Biografie

Let’s look back now at these three goals.

C.1 Monism

The Good of the Platonic tradition, (Chapter 1), or the ´Siva tattva of the Trika cosmos (Chapter 2), is to be understood as the source and creator of the universe, outside of space and time. In the monist traditions of both East and West, the source precedes the creation of space and time. In our model this role is played by a monster dynamical cellular network, QX∗.

In Kashmiri Shaivism, they say that the creation of the universe in 36 tattvas was accomplished in a blink, as ´Siva opened his eyes. Even so, we think of the 36 tattvas as unrolling in an ordered sequence, and the ”density” as it were increasing step-by-step until, at the end, the levels of matter, energy, spacetime, and physical forces condense. Space and time are tattvas coming at the tail end of the process.

Although explicitly nondualist, these traditions seem to lack any interconnection scheme. Our model provides a dynamical cellular network that contains and connects all levels of consciousness.

96See Appendix 2 for some background of this.

Abraham and Roy 75

C.2 Discrete space and time

Chapters 3 and 4 of Part One, and Chapters 6 and 7 of Part Two, provide a basis for discrete space and time all the way from ancient Greece to the latest developments of quantum physics. There seems little doubt now that discrete space and time are fundamental to our conception of the universe.

It is the argument of Steven M. Rosen, in The Dimensions of Apeiron, that ancient concepts of space and time were originally discrete.97 After Plato, the continuum emerged into cultures worldwide, and this process concluded with the denial of atomism in the works of Descartes, the triumph of Maya. It is not unlikely that some of the problems of our time derive from this illusion of continuity, and thus, there is a need to revive atomistic concepts of space and time. In modern science, especially in the process physics of Whitehead, this revival is under way.98 The demise of the continuum models began in 1839, according to Rosen.99

Our discrete network models, from which space and time emerge from a process of condensation and smoothing, support this revival. Further, they suggest a similar model for cosmic consciousness, and a possible resolution of the mind/body problem.

C.3 The interconnection of the extended universe

Supposing a model of the extended universe in several levels — for example, ﬁve koshas or the thirty-six tattvas — we propose a QX network and an AR process of condensation for each level. The interconnection or entanglement of adjacent levels is accomplished via mathematical connections (bonds between nodes). Thus there is no need to search for physical forces to connect, for example, collective consciousness and the physical universe, or the individual mind and body.

In fact, it makes no sense to propose physical forces, even quantum forces, as a mechanism connecting adjacent tattvas. This could work only under the materialist paradigm, in which a thought is a biochemical state of the physical brain, as in the models of Eccls, Sperry, Stapp, Penrose, Hameroﬀ, et al.

Instead, we propose only mathematical connections between tattvas, that all tattvas are realizations of dynamical cellular networks, and the carriers of inﬂuence between them are mathematical. That is, dynamical links pop up between nodes on diﬀerent levels of creation according to rules connect the whole shebang. Although we propose mathematical connections between each tattva and its sequel, we might focus now on the bottom of the chain. Let us image just three levels, as in Plato’s cosmology below The Good. Thus,

97(Rosen, 2004) 98See Figure 1 in Chapter 4. 99(Rosen, 2004; p. xxx)

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• MX = the mental sphere, consciousness (a network) • QX = the terrestrial sphere, the universe (a network) • ST = the universe of matter and energy (the classical continuum)

We have proposed: MX → QX by links, and QX → ST by condensation. Of course, we could easily interpose another network, the quantum vacuum, QV , so: MX → QX → QV → ST or similarly, identify QX = QV as in RR, hence MX → QV → ST as in the models of Eccles, Sperry, Stapp, Penrose, and Hameroﬀ discussed previiously.

In any case, we point out that QV → ST. the quantum mechanical model for mental states, does not solve the mind/body problem. Our proposal goes to the heart of the mind/brain problem in a monistic way, consistent with the cosmologies of Plato and the Trika philosophy

In any case, what are space and time, really? We are proposing that the real spacetime is a construction (condensation) from a dynamical mathematical object ”outside”, hidden by Maya, and appearing to us as a submicroscopic granulation, smoothed over by the limitations of our perceptual organs.

This model easily accommodates the the measurement problem of quantum mechanics (Chapter 4 of part One), the big ﬁve paranormal phenomena (Chapter 5 of Part One), and the most important target of all, the mind/body problem (Chapter 8 of Part Two).

As the nondualists like Plato and Abhinava have been tellimg us, this mathematical consciousness contains our illusion of ordinary reality, mind contains both, and the superreal is primary — demystifying the ¯ak¯a´sa.

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Appendix 1. (Quantum) Space-Time) as a Statistical Geometry of Fuzzy Lumps and the Connection with Random Metric Spaces

By Manfred Requardt (email : requardt@theorie.physik.uni-goettingen.de), Institut fur Theoretische Physik, Universitat Gottingen,, Germany, and Sisir Roy, (email : sisir@isical.ac.in), Indian Statistical Institute, Calcutta - 700 108, India,

Abstract

We develop a kind of pregeometry consisting of a web of overlapping fuzzy lumps which internal with each other. The individual lumps are understood as certain closely entangled subgraphs (cliques) in a dynamically evolving network which in a certain approximation, can be visualized as a time-dependent random graph. This strand of ideas is merged with another one, deriving from ideas, developed some time ago by Menger et al. that is, the concept of probabilistic- or random metric spaces, representing a natural extension of the metrical continuum into a more microscopic regime. It is our general goal to ﬁnd a better adapted geometric environment for the description of microphysics. In this sense one may it also view as a dynamical randomisation of the causal-set framework developed by e.g. Sorkin et al. In doing this we incorporate, as a perhaps new aspect, various concepts from fuzzy set theory.

A1. Introduction

There exists a certain suspicion among the scientiﬁc community that nature may be discrete or rather “behaves discretely” on the Planck scale. But even if one is willing to agree with this “working philosophy”, it is far from being evident what this vague metaphor actually means or how it should be implemented into a concrete and systematic inquiry concerning physics and mathematics in the Planck regime.

There are basically two overall attitudes as to “discreteness on the Planck scale”, the one starts (to a greater or lesser degree) from continuum concepts (or more speciﬁcally : concepts being more or less openly inspired by them) and then tries to detect or create modes of “discrete behavior” on very ﬁne scales, typically by imposing quantum theory in full or in parts upon the model system or framework under discussion. We call this the “top down” approach.

There are prominent and very promising candidates in this class like e.g. string theory or loop quantum gravity with e.g. Spin networks emanating from the latter approach. As these

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approaches are widely known we refrain from citing from the vast corresponding literature. We recommend instead two more recent reviews dealing with the latter approach but containing also some remarks about the former one ([1] and [2]). As a beautiful introduction to the conceptual problems of quantum gravity in general may serve e.g. [3].

In the following investigation we undertake to describe how macroscopic space-time (or rather, its underlying mesoscopic or microscopic substratum) is supposed to emerge as a superstructure of a web of lumps in a ﬂuctuating dynamical cellular network. One may call this the “bottom up” approach. In doing this, two strands of research are joined, which, originally, started from diﬀerent directions. The one is the cellular network and random graph approach, developed by one of the authors (M.R.), the other the statistical geometry of lumps, a notion originally coined by Menger and coworkers and being further developed by various groups (see e.g. [17], [18], [19], [20]). It is worth mentioning that Einstein himself was not against such a grainy substratum underlying our space-time continuum (see the essay of Stachel in [21]).

The point where these diﬀerent strands meet is the following. In the dynamical network approach of (M. R.) macroscopic space-time is considered to be a coarsegrained emergent phenomenon (called an orderparameter manifold in [8]). It is assumed to be the result of some kind of geometric phase transition (very much in the spirit of the physics of self-organisation). This framework was developed in quite some detail in e.g. [8]. We argued there that, what we consider to be the elementary building blocks of continuous space-time, i.e. the so-called physical points, are on a ﬁner scale actually densely entangled subclusters of nodes and bonds of the underlying network or graph. In [8] we called them also cliques (which denote in graph theory the maximal complete subgraphs or maximal subsimplices of a given graph).

We further argued there that the substructure of our space-time manifold consists in fact of two stories, the primordial network, dubbed by us QX, and, overlying it, the web of lumps or cliques, denoted by ST, which can also be viewed as a coarser mesoscopic kind of network with the cliques or lumps as supernodes and with superbonds which connect lumps having a non-void overlap. This correspondence raises the possibility to relate the lumps or cliques of [8] with the lumps occurring in the approach of Menger et al.

One should however note that the two philosophies are not entirely the same. In [8] and related work the lumps emerge from a more primordial discrete dynamical substratum and consequently have a speciﬁc internal structure. In the approach of Menger et al. (at least as far as we can see) they ﬁgure as the not further resolvable building blocks of space-time if one approaches the so-called Planck-regime from above, i.e. from the continuum side. In other words, the former approach is more bottom-up oriented while the latter one is more top-down. About such not further resovable scales of space-time (which the ordinary continuum picture ends) was of course also speculated by quite a few other people, most notably Wheeler (see e.g. [9], p. 1203 ﬀ.).

Our personal working philosophy is that space-time at the very bottom (i.e. near or below the notorious Planck scale) resembles or can be modeled as an evolving information processing

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cellular network, consisting of elementary modules (with, typically, simple internal discrete state spaces) interacting with each other via dynamical bonds which transfer the elementary pieces of information among the modes. That is, the approach shares the combinatiorial point of view in fundamental space-time physics which has been advocated by e.g. Penrose. It is a crucial and perhaps characteristic extra ingredient of our framework that the bonds (i.e. the elementary interactions) are not simply dynamical degrees of freedom (as with the nodes their internal state spaces are assumed to be simple) but can a fortiori, depending on the state of the local network environment, be switched on or oﬀ, i.e. can temporarily be active or inactive! This special ingredient of the dynamics hopefully allows the network to perform geometric phase transitions into a new ordered phase displaying a certain twostory structure to be explained below. This conjectured emergent geometric order can be viewed as kind of a discrete proto space-time or pregeometry carrying metrical, causal and dimensional structures (as to the cellular network approach ef. [4] to [8], the other point of view is expounded in e.g. the book of Roy; [19] to which we also refer for a more complete list of references).

We will see that several types of distance concepts do emerge in our analysis on the various scales of resolution of space-time. On the in our scheme most fundamental level, that is, the primordial network (or graph), we have a natural distance concept (node distance). The same holds for the network of lumps, derived from it (the clique graph : see below). If we leave these fundamental levels (of relatively discrete behavior) and enter the realm of quasicontinuity, other possibilities do emerge. It is interesting to relate these various concepts to each other and to distance concepts discussed e.g. in [20] or [40]. We show in particular that our network of fuzzy lumps leads naturally to the concept of random metric spaces (see section (7.2)). Spaces of this kind should emerge at the interface between the presumed discrete Planck scale scenario and a perhaps quasi-continuous, that is, coarse-grained spatial environment, connecting this primordial level with the ordinary continuous regime.

Our overall working philosophy is that geometry has to emerge from a purely relational picture. We hence embark on a reconstruction of the continuum concepts of ordinary physics and/or mathematics from more primordial pregeometric (and basically discrete) ones which, in our view, hold sway on the Planck scale. We presume that this may also shed some light on the better understanding of quantum theory as such and on the various top-down approaches mentioned above. We note in passing, that this point of view has a venerable history of its own, beginning with e.g. Leibniz. A perhaps related but technically diﬀerent approach was developed by Isham et al. some time ago (see e.g. [22] or [23]), who chose to quantise metrical concepts (a more recent approach is [24]). It would be an interesting task to relate these two approaches to each other.

As another precursor may be considered the causal-set approach of Sorkin et al. see e.g. [31] or [32]). In a sense our framework adds dynamics, viz. interaction, and random behavior to this picture. subsection*A2. The Cellular Network Environment Motivated by the above working philosophy, we emulate the underlying substratum of our world, or, more speciﬁcally, of our space-time (quantum) vacuum (containing however in addition all the existing quantum and macro objects as extended excitation patterns!) by what we call a cellular net

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work. This discrete structure consists of elementary nodes, ni, which interact (or exchange information) with each other via bonds, bik, playing the role of (in this context) not further reducible elementary interactions. The possible internal structure of the nodes (modules) or bonds (interaction channels) is emulated by discrete internal state spaces carried by the nodes/bonds. The node set is assumed to be large but ﬁnite or countable. The bond bik is assumed to connect the nodes ni,nk. the internal states of the nodes/bond are denoted by si,Jik respectively. As our philosophy is, to generate complex behavior out of simple models we, typically, make simple choices for them, one being e.g.

si ∈ q·Z,Jik ∈{−1,0,+1} (1)

with q an elementary quantum of information.

As in our approach the bond states are dynamical degrees of freedom which, a fortiori, can be switched oﬀ or on, the wiring, that is the pure geometry of the network is also an emergent, dynamical property and is not given in advance. Consequently, the nodes and bonds are typically not arranged in a more or less regular array, a lattice say, with a ﬁxed nea-/far-order. This implies that geometry will become to some extent a relational (Machian) concept and is no longer an ideal element (cf. the more detailed discussion in [12], which deals primarily with the emergence of quantum theory as a consequence of the geometric ﬁne structure of such a network).

On the other side, as in cellular automate, the node and bond states are updated (for convenience) in discrete clock time steps, t = z ·τ,z ∈ Z and τ being an elementary clock time interval. This updating is given by some local dynamical law (examples given below). In this context local means that the node/bond states are changed at each clock time step according to a prescription with input the overall state of a certain neighborhood (in some topology) of the node/bone under discussion. We want however to emphasize that t is not to be confounded with some physical time, which, for its part, is also considered to be an emergent coarse grained quantity. The well known problem of time is, for the time being, not treated in detail in the following, as it presents a big problem of its own, needing a careful and separate analysis (see e.g. [10 or [11]). That is, at the moment the above clock time is neither considered to be dynamical nor observer dependent. We discussed however the presumed emergence of a new primordial time scale which sets the scale for the regime where quantum ﬂuctuations hold sway in [12].

A simple example of such a local dynamical law we are having in mind is given in the following deﬁnition.

Deﬁnition 2.1 (Example of a Local Law) At each clock time step a certain quantum q is exchanged between, say, the nodes ni,nk, connected by the bond bik such that

Abraham and Roy 81 si(t + τ)−si(t) = q·X k Jki(t) (2)

(i.e. if Jki = +1 a quantum q ﬂows from nk to ni etc.)

The second part of the law describes the back reaction on the bonds (and is, typically, more subtle.) Thus is the place where the so-called ‘hysteresis interval’ enters the stage. We assume the existence of two ‘critical parameters’ 0 ≤ λ1 ≤ λ2 with :

Jik(t + τ) = 0 if | si(t)−sk(t) |=:| sik(t) |> λ2 (3)

Jik(t + τ) = ±1 if 0 < ± sik(t) < λ1 (4)

with the special proviso that

Jik(t + τ) = Jik(t) if sik(t) = 0 (5)

On the other side

Jik(t + τ) =  

± 1 Jik(t) 6= 0 if λ1 ≤±sik(t) ≤ λ2 0 Jik(t) = 0

(6)

In other words, bonds are switched oﬀ if local spatial charge ﬂuctuations are too large, switched on again if they are too small, their orientation following the sign of local charge diﬀerences, or remain inactive.

Another interesting law arises if one exchanges the role of λ1 and λ2 in the above law, that is, bonds are switched oﬀ if the local node ﬂuctuations are too small and are switched on again if they exceed λ2. We emulated all these laws on a computer and studied a lot of network properties. The latter law has the peculiar feature that it turned out to have very short transients in the simulations, i.e. it reaches an attractor in a very short clock time. Furthermore these attractors or state-cycles turned out to be very regular, that is, they had a very short period of typically six, that is, the whole network returned in a previous state after only six clock time steps, which is quite remarkable, given the seeming complexity of the evolution and the huge phase space([13]).

Remarks :

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1. It is important that, generically, such laws, as introduced above, do not lead to a reversible time evolution, i.e. there will typically exist attractors or state-cycles in total phase space (the overall conﬁguration space of the node and bond states). On the other hand, there exist strategies (in the context of cellular automata!) to design particular reversible network laws (cf. e.g [14]) which are, however, typically of second order. Usually the existence of attractors is considered to be important for pattern formation. On the other side, it may suﬃce that the phase space, occupied by the system, shrinks in the course of evolution, that is, that one has a ﬂow into smaller subvolumes of phase space.

2. In the above class of laws a direct bond-bond interaction is not yet implemented. We are prepared to incorporate such a (possibly important) contribution in a next step if it turns out to be necessary. In any case there are not so many ways to do this in a sensible way. Stated diﬀerently, the class of possible physically sensible interactions is perhaps not so numerous.

3. As in the deﬁnition of evolution laws of spin networks by e.g. Markopoulou, Smolin and Borissov (see [15] or [16]), there are in our case more or less two possibilities : treating evolution laws within an integrated space-time formalism or regard the network as representing space alone with the time evolution being implanted via some extra principle (which is the way we have chosen above). The interrelation of these various approaches and frameworks, while being very interesting, is however far from obvious at the moment and needs a separate detailed investigation.

Observation 2.2 (Gauge Invariance) The above dynamical law depends nowhere on the absolute values of the node charges but only on their relative diﬀerences. By the same token, charge is nowhere created or destroyed. We have ∆(X QX s(n)) = 0 (7)

To avoid artiﬁcial ambiguities we can e.g. choose a ﬁxed reference level and take as initial condition respectively constraint X QX s(n) = 0 (8)

There are many diﬀerent aspects of our class of cellular networks one can study in this context. One can e.g. regard them as complex dynamical systems, or one can undertake to develop a statistical or stochastic framework etc. In a purely geometric sense, however, they are evolving graphs. As we are in this paper primarily concerned with the analysis of the microstructure of (quantum) space-time, it seems to be a sensible strategy to supress, at least

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in a ﬁrst step, all the other features like e.g. the details of the internal state spaces of nodes and bonds and concentrate instead on their pure wiring diagram and its reduced (graph) dynamics. This is already an interesting characteristic of the network (perhaps somewhat reminiscent of the Poincar´e map in the theory of chaotic systems) as bonds can be switched on and oﬀ in the course of clock time so that already the wiring diagram will constantly change. Furthermore, as we will see, it encodes the complete near- and far-order structure of the network, that is, it tells us which regions are experienced as near by or far away (in a variety of possible physical ways such as strength of correlations or with respect to some other physically meaningful metric like e.g. statistical distance etc.). Evidently this is one of the crucial features we expect from something like physical space-time. In the above simple scenario with Jik = ± 1 or 0 one can e.g. draw a directed bond, dik, if Jik = + 1. with Jik = −Jki implied, and delete the bond if Jik = 0. This leads to a (clock) time dependent graph, G(t). or wiring diagram. In other words, we will deal in the following mainly with the evolution and structure of large dynamical graphs.

We close this section with a brief r´esum´e of the characteristics an interesting network dynamics should encode (in our view).

R´esum´e 2.3 Irrespectively of the technical details of the dynamical evolution law under discussion it should emulate the following, in our view crucial, principles, in order to match certain fundamental requirements concerning the capability of emergent and complex behavior.

1. As is the case with, say, gauge theory or general relativity, our evolution law on the surmised primordial level should implement the mutual interaction of two fundamental substructures, put sloppily : “geometry” acting on “matter” and vice versa, where in our context “geometry” is assumed to correspond in a loose sense with the local and/or global bond states and “matter” with the structure of the node states.

2. By the same token the alluded selfreferential dynamical circuitry of mutual interactions is expected to favor a kind of undulating behavior or selfexcitation above a return to some uninteresting ‘equilibrium state’ as is frequently the case in systems consisting of a single component which directly acts back on itself. This propensity for the ‘autonomous’ generation of undulation patterns is in our view an essential prerequisite for some form of “protoquantum behavior” we hope to recover on some coarse grained and less primordial level of the network dynamics.

3. In the same sense we expect the overall pattern of switched-on and-oﬀ bonds to generate a kind of “protogravity”.

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A3. The Network of Lumps

In [8] we argued that our microscopic cellular network, QX, may be capable of performing a geometric phase transition into a more ordered phase, dubbed QX/ST, with elementary building blocks the maximal connected subgraphs or cliques (maximal subsimplices) occurring in the primordial graph, belonging to QX. We further argued that this emergent orderparameter manifold (as it plays a role similar to ordinary orderparameters in the statistical mechanics of phase transitions) may constitute a stochastic protoform of our ordinary space-time or quantum vacuum with the cliques as protoforms of physical points (lumps), having an inner dynamical structure and being entangled with the other (overlapping) lumps via common nodes or bonds.

The stochastic aspects are brought in by the underlying dynamical network law, which induces, among other things, a certain amount of creation and annihilation of bonds among the microscopic nodes. As a consequence the size and shape of the cliques or lumps ﬂuctuates in the course of network evolution. These aspects have been analyzed in quite some detail in [8] within the framework of random graphs.

This derived coarser network, viz. the clique graph or web of lumps, is deﬁned as follows. The cliques are represented by new meta-nodes, the respective metabonds represent overlap of cliques (in form of common nodes).

Observation 3.1 Note that, while this new network may be regarded as being coarser in some sense, it may nevertheless in general consist of much more nodes and bonds than the underlying primordial network. Usually there are much more maximal subsimplices than primordial nodes, as a given node will typically belong to quite a few diﬀerent maximal subsimplices (cf. the estimates in [8]). This array of intersecting maximal subsimplices has the natural structure of a simplicial complex with the smaller simplices as faces of the maximal ones, viz. the cliques (as to this notion see any textbook on algebraic topology like e.g. [25], ef. also section 3.2 of [6]). If we represent this simplicial complex by a new (clique) graph with only the maximal simplices occurring as meta-nodes we loose, on the other side, some information, as we do not keep track of, to give an example, situations where, say, three lumps or cliques have a common overlap. This situation is not distinguished from the scenario where only each pair of the triple has a common overlap.

Remark 3.2 On the other hand, it may be physically advantageous to loose microscopic information in a controlled way. We will show below that this possibility to consider the network as a simplicial complex stops at this ﬁrst level, the coarser picture of a clique graph, however, allows on the other side for a geometric renormalization group procedure as we can repeat this process on each level, thus creating a whole tower of such metagraphs, one lying over the other, leading in the end to webs which resemble more and more our ordinary space-time.

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We now have to make the physically motivated assumption that our network of lumps, which is assumed to be in the phase QX/ST, is not ﬂuctuating too wildly, put diﬀerently, that the eﬀective dynamics, induced by the underlying microscopic dynamics deﬁned above, leaves the individual cliques. Ci, suﬃciently stable. We hence assume that they do not change their form too much from clocktime step to clocktime step, so that we are able to keep track of the history of the individual cliques over an appreciable amount of time steps.

Ci(t)∩ Ci(t + 1) ≈ Ci(t) (9)

This means that only a small portion of nodes enters or leaves each of the cliques in every time step. We make this assumption so as to be able to perform some sort of assemble averages over ﬂuctuating but individual (that is, labelled) cliques in the next section and identify them with fuzzy lumps.

A4. Cliques as Fuzzy Sets

We now concentrate on the (clock-time) evolution of an arbitrary (generic) but ﬁxed clique, denoting its time sequence by

C(t0),C(t1),···,C(tN) (10)

We argued above that the clique under observation changes its shape and size only mildly from time step to time step. As it is frequently done in physics, we want to replace the above time series of a generic clique by an ensemble. To this end we deﬁne C :=[ C(ti) , C :=\ C(ti) (11)

We now deﬁne a so-called membership function, m(x), for an arbitrary node, x, in the graph, G. Traditionally, the usual habit in fuzzy set theory is it, to speak of fuzzy sets, but actually they do exist only via their membership functions (see below). That is, we relate to each cluster of sharp cliques, deﬁned above, a so-called fuzzy clique, deﬁned by its membership function. We denote the fuzzy clique, belonging to the above ensemble of sharp cliques by C and deﬁne its membership function, mC, as follows :

Deﬁnition 4.1 (Fuzzy Clique) For each node, x, we deﬁne

86 The Digital Akasha mC :=   0 x / ∈ C 0 ≤ p ≤ 1 x ∈ C 1 x ∈ C (12) with p denoting the relative frequency of occurrence of node x in ∪C(ti),∪ the disjoint union, Viz.

p := N(x)/N (13)

N(x) the number of occurrences of x in ∪C(ti).

Remark : We want to remind the reader of the assumptions we have made above. Our fuzzy clique or fuzzy lump is a set with the individual nodes carrying a certain weight, viz. its degree of membership with respect to the fuzzy clique. C, which, on its side, is given by the corresponding membership function, mC(x). Note that the underlying philosophy is quite modern as it replaces an underlying space (a set of points) by a class of functions on the space. A similar philosophy is e.g. hold in non-commutative geometry and related ﬁelds.

We should add at this place some remarks of precaution. We studied in quite some detail in [8] the statistical distribution of the number and size of occurring cliques in, say, a random graph. It turns out that most of the cliques have a typical (generic) size. On the other hand there exists a (possibly small) fraction of degenerated, i.e. very small ones. These small ones may even happen to vanish and/or emerge in the envisaged clock-time interval. That, is, we expect our general picture to be true modulo some stochastic noise, which we choose to neglect at the moment.

Conclusion 4.2 Under the assumption, being made, we can, in a certain approximation, replace the clique graph introduced above by a net of overlapping fuzzy lumps. Mathematically this net is implemented by the corresponding class of membership functions, {mi(x)}, with i labelling the fuzzy lumps and where the overlap of cliques is now encoded in the overlap of the supports of the functions, mi(x), viz.

x ∈ ∼ Ci if mi(x) > 0 (14)

We see from the above that, to put it brieﬂy, fuzzy-set theory consists of replacing the ground set, P(X), X some space of points, P(X) the set of subsets of X, and the respective operations on it, by the set of functions, IX,I the unit interval, and corresponding operations on it. P(X) can be made into a Boolean algebra or a lattice with the help of the operations ∩,∪ or ∩ as multiplication and the symmetric diﬀerence

Abraham and Roy 87

∆(A,B) := (A\B)∪(B\A) (15)

as addition.

Corresponding relations can be given for fuzzy sets, but there are a lot of diﬀerent possibilities to implement or mimik the above set-theoretic operations. The perhaps most straightforward ones are

∩→ min(∼ A(x), ∼ B(x)) ∪→ max(∼ A(x), ∼ B(x)) (16)

where, for brevity, we use from now on the same symbol for the fuzzy set and its membership function. Some more remarks on these matters can be found in the following section. (A nice introduction to fuzzy set theory on a mathematically satisfying level is e.g. [26], including a huge bibliography. A brief but concise representation can also be found in [27]). Some slightly more advanced relations between fuzzy sets, being of potential use in our geometric enterprise are the “degree of being a subset” and the “degree of similarity” between two fuzzy sets. Deﬁning the size of the fuzzy set by

[∼ A] :=X x

∼ A(x) or Z ∼ A(x)dnx (17)

(existence of the rhs being assumed), we deﬁne e.g. the degree of being a subset as

sub(∼ B; ∼ A) := [∼ A∧ ∼ B]/[∼ A] (18)

and the degree of similarity of two sets by

sub(∼ A, ∼ B) := [∼ A∧ ∼ B]/[∼ A∨ ∼ B] (19)

with extending the notions of∩,∪; one may take e.g. the above (min, max)- implementations.

The above binary relations, given e.g. by (min, max), fullﬁl the criteria of so-called (t,s)norms (see the next section). There are other such functions in use, which have perhaps nicer properties. One is the so-called Lukasiewicz-(t,s)-norm :

tL(∼ A, ∼ B) := max{0, ∼ A + ∼ B−1} sL(∼ A, ∼ B) := min{1, ∼ A + ∼ B} (20)

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leading to (for more details see e.g. [28]). The advantage of the Lukasiewicz-operations is that they induce a metric on fuzzy-sets. We state without proof.

Observation 4.3 1−simL(∼ A, ∼ B) deﬁnes a metric on fuzzy-sets.

Remark 4.4 We rediscover exactly this t-norm in section (7.2) in quite a diﬀerent context (E-spaces).

As to this observation, note that for ordinary (ﬁnite) sets, ∆(A,B) induces also a metric.

Observation 4.5 (Hamming-distance) | ∆(A,B) | deﬁnes a metric on ﬁnite sets.

Proof : The only non-trivial property is the triangle inequality. As a direct approach is perhaps a little bit tedious, we give a perhaps more pedagogical proof using graph theory. Subsets of a ﬁnite set, X, can be represented by functions f ∈{0,1}X. These functions, on the other side, can be represented as the vertices of a hypercube, Q|X|. | ∆(A,B) | is now the minimal length of a path between fA,fB on Q|X|, i.e., the minimal number of ﬂips of (0,1) along a path connecting fA and fB. This distance on graphs is however a metric.

After this short digression we want to return to our original enterprise, i.e. to relate our approach to the probabilistic-metric-approach, mentioned above. We started from an underlying graph, G(t), and a superimposed clique-graph. Gcl(t), both carrying a natural distance function, i.e.

d(ni,nk) , dC(Ci,Ck) (21)

that is, the minimal length of a path, connecting the given nodes (cf. [4] to [8]). Keeping the labelled nodes or cliques (supernodes) ﬁxed, both these distances ﬂuctuate in the course of clock-time evolution as bonds are switched on or oﬀ according to one of the microscopic dynamical laws, given above. The clique-metric will ﬂuctuate since the cliques change their shape and size, viz. also their degree of overlap. On the other side, its ﬂuctuations are supposed to be less erratic as a non-void overlap means usually that several nodes and bonds are involved, hence clique-distance should be more stable.

When we switch from this dynamical picture of a time-dependent graph, G(t), to the ensemble picture of fuzzy cliques or lumps, our point of view changes to a static but, on the other side, probabilistic one. This latter point of view is more in the spirit of the framework of Menger et al. That is, the structure of the space under study is no longer time dependent while its largely hidden dynamics is now encoded in various probabilistic notions like e.g.

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a random metric (which we would like to derive from some underlying principles). We will make the connection to this other framework after the following sections, introducing some technical material about fuzzy-set theory and probabilistic metric spaces. Furthermore we plan to relate in this ﬁnal section our ap0proach to the complex of ideas developed by e.g. Sorkin et al. (see e.g. [31] or [32]; we rocommend in particular the latter reference as a thoughtful introduction into this particular bundle of ideas).

A5. Some Concepts from Fuzzy Set Theory

We give in this section a brief introduction into the ideas of fuzzy set theory. Let A be a fuzzy set on X. Then by deﬁnition A(x) is interpreted as the degree to which x belongs to A. The fuzzy operations like fuzzy unions, intersections and complements are certain (context dependent) generalizations of the corresponding classical set operations. Let e.g. B be another fuzzy set and A be the complement of A. A particular variant of such fuzzy operations are the following :

A(x) = 1−A(x) (22)

(A∩B)(x) = min[A(x),B(c)] (23)

and

(A∩B)(x) = max[A(x),B(c)] (24)

In the literature, fuzzy intersections and unions are in general deﬁned via so-called t-norms (triangular norms) and t-conorms respectively. However, it should be noted that such fuzzy operations are by no means unique. Diﬀerent choices of such t-norms may be appropriate to represent these operations in varying contexts and hence they are known as context dependent operations. Essentially fuzzy set theory provides us with an intutive notion of uncertainty. Subsequent to the development of fuzzy set theory, fuzzy measure theory was developed (see e.g. [29]). The concept of fuzzy measure theory is conceptually an important step in understanding the foundational issues related to fuzzy set theory. It provides a broader framework which allows to introduce something like possibility theory.

Crucial in this ﬁeld as well as in the ﬁeld of probabilistic metric spaces is the above mentioned notion of t-norm (more about the history of this important concept can be found in the book of Schweizer and Sklar, [20]). The (generalized) intersection of two fuzzy sets A and B is deﬁned by a binary operation on the unit interval as

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t : [0,1]×[0,1] → [0,1] (25)

Deﬁnition 5.1 (T-Norm)

t(a,1) = a (boundary condition) (26)

b ≤ d implies t(a,b) ≤ t(a,d) (monotonicity) (27)

t(a,b) = t(b,d) (commutativity) (28)

t(a,t(b,d)) = t(t(a,b),d) (associativty) (29)

It is instructive to compare ordinary probability theory with possibility theory and in particular, probability theory with fuzzy set theory. In some sense a fuzzy measure is the dual concept to the concept of fuzzy sets. It encodes the possibility of a given ﬁxed (fuzzy) object to belong to the respective sets C (cf. also [30]).

Deﬁnition 5.2 Given a set X and a nonempty family C of subsets of X, which, for convenience, we take to ba a σ-algebra. A fuzzy measure on (X,C) is a function g : C→ [0,1] that satisﬁes the following requirements :

1. g(θ) = 0 and g(X) = 1 (boundary requirements). 2. for all A,B ∈C, if A ⊆ B, then g(A) ≤ g(B) (monotonicity) 3. for any increasing sequence, A1 ⊂ A2 ⊂ ... in C so that S∞ i=1 Ai ∈ C we havelim i→∞g(S∞ i=1 Ai) (continuity from below) 4. for any decreasing sequence A1 ⊃ A2 ⊃ ... in C,limi→∞g(Ai) = g(T∞ i=1) (continuity from above).

In case of a probability measure we have for A∩B = θ :

P(A∪B) = P(A) + P(B) (30)

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From this one sees that ordinary probability measures are a true-subclass of possibility or fuzzy measures. As we do not use these more advanced concepts at the moment, we refrain from giving more details which can be found in the mentioned literature.

A6. Concepts from the Theory of Probabilistic

In the statistical geometry, developed by Menger et al. points are no longer considered as the elementary building blocks. In some sense lumps play now their role as primordial, not further resolvable elements. Various concepts of a probabilistic nature are introduced which allow to quantify varying degrees of (in) distinguishability of objects. In this way Menger solved Poincare’s dilemma of having, on the one side, a transitive mathematical and, on the other side, a possible intransitive physical relation of equality (cf. [41] and further remarks in section (7)). In this geometrical framework we have tow basic ingredients :

1. The concept of hazy or fuzzy lumps

2. The “randomisation” of various geometrical or metrical concepts

Frechet ([33]) gave an abstract formulation of the notion of distance in 1906. Hausdorﬀ ([34]) proposed the name metric space and introduced the function d that assigns a nonnegetive real number d(p,q) (the distance between p and q) to every pair (p,q) of elements (points) of the set S. Its properties are

d(p,q) = 0 ↔ p = q (31)

d(p,q) = d(q,p) (32)

d(p,r) ≤ d(p,q) + d(q,r) (33)

In 1942 Menger ([35]), guided by the experimental situation in the natural sciences, proposed to replace the “deterministic” function d(p,q) by a more probabilistic concept. He introduced the probability distribution. Fpq, whose value Fpq(x) for any real number x, is interpreted as the probability that the distance between p and q is less than x.

Remark 6.1 We assume that Fpq is continuous from the left. On the other side, we may nevertheless encounter the situation that F+ pq(0) > 0. This means that there may be a nonvanishing probability for two diﬀerent lumps to have a vanishing distance. This particular

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point is discussed by Menger in [40] and implements Poincare’s observation of a possible non-transitive behavior.

Since probabilities can neither be negative not be greater than 1, we have

0 ≤ Fpq ≤ 1 (34)

for any real x. Menger deﬁned a statistical metric space as a set S with an associated set of probability distribution functions Fpq which satisfy the following conditions.

Fpq(0) = 0 (35)

If p = q then Fpq(x) = 1 for all x > 0 (36)

If p 6= q then Fpq(x) < 1 for somex > 0 (37)

and

Fpq(x + y) ≥ T(Fpr(x),Fqr(y)) (38)

for all p,q,r in S and all real numbers x,y. Here T is a function from the closed unit square [0,1] × [0,1] into the closed unit interval [0,1] (called a triangular function or triangular norm, see below).

One may view these probabilistic metric spaces as being derived from an underlying fully probabilistic model system, living on a probability space and with the physical observables being random variables. Spacek [36] was the ﬁrst to look at the subject from this point of view. He proposed the name random metric space instead of probabilistic metric space and discussed the relationship between these two notions. Stevens [37] in his doctoral dissertation tried to modify Spacek’s approach. The main idea behind Steven’s approach lies in the fact that one has a set S and a collection P of measuring rods. One chooses a measuring rod d from P at random to measure the distance between two given points, p and q, of S. With the help of this idea Stevens deﬁned the distribution function Fpq and showed that the metrically generated space so obtained is a Menger space.

Menger, on the other side, started with a probability distribution function instead of random variables. This is related to the fact that the outcome of any series of measurements of the values of a nondeterministic quantity is a distribution function and the probability space may

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be unobservable in principle. This point of view has its roots in the positivistic philosophy (ef. e.g. [40]) and is in line with a possible nonclassical behaviour.

Sherwood [38] approached the problem from a diﬀerent point of view. Following the concept of distribution-generated spaces as introduced by Schweizer and Sklar ([39]). Sherwood proposed the concept of E-space. In an E-space, the points are functions from a probability space (Ω, A, P) into a metric space (M,d). For each pair, (p,q), of functions in the space, the function d(p,q), deﬁned as

(d(p,q))(ω) = d(p(ω),q(ω)) (39)

for all ω in Ω is a random variable on (Ω, A, P). The function Fpq is then the distribution function of this random variable, i.e. we have

Fpq(x) = P((d(p,q))(ω) < x) (40)

In this way, Fpq can be regarded as the probability that the distance between p and q is less than x. Sherwood showed that every E-space is a Menger space (cf. the subsection (7.2) below).

Another subclass of probabilistic metric spaces are the so-called distribution generated spaces. The main idea is roughly as follows : Let S be a set. With each point p of S associate an n-dimensional distribution function Gp and with each pair, (p,q), a 2n-distribution function Hpq so that it holds :

Hpq(→ u → v = (∞,......,∞)) = Gp(→ u) (41)

Hpq(→ u = (∞,......,∞),→ v) = Gp(→ v) (42)

for any → u = (u1,······,un) and → v = (v1,······,vn) in Rn. Let now Z(x) be a cylinder in Rn with Z(x) :=n(→ u, → v) ∈ R2n;|→ u −→ v |< xo (43)

for any x > 0. This deﬁnes a distance distribution function. Fpq. via

94 The Digital Akasha Fpq(x) =ZZ(x) dHpq =: PHpq(Z(x)) (44)

The technical details can be found in e.g. [20]. Note that, in general, the pairs of S need not be independently distributed.

In one interpretation one may view the elements of S as “particles”. Then for any Borel set A in Rn, the integralRA dGp can be interpreted as the probability that the particle p can be found in the set A and Fpq(x) as the probability that the distance between the particles q and p is less than x. This then yields another type of probabilistic metric space.

A subclass consists of models where the members in S behave independently of each other. Such spaces are called cloud-spaces (C-spaces). A function g from Rn into R+ is an ndimensional probability density if the function G deﬁned on Rn by

G(→ u) =Z((−∞,......,−∞),→ u))

g(→ v)d→ v (45)

is an n-distribution function. If p is a point in a distribution-generated space over Rn such that Gp is absolutely continuous, then the corresponding density gp of Gp may be visualized as a “cloud” in Rn - a cloud whose density at any point of Rn measures the relative likelihood of ﬁnding the particle p in the vicinity of that point.

Another, equally natural interpretation (which is perhaps more in line with our concept of fuzzy lumps; see section (4)) is to visualize the space as an aggregate of clouds or fuzzy points. To make life simple we may assume that to all points, p, is associated a vector, → cp in Rn so that

gp(→ u) = g(→ u −→ cp) (46)

with g specially symmetric. We may then replace the points by clouds in Rn and get a notion of homogeneity. This model is perhaps the most straightforward extension of ordinary (Euclidean) space and resembles our network model of fuzzy lumps (viewed as fuzzy sets) if we chose to embed it into ordinary Rn.

It can be shown that by taking the convolution product of g(→ u−→ cp) and g(→ v −→ cq), one can generate a particular type of random metric space. By applying then a further amount of probabilistic machinery one can create what is called Frechet-Minkowski-metrics, dβ on the original space S, which may e.g. be classical Euclidean space. These metrics are related with and derived from the underlying random metric space (cloud space). The to some extent intricate calculations can be found in [20], chapter 10.

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These Frechet-Minkowski-metrics, dβ, associated with (semihomogeneous) C-spaces over Rn have a remarkable structure. At small distances this metric is non-Euclidean and the distance between two distinct points p,q is bigger than a ﬁxed positive constant, associated with g(→ u). On the other hand, it becomes euclidean in the asymptotic region. It appears from the above picture that if we consider a C-space as a space of clouds (which may move around), the “haziness” of the distance between p and q, which is predominant when the clouds are close together, becomes more and more insigniﬁcant when the clouds are suﬃciently far apart. In this sense Frechet-Minkowski metrics become asymptotically euclidean.

A7. Random Metrics on our Networks of Fuzzy Lumps

In this section we want to relate the various complexes of concepts and techniques, developed or presented above, with each other. That is, we view on the one side the (dynamically) ﬂuctuating cliques as smeared out lumps, viz. as fuzzy sets. On the other side, we want to construct so-called probabilistic or random metric spaces over this space of fuzzy lumps.

Note that this task is however not entirely straightforward as within the mathematical framework of the latter approach the underlying space is usually treated as a more or less structureless space of simple points with the underlying possible causes of the ﬂuctuations in the metrical distance usually not being openly discussed. This has however to be done in the concretely given model systems.

We should ad the remark that there are also diﬀerent scientiﬁc philosophies behind these various points of view (cf. [20] p.17). There exists a tradition to keep the (or a) underlying concrete probability space out of the game and regard is as secondary, while concentrating on the class of (phenomenologically given) distribution functions.

On the other side, a concretely given (and, what is even better, physically motivated) microscopic probabilistic ground space provides (among other things) a common and unifying reference frame. In other words, we have a canonical deterministic metric on the underlying dynamical graph, G(t) or its clique graph, Gcf(t) (cf. (21)). We have argued that our cliques or lumps are ﬂuctuating as a result of the imposed dynamical laws (discussed in section 2). From this input we have to infer the concept of a probabilistic distance between two given labelled (fuzzy) lumps. We divide our section into two subsections. In the ﬁrst subsection we introduce distance concepts on the space of overlapping (static) fuzzy lumps as they are given in the context of fuzzy set theory as discussed in section 4. In this context ﬂuctuations are only implicitly included and the model theory has (one may say) the status of kind of a mean ﬁeld theory. On the other side, the metrics, we introduce in the following subsection are related in spirit to distance concepts introduced by Menger in his interesting essay [40]. More fundamental are in our view the concepts we develop in the other subsection. We construct an explicit probability space, introduce metrical distance as an explicitly given random variable, ﬁnd the corresponding triangular norm and show that the space we get is a model of a so-called E-space, introduced in Section 6. These E-spaces are a subclass of

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Mengers probabilistic metric spaces.

7.1 Metrics on Fuzzy Lumps, the Mean Field Picture

The perhaps most immediate method to impose some metric distance on our space of fuzzy lumps is the following one. In a ﬁrst step we deﬁne a distance concept for the immediate (“inﬁnitesimal”) neighborhood of a given lump and then proceed (for ﬁnite distances) by a chain method, viz. paste inﬁnitesimal distances together. We start with two fuzzy lumps, ∼ C0, ∼ C1, which overlap as fuzzy sets. We hence have (see section 4) :

0 ≤ p(∼ C0, ∼ C1) := 1− sim (∼ C0, ∼ C1) < 1 (47)

We have now several possibilities to proceed. We can e.g. ﬁx a certain scale resolution and state that two fuzzy lumps are indistinguishable if

p(∼ C0, ∼ C1) ≤ ε (48)

This is in accord with already classical ideas of Poincar´e, developed in [41], p.31f. Put diﬀerently, in the physical world we may have a non-transitive inﬁnitesimal distance concept or, more generally, a non-transitive concept of identity so that

A = B , B = C , A 6= C (49)

d(A,B) = 0 , d(B,C) = 0 , d(A,C) 6= C (50) Deﬁnition 7.1 (Distance, Variant 1) Denoting by γ a path connecting ∼ Ci and ∼ Cj, which consists of a chain s.t.

∼ Ci =: ∼ Ck0 , ∼ Cj =: ∼ Ckn (51)

and

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p(∼ Ckj, ∼ Ckj+1) ≤ ε (52)

we deﬁne

distε(∼ Ci, ∼ Cj) := inf γ

n (53)

(distε := ∞ if there is no such chain).

It can be easily shown that this deﬁnition fulﬁls the axioms of a metric. Note however that the validity of the triangle inequality is enforced by attributing the distance ‘one’ to two “indistinguishable” lumps. This is a little bit unpleasant but inescapable if we want to base our macroscopic distance concept on the concatenation of “inﬁnitesimal” distances. On the other side, this kind of distance is probabilistic in a very restricted sense and yields only distances which are multiples of a ﬁxed given distance element.

We can improve the situation by taking as inﬁnitesimal distance elements the above measures of overlap, p(∼ CKj, ∼ Ckj+1), and deﬁne

Deﬁnition 7.2 (Distance, Variant 2)

d(∼ Ci, ∼ Cj := inf γ Xp(∼ Ckj, ∼ Ckj+1) (54) Note that in contrast to the preceding deﬁnition now the inﬁmum need not be taken at a path of minimal canonical length. Quite the contrary, in exceptional situations the canonical length of a path leading to a very short distance may be quite large, whereas this is not expected to be the ordinary situation. Anyway, the second distance concept is a little bit more probabilistic.

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7.2 The Space of Cliques as a Probability Space and the Connection with Random Metric Spaces

We repeat brieﬂy what we have said in preceding sections. We have assumed that we have a network of a more or less ﬁxed number of, however, ﬂuctuating cliques. That is, their degree of overlap may vary in the course of time. At each discrete time step during a suﬃciently long time interval, i, (so that we can exploit probabilistic concepts in, at least, a certain approximation) we have a deﬁnite array of overlapping cliques, Gcl(t). As ground set for our probability space we take this set, i.e.

X := {Gcl(t),t ∈ I,[I] = N,N some suﬃciently large number} (55)

The simplest assumption is to assume that all conﬁgurations have the same probability, P(Gcl(t)) := 1/N, say

Remark 7.3 On physical grounds a more sophisticated probability density may suggest itself. Our results, derived below, are however independent of the particular choice of P.

Observation 7.4 The above deﬁnes a discrete probability space, X, on which random variables can be introduced, their expectation value being denoted by <> .

Example 7.5 We previously introduced the membership function, ∼ Ci(x), of a fuzzy lump, ∼ Ci. Deﬁning the elementary random variable (x a node of the primordial underlying graph) xCi(Gcl(t)) =1 x ∈ Ci(t) 0 else (56)

we have

∼ Ci(x) = hxCii =: x(∼ Ci) (57)

(cf. the section about fuzzy lumps).

In the same sense we can deﬁne the distance between two ﬁxed lumps, Ci,Cj, as a random variable on our probability space, taking the values d(Ci,Cj)(Gcl(t)). In contrast to our ordinary fuzzy lumps, all these random variables do ﬂuctuate, the ﬂuctuations given by

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h(xC−i −hxCii)2i or h(d−hdi)2i (58)

For simplicity reasons we write from now on the metric as d(p,q),p,q denoting some cliques, and with the understanding that it represents the above random variable.

If we want to relate our approach to the probabilistic metric framework laid out in e.g. [20], we have to inspect the transitivity properties of our distance concept.

Observation 7.6 With p,q,r three points (cliques), we have for the respective random variables

dp,r ≤ dp,q + dq,r (59)

as they are evaluated on a ﬁxed network, Gcl(t), where the ordinary triangle inequality of the canonical graph distance holds. Hence

hdp,ri≤hdp,qi+hdq,ri (60)

On the other side, probabilistic metric spaces are deﬁned via the probability distributions, Fpq(x), denoting the probability that the distance between p,q is smaller than x. The ordinary triangle inequality is replaced in this framework by a more complicated estimate (cf. section 6) :

Fpr(x + y) ≥ T(Fpq,Fqr) (61)

with T a triangular norm.

Our task consists now in determining this function T within our own stochastic framework. To this end we employ the following formula which holds for probability measures (i.e. P(X) = 1).

Lemma 7.7 With A,B measurable sets in X it holds

P(A∩B) ≥ P(A) + P(B)−1 (62)

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Proof : This result has also been employed in[20], p.26. For completeness sake we repeat it here. We have

P(A∩B) = P(A) + P(B)−P(A∪B) ≥ P(A) + P(B)−P(X) = P(A) + P(B)−1 (63)

and therefore

P(A∩B) ≥ max(P(A) + P(B)−1,0) (64) This proves the lemma.

We have by deﬁnition :

Fpr(x + y) = P(dpr < x + y) (65)

which, for the simplest choice of P, is just the fraction of conﬁgurations, Gcl(t), with the canonical graph metric fulﬁlling this inequality. As the triangle inequality holds for the canonical graph metric we have

P(dpq < x,dqr < y) ≤ P(dpr < x + y) (66)

and hence, with the help of the above lemma :

Fpr(x + y) = P(dpr < x + y) ≥ max(Fpq(x) + Fqr(y)−1,0) (67)

Observation 7.8 We proved that our random metric, introduced above, fulﬁlls a triangular inequality

Fpr(x + y) ≥ W(Fpq(x),Fqr(y)) (68)

with W the triangular norm

W(a,b) := max(a + b−1,0) (69)

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Hence our above probability space is a Menger space.

(That W is in fact a triangular norm is easy to show and is also employed in e.g. [20]).

A special class of probabilistic metric spaces are the so-called E-spaces (cf. the section 6 and [20], chapt. 9). To complete our analysis we can reframe our model so that it becomes such an E-space. The ground space, X, is our probability space. The metric target space, M, is a new space-time supergraph, deﬁned as follows :

Deﬁnition 7.9 (Space-Time Picture of Clique Graph) M :=[ t∈I Gcl(t) (70) with the understanding that exactly the nodes Ci(t) and Ci(t + 1),t,t,+1 ∈ I, are connected by additional bonds. That is, Ci(t) describes the orbit of the ﬁxed but ﬂuctuating clique, Ci, in the course of time.

We can now relate bijectively the cliques, Ci, and certain functions from X to M.

Observation 7.10 With the class of functions, Ci,

Ci(Gcl(t)) := Ci(t) ∈ M (71)

and their distance

dpq(Gcl(t)) := d(Cp(t),Cq(t)) (72)

the above probability space can be considered as an E-space.

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