending on
the details of the model and the context.

Our network (graph) () carries a natural metric which makes it into a ’metric space’. This was already employed in [1], to which the reader is referred for further details. Assuming that is connected (i.e. every two nodes can be connected by a bond sequence) there exists a ’path’ of minimal ’length’ (i.e. number of bonds). Then we have:

(20) |

defines a metric on .

With this metric becomes a discrete topological space (even a ’Hausdorff space’) with a natural neighborhood structure of a given node, i.e:

(21) |

The topology as such is however relatively uninteresting since in a discrete, finite Hausdorff space each one-element set is necessarily both open and closed. The simple proof runs as follows:

(22) |

is finite hence is a finite intersection of open sets and hence again open. By construction contains no points other than , that is is open. On the other side, with open, is also closed.

But despite of this, ’finitary’ topological spaces are not(!) automatically trivial as can be seen from the beautiful analysis performed by Sorkin [3]. However they are typically only socalled -spaces (as to these topological notions cf. any good textbook on topology as e.g. [14]).

At this place we want to make a proviso. The existence of a (physically) natural metric on and a corresponding natural neighborhood structure neatly implementing the physically important concepts of ”near by” and ”far away” is perhaps much more important than the existence of a (mathematically) non-trivial topological structure. It may well be that on such genuinely discrete sets like networks or graphs natural(!) topological/geometric concepts should rather be adapted to the real discreteness of the substratum and perhaps not so much to an abstract axiomatic system defining a topology. This point of view will be more fully developed in the following section, to which we postpone the details of the (partly intricate) topological/geometric analysis.

In the rest of this section, however, we will try to sketch the expected qualitative behavior of our network by employing both statistical concepts and the concrete local law introduced above. Of particular interest in this respect are the initial phase and the scenario when it starts to leave this (possibly ’metastable’) phase due to some sort of phase transition. The analysis is, for the time being, necessarily of a qualitative/statistical nature as only generic aspects should matter, especially as we are at the moment mainly interested in deriving observable consequences on much coarser scales which should not depend to much on microscopic details. For another - as is the case in statistical mechanics - it would be technically impossible (at least at the moment) to justify too detailed assumptions and draw precise conclusions from them.

Let us briefly recall the assumptions made about the phase :

(23) |

i.e. is almost a simplex or complete graph.

ii) The local state space at node consists of multiples of an
elementary quantum , i.e:

(24) |

ranging from to an appropriately large number or from to
(cf. section 2). If we do not want to bother about boundary
conditions we assume the accessible local state space to be the full
.

iii) The bonds carry the valuation modelling the
elementary interactions .

iv) At each clock time step either an elementary quantum is
transported via depending on the sign of or the bond
is ”dead” if . The bond states at clock
time are dynamically coupled with the ”charge difference”
of the incident site states at clock time .

v) Of crucial importance for pattern formation is the built-in
”hysteresis law” and the hysteresis interval

(25) |

For

(26) |

the corresponding bond becomes extinct in the next step or becomes alive again if it had been extinct before, respectively the other way around in case of law 2.4 C). In the following we discuss law 2.4 B). The qualitative reasoning in case of C) would follow similar lines.

If the local fluctuations are sufficiently large on average, i.e:

(27) |

the corresponding phase is more or less stationary (a slightly more
detailed analysis can be found below). Inspecting the
situation assumed to prevail in the phase with
extremely large, we may infer that also the
average fluctuation of the local charge at an arbitrary node
is typically very large at each clock time step.

3.7 Conjecture/Assumption: We conjecture that in the (chaotic)
regime correlations have typically an extremely short range due
to the enormous number of links per node and the character of the
local law.

Assuming then that the local ”orientations” of the incident bond
variables are almost statistically independent, both the fluctuations
of the local charge at a typical node and of the charge
difference with respect to neighboring nodes can be
inferred from the ’central limit theorem’(see e.g. [15]),
yielding among other things:

(28) |

(29) |

if we neglect possible phase boundaries (). In
other words, local fluctuations are expected to be enormous both with
respect to (clock) time at an arbitrary but fixed node and among nearest
neighbors as is assumed to be extremely large.

Remark: Note that the following holds (assuming for simplicity that
):

(30) |

which can be written as

(31) |

with denoting the ’probability distribution’ under discussion, the corresponding ’probability density’. For convenience and for the sake of greater generality we write everything continuously while both our ’sample space’ and the occurring ’random variables’ are, by construction, discrete. (Being sloppy, we identify the random variable, say, with the respective values it can acquire).

As, by assumption, the random variables are independent we have:

(32) |

and hence get with:

(33) |

(34) |

i.e., up to a trivial factor the same kind of distribution.

The soundness of the above conjecture has to be confirmed by performing simulations with a variety of local configurations around a given typical node. Qualitatively one could test this conjecture as follows: Let us e.g. assume that at node happens to deviate from the surrounding in a systematic way at time ; to be specific, we assume for most of the neighboring nodes.

This local configuration makes it highly probable that, due tu part B)
of the particular local law introduced above, most of the bonds are ”pointing” from
to (more precisely: the corresponding will be
positive) after one time step , which, after another step
results via part A) in a huge discharge of node . As a
consequence the charge happens to be far below the typical
charge of the neighboring sites. This then forces most of the incident
bonds respectively to reorient themselves which
results in a huge surplus charge of node after another two time
steps and so on. Given that all the nodes under discussion are densely
entangled with each other the local dynamics in the phase may
indeed be sufficiently chaotic to justify our conjecture. What is
however not entirely clear is the statistical weight of such a systematic deviation from
a more or less random distribution of the local charges (see the following
remarks). Furthermore complex systems are capable of performing a lot
of very surprising things (e.g. approaching attractors very quickly
irrespective of their seemingly chaotic behavior; an observation made
by e.g. Kauffmann in his study of ’switching nets, see [17])

3.8 Some Annotations to the Probalisitic Framework: As to the
soundness of the above conjecture some more remarks are appropriate:

i) This is one of the points, mentioned in the introduction, which
comprises in effect a whole bunch of important questions which have a
relevance of their own. The (partly intricate) technical details
coming up in this context shall be postponed for the main part to
forthcoming work (apart from some preliminary remarks following below)
as we consider it to be our main task at the moment to develop in the
rest of the paper a scenario in which space-time is to emerge as a
kind of (coarse grained) order parameter manifold floating in the
discrete network and to establish geometric/causal notions like
”nearby” or ”far away”.

ii) Our philosophy is supported by observations made by
e.g. S.Kauffmann and reported in ref. [17], p.109 or 112,
viz. that too densely connected networks seem to support chaotic
behavior while more sparsely connected ones are capable of generating
complex behavior. The networks discussed there are however more rigid
than ours in several respects and of a markedly different nature as to
the details of their dynamical laws (’switching nets’).

iii) It is of course not really crucial that something like the ’central limit theorem’ does strictly hold. What we actually do need
is a guarantee that fluctuations tend to be very large and incoherent
and depend to some extent on the density of the wiring. Note that
e.g. for not necessarily independent random varariables (with,
for simplicity, vanishing mean and common variance ) the
following holds:

(35) |

The first term on the rhs goes as . If the are ’uniformly weakly correlated’ in the following sense:

(36) |

uniformly in and with , we have with :

(37) |

that is

(38) |

Note that the above assumption is not particularly far fetched as
products like , typically oscillate around
zero. Furthermore, similar relations can be established for random
variables which display a certain ’cluster behavior’ with
respect to space or/and time (as e.g. in ordinary statistical
mechanics).

iv) A careful treatment of many facetts of the central limit theorem
can be found e.g. in [15]. Note that in our case the number of
involved random variables is large but not really infinite. In this
situation the ’Berry-Essen-Theorems’ can be applied
(cf. e.g. [19]). Furthermore, in [20] some additional
interesting remarks concerning the possible extension of the central
limit theorem to weakly dependent random variables can be found. Note
in particular that one consequence of the central limit theorem,
viz. the normal distribution of the local fluctuations at, say, a
typical node , is not necessarily needed in our scenario,
i.e. the fact that the small fluctuations are the most probable ones
and dominate the behavior.

v) Quite the contrary, the particular kind of local fluctuations we
discussed above may rather be an indication for a much more
interesting behavior (not so frequently found in the usual examples of
statistical mechanics), i.e. the tendency of amplification of small
deviations, viz. a kind of ’positive feedback behavior’, which
may play perhaps an important role in stabilizing certain phases or, alternatively, drive the system away from perhaps only ’metastable’
phase towards certain ’attractors’. It is however not clear at the moment how frequently these
special fluctuations actually do occur; put differently: how large
their statistical weight is compared to the more chaotic fluctuations
which would rather support a kind of gaussian behavior. This leads to
the last point we want to mention.

vi) The above discussion shows that a more elaborated kind of
statistical or stochastic theory is called for if one is dealing with
such peculiar systems (some steps have already been taken in
e.g. refs. [13] and [18] for a however simpler class, the
cellular automata). Note that, in contrast to e.g. Gibbsian
equilibrium statistical mechanics, many pieces of an a priori
framework are missing as a natural probability measure on a suitable
sample space, criteria concerning the relevance and statistical weight
of the various initial configurations, their long-time effects (which
are notoriously difficult to forecast in complex systems) and the
like.

For that reason we made, for the time being, the above heuristic
assumptions and suggest to calculate the various averages and probability
distributions in a more practical way by assuming that the system
behaves reasonable and that the states we are dealing with are
sufficiently generic so that averages and probabilities taken with
respect to e.g. ”space” and/or ”time” over, say, one concretely
given actual state of the network (or a time sequence of actual
states) give sensible results (due to the assumption that the huge
numbers of involved nodes and bonds, may serve as a
substitute for ensemble averages). To give an exemple:

vii) The local law shows that fluctuations of the charge at a
given node during one clock time step are given by , the sum extending over the neighboring nodes. If one wants
to make statistical statements about fluctuations at a typical node,
without having an allcomprising statistical framework at ones
disposal, one can proceed as follows:

One chooses e.g. to concentrate on the situation at a fixed clock time
, i.e. make the statistics over the distribution of node and bond
states at a fixed time.

Definition: a) The points of the local ’sample space’
at an abstract typical node are all the possible ’bond configurations’

(39) |

with the degree of the node, i.e. the number of incident bonds
, running from to .

b) Probabilities of ’elementary events’ (i.e. a given
configuration) are extracted simply from a frequency analysis over the
array of nodes (frequency of occurrence of the various bond
configurations under discussion) at time with a suitable but more or less
arbitrary numbering of bonds being implied. More general events can
then be constructed by the usual additivity properties of measures.

c) The themselves are ’elementary random variables’
with

(40) |

With the help of the elementary probabilities calculated in b) each of the random variables has a (discrete) distribution and we can form corresponding sums, i.e:

(41) |

As charge fluctuations at a node have been linked with the above
which is, on the other side, a sum over elementary random
variables ( assumed to be very large)
we can make the following observation:

Observation: It is and the to which
our above assumptions like e.g. the central limit theorem do apply.

3.9 Corollary to 3.7: In the perhaps more realistic case and large but nevertheless
, which we think is natural, given that we assume our
whole universe to emerge from such a , the above estimate shows
that in the regime the entire local state spaces
are covered by the expected local fluctuations of with
essentially equal probability .

In other words, one may say that the ’local entropy’ at each
node

(42) |

is maximal in the phase , thus reflecting the absence of any
stable pattern.

Remarks:i) Such (information) entropy concepts may turn out to be quite
useful in analyzing systems like our one (see e.g. [13] or [18]
where related phenomena were analyzed in cellular automata).

ii) We would like to note that (vague) resemblances to the scenarios
discussed in, say, ’synergetics’ and related fields are not
accidental in our view (cf. e.g. [16] or the beautiful book about
the working philosophy of the Santa Fe Institute; [17] and the
references therein). To various paradigmatic catch words like ’order parameter’, ’slaving principle’ or ’selforganized
criticality’ we hope to come back in forthcoming work.

## 4 The Transition from the Phase to and the Emergence of Space-Time as an Order Parameter Manifold

We provided arguments in the previous section that fluctuations tend to be extremely large in the primordial phase and correlations so short lived that any kind of pattern formation will be obstructed as long as the fluctuations are on average substantially larger than the lower critical parameter or, even better, larger than (version 2.4 B) of the local law):

(43) |

(the average taken with respect to space, time or both).

Let us now assume that, by chance, a sufficiently extended and
pronounced spontaneous fluctuation happens to be created in a ’subgraph’ by the network dynamics around
clock time .

4.1 Definition: With a graph, , its sets of
vertices (nodes) and edges (bonds) is called a subgraph if

(44) |

It is called a section graph if for every pair of nodes

(45) |

We assume this fluctuation to consist of an array of anomalously small charge differences

(46) |

( meaning that the charge differences are typically smaller
than or approximately equal to ).

Remark: Note that the subgraph need not be connected! Quite the
contrary, it may well be that a subgraph consisting of an array of
effectively distributed disconnected subclusters will turn out to
serve its purpose much better.

If the surrounding network environment is favorable this fluctuation may then trigger an ’avalanche’ of rapidly increasing size in the following way:

(47) |

(the index in denoting a spacial average) will have the effect that a possibly substantial fraction of bonds become temporarily inactive after one clock time step (and, possibly, for a longer lapse of time ).

By assumption we are still in the chaotic phase . So we expect that such local temporal deviations from the overall chaotic ”equilibrium state” will typically quickly dissolve in the rapidly fluctuating background (at least for the specific network dynamics we proposed above and for most of the possible scenarios). If, however, the drop of the average node degree in happens to be pronounced enough so that

(48) |

where the lhs means the generic(!) amount of fluctuations which can be
expected from the assumptions about the environment
(cf. Conjecture/Assumption 3.7), i.e. almost statistical independence
of the orientations of the bonds being incident with the nodes
belonging to , we may have an entirely new situation!

Remark: , i.e. the accidental particular
fluctuation which happened to emerge spontaneously at time ,
should not be confused with the above ’typical’ degree of
fluctuations which one has to expect from probability theory. As to
the order of their respective magnitudes we suppose that:

(49) |

even after a certain amount of bonds in have already died off in
the course of the phase transition.

4.2 Supposition: We expect that in the course of the phase
transition, which started around clock time the amount of
fluctuations in will typically lie below the upper critical
parameter but may on average lie above the lower critical
parameter .

If the above described scenario actually happens to take place, then it may occur that the fluctuation pattern prevailing in will not be reabsorbed in the background after a few clock time steps (as has been probably the case many times before) but may represent the seed for a new and different kind of evolution. Given that the situation is as being described in Supposition 4.2 there is a realistic chance that on average more bonds are switched off in and around than are switched on again, i.e:

(50) |

It is obvious that the details of this process will depend on the details of the
probability distribution (probability density) for the random variable
restricted to , this distribution being assumed to be
constructed according to the principles described above. To be more
specific, this kind of evolution will hold under the following
proviso:

4.3 Observation: With being the spacial
probability density for the random variable for clock time
, the unfolding process (i.e. continuing annihilation of bonds)
will go on if

(51) |

Remarks: i) As we already mentioned at the end of section 3 such notions are not entirely easy to define rigorously. Such a probability distribution should rather be understood in the following way. On the one side we have a completely deterministic process, a result of which was (by assumption) this peculiar fluctuation phenomenon at clock time . As it is practically impossible to follow the process in every detail step by step for one has to resort to probabilistic arguments.

On the basis of a given average node degree and the
assumed statistical independence of bond orientations (in ) one
can e.g. calculate (as was shown in section 3) the probability
distribution of between two typical neighboring
nodes in . If this fulfills the above inequality for
one has reason to expect that the unfolding process will go on
(at least for a while).

ii) On the other hand, less stringent assumptions about the stability
of the spontaneous fluctuation at will perhaps already
suffice. What is actually important is that more bonds are annihilated
on average than are created again. This will presumably both depend on
the geometric structure of the subgraph and the overall network
state of the graph around that critical time . We will
however postpone a more detailed analysis in this direction and enter
in a description of the new phase which we expect to emerge
from this phase transition.

In a first step we will simplify our task and concentrate solely on the geometry of the wiring diagram of the underlying graph , i.e. neglect the details of the internal states at the nodes and bonds apart from a bond being dead or alive (as was already done before to some extent), viz. a bond occurs in the wiring diagram of if it is alive at time . This will enable us to extract more clearly the underlying geometric content being encoded in the network. Furthermore, in order to limit the amount of technical notation, we will not always explicitly mention the statistical or fluctuating character of the various occurring quantities or notions. The correct meaning is assumed to be tacitly understood in the respective cases without extra mentioning.

What we are actually after is exhibiting the process in the course of which what we will later call ’physical points’ gradually emerge from certain protoforms. We assumed that in the primordial phase almost all nodes have been connected with each other, viz. the number of bonds in is roughly:

(52) |

with some huge number. Let us now assume that at the onset of the phase transition a certain (possibly small) fraction of bonds have become annihilated, say

(53) |

We make, in addition, the idealisation that is (for convenience, i.e. not only approximately) a complete graph (simplex). We have then that arbitrarily selected bonds can at most connect different nodes, hence there still exist at least nodes which are maximally connected, viz. they are spanning a still huge subsimplex . On the other hand there are at most nodes with one or more incident bonds missing.

This picture will become increasingly intricate if more and more bonds are switched off. To keep track of a possible emerging ’superstructure’ we will proceed as follows. We want to cover the graph (rather its node set) with a class of particular subgraphs , constructed according to the following rule:

Starting from an arbitrary node, say , we choose a node being connected with by a bond, and in the i-th step a node being connected with all(!) the preceeding ones; pictorially:

(54) |

The extension process stops if there is no further node
being connectable with all the preceeding ones.

4.4 Definitions: i) A subgraph is called a ’maximal
subsimplex’ () if there is no other simplex
contained in with .

ii) A subgraph is called to cover if and at every
node there exists at least one bond .

Remark: Such are called in combinatorics ’cliques’ as we
learned recently.

This class of , , and their mutual entanglement
will be a key concept to characterize the geometrical (large scale)
structure of the unfolding graph .

4.5 Definition: Let be a class of subgraphs of .

i) is the graph with if
every ,

if every

ii) is the graph with if for at least one

if for at least
one .

4.6 Observations: i) Starting from an arbitrary node and
performing the above described steps we get a certain being
contained in ; described pictorially as

(55) |

With given, each permutation will yield the same , i.e:

(56) |

Furthermore each can so be constructed, starting from one of its nodes. Evidently this could be done for each node and for all possible alternatives as to the choice of the next node in the above sequence.

It is however important to note that, starting e.g. from a given node,
there may be several alternatives at each intermediate step, leading
in the end to different(!) having some (or even many) nodes and/or bonds in
common! We note in passing that

ii) covers with the class of . Note
however that in general is only a true subgraph of ,
i.e.

(57) |

4.7 Example:

(58) |

There exist two

(59) |

where at two alternative choices can be made.

It turns out to be an ambitious task to calculate the cardinality of
in a given graph as a function of the, say, missing
bonds and their distribution in . the reasons for this are
manifold. For one, this number depends sensitively on the way the
missing bonds are distributed in . For another, as can most easily
be seen by studying examples, the prescription how the can be
(re)constructed (cf. Observation 4.6) is strongly ”path dependent”
in the sense that they may be (in particular if they are large)
intricately entangled. which makes an easy counting quite
delicate. Nevertheless this is an important task.

4.8 Task: Estimate the number of spanning by using only a
few characteristics like the number of missing bonds and the like. It
may well be that also in this situation a statistical approach would
be the most appropriate.

Remark: This problem is presently under study by ourselves. We suppose
that there is some veritable and interesting piece of mathematics
buried in it which goes far beyond the particular context of our
investigation and which would be of relevance in its own right. (Such
problems are typically adressed in [23] which, we think, will
be of help in this respect). We
conjecture that there are, among other things, structural similarities
to something like (co)homology theory to be built over such discrete
structures.

While a general complete solution is at the moment not at our disposal, we are going to present some preliminary steps and estimates in that direction. As we already remarked, one might have the idea that it be possible to provide both a general and sensible upper bound on the number of in a graph with, say, nodes and bonds. This turns out to be a very difficult endeavor. It seems to be easier to shift, in order to get a better feeling for the crucial points of the problem, the point of view a little bit. That means we want to construct (typical) examples where the number of can be given, thus showing what order of magnitude one has to expect.

We construct a special graph as follows: Take nodes, choose a subset consisting of exactly nodes , make a simplex. With the remaining nodes we proceed in the same way, i.e. we now have two subsimplices .

We now choose a one-one-map from to , say:

(60) |

We now connect all the with the except for the pairs . The graph so constructed has

(61) |

We see from this that, as in our network scenario, the number of missing bonds is a relatively small fraction, hence, the example may be not so untypical.

We can now make the following sequence of observations:

4.9 Observations i) is already a as each
has one bond missing with respect to .

ii) One gets new by exchanging exactly one with its
partner , pictorially:

(62) |

yielding further .

iii) One can proceed by constructing another class of , now
deleting and adding their respective partners, i.e:

(63) |

iv) This can be done until we end up with the

(64) |

The combinatorics goes as follows:

(65) |

i.e., our node-graph (with bonds missing) contains exactly
.

4.10 Some Comments on Combinatorics: That the combinatorical
problems we have adressed above are really ambitious can be seen with
the help of the following argument. As a ”warm-up exercise” one can
tackle a seemingly much simpler problem, namely trying to find
sensible bounds for the cardinality of certain coverings of a given finite (but, in general, large) set with
being a covering of , , the ’power set’ of . The constraint is that in a given covering no subsets of
already occurring sets are to be allowed, i.e.,

(66) |

The reason for this particular constraint stems from our (see Corollary 4.12 below).

We are e.g. interested in bounds of the following nature:

Give an
effective bound for , i.e. a bound which is
better than , by employing the above constraint.

As for our graph problem discussed above, this appears to be still
quite difficult if one starts from first principles. Shifting the point of view one can instead try to
construct coverings with a large in order to get at least a
sensible lower bound on . It turns out to be effective to
choose coverings with sets with for all and
then take if with being even. (This idea we owe
to our coworker Th. Nowotny; note that in that case the above
constraint is automatically implied). Then we have (
large):

(67) |

with the help of Stirling’s formula. Note that a covering with non-overlapping sets (i.e. a ’partition’) has always a cardinality .

A little bit surprisingly, this construction happens already to be
optimal, as we found out recently when we stumbled by chance over
an old but noteworthy paper by E.Sperner, [21] (see also the beautiful book
of H.Lueneburg, [22] chapt.XVI and the remark on p.492 or [26]). In
this paper the following result has been proven (which, by the way
gave rise to a full-fledged subtheory in combinatorics);
(in our notation):

Sperner’s Theorem:

(68) |

and is attained for , a covering with sets, if
is even, and if is odd.

Remark: It may well be that the methods developed there may be also
helpful in our more general graph-problem.

Returning to our concrete network scenario we can now proceed as
follows:

can be split in the following way:

(69) |

with the set of nodes with some of the bonds among them missing, the set of nodes being maximally connected (see the remarks after formula (53))

(70) |

Almost by definition, generate a simplex .

4.11 Observation: i) The simplex is contained in each of
the maximal subsimplices , i.e:

(71) |

ii) itself is not(!) maximal as is always a larger
simplex with and being the section graph
spanned by and .

iii) To each maximal simplex belongs a unique
maximal subsimplex with

(72) |

4.12 Corollary: From the maximality of the follows a general structure relation for the and :

(73) |

and neither

(74) |

viz. there always exists at least one
s.t. and vice versa.

Proof of Observation 4.11: i) Starting from an arbitrary node , it is by definition connected with all the other nodes in ,
since if say are not connected they both belong to (by
definition). I.e., irrespectively how we will proceed in the
construction of some , can always be added at any
intermediate step, hence . On the other side
one can easily construct scenarios where .

ii) As is connected with each (by definition of
and ), the section graph is again a (larger)
simplex.

iii) We have for all , hence

(75) |

with the corresponding section graphs in .

With being a simplex, is again a subsimplex which
is maximal in . Otherwise would not be maximal in .

On the other side each is uniquely given by
a maximal in as each node in is connected with all
the nodes in .

We see from the above that as long as , the number of dead
(missing) bonds, is much smaller than the number of bonds in the
initial simplex , there does exist a considerable overlap , among the class of . This
overlap will become smaller with increasing with clock time
; by the same token the number of will increase. To describe
this unfolding process we make the following pictorial abbreviations
and observations:

4.13 Abbreviation: We abbreviate (not) connected by
a bond by

(76) |

We then have:

4.14 Observation: i) implies that they are
lying in different ’s.

ii) are disjoint, i.e. iff

(77) |

or vice versa.

4.15 Consequence: This shows that it may well be that
while the two have still a lot of
’interbonds’, i.e. bonds connecting the one with the other. The
guiding idea is however that and , taken as
a whole, will be generically considerably less strongly entangled with
each other than the nodes within or among
themselves after the unfolding process is fully developed.

4.16 The Physical Picture: For small, viz. still large, we regard the emerging ’s as ’protoforms’ of ’physical points’. We suppose that
this picture becomes more pronounced with increasing ,
i.e. increasing clock time, when the entanglement between different
proto points becomes weaker. For sufficiently small all these
proto points are hanging together via the non-empty(!) while for
sufficiently large (i.e. clock time large) and the dead bonds
appropriately distributed over it may happen that
and that, furthermore, a pronounced far- and near-order among the
grains is established via their (varying) ’degree of
connectedness’.

This is the scenario which, we hope, will support a certain ’superstructure’ we dub (space-time), the whole complex we like to call , i.e. a still wildly fluctuating ”quantum underworld” with a both coarser and more smoothly behaving ’order parameter manifold’ being superimposed.

One can supply a rough estimate as to the threshold which (under certain conditions) may divide these two scenarios. With the primordial graph being nearly a simplex of node number, say, , edge number and bonds already annihilated, we had the result (see above) that consists at least of

(78) |

nodes as long as the rhs is positive. Hence:

4.17 Observation: There is a chance that , i.e an
effective distribution of the annihilated bonds being assumed, when

(79) |

We conjecture that this is a characteristic number which indicates where the transition zone will probably lie which divides the two regimes described above. The threshold value yields:

(80) |

bonds being still alive, which implies an average node degree:

(81) |

in other words, on the average there is only one bond missing per
node!

Remark: The content of Observation 4.17 makes it clear again that
relatively advanced stochastic graph concepts are called for, of a
kind as they are developed in the beautiful book of Bollobas about
’random graphs’ ([23]), i.e. it has to be clarified
whether e.g. is actually representing what is
called a ’threshold function’ for random graphs. These extremely
important concepts will be discussed in more detail elsewhere, as we
want to enter in the remaining part of this paper into a more thorough
analysis of the nature of ’physical points’ we have introduced
above and their mutual entanglement, in particular for large
i.e. far away from the phase transition regime.

Our strategy to relate the with what one may call on a
much more coarse grained level physical points suggests the
introduction of the following mathematical concepts, being of wide use
in combinatorial mathematics and related fields
(cf. e.g. [24] or [25]).

4.18 Definition: Over the ’ground set’ we construct an
’incidence structure’ or ’block space’ in
the following way:

i) The ’blocks’ are the

ii) The points of the incidence structure are the nodes

iii) The ’incidence relation’ is .

In other words we say a node is incident with a block or
belongs to . In our scenario the blocks are elements of the
power set (which needs not always be the case).

iv) With we denote the blocks which are incident with
, i.e. . In the general theory would
denote the nodes being incident with , in our case where
we identify and ;
are their respective cardinalities.

v) The above is generalized in an obvious way to
or .

4.19 Definition: i) An ’automorphism’ of a block space
is a bijective map s.t.

(82) |

ii) The automorphism is called ’inner’ if it acts within the respective blocks, i.e:

(83) |

4.20 Definition: With respect to the above block space we can
speak of an

i) ’interior bond’ of a given , i.e:

(84) |

ii) ’exterior bond’ with respect to a given , i.e:

(85) |

iii) an ’interbond’, i.e:

(86) |

iv) a ’common bond’ of , if is an
interior bond both of and .

v) a ’true interbond’ if for :

(87) |

vi) We then have the relation for given :

(88) |

Remarks: i) The relation between these classes describes the (time
dependent) degree of entanglement among the blocks ,
viz. among the physical proto points and, as a consequence, the
physical near- and far-order on the level of , the macroscopic
causality structure of space-time and the non-local entanglement we
observe in quantum mechanics.

ii) In Definition 2.1 we defined the notion of a ’simple graph’ and
related it with the concept of a ’homogeneous’, ’non-reflexive’
relation. Note that, on the other side, a graph is a particular
example of a block space, with the class of bonds being the blocks and
the obvious incidence relation between nodes and bonds.

This identification generalizes easily to more complex scenarios in the following way: Following Bollobas in [26] we define a ’set system’ over, say, to be a subset of the powerset , i.e:

(89) |

Such a set system is by the same token a block space, both being, on the
other side, examples of a ’heterogeneous relation’ between
and ( a homogeneous relation corresponding to blocks
).

4.21 Definition: i) is the class of in

ii) A homogeneous relation is a subset of or, by the same
token, a subset of

iii) A heterogeneous relation is a relation between between two sets
of, typically, different nature, i.e. a block space. If the set of
blocks is contained in one may also call it a ’hyper
graph’ and its blocks, i.e. the occurring tuples of nodes its ’hyper edges’. If the hyper edges are uniformly taken from
it is called a ’k-uniform’ hyper graph (see
e.g. [26])

iv) As is the case for simple graphs and block spaces, one can now
talk of the ’degree of a node’ in a hypergraph, i.e. the number
of incident hyper edges, i.e. or the rank of a hyper edge,
which is in our example simply the number of nodes belonging to
it.

Remark: We would like to emphasize that, while this various
definitions and introduced concepts seem to be, at first glance, more
or less straightforward and related, they nevertheless turn out to be
very efficient and economical tools to deal with a number of
surprisingly deep questions in discrete mathematics from different
perspectives (e.g. graphs, relational mathematics, incidence
structures or block spaces; see the above mentioned
literature).

4.22 Observation: As a simplex is uniquely given by its set of nodes,
i.e. a set , we can associate our complex ,
i.e. the underlying graph and the array of , with a
hypergraph by identifying the ’s with its hyperbonds. This
hypergraph is nothing but (introduced in
Observation 4.6), viz. it exhibits in general not the full wiring of
or but only the bonds occurring in the ’s, the true
interbonds, however, are missing in .

In a next step one can go to a coarser level of resolution by defining
the ’intersection graph’ of the set ; henceforth
(as the notion hypergraph is already fixed in graph theory in the
sense defined above) we prefer to call it the associated ’skeleton
graph’ or ’super graph’.

4.23 Observation: By shrinking the blocks to ’super nodes’ one gets another graph by saying that two super nodes
, (), are linked by a ’super
bond’ if their intersection . This ’super graph’ may then be associated
with the manifold of ’unresolved’ physical points, , if we
neglect their internal complexity (cf. this with what we said in 4.15,
4.16 above).

The above train of thought shows that one may impose on the underlying network a certain hierarchy of levels of varying resolution of the physical landscape which becomes more and more pronounced with increasing number of turned-off bonds, viz. with increasing clock time. Given an arbitrary but fixed block , one can define its infinitesimal neighborhood as consisting of the with

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in other words, the nodes adjacent to in the super graph of 4.23.

There may then exist