Simple indicators for Lorentzian causets
SSimple indicators for Lorentzian causets
Tommaso Bolognesi and Alexander Lamb CNR/ISTI, Pisa - [email protected] Santa Cruz, CA - [email protected]
Abstract
Several classes of directed acyclic graphs have been investigated in the last two decades,in the context of the Causal Set Program, in search for good discrete models of spacetime.We introduce some statistical indicators that can be used for comparing these graphsand for assessing their closeness to the ideal Lorentzian causal sets (’causets’) – thoseobtained by sprinkling points in a Lorentzian manifold. In particular, with the reversedtriangular inequality of Special Relativity in mind, we introduce ’longest/shortest pathplots’, an easily implemented tool to visually detect the extent to which a generic causetmatches the wide range of path lengths between events of Lorentzian causets. This tool canattribute some degree of ’Lorentzianity’ - in particular ’non-locality’ - also to causets thatare not (directly) embeddable and that, due to some regularity in their structure, wouldnot pass the key test for Lorentz invariance: the absence of preferred reference frames. Wecompare the discussed indicators and use them for assessing causets both of stochastic andof deterministic, algorithmic origin, finding examples of the latter that behave optimallyw.r.t. our longest/shortest path plots.
A rather direct way to obtain a discrete model of spacetime from a continuous one - i.e. froma Lorentzian manifold that satisfies the Einstein field equations - consists in applying to thelatter the sprinkling technique and building a causal set .A causal set , or causet [13, 30, 26], is a partially ordered set (’poset’) of atomic spacetimeevents in which the partial order relation ’ (cid:22) ’ expresses causal dependencies between events.Beside being reflexive, antisymmetric and transitive, relation ’ (cid:22) ’ is required to be locally finite ,or finitary , that is, for any pair ( s, t ) of events, the set of points between them must be finite: |{ x | s (cid:22) x (cid:22) t }| < ∞ .A causet can be represented by a DAG (Directed Acyclic Graph), whose edges reflect thepartial order relation. Any such DAG is transitively closed, by definition, but we shall alsoconsider transitive reductions of these graphs (their ’Hasse’ diagram), from which the originalcauset is recovered by transitive closure. The edges of a causet - of the transitively closed DAGthat represents it - are often called ’relations’, while those of its transitive reduction are called’links’, and identify the nearest neighbours of each event/node.In light of our interest here for discrete spacetime modeling, in the sequel we shall sometimesabuse the term ’causet’ and sloppily use it even when the generic term ’DAG’ would be more1 a r X i v : . [ g r- q c ] J u l ppropriate. For example, we shall consider procedures that build ”causets” which are neitherclosed nor reduced - these are called ’raw’ causets. The statistical indicators that we discussin the paper, however, shall always refer either to the transitive closure or to the transitivereduction of these graphs.By the sprinkling technique [13, 30, 26] one can derive a causet from a Lorentzian manifold M provided with a volume measure, in two steps. First one considers a uniform, Poissondistribution of points - to become the causet nodes - in a finite region of M , with density δ ,so that the expected number of points in a portion of volume V is δV and the probability tofind exactly n points in that portion is: P ( n ) = ( δV ) n e − δV n ! , (1)Then the causet edges are created by letting the sprinkled points inherit the causal (lightcone)structure of M . In the sequel we shall conveniently call these objects sprinkled causets .A challenging goal of causet-based quantum gravity research is to reverse the above logic,and to try and build causets of physical significance without resorting to an underlying con-tinuum. Several techniques for doing this have been investigated in the last two decades, andsome of them will be described in Section 2. Under this perspective, the manifold is onlyobtained a posteriori, as an asymptotic approximation.One way to assess a causet C obtained by some of these techniques is to check whether C is faithfully embeddable in some manifold M of appropriate dimension, that is, whether itcould have arisen with high likelihood from sprinkling events in M . In [19] Henson introducesa method for building an actual embedding of a given causet in 2D Minkowski space, butrecognizes the need of additional, complementary, possibly more efficient criteria for definingscales of causet ’manifoldlikeness’. Some of these efforts are mentioned in the conclusive section.In the approach described here, we are interested in extracting some distinguishing featureof sprinkled causets other than , or weaker than faithful embeddability . One simple reasonis that embeddability is hard to test; furthermore, it might fail on the original causet whilesucceeding with a coarse-grained version of it. Hence we shall try to identify statistical features,in the discrete setting, that could reflect the peculiarities of the Lorentz pseudo-metrics in thecontinuum, which is responsible for the reversed triangular inequality and the twin paradox ofSpecial Relativity. We shall then use these indicators as a benchmark for the graphs obtainedby other techniques. While it is today sufficiently understood that ’ a fundamental spacetimediscreteness need not contradict Lorentz invariance ’ [16], the toolkit for assessing causets interms of Lorentzianity is still quite poor.In Section 2 we briefly recall four stochastic causet construction techniques, including sprin-kling in flat (Minkowski) and positively curved (de Sitter) manifolds.In Section 3 we concentrate on statistical properties based on counting relations . Keepingin mind that sprinkled causets are transitively closed by construction, the counts shall beconcerned with the edges of the transitively closed versions of the graphs obtained by theother techniques. First we analyse the distributions of node degrees - the number of emanatingrelations -, finding that power-law distributions are quite common. Then we collect otherstatistical information relevant to causet dimensionality estimation, and visualise it in whatwe call ordering fraction spectra . 2n Section 4 we focus on the transitive reduction of the considered graphs, thus we count links . This turns out to be a more discriminative criterion for addressing Lorentzianity issues.We first analyze the (link-)degree of the root node of an interval as a function of the numberof nodes in the interval. Then we introduce a special type of diagram - longest/shortest pathplots - meant to expose the peculiar wide range of path lengths between two generic nodes, afeature of Lorentzian sprinkled graphs sometimes referred to as ’non-locality’. We illustrate,again, how the various causet classes perform with respect to these indicators.In Section 5 we introduce deterministic causet construction techniques based on permuta-tions of tuples of natural numbers and on their manipulation by stateless or stateful controlunits; we show that one instance of this model (out of 65,536) yields a pseudo-random causetwith excellent non-locality properties comparable to those of sprinkled causets.In Section 6 we summarize our results and mention some related work.Let us stress again that the proposed indicators reflect a concept of ’Lorentzianity’ weakerthan faithful embeddability in a Lorentzian manifold; they do not provide necessary and suf-ficient conditions for this strongest form of Lorentzianity. The idea is to investigate otherproperties of sprinkled Lorentzian causets (embeddable by definition), and select some thatcan be regarded as sharply characterising these causets of reference. In this section we introduce the four stochastic causet construction techniques to be analyzedin the rest of the paper. In addition, and for the sake of comparison, we shall occasionallyconsider regular lattices. In the sequel we use the concept of order interval , often abbreviatedto ’interval’: if s and t are two points of a discrete or continuous set with partial order (cid:22) , thenthe order interval between them, denoted I [ s, t ], is the set { x | s (cid:22) x (cid:22) t } , which includes s and t , by the reflexivity of the order relation. A Minkowski sprinkled causet is obtained by applying the already mentioned sprinkling tech-nique [13, 30, 26] to a portion of d -dimensional Minkowski space. For example, if M (1 , is3D Minkowski space with time dimension t and space dimensions x and y , L is the squaredLorentz distance L ( p ( t p , x p , y p ) , q ( t q , x q , y q ) = +( t p − t q ) − ( x p − x q ) − ( y p − y q ) , and C is a bounded, connected subset of it, e.g. a unit cube, then a set S of points Poisson-distributed in C yields a sprinkled causet graph G ( S, E ), where the set of graph nodes is S itself and the edges E are the ordered pairs of nodes ( p, q ) such that q is in p ’s future lightcone: E = { ( p, q ) ∈ S ) | L ( p, q ) ≥ ∧ t p < t q } . Once the graph is constructed, node coordinatesbecome irrelevant. In general G ( S, E ) may have several sources - nodes whose in-degree is zero- and sinks - nodes whose out-degree is zero; however, given two nodes s and t of it, the orderinterval I [ s, t ] has, by definition, only one source ( s ) and one sink ( t ). De Sitter spacetime is an exact solution of the Einstein field equations of General Relativity,conjectured to model the universe both at the Plank era (at time t < − sec.) and in the far3uture, at thermodynamic equilibrium. De Sitter spacetime has constant positive curvature,and describes the universe as an empty and flat space - one with zero curvature - which growsexponentially under the effect of a positive cosmological constant.Sprinkling in de Sitter spacetime, up to four dimensions, is discussed in [20], where variousproperties of the corresponding causets are studied both analytically and by simulation, andcompared with properties of various other types of network.Here we restrict to two-dimensional de Sitter spacetime DS (1 , , for which a well knownrepresentation as a one-sheeted, 2D hyperboloid embedded in flat 3D Minkowski space M (1 , is available, and, following in part [20], we present the simple analytical steps behind theimplementation of 2D de Sitter sprinkling we have used for our experiments.Let t , x , y be the axes of M (1 , , where t is the vertical time axis around which a hyperbolarotates to create the hyperboloid, and let τ and θ be the time and space coordinates in DS (1 , :in the embedding, τ flows vertically while angular coordinate θ spans space - a circle of constantcoordinate t (and τ ). Given the hyperboloid equation x + y − t = 1, the differential elementof Lorentzian length ds = dt − dx − dy , and its vertical projection dτ = dt − dx , one canreadily derive, by integration of the latter along a vertical hyperbolic segment, the functionaldependences τ ( t ) = ArcSinh ( t ) and t ( τ ) = Sinh ( τ ). Similarly, the radius of the space circle attime τ is r ( τ ) = Cosh ( τ ), hence the size of de Sitter space is 2 πCosh ( τ ) = π ( e τ + e − τ ) . Thus,space grows exponentially with time. (For a visualization of this result, see the interactivedemo [11].) It is finally easy to see that the differential element of Lorentzian length in DS (1 , is ds = dτ − Cosh ( τ ) dθ . For implementing de Sitter sprinkling we need to know how to distribute points on thesurface of the hyperboloid, and how to build causal edges. If δ is the desired uniform density,the expected number of points on the circular hyperboloid section S between τ and τ + dτ is δArea ( S ) = δ πCosh ( τ ) dτ . Thus, if we uniformly sprinkle in a section of the hyperboloid with0 ≤ τ ≤ τ max , the time coordinate of these points is a random variable τ whose (normalized)density is: f τ ( x ) = Cosh ( x ) / Sinh ( τ max ) , (2)(as already established in [20], equation (3)), where we use the fact that (cid:82) τ max Cosh ( x ) dx = Sinh ( τ max ).The Minkowski coordinate t of these points is itself a random variable t . Keeping in mindthe above functions t ( τ ) and τ ( t ), and using a fundamental theorem on functions of randomvariables, we find that the density f t of t is constant: f t ( x ) = f τ ( τ ( x ))) t (cid:48) ( τ ( x )) = Csch ( τ max ) , (3)where ’ Csch ’ is the hyperbolic cosecant. Then, in light of the circular symmetry of the distri-bution, implementing a Poisson sprinkling in DS (1 , , with τ ranging in [0 , τ max ], is straightfor-ward. We create points with coordinates ( r, θ, t ), where polar coordinates ( r, θ ) replace ( x, y ),such that: • t is distributed uniformly in [0 , t max ], where t max = Sinh ( τ max ), • r = √ t + 1, 4 θ is distributed uniformly in [0 , π ).Once the points are uniformly distributed in DS (1 , as described, causal edges can beestablished among them by referring directly to their Lorentz distance in M (1 , : although thesquared Lorentz distance changes when moving from p to q on a geodesic in DS (1 , or on one in M (1 , , the signs of these two distances always agree, thus yielding the same causal structure. Transitive percolation dynamics has been widely studied in the context of the Causal SetProgramme [26, 1, 24, 27]. An N -node percolation causet is built by progressively numberingnodes 1, 2, ..., N , and by creating an edge i → j , for i < j , with fixed probability p (typically p (cid:28) originary percolation , which guarantees the existence of a single sourcenode, or root , is studied in [1], where it is also observed that any subset S of a standard,transitive percolation causet C , consisting of a node x and all nodes in its future, is an instanceof an originary percolation causet. This technique is described in [20], where it is also shown that the growth dynamics is asymp-totically identical to that of sprinkled causets from de Sitter space. Each node is assigneda progressive natural number n , as in percolation dynamics, and is placed in 2D Euclideanspace, with polar coordinates ( r, θ ), where r is a monotonic, increasing function of n (whoseprecise nature is not essential for building the causet) and θ is a random angle in [0 , π ). Whennew node n is created, a fixed number m of edges reaching n from previously created nodesis added: s i → n , with s i ∈ { , , . . . ( n − } and i = 1, 2, ..., m . Typically, m = 2, and thisis the value we use throughout the paper, unless otherwise stated. Nodes s i are those thatminimize the product s i ∆ θ i , where ∆ θ i is the angular distance of node s i from node n .The name ’popularity/similarity’, abbreviated as ’pop/sim’ in the sequel, reflects the trade-off between popularity and similarity that determines the structure and drives the growth ofvarious complex networks, including the Internet [23]. Node labels (birth dates) represent’popularity’, while angular distances express ’similarity’: when the from-nodes s i are chosenfor being connected to new node n , older nodes - with smaller labels - are preferred, and inthe long run they become more and more ’popular’, thus increasing their out-degrees; at thesame time, the chosen nodes must be as ’similar’ as possible (small angular distance) to thetarget node n . For the sake of comparison we shall also consider directed regular grids, in particular squaregrids whose edges are oriented parallel to the cartesian axes of their embedding euclidean 2Dspace. These graphs manifest maximum locality, as opposed to the non-locality typical ofsprinkled causets from a Lorentzian manifold.Randomized versions of maximally local graphs can be obtained by sprinkling points inEuclidean space of some dimension, where one of the dimensions is time, and by creating an5dge from node p to node q whenever q has higher time coordinate than p and the Euclideandistance between p and q is smaller than a given threshold. In this section we try to characterize causets by collecting statistical information based onedge counts for transitively closed graphs. We focus on two types of statistical indicator: nodedegree distributions and ordering fraction spectra . These are applied to our reference causets- sprinkled Minkwski and de Sitter - and to the other causet classes: since the former aretransitively closed by construction, we shall take transitive closures of the latter too, beforecounting edges.
Minkowski space M (1 , represents a spacetime that extends to infinity in both dimensions; afinite causet is obtained by sprinkling points in a bounded region of it. However, dependingon the observed variables, the statistics of such causets may be affected by the shape of theregion border. One way to avoid this problem is to analyze causet intervals , as defined above,although we shall not ignore other types of causets whose analysis we find useful, for reasonsto be explained as these are introduced. Intervals: sprinkled Minkowski, square grid, sprinkled de Sitter
Thus, let us start by investigating the distributions of node degrees for causet intervals , wherethe degree of a node is the number of edges emanating from it (out-degree). We shall representthese distributions by histograms, built by segmenting the range of possible degrees into binsof fixed or variable width.The four plots in Figure 1 show histograms for the node degrees of four types of intervalcauset. The histograms contain 50 points each; they were obtained by partitioning the rangeof possible degrees into 50 slots of equal width and by counting the nodes whose degrees fall ineach slot. For the three cases corresponding to sprinkling in Lorentzian manifolds (Minkowskiand de Sitter), each histogram has been obtained by averaging over 30 causets. For each slotthe plot provides three points, that indicate the averaged value of the node count and thestandard deviation.In the upper-left plot, referring to thirty 8000-node sprinkled 2D Minkowski intervals,the continuous line represents the theoretical density − Log ( z ), appropriately scaled. Thisnegative logarithm density can be derived by assimilating the node degree to a random variable(r.v.) r = xy , where r.v.’s x and y have uniform densities in the unit interval: f x ( z ) = 1, f y ( z ) = 1, z ∈ [0 , , × [0 ,
1] in Euclidean 2D space, and modelthe degree of the generic sprinkled point p (1 − x, − y ) as the area xy of the rectangle ofdimensions x and y . The distribution function for r is: F r ( z ) = P rob [ xy ≤ z ] = z (1 − Log ( z )),yielding, by derivation, the density function f r ( z ) = − Log ( z ).The analysis above is essentially valid also for the 8004-node regular square grid - an intervaltoo - whose node degree histogram is shown in the upper-right plot. (The grid has integer node6igure 1: Node degree histograms for transitively closed interval causets. Each histogram refers to 50 slots ofequal width. For the three sprinkled cases the histogram was obtained by averaging over thirty interval causets;for each slot, standard deviations from the mean node count are also shown. To reduce graphical cluttering thevertical bar representation of the bins is avoided. Solid lines represent fitting negative logarithm functions. coordinates, source s (1 , t (92 , τ ranging, respectively, in [0 ,
1] and [0 , τ max the de Sitter and Minkowski intervals are quite similar, as reflected in their node degreedistributions. As the width of the time window increases, the effect of curvature becomes moresensible and the plot departs from the logarithmic behaviour.A first conclusion can be already drawn by a qualitative analysis of the plots in Figure 1:node degree distributions of transitively closed graph intervals fail to characterise Lorentzian-ity, since they do not distinguish between a sprinkled Minkowski or de Sitter interval – theLorentzian causets of reference – and a regular grid, which is the typical example of a non -Lorenzian causet. Full circular causets: sprinkled Minkowski, square grid, sprinkled de Sitter
In spite of the poor performance of node degree distributions as Lorentzianity indicators,relative to causet intervals , we are interested in investigating a bit further their application to full causets. This step is mainly suggested by the observation that the infinite hyperboloidrepresentation of 2D de Sitter spacetime in 3D Minkowski spacetime (subsection 2.2) lendsitself to an additional, natural ’cutting’ operation for obtaining finite portions of the manifold,beside the extraction of diamond-shaped intervals, namely the selection of the full slice of7he curved surface between two parallel, horizontal planes in the embedding 3D Minkowskispacetime, with time parameter τ ranging between, say, 0 and τ max . Unlike in an interval, thisfull slice spans (finite) space completely. (In Minkowski spacetime we need to explicitly boundboth time and space, for getting a finite region.) These precise full slices of de Sitter spacetimehave been considered in [20], where degree distributions are found to follow a power law. Weare interested in comparing those results with cylindrical counterparts of sprinkled Minkowskiand square grid causets, for further assessment of the indicator.Thus, we have considered two full slices of de Sitter spacetime, with time bounds [0 ,
1] and[0 , LogLog plots confirm the power-law character of these distributions.Figure 2:
Node degree histograms for transitively closed causets with rotational symmetry. Similar to Figure1, each histogram refers to 50 slots of equal width. For the three sprinkled cases the histogram was obtainedby averaging over ten interval causets; standard deviations from the mean node count are also shown.
LogLog plots are used here, for highlighting the power law character of the distributions.
Inspired by the rotational symmetry of the two de Sitter full slices, and for the sake ofcomparison, in the upper row of Figure 2 we have considered the cylindrical counterparts ofthe interval causets covered in the upper row of Figure 1 - sprinkled Minkowski and squaregrid.The upper-left plot of Figure 2 shows the node degree histogram for ten causets obtained byuniformly sprinkling points in a rectangular portion of M (1 , that wraps completely around All four causet types presented in Figure 2 can be though of as embedded in a 3D space: the term ’rotationalsymmetry’ refers to this external viewpoint. Under an internal viewpoint, the symmetry is translational. The out-degree of a node in theMinkowski cylinder causet is equated to the area of the future lightcone of that node, whosevalue ranges from 0 to 1. The distance h of the generic node p from the upper border of thecylinder is a r.v. with uniform density: f h ( x ) = 1, for x ∈ [0 , p ’sfuture lighcone is r.v. A = h . By applying well known results on functions of r.v.’s, we obtain f A ( y ) = y − / , y ∈ [0 ,
1] for the density of r.v. A . Thus the density is a power-law, and the LogLog plot in Figure 2 (upper-left) is correspondingly linear. In the histogram, node degreesare partitioned into 50 slots of equal width.The upper-right plot of Figure 2 refers to a square grid arranged around a cylinder (’cylin-drical grid’), with edges at +45 and -45 from the cylinder axis - a regular counterpart of thesprinkled Minkowski cylinder. The analysis of the node degree density is somewhat analogousto that of the Minkowski cylinder. Similar to the previous case, the histogram is obtained bypartitioning the degrees into 50 equal slots, and reveals a power-law distribution. The verticalquantisation is due to the regularity of the graph, which is formed by 2 n − n − n = 159. Note that nodesare aligned vertically only at alternate rows. All 317 nodes of each row have the same degree,hence the populations of degrees in each of the 50 slots of the histogram necessarily differ bymultiples of 317. Full pop/sim causet
The histogram in Figure 3 refers to ten (transitively closed) 8k-node pop/sim causets, withfixed node in-degree m = 2. For improved smoothness we have used geometrically increasingbins, thus the number of nodes for each slot has been divided by the bin width, providing adegree density . Again, a LogLog plot was chosen for highlighting that a power law is in action.Note that the pop/sim causet, described in terms of polar coordinates ( r, θ ), could alsobe regarded as rotationally symmetric; one difference with the previous causets of this type,however, is that the r ’coordinate’ plays here a somewhat abstract role, being used purely forordering nodes as they are created, not for defining a particular metric of the manifold wheresprinkling takes place.The fact that the plot in Figure 3 reflects a power law, like the plots in Figure 2, isnot surprising since the growth dynamics of the pop/sim procedure, relative to node degreedistribution, is shown in [20] to be asymptotically identical to that of de Sitter sprinkledcausets.The results just illustrated for full, circular causets neatly confirm the conclusion we haveanticipated about node degree distributions as potential Lorentzianity indicators: these distri-butions do not discriminate between sprinkled causets - our reference - and grids, neither whenlooking at intervals (Figure 1) nor when considering full circular instances (Figures 2 and 3).Having ruled out this indicator, we would not need to further analyse the only case left- percolation causets. Yet, the analysis of their degree distributions may be of some interest This is essentially analogous to sprinkling in a finite, rectangular, spacelike strip of M (1 , , with theadvantage of avoiding the (slight) boundary effects on the statistics induced by the two vertical cuts. Node degree histogram for ten, 8k-node transitively closed pop/sim causets. Bins of geometricallyincreasing width have been used, and node degree values have been normalised accordingly, providing a densitythat may be smaller than 1. Standard deviations from the mean are shown. Lower standard deviations for someof the rightmost points fall below zero, thus escaping representation on the
Log scale. in itself, due to the peculiar role played by the edge probability parameter, and is presentedin Appendix A. Interestingly, a power law distribution essentially emerges also in that case,providing further evidence of the ’flattening’ effect of transitive closure, an operation whichseems to obscure important structural differences among ’raw’ causets of different kinds. Counting the relations of causet intervals is also at the basis of the Myrheim-Meyer dimensionestimator. Let I Dk [ s, t ] be a k -node interval with source s and sink t , obtained by sprinkling in D -dimensional Minkowski space. As k grows, the expected number R ( D, k ) of edges in I Dk [ s, t ]quickly approaches (from above) the value [22, 30]: R ( D, k ) = f ( D ) (cid:32) k (cid:33) where f ( D ) = 32 (cid:0) D/ D (cid:1) . In general, the ordering fraction of a k -node causet (interval) is defined as the ratio R/ (cid:0) k (cid:1) between the number R of edges in it and the maximum number (cid:0) k (cid:1) of directed edges thatcould connect so many nodes. Thus, the expected ordering fraction of a sprinkled , k -node, D -dimensional Minkowski interval must quickly approach f ( D ), as k grows. We can then Indeed, the analysis and comparisons in [20] deal also with the actual power law exponent − γ . For the deSitter universe, γ is found to evolve on a cosmological time scale from value 3/4 to an asymptotic value 2. Thelatter value is found to characterise also several networks of different origin. D of a generic causet by counting itsnodes ( k ) and edges ( R ) and numerically inverting function f ( D ): D = f − (cid:32) R (cid:0) k (cid:1) (cid:33) The upper-left plot of Figure 4 was obtained by creating 100 intervals of random volume(number of points) lower than 1000 in 2D-, 3D- and 4D-Minkowski space, and by plotting, foreach interval I , the point ( Volume ( I ), OrderingFraction ( I )).Figure 4: Ordering fraction spectra. Upper row: sprinkled causets. Lower row: grid, percolation and pop/simcausets. As k grows, the points nicely align to the expected values - the horizontal gridlines - whichmark the values of f ( D ) for the indicated dimensional values. We call these diagrams orderingfraction spectra . Let us stress that this indicator focuses on intervals extracted from graphsof arbitrary kind, and is therefore unaffected by the overall shape or ’border’ (if any) of thelatter.The upper-right plot of Figure 4 was obtained by creating 50 intervals of random volume –again less than 1000 points – in 2D de Sitter spacetime; for the lower set of points in the plot,that matches well the 2D ordering fraction value 0.5, time variable τ ranges in [0, 1], whilefor the upper points the range is [0, 5]. Note that both point sets are derived from sprinklingin (the hyperboloid model of) 2-dimensional de Sitter spacetime. This plot thus indicatesthat, while the Myrheim-Meyer dimension estimator operates correctly for flat, Minkowskispacetime, it becomes unreliable when sprinkling in curved manifolds, becoming sensitive tothe interval time span. This circumstance was already observed in [21]. In any case, based onthe above definition of ordering fraction, the flat nature of the plots for sprinkled Minkowskiand de Sitter causets implies that in these cases the growth rate of the edge count function R ( k ) is O ( k ). 11n the lower part of Figure 4 we show three more ordering fraction spectra. The firstdiagram plots the ( Volume ( I ), OrderingFraction ( I )) pairs for all intervals I of a 32 × n × m directed square grid interval is: (cid:80) mx =1 (cid:80) ny =1 xy − mn (cid:0) mn (cid:1) . Similar to the sprinkling case, this plot reveals that the Myrheim-Meyer dimension estima-tor is valid asymptotically.The second plot shows the typical shape of the ordering fraction spectra of causets obtainedfrom percolation dynamics using constant edge probability. The plot was obtained by randomlysampling 500 intervals from the transitive closure of a 1000-node causet with edge probability0.05. On the large scale, percolation causets tend to be one-dimensional.The poor ordering fraction spectrum in the third plot was obtained by sampling 500 inter-vals from a 5000-node pop/sim causet. The repertoire of possible (
Volume ( I ), OrderingFrac-tion ( I )) pairs is very limited here, with volumes below 6 nodes, and the single value 1 for theordering fraction. The preferential attachment policy of the pop/sim algorithm – new nodesprefer to connect with the popular nodes with lowest birth times – yields causets with veryshort chains, thus small intervals, that are totally ordered (a necessary and sufficient conditionfor the ordering fraction to be 1) or almost totally ordered. An exhaustive search of all theintervals from this specific 5000-node causet reveals that the lowest possible ordering fractionvalue is indeed 21/22, achieved by only three intervals of volume 12.In light of the fact, claimed in [20], that pop/sim and sprinkled deSitter causets sharethe same asymptotic dynamics, we would be interested in comparing their ordering fractionspectra, shown, respectively, in the upper-right and lower-right plot of Figure 4. Unfortunately,the limited interval volumes that we can attain for pop/sim causets makes the comparisonunfeasible. However, much lower ordering fraction values are achieved for whole pop/simcausets, rather than for their intervals. In particular, we experimentally find that the orderingfraction of a k -node pop/sim causet is not constant: it decreases with k , and is an O ( k − )function. Since the ordering fraction is R ( k ) / (cid:0) k (cid:1) and (cid:0) k (cid:1) is O ( k ), this result indicates that R ( k ) – the number of edges in the transitively closed causet – is O ( k ), like the number RR ( k )of edges in the original, ’raw’ causet. This linear growth is illustrated in Figure 5-left. (Notethat the linearity of RR ( k ) is a direct consequence of adding a fixed number of edges with eachnew node.)On the other hand, we also find experimentally that the growth rate of the edge countfor whole k -node de Sitter sections is O ( k ) (Figure 5-right), like that indirectly revealed byFigure 4 above for k -node de Sitter intervals . Thus, in spite of the asymptotic similarity offull de Sitter causets and pop/sim causets relative to node degree distributions, these causetclasses differ in the growth rates of their edges, which we found to be, respectively, O ( k ) and O ( k ).The conclusion we can draw from the inspection of the above plots is that ordering fractionspectra, while useful for possibly detecting causet dimensionality (limited to sprinkled causetsfrom flat spacetime, according to [21]), are not relevant for revealing causet Lorentzianity, since,again, they do not separate sprinked causets (Lorentzian) from a regular grid (non-Lorentzian).12igure 5: Growths of the number of relations (edges) in pop/sim and de Sitter causets as a function of causetsize, ranging from 100 to 2000 nodes, in steps of 100. For each size we compute the number of relations for 10causets, and plot the mean and standard deviation. These data are matched against their linear and quadraticfitting functions.
In this section we try to characterise causets by collecting statistical information on edgecounts and paths for their transitively reduced forms. ’Links’ is the name commonly adoptedfor indicating the essential edges left in the causet after transitive reduction. (We shall onlydeal with finite causets, thus finite partial order relations. Recall that the transitive reductionof a finite relation R is the smallest relation that admits the same transitive closure of R ; when R is acyclic, its transitive reduction is unique.)A first simple indicator is suggested by this quote from D. Rideout [25]:” The ’usual’ discrete structures which we encounter, e.g. as discrete approxi-mations to spatial geometry, have a ’mean valence’ of order 1. e.g. each ‘node’ ofa Cartesian lattice in three dimensions has six nearest neighbors. Random spatiallattices, such as a Voronoi complex, will similarly have valences of order 1 [...].Such discrete structures cannot hope to capture the noncompact Lorentz symme-try of spacetime. Causal sets, however, have a ‘mean valence’ which grows withsome finite power of the number of elements in the causet set. It is this ’hyper-connectivity’ that allows them to maintain Lorentz invariance in the presence ofdiscreteness. ”It should perhaps be clarified that we cannot meaningfully apply the concept of Lorentzinvariance directly to a causet C , simply because there are no coordinates to be Lorentz-transformed. But when C comes already embedded in some manifold M , with its coordinatesystem, we may keep the information on node coordinates, apply the transformation to M ,and see its effect on C , which is now dragged and reshaped by the operation. When C isobtained by a Poisson sprinkling in M it can be shown that in the equivalence class of allLorentz-transformed embedded versions of C there is no preferred element that we can pickout, while if we embed, for example, a regular grid, this is not the case. In the above The sprinkled causet vs. grid (or ’diamond lattice’) example is often mentioned in the literature, e.g.in [30] and [20]. Of course, a typical, random-looking sprinkled causet gets Lorentz-transformed into another C , without worrying about its embeddability. Asa first property, we look at the growth rate of node degrees in transitively reduced sprinkledcausets, keeping in mind that the first concern will be to detect an actual growth of nodedegrees with graph size, i.e. to exclude the above mentioned O (1) growth. In [14] Bombelli et al. mention that, considering the causet C [ s, t ] obtained from uniformlysprinkling points in an order interval I [ s, t ] of height T of d-dimensional Minkowski space ( T being the Lorentz distance between s and t ), the number of nearest neighbors of root s in theinterval – the number of outgoing links – grows like Log ( T ) for d = 2, and like T d − for d ≥ k of nodes sprinkled in it (thus, the density),in which case the root degree deg ( k ) has growth O ( Log ( k / )) = O ( Log ( k )) for d = 2, and O ( k (( d − /d )) for d ≥ In this subsection we carry out explicitly the analysis of cases d = 2 and d = 3 (whichis not given in [14]), obtain accurate asymptotic expressions for deg ( k ), not just their growthrate, and verify their agreement with data from simulations.
2D sprinkled Minkowski intervals
For the analysis of the two-dimensional case we find it convenient, as done before, to representthe sprinkled interval as a unit square [0 , × [0 ,
1] in Euclidean 2D space, where s (0 ,
0) is theinterval root; k points are uniformly distributed inside the square (see Figure 6, where k = 11- source and sink are excluded from the count).By the definition of ’link’, deg ( k ) corresponds now to the number of points, out of k , thatform empty rectangles R i ( s, p i ); the latter are identified by their lower-left and upper-rightvertices. Three such rectangles are depicted in Figure ?? -left, with dotted lines; their purplediagonals are the only links from the root s , for the causet (not shown) associated with thesprinkling.Finding the expected value E [ deg ( k )] of random variable deg ( k ) is equivalent to finding the random-like causet which is, strictly speaking, different from the original. Their indistinguishability is to beintended in statistical sense. For example, the density of points, or the expected distance and angle distributionfrom a generic point to its closest neighbor are unaffected by the transformation. Under a statistical mechanicsmetaphor, they are two equivalent micro-states yielding the same perceived macro-state. The case of theregular ’diamond lattice’ is clearly different, due to the macroscopic ’polarisation’ induced in it by the Lorentztransformation. We detect a problem in [17], equation (8), which indicates that the number of links from the source of a( d + 1)-dimensional interval with N nodes is proportional to N ( d − / for d >
1. In light of the above results,the formula should read N ( d − / ( d +1) . Left: sprinkling 11 points in the unit box and finding the three links from the root. Right: O ( log ( k ))growth of the expected root degree of a 2D Minkowski causet interval. For each value of k , ranging from 100 to3000 in steps of 100, the expected degree was obtained by averaging over 100 sprinklings in the unit box, eachconsisting of k points. Standard deviations are show, as well as the match between averaged data and analyticalexpression. expected number of so called ’Pareto-optimal’ elements for the same set of points. In brief,the analysis is as follows. The probability of point p i to be the endpoint of a link s → p i , underthe condition that its coordinates are ( x i , y i ), depends on these coordinates, and correspondsto the probability of all other k − outside rectangle R ( s, p i ): P rob ( s → p i is a link | p i ’s coordinates are ( x, y )) = (1 − xy ) k − , (4)since each of those points has probability xy to fall inside the rectangle. For finding theunconditioned probability of p i yielding a link, we let its coordinates range over the unitsquare: P rob ( s → p i is a link) = (cid:90) (cid:90) (1 − xy ) k − dxdy = H ( k ) /k, (5)where H ( k ) is the Harmonic Number function: H ( k ) = (cid:80) ki =1 1 i . The expected number of linksis then obtained by adding the contributions of all k points: E [ deg ( k )] = H ( k ) . (6)Figure ?? -right is a plot of experimental data for the root degree of sprinkled, transitivelyreduced 2D Minkowski interval causets, for densities from k = 100 to k = 3000, in steps of100 (the dots). Each data point was obtained by averaging over 100 intervals of fixed density.These data are matched by the O ( log ( k )) theoretical expected degree E [ deg ( k )] of equation 6(solid line). Recall that H ( k ) ∈ O ( log ( k )) since it asymptotically approaches log ( k ) + γ , where γ is the Euler-Mascheroni constant 0.5772.
3D sprinkled Minkowski intervals
For the analysis of the 3D case we use a similar approach. We sprinkle a set S of k points in a3D interval I [ s, t ] of Minkowski space M (1 , , assuming, w.l.o.g., s = (0 , ,
0) and t = (0 , , http://math.stackexchange.com/questions/206866/expected-number-of-pareto-optimal-points. Interval I [ s, t ] in Minkowski space M (1 , (in yellow), and a sub-interval X [ s, u ], for a point u inside I [ s, t ]. The analysis of the degree of s makes use of the volume of X [ s, u ] as a function of the position of u . Any point u ( r, θ, z ) - using cylindrical coordinates - inside I [ s, t ] identifies a sub-interval X [ s, u ], delimited, in Figure 7, by the lower yellow cone and the blue cone. The probabilityof a sprinkled point to fall inside X [ s, u ] is vol ( X [ s, u ]) /vol ( I [ s, t ]), where vol ( I [ s, t ]) = 2 π/ X [ s, u ] depends only on u ’s coordinates r and z , and can be calculated to be: vol ( X [ s, u ]) = π
12 ( z − r ) / . (7)Then, analogous to the 2D case (equation (4)), we obtain the conditional probability of u totop an empty subinterval, thus yielding a link: P rob ( s → u is a link | u ’s coordinates are ( r, θ, z )) = (1 −
18 ( z − r ) / ) k − . (8)Let P link ( r, z ) concisely denote the above conditional probability. Analogous to equation5, the probability of the generic point u to be the endpoint of a link s → u is given by a tripleintegral in cylindrical coordinates, divided by the volume of the region I [ s, t ] of integration: P rob ( s → u is a link) = (cid:82) I [ s,t ] rP link ( r, z ) dr dθ dz π ( z − r ) / . (9) A quick and elegant way to carry out this calculation was suggested by an anonymous referee. The element dV of volume in Minkowski space is Lorentz invariant, hence vol ( X [ s, u ]) must be a function of ( z − r ).Furthermore 3D volume must scale as the cube of the linear measure, hence it is proportional to ( z − r ) / .The multiplicative constant π/
12 is obtained by explicitly computing a specific volume, e.g. for z = 2 and r = 0. k points: E [ deg ( k )] = k P rob ( s → u is a link) . (10)For the calculations it is now convenient to distinguish between lower cone and upper cone ,corresponding to z ∈ [0 ,
1] and z ∈ [1 , P rob ( s → u is a link) = 12 P rob ( s → u is a link | u ∈ lower cone)+ 12 P rob ( s → u is a link | u ∈ upper cone) , (11)For the lower cone, of volume π/
3, we obtain:
P rob ( s → u is a link | u ∈ lower cone) = (cid:82) z =0 (cid:82) πθ =0 (cid:82) zr =0 rP link ( r, z ) dr dθ dzπ/
3= 2 − k k − k + 3 F ( 23 , − k, ,
18 ) , (12)where F is the Gaussian, ordinary Hypergeometric function.Integration for the upper cone is harder, however we can safely ignore this component inlight of the fact that the integrand function P link ( r, z ) basically vanishes in the upper cone, asillustrated by the density plot in Figure 8-left.Figure 8: Left: Density plot for the probability of a point u ( r, θ, z ) to be the endpoint of a link s → u , asa function of r and z : the contribution from the upper cone is almost null. Right: Growth of the expecteddegree E [ deg ( k )] of the root node of a causet obtained by sprinkling k points in 3D Minkowski intervals; matchbetween experimental data for E [ deg ( k )] and the analytic expression of eq. (13). Data points refer to values of k from 100 to 3000, in steps of 100. Each point was obtained by averaging over 50 intervals of fixed density k . In conclusion, in light of equations (10)-(12) we derive the following asymptotic expressionfor the root degree as a function of k : E [ deg ( k )] ≈ / k −
1) + (3 / k ∗ F ( 23 , − k, ,
18 ) . (13)In Figure 8-right we show the match between the analytic expression of eq. (13) and experi-mental data. 17n spite of the many thousand simplified forms for special cases of the Hypergeometricfunction , we could not find a simplified expression for the specific parameter setting of eq.(13), from which to explicitly detect the O ( k / ) growth rate of the expected node degree, asmentioned in [14]. However, an accurate verification of this growth rate for the function at ther.h.s. of eq. (13) is readily obtained in Mathematica , by using function fitting facilities and theavailable implementation of the Hypergeometric function (details are omitted).On the other hand, an explicit derivation of the O ( k / ) result is achieved by approximatingthe probability in eq. (8) by its negative exponential Poisson form.The Poisson probability to find h points inside interval X [ s, u ], out of k − I [ s, t ], is λ h h ! e − λ , where λ is the expected value of h : λ = E [ h ] = ( k − vol ( X [ s, u ]) vol ( I [ s, t ]) = ( k −
1) 18 ( z − r ) / , (14)and the probability to find zero points inside X [ s, u ] reduces to e − λ . This yields a revisedexpression for the conditional probability of (8): P rob ( s → u is a link | u ’s coordinates are ( r, θ, z )) = e − ( k − z − r ) / . (15)Let P (cid:48) link ( r, z ) denote the above conditional probability. As before, the unconditionedprobability of a generic point to yield a link is obtained by integration, and, again, we restrictto the lower cone. P rob ( s → u is a link | u ∈ lower cone) = (cid:82) z =0 (cid:82) πθ =0 (cid:82) zr =0 rP (cid:48) link ( r, z ) dr dθ dzπ/ − − e − k + ( k − E / ( k − )) k − / k − / , (16)where E n ( x ) is the exponential integral function. Again we obtain an approximation of theexpected degree by multiplying the above probability by k/
2, which can be thought of also asthe expected number of points that fall in the lower cone. A little manipulation yields: E [ deg ( k )] ≈ k ( 8( e − k − k − − E / ( k −
18 ) + 6 Γ[5 / k − / ) . (17)The first and second terms converge, respectively, to -8 and 0. The leading, third termexplicitly reveals the O ( k / ) growth, confirming, for dimension d = 3, the O ( k d − d ) growthrate of node degrees mentioned in [14]. (Integration relative to the upper cone yields slightlymore complicated terms, but one finds that they all vanish as k → ∞ .)How about the other causet classes? See, for example, http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/03/ whichlists 111,271 formulas. D sprinkled de Sitter intervals
Our analysis of the the expected node degree growth for sprinkled de Sitter intervals is limitedto the experimental approach, and to the sole 2D case.We considered a fixed interval I [ s, t ] of the hyperboloid representing 2D de Sitter spacetime(see subsection 2.2), with time parameter τ ranging between τ min = 0 at s to tau max = 5 at t . Then, for each considered value k ( k ranging from 100 to 2000 in steps of 100) we built100 directed graphs, each obtained by sprinkling k points in I [ s, t ], finding causal edges, andapplying transitive reduction; the latter was the bottleneck of the whole computation. Theexpected root degree E [ deg ( k )] was obtained, for each k , by averaging over the 100 samples.The expected degree growth appears to be O ( Log ( k )) as for the Minkowski case, as illus-trated in Figure 9.Figure 9: Growth of the expected degree E [ deg ( k )] of the root of a causet obtained by sprinkling k points ina 2D de Sitter interval spanning time interval [0 , k from100 to 3000, in steps of 100. Each point was obtained by averaging over 100 causets with a fixed value of k .Standard deviations are shown. In fact, there is no reason to expect sprinkled de Sitter causets to manifest a behaviourqualitatively different from that of sprinkled Minkowski causets, in the sense that we canstill exclude the extreme O (1) scenario. An informal justification for this claim is obtainedby considering the ultimate reason why node degrees steadily grow in Minkowski sprinkledcausets: as the sprinkling density increases, new nodes x will indefinitely appear that are closeenough to the future lightcone of the interval root s to create a link s → x that contributesto the growth of the degree of s . This argument should not be affected when introducingcurvature. Percolation causet intervals
The node degree growth scenario for percolation causet intervals is quite different, and isultimately determined by the peculiar phenomenon of posts . A post in a partial order is anelement that happens to be related with all other elements. In other words, a post x in acauset creates a bipartition of the whole set of nodes into the future and the past lightcone of x . In [3] it is established that, when the edge probability p is constant, almost surely postskeep appearing indefinitely, as the number n of elements grows. This yields a ’bouncing If the edge probability p is allowed to vary as a function of n , then [7] proves that almost surely posts arisewhen np − e − π p → ∞ , as n → ∞ , while they almost surely do not arise when this quantity tends to 0. In this p , p , ...p n , ... represents a Big Crunch/BigBang event.Let C be a percolation causet built by using a fixed edge probability p , and let C i denotethe interval I [ p i , p i +1 ] between two adjacent posts. An interval I [ x, y ] of C of growing size n will eventually span across multiple posts, i.e. x and y will fall in separate C i ’s. The expecteddegree of x will be bounded by the expected size of the C i where it falls, which does not dependon n . Thus, the growth of the expected degree of x as a function of n is in O (1).Note that this (null) growth rate is consistent with the asymptotic dimension 1 suggestedby the ordering fraction spectrum in the lower-central plot of Figure 4.Figure 10 plots data for the root degree of percolation causet intervals. This plot is providedFigure 10: Histogram of the expected root degree E [ deg ( k )] for percolation intervals of various sizes. 10,000intervals have been used, each picked at random from a different 1000-node percolation causet with edge prob-ability p = 0 .
01. The intervals were grouped into 15 slots of equal width according to their sizes (number ofnodes). For each slot, the mean root degree of the corresponding intervals is shown, with standard deviations. mainly for uniformity with our treatment of the other causet cases, but the data we could collectby simulation does not provide a clear indication of the anticipated O (1) behaviour. This isnot surprising since the O (1) result crucially depends on the existence and inter-distances ofposts, and the occurrence of these special nodes is extremely rare.To get a feel for their rarity we have collected some statistical data on their frequency,following two alternative criteria.Figure 11-left shows the cumulative number of posts achieved by 10,000 n -node percolationcausets for various values of n . The edge probability p here depends on n , and is defined as h/n , where the four cases h = 5 , , ,
50 are considered. It is trivial to see that this choice ofvariable edge probability keeps the expected node degree (in-degree + out-degree) constant.Note that we include in the count of posts the degenerate cases of node 1 and node n , wheneverthe degree of these nodes in the transitively closed graph happens to be n − p , andconsider the cases p = 0 . , . , . , . , .
1. In the limit case of p = 1 the order becomes totaland all nodes are posts. But as p decreases, post occurrences are dramatically reduced. When p = 0 .
1, for most values of n the number of posts found in 10,000 causets drops to zero, failingto be represented in the Log plot. paper we only consider the case of a constant p , which implies that the first case applies. Left: Number of posts cumulatively found in 10,000 n -node percolation causets for various valuesof n . The edge probability is p = h/n for h = 5 , ,
20. Right: Number of posts cumulatively found in 10,000 n -node percolation causets for various values of n , and for edge probabilities p from 0.1 to 0.5 in steps of 0.1.In both plots the points below the horizontal line at 10 correspond to settings of the n and p parameters forwhich the causet is expected to include less than 1 post. Both plots include a horizontal line at ordinate 10 : since we use 10 raw percolationcausets for each n , any point in the plot that falls below the line corresponds to causets forwhich the expected number of posts is less than 1.Let us now go back to the histogram of Figure 4. The causet intervals considered in thatfigure were derived from 1000-node percolation causets with edge probability p = 0 .
01, whichyields np = 10: for this parameter setting, the diagram of Figure 11-left clearly indicates thatthe chances to find even a single post in a 1000-node causet are vanishingly small, and weconclude that posts are not involved in the shaping of that histogram. Pop/sim causet intervals
For pop/sim causets (see Subsection 2.4), the expected degree (counting outgoing links) of theroot s of a k -node interval I [ s, t ] does not grow with k , i.e. it is an O (1) function, regardlessof the value of the parameter m representing the number of new edges x → n ... x m → n contributed by each new node n to the growing raw causet. The reason is as follows.Consider an i -indexed family of independent , pop/sim k -node intervals I i [ s i , t i ], with i ranging in some (large) index set M , and where, for independence, each interval is extractedfrom a separate raw pop/sim causet G i , i ∈ M . For a generic raw pop/sim DAG G ( E, N ) withfixed in-degree m , each newly added node x contributes exactly m edges - those that reachthe node -, except for the initial m nodes, that have no incoming edge. Thus | E | < m | N | ,which implies, for out-degrees, that M ean { outDegree ( x ) | x ∈ N } < m . This expectationmust also be valid for the set of interval roots s i : M ean { outDegree ( s i ) | i ∈ M } < m . Each outDegree ( s i ) is computed relative to the complete corresponding raw graph G i . Letting outDegree (cid:48) ( s i ) and outDegree ”( s i ) denote the out-degree of s i relative, respectively, to interval I i [ s i , t i ] and to the transitive reduction of the latter (which still must have k nodes), we have: outDegree ( s i ) ≥ outDegree (cid:48) ( s i ) ≥ outDegree ”( s i ). We are using here the obvious fact thatthe links are a subset of the raw edges. This allows us to conclude, restricting to intervals andlinks, that M ean { outDegree ”( s i ) | i ∈ M } < m . Then, the fact that the above inequalities donot depend on k establishes the O (1) growth result.21 egular grid interval The case of regular grids trivially falls into the O (1) growth scenario. As pointed out in theabove quote [25], in these graphs (or even in their randomized variants) node degrees do notgrow with the number of nodes. Assessing the node degree growth indicator: the case of the irrational grid
Looking at node degree growth is a useful way to investigate causet Lorentzianity. In particular,avoiding an O (1) growth for interval root degrees is a necessary condition for achieving thehyper-connectivity of Lorentzian, sprinkled causets. This allows us to rule out percolationcausets, grid causets, and pop/sim causets.Furthermore, once an unbounded growth rate is detected, a more accurate characterisationof the growth - whether logarithmic or polynomial of certain degree - might provide us witha dimensionality estimate for the causet under study, based on the well characterised growthrates for sprinkled causets.Then, in order to come up with some reasonable definition of Lorentzianity in the discretesetting we might be tempted to promote unbounded node degree growth to the status of anecessary and sufficient condition. For this to be a good choice, any transitively reduced causetmanifesting an unbounded node degree growth should appear to us as a ’good’ Lorentizancauset. The following counter-example suggests that this is not the case.Consider a conceptually simple variant of the square grid, that we call ’irrational grid’, inwhich the points in 2D Minkowski spacetime M (1 , have coordinates ( x √ , t ), with integers x and t ranging in ( −∞ , + ∞ ). It is easy to establish that the degree of any node in thetransitively reduced causet associated with these points is infinite.More precisely (see Figure 12-left) if I [ s, t ] is a diamond interval in M (1 , between s (0 , t (0 , n √
2) (not a grid point), where positive integer n is the index of thecolumn of grid points delimiting the diamond at the r.h.s., and S ( n ) is the set of irrationalgrid points that fall inside I [ s, t ], then we can readily estimate the growth of | S ( n ) | to be,asymptotically, | S ( n ) | ≈ n √ . Figure 12:
Left: irrational grid and diamond for n = 5. Center: transitively reduced causet for the diamondpoints. Root degree is 5. Right: logarithmic growth of the root degree as a function of diamond size (log-linearplot)
On the other hand, letting C ( n ) denote the transitively reduced causet with root s derived22rom the points of set S ( n ) (Figure 12-center), and deg ( n ) denote the degree of s in C ( n ), wefind that the precise initial values of | S ( n ) | , and the values of n that mark an increment of deg ( n ), are as listed in Table 1 below. n | S ( n ) |
46 71 101 138
230 283 342 408
556 ... deg(n) Table 1:
Data for the irrational grid. First row: Semi-width of diamond interval (/ √ C ( n ). Third row: degree of causet root. Degree values, as shown in the third row, are always odd and grow by +2 increments, forthe symmetry of the graph. Furthermore, the analysis of the lenghts (cid:96) i of the ’runs’ of equalvalues in the degree sequence reveals that (cid:96) i = (cid:96) i +1 for all odd i ’s - e.g. (cid:96) = (cid:96) = 1, or (cid:96) = (cid:96) = 5 - and that the ratio (cid:96) i /(cid:96) i − , for i ≥ √ . This growth is illustrated in Figure 12-right, which plots the ( | S ( n ) | , deg ( n )) pairs listed inTable 1 and beyond. It turns out that this degree growth and the corresponding growth forsprinkled 2D causets (Figure 6-right) are not only analogous in their logarithmic character, butalso very close numerically. In this respect, the irrational grid could be seen as an adequate regular counterpart of the random -looking, sprinkled 2D causet.In summary, while the node degree growth indicator correctly induces us to rule out thecauset obtained from the original, square grid (or from any rational grid, for that matter),it fails to rule out the irrational grid, although, under the strong criterion for Lorentz in-variance that bans preferred reference frames, almost all types of grid should be ruled out.(We write ’almost’ in light of an interesting, recently identified class of regular
2D lattices,called ’Lorentzian lattices’ [29], which seems to partially satisfy the ’no preferred frame’ cri-terion: these structures are invariant under a discrete subgroup of the Lorentz group, whileadditionally offering a very good number-volume correspondence.)The useful lesson that comes from the irrational grid counter-example is that, when tryingto characterise Lorentzianity, we should decouple the regularity/irregularity issue - with itsimpact on the presence/absence of preferred frames - from other aspects, e.g. involving nodedegrees or path lengths (to be discussed next), which are nevertheless equally relevant toLorentzianity. Indicators for the latter aspects are unlikely to tell apart order from disorder,but this fact may indeed turn into an advantage, when one is interested (as we are) in theanalysis and synthesis of algorithmic causets, which, unlike stochastic causets, span the wholerange between fully regular and (pseudo-)random patterns. In other words, we certainly valuea scenario in which some degree of Lorentzianity can be attributed to a causet with regularcomponents. We thank an anonymous referee for suggesting the simple expression 3 + 2 √ √ .2 Longest/shortest path plots The new indicator we introduce now is still based on link counts, but is concerned with longestand shortest paths between nodes. Similar to the indicators based solely on intervals, this oneis not affected by the overall shape of the graph, or by ’border’ considerations.We shall try to characterise the extent to which a causet succeeds in reproducing a keyproperty of Lorentzian manifolds: the reversed triangular inequality . Consider two events p and p in flat Minkowski space M , , with p in the future lightcone of p . The (+ - - -)signature of the Lorentz metric is at the basis of the inequality, which in turn leads to the twinparadox: the time delay experienced by the first twin, who travels from p and p followinga straight line (a geodesic), is maximum; any time-like trajectory p → x → p taken by thesecond twin via some intermediate spacetime point x registers a shorter time delay. In termsof Lorentz distance L : L ( p , p ) ≥ L ( p , x ) + L ( x, p ) . (18)The limit case - zero time - is registered when p → x and x → p are light-like segments thatform at x a π/ C is derived by sprinkling in Minkowski space M , we can approximate theLorentz distance in M between two timelike related points p and p by the length (numberof edges) of the longest chain P between them in C [22], which represents a geodesic (thiscorrespondence is conjectured to hold also for sprinklings in curved manifolds). We denotethis length by lpl ( p , p ), for ’longest path lengh’. Note that all edges of path P must be links,that is, edges of the transitive reduction C red of C : if one were not a link, it could be replacedby two or more links, yielding a longer path and conflicting with P being the longest. Thus C red , which is unique, codes all the necessary information for measuring the Lorentz distancebetween any two events. In the sequel we shall drop the ’red’ subscript.How could we then statistically characterise the extent to which a causal set C reproducesthe inequality of equation 18? In light of the correspondence between L ( p , p ) and lpl ( p , p ),and just sticking to triangles, the most direct approach would be to compile statistics on, say,the difference or ratio between lpl ( p , p ) and lpl ( p , x ) + lpl ( x, p ), for all x between p and p .Following this line of thought, we can compute the expected gain r of path p → x → p over p → p when x is chosen uniformly at random in a 2D or a 3D Minkowski interval I [ p , p ].We find: r = (cid:82) x ∈ I [ p ,p ] ( L ( p , x ) + L ( x, p )) /L ( p , p ) dxV ol ( I [ p , p ]) = (cid:40) / / lpl and summation inplace of, respectively, function L and integration. Additionally, these constants might proveuseful as dimension estimators, analogous to ordering fractions for the well known Myrheim-Meyer estimator.However, we have an even simpler way to characterise the variety of path lengths between p and p . Since in a Lorentzian manifold M we find infinitely many trajectories between p and p that are shorter, or even much shorter than the geodesic, we require that the finite24umber of alternative paths between two related nodes in C widely range in length too, fromvery long to very short. For further simplicity, we concentrate on the extreme cases of thelongest and shortest paths between event pairs.The interplay between these lengths is represented in a convenient compact way by usingwhat we call the longest/shortest path plots . The function f C,s ( l ) depicted in these plots isdefined relative to a node s - typically a root - of a causet C , refers to all and only those pathsthat start from s , and yields the mean shortest path length corresponding to longest pathlength l . Formally: f C,s ( l ) = M ean { spl ( s, x ) | x ∈ N odes ∧ lpl ( s, x ) = l } , (20)where spl ( s, x ) and lpl ( s, x ) denote, respectively, the lengths of the shortest and longest pathsfrom s to x .Figure 13 shows longest/shortest path plots for four different types of transitively reduced,sprinkled causets. The upper-left plot refers to the 2D Minkowski case. For building this plotFigure 13: Longest/shortest path plots for four different types of sprinkled causet. Each plot is obtained byconsidering four 5000-node interval causets of the corresponding type, by collecting longest and shortest pathlengths from the root to all nodes, and by computing the average shortest path length (and standard deviation)associated with each possible longest path length. Upper-left: 2D sprinkled Minkowski intervals. Upper-right:2D sprinkled de Sitter diamond interval, with time span (0, 5). Lower row: 3D and 4D sprinkled Minkowskiintervals. we considered four distinct 5000-node Minkowski intervals and computed, for each of them,the longest and shortest path length pairs from the root to any node. Then we aggregated the20,000 pairs in the plot, according to function definition (20).25he other three plots of Figure 13 have been build analogously. The upper-right plotrefers to 2D sprinkled de Sitter intervals; in spite of the presence of curvature, it appears quitesimilar to the 2D Minkowski plot. The lower plots refer to 3D and 4D sprinkled Minkowskiintervals, and reveal two facts: if the the number of nodes in the graph is kept constant anddimensionality increases, then (i) the range of longest path lengths reduces, and (ii) shortestpaths associated with a fixed longest path length get shorter.We believe that the systematic presence of very short shortest paths between nodes thatare separated by increasingly long longest paths, as neatly represented in these plots, reflects,in the discrete setting, a key feature of Lorentzian manifolds. We shall therefore consider thepresence of longest/shortest path plots qualitatively similar to those in Figure 13 as a necessaryrequirement for causets aiming at ’Lorentzianity’.Let us now focus on the slow growth of the above longest/shortest path plots for the 2Dcases. For doing this, let us consider longest and shortest paths separately. With slightabuse of notation, we now let lpl ( k ) and spl ( k ) define, respectively, the average longest and average shortest path length from source to sink of a sprinkled 2D Minkowski interval asa function of the number k of sprinkled points - the average being taken over large sets ofintervals. Longest path length
In [15] and [6] it is shown that lpl ( k ) k − /d converges to some constant m d as the number k ofpoints sprinkled in a d -dimensional Minkowski interval grows to ∞ . The value of m d , however,has been calculated only for d = 2, and found to be m = 2. This immediately yields anasymptotic estimate of 2 (cid:112) ( k ) for lpl ( k ).The LogLog plot in Figure 14-left shows the match between this theoretical prevision andexperimental data. The latter was obtained by considering four independent 50,000-nodeMinkowski intervals, then picking at random 5,000 intervals from each of them, then computingfor each of these the pair ( k, l ( k )), where k is the number of points in the interval (its volume)and l ( k ) is the length of the longest path from source to sink, and then computing lpl ( k ), theaverage of all the l ( k )’s for a given k . Shortest path length
The
LogLinear plot of Figure 14-center shows the mean shortest path lengths spl ( k ) obtainedfrom the same experiments used for the plot at its left. The plot is indeed a histogram withvariable bin size: the 23 data points correspond to 23 bins of geometrically growing size. Foreach bin [ k min , k max ] we collect all shortest path lengths corresponding to intervals whosevolume k falls in that range, and compute their average. Note that this is now a LogLinear plot: only the x -axis is logarithmic. The linear pattern indicates that spl ( k ) has O ( log ( k ))growth. The matching function (solid line) is 0 . log ( k ) + 2 . √ k for lpl ( k ) and the estimate 0 . ∗ log ( k ) + 2 .
68 for spl ( k ) - for fitting experimental longest/shortest path plots of sprinkled 2D The reduced growth rate of these plots for higher dimensional cases is an interesting phenomenon that weleave for further study.
Left:
LogLog plot for the mean longest path length of transitively reduced sprinkled Minkowski2D interval causets as a function of the sprinkling density k , matched against theoretical growth function 2 √ k .Center: LogLinear histogram for mean shortest path length of transitively reduced sprinkled Minkowski 2Dinterval causets as a function of the sprinkling density k . Fit by natural logarithm. Experimental data for thesetwo plots was derived from four 50k interval causets. Right: Function f ( x ) = 0 . ∗ log ( x / .
68 approximatesthe longest/shortest path plot for 2D Minkowski sprinkled causets under the assumptions lpl ( k ) = 2 (cid:112) ( k ) and spl ( k ) = 0 . log ( k ) + 2 . Minkowski intervals. This is done via a ’hybrid’ function f that combines the empirical spl ( k )and the theoretical lpl ( k ) by eliminating variable k : f ( x ) = 0 . log ( lpl − ( x )) + 2 .
68 = 0 . log ( x .
68 = 0 . log ( x/
2) + 2 . . (21)The plot of function f is shown in Figure 14-right (solid line), matched against the longest/shortestpath plot derived from the four 50,000-node intervals used for the first two plots. Longest/shortest path plots for the other causet classes
Figure 15 provides plots for grids and pop/sim causets. Actually, these are longest/shortestpath arrays that present directly the raw data used for building the shortest/longest path plotsas defined in equation (20): in each of the three rectangular arrays the grey level of the entryat column lpl and row spl represents the relative frequency of the ( lpl, spl ) pairs found in thecorresponding causet (thus, the longest/shortest path plot is obtained by simply averaging thedata in the array, column by column).The longest/shortest path array for a grid graph, in which the lengths of the longest andshortest paths from the root s to any node x trivially coincide, is shown in Figure 15-left.The diagram provides now a particularly effective visual account of the fact that these graphsoccupy the opposite extreme of the spectrum (maximum locality), relative to sprinkled causets(maximum non-locality).For the sake of comparison, we also consider a randomized version of the regular grid, a’proximity’ graph obtained by sprinkling points in a 2D unit square and using the Euclideandistance d and a threshold δ for creating edges: a directed edge from point p ( x , y ) to point p ( x , y ) is created whenever y > y and d ( p , p ) < δ . The longest/shortest path array forsuch a graph, with threshold δ = 0 .
1, is shown in Figure 15-center.Finally, Figure 15-right shows the longest/shortest path array for a 10k-node pop/simcauset. While the asymptotic behaviour of this class of causets approximates that of de Sitter27igure 15:
Longest/shortest path arrays for three causets. Left: 32 × sprinkled causets in terms of node degree distributions [20], our analysis reveals, again, aremarkable difference between the two causet types, at the finite scale, with pop/sim causetsunable to develop short paths as alternatives to long paths to the same node. No clue ofnon-locality seems to emerge in these experiments with pop/sim causets.In the case of graphs from percolation dynamics, relatively high ratios between longest andshortest path lengths can be achieved. This is documented by the experimental longest/shortestpath plot of Figure 16 which, similar to the plots in Figure 13, was derived from four 5k-nodecausets. The edge probability we used for building them is p = 0 .
01, which yields np = 50 (for n = 5000 nodes).Figure 16: Longest/shortest path plot from four 5k-node percolations causets with fixed edge probability 0.01.
When keeping the edge probability p constant, these graphs tend to become 1D structuresas n grows. This fact was already revealed by the ordering fraction spectrum of Figure 4,lower-central plot, which is characterised by the same value of np = 50. As a consequence,both the longest and the shortest paths must exhibit a roughly linear growth with respect tothe number k of nodes, although the two multiplicative factors may largely differ. This impliesthat the longest/shortest path plot eventually assumes a linear character too, as apparent inthe plot of Figure 16.On the largest scale, the emergent 1D graph structure is related to the discussed phe-nomenon of ’posts’. However, for np = 50, as in the percolation causet under discussion, the28lot in Figure 11-left indicates that the expected number of posts drops below 1 as the numberof nodes grows above value n = 150, which means that the linear growth of the longest/shortestpath plot of Figure 16 is not due to the presence of posts.In conclusion, we observe that none of the causet classes covered in Figures 15 and 16achieves plots comparable to the one obtained for the sprinkled causets of Figure 13. In thisrespect, the longest/shortest path plot indicator - an analytical tool of easy implementation- appears to exhibit a discriminative power equivalent to that of the previous node degreegrowth indicator, at least relative to the considered causet classes.What about the regular, irrational grid example of the previous subsection? Figure 17shows the longest/shortest path plot for a 4524-node irrational grid diamond causet.Figure 17: Longest/shortest path plots for a 4524-node diamond causet from the irrational grid.
In analogy with what was observed w.r.t. node degree growth indicators, the plot forthis highly regular directed graph appears much closer to the longest/shortest path plots forthe Lorentzian, stochastic, sprinkled causets, than to those of the other classes,+ and yetthe causet violates the no-preferred-frame requirement for Lorentz invariance, like the othergrid cases. This circumstance further confirms our previous remark on the opportunity, whendealing with Lorentzianity, to separate the regularity/irregularity issue from the detection ofother statistical aspects involving node degrees or path lengths.
In [9, 8, 10] the first author has described various deterministic, algorithmic causet construc-tion techniques based on simple models of computation and completely independent from anunderlying manifold. In this section we introduce a new family of deterministic techniques thatmediates between the manifold-dependent stochastic technique of sprinkling and the purely ab-stract (manifold-independent) algorithmic approach, in an attempt to achieve longest/shortestpath plots – our main indicator of Lorentzian non-locality – comparable to those obtained forsprinkled causets, while retaining the benefits of the deterministic approach. The main benefitexpected from deterministic over stochastic techniques is the emergence of structure, or a mixof structure and (deterministic) chaos, as widely discussed and shown in [31, 9, 8, 10]. Thisvariety of emergent properties is also the reason why we prefer causets produced by deter-ministic, dynamical, computational systems over ones, still deterministic, obtained by direct29athematical definitions, such as the irrational grid introduced at the end of Subsection 4.1.(Another reason for this preference is that dynamical systems appear to nicely fit a ’computa-tional cosmology’ perspective.)We introduce a family of automata that we call permutation ants (PA), in which the controlunit (the ’ant’) moves on a finite array A of cells by short steps or jumps while sequentiallyperforming simple operations on them, such as reading, writing, comparing, swapping cells oradding new ones. At each step the cell array A of length n contains a permutation π of thefirst n integers.The n -element permutation π is directly transformed into an n -node causet as follows. Eachnode is labeled by the pair ( i, π ( i )), which can be understood as a pair of integer coordinates;a directed edge ( x , y ) → ( x , y ) is created between two nodes if and only if x < x and y < y . In doing so, we are essentially still reasoning in terms of lightcones, whose bordersare now parallel to the cartesian axes. An important difference with sprinkling is that the ant,in its journey, may go back and modify previously visited sites of the growing causet.In the next two subsections we describe two types of PA automaton. The three pieces ofinformation that describe them are: (i) the data structure on which the ant operates; (ii) theset S of situations that are recognized by the ant; and (iii) the set R of possible ant reactions,that depend on the situation. Following the approach of [31], we are interested in enumeratingand exploring exhaustively the complete space of instances of each automaton. If all reactionsare applicable to any situation, the size of this space is | R | | S | . In the stateful PA automaton the ant can be in a finite number of states, like in Turingmachines. We shall restrict to the set of states {
0, 1 } . Data structure
The ant operates on the cells of array A , which keeps permutation π as described above. Situation
The situation is coded by 2 bits, b and b , yielding 4 cases: b - This bit represents the current state of the ant, namely 0 or 1. b - With the ant positioned at cell c with content x , b detects whether x ≤ c or x > c . Reaction
The reaction is coded by 4 bits, b , ..., b , yielding 16 cases. b and b - these 2 bits identify 4 possible reactions, numbered from 0 to 3:0 - Swap contents of cells c and c − c − c and c + 1 (fails if c + 1 does not exist);2 - Create new cell at the right of cell c , with value max +1, where max is the currentnumber of cells;3 - Add new cell at the left of cell c , with value max +1. b - This bit defines the new state of the ant.30 - This bit defines the ant’s move:0 - Move one step to the left (fails if cell c − c + 1 does not exist).For each of the 4 situations there is a choice among 16 reactions, thus there are 16 = 65536different automaton instances, that we number by the decimal representation of the 16 bitsthat characterize each of them (4 reaction bits per situation). We have simulated and inspectedall of them, starting from initial configuration A init = (1 ,
2) and the ant in state s init = 0,positioned at cell 1. Note that when the reaction fails - the ant attempting to access cellsbeyond the array limits - the whole computation is aborted. Out of the 12278 automaticallyselected cases that survive after 100 steps, we have manually selected two interesting cases.(’Manual’ selection consisted in displaying on the computer screen large arrays of thumbnailplots, each showing the ant dynamics for the first 100 or so steps, and in spotting the very fewnon-regular cases, an easy and relatively fast job for the human eye.)Automaton 1925 has the irregular and quite remarkable behaviour documented in Figure18.Figure 18: Stateful PA automaton n. 1925. Ant trajectory, final permutation (inset), ordering fractionspectrum, node degree growth (counting links), longest/shortest path plot.
The upper-left diagram plots the positions occupied by the ant on the growing array A during a one-million-step computation. Quite remarkably, the ant keeps sweeping the full rangeof available cells from one extreme to the other, without ever entering a regular behaviour and,more surprisingly, without ever attempting to cross the boundaries. The final permutation isrepresented in the inset plot, as the set of points ( i, π ( i )), i = 1, ..., 2811. The diagonal shadowindicates that the permutation is a sort of compromise between the identity function and a31andom scattering - a mix of order and randomness. These two diagrams illustrate what wemean when we claim that algorithmic, deterministic causet construction techniques can offersurprising emergent properties that cannot be expected from stochastic techniques.The upper-right plot of Figure 18 provides the ordering fraction spectrum for 500 intervalsof the final causet - a graph with 2811 nodes and 2,456,824 edges, that decrease to 18,531 aftertransitive reduction - and suggests a Myrheim-Meyer dimension slightly less than 2D.The lower-left plot shows the averaged node degree (counting links) for the root nodesof intervals of variable size. We have generated 3000 intervals I [ s, t ] - subgraphs of the finalcauset - each time picking a random source node s and a random sink t visible from s . For eachinterval we computed the pair ( k, deg ), where k is the interval volume - the number of nodes -and deg is the degree of s . The histogram, with geometrically increasing bins, is derived fromall these pairs: for each slot we plot the average deg , with standard deviation, relative to allpairs for which volume k falls in the slot. The degree growth in this case is slightly weakerthan that observed with the 2D Minkowski sprinkling case (compare with Figure 6-right).The lower-right diagram is the longest/shortest path plot, shown with standard deviations.Note that we cannot exclude a priori the existence of multiple sources (and sinks) in the causetobtained from a permutation as described: the plot is build by considering the lengths of thelongest and shortest link-paths from node 1 to all the nodes reachable from it, which may beless than the total number of nodes. Recall that the average and the standard deviation referto the set of shortest path lengths corresponding to the same longest path length.Interestingly, the longest/shortest path plot for the causet built by PA automaton 1925(Figure 18-lower-right) is roughly similar, numerically, to the analogous plot for 2D sprinkledMinkowski intervals (Figure 13-upper-left). A possibly counter-intuitive feature of this plot isthat it exhibits some decreasing parts. How is it possible that shortest path lengths decreaseas longest path lengths increase? The reason is, roughly, as follows. When the longest pathsfrom the root are below a certain threshold, they can only reach a portion A of the causetfor which shortest paths offer limited distance reduction over the longest ones. But as longestpaths grow longer, they may eventually reach a portion B of the graph that admits fastconnections from the root, via a ’short-cut region’ X that could not be exploited for reachingnodes in A , e.g. for the absence of paths from X to A . Emergent macroscopic phenomenaof this type are not possible in purely stochastic causets, and remind us of the phenomenonof ’compartmentation’ that we have observed in other algorithmic, pseudo-random causets [8].Note that some decreasing segment is observed also in the longest/shortest path plot for theirrational grid (Figure 17).The stateful PA automaton 1929 has a definitely more regular behaviour, illustrated inFigure 19.The ant trajectory is shown in the upper-left plot for 10,000 steps (it proceeds similarlyup to at least one million steps). The final permutation is represented in the inset plot. Theupper-right plot shows the ordering fraction spectrum for 500 intervals of the final causet - agraph with 2969 nodes and 2,566,500 edges, that decrease to 5933 after transitive reduction;this plot differs considerably from that of the previous automaton n. 1925, providing a vagueindication of a low, non-integer Myrheim-Meyer dimension.The lower-left diagram plots averaged node degrees (counting links) for the root nodes ofintervals of variable size. Analogous to the case of the previous automaton, the histogram uses32igure 19: Stateful PA automaton n. 1929. Ant trajectory, final permutation (inset), ordering fractionspectrum, node degree growth (counting links), longest/shortest path plot. bins of geometrically increasing size, and reflects a slightly weaker growth than what observedin the analogous plot for 2D Minkowski sprinkling (Figure 6-right).In the lower-right diagram of Figure 19 we provide the longest/shortest path plot. Note thatlongest path lengths hit here record values close to 1000, and yet the corresponding averageshortest path lengths exhibit moderate growth, yielding a plot that appears compatible withthat for 2D sprinkled Minkowski intervals (Figure 13-upper-left).
The second type of algorithm that we consider is a stateless PA automaton . The control head,or ant, is now stateless, but this simplification is compensated by a slightly more complex arraystructure, and a type of reaction similar to a GOTO statement.
Data structure
Array cell A ( i ) is now a pair ( bit ( i ) , π ( i )), where bit ( i ) is a bit and π ( i ) is the permutationelement, as before. Situation
Coded by 2 bits, b and b , yielding 4 cases: b = bit ( c ) - Cell c is where the ant is currently positioned. b = bit ( c −
1) - (Fails if cell c-1 does not exist).33 eaction
Coded by 3 bits - b , b , b - yielding 8 cases. b = 0 ⇒ Write b in cell c and swap cells c and c − b = 1 ⇒ Create a new cell ( b , max + 1) and insert it at position c.( max is the current number of cells). b - This bit defines the ant’s move.0 - Ant does not move.1 - Ant jumps to cell π ( c ) (GOTO statement; c is the current ant position).Reasoning as for the stateful ant, we now obtain 8 = 4096 automaton instances. We havesimulated and inspected all of them, starting from an initial two-cell configuration A init =((0 , , (0 , Stateless PA automaton n. 2560. 3000-step computation. Ant trajectory, final permutation (inset),ordering fraction spectrum, node degree growth (counting links), longest/shortest path plot.
34t first sight this automaton appears as a ’perfect sprinkler’ since its ordering fraction spec-trum, node degree growth and longest/shortest path plot are basically indistinguishable fromthose of sprinkled 2D Minkowski causets; a closer analysis, however, reveals some slight depar-ture from pure sprinkling. We have estimated function lpl ( k ) three times, relative to three dis-tinct final causets obtained by letting the automaton run, respectively, for n = 1000 , , vol , lpl ) pairs, where vol is thevolume of a subinterval I [ s, t ] of the causet - s being the causet root - and lpl is the lengthof the longest path from s to t , and then we have computed the best fit of these data againstfunction a ∗ k b , keeping in mind the reference values a = 2 and b = 1 / n of nodes of the causet grows ( n = 1000 → → a and b tend to drift away from the above reference values: a = 1 . → . → . b = 0 . → . → .
54. It would be interesting to find whether this tendencypersist with much higher values of n - an analysis which is unfortunately limited by the severecomputational bottleneck of transitive reduction.Another interesting feature of this automaton is that its definition can be greatly simplifiedwithout affecting its behaviour. We noticed that the final causet built in n steps has n + 2nodes. This means that, starting from the two-cell initial array, the ant reaction is always of onetype: create a new cell at each step. A closer look at the behaviour of the computation revealsthat several options in the algorithm are never used by specific instance n. 2560. In particular,even the bits that decorate array cells can be eliminated! We can then provide a very concisealgorithm that performs exactly the same computation. Here is the tiny Mathematica code forthe simplified ant step: step[{array_, pos_}] := {Insert[array, Length[array] + 1, pos], array[[pos]]}
The above function ’step’ takes a pair { array, pos } , where array represents a permutationof the first n positive integers and pos is an integer identifying a position in it, and returns anew pair { array (cid:48) , pos (cid:48) } , where array’ is the result of inserting value n + 1 at position pos of array (so that array’ is now a permutation of the first n + 1 positive integers), and pos’ is theinteger found at position pos of array . By iterating the function call 3000 times with initialarray (1, 2) and initial ant position 2, one obtains exactly the same results of Figure 20.We believe that this concise randomization algorithm might be of interest, independent ofthe application to causets; it would be interesting to further investigate its statistical qualitiesor its possible relation with other such generators.In Figure 21 we present just the ant trajectory for instance 3593 of the stateless PA automa-ton. The causet obtained by this permutation is one-dimensional, thus uninteresting in termsof ’Lorentzianity’ and non-locality. The reason for presenting the plot is to show yet anotherinstance of the typical, triangle-based self-similar patterns that emerge in cellular automataand other models, as widely illustrated in [31]: it is precisely this richness of emergent phe-nomena, from regular to pseudo-random patterns, that motivates our interest for algorithmiccausets. 35igure 21: Ant trajectory for stateless PA automaton n. 3593.
In this paper we have introduced statistical indicators for the assessment and comparison ofcausal set classes, meant to focus specifically on their ’Lorentzianity’, intended as the manifes-tation, in the discrete setting, of the ’non-local’ nature of Minkowski space, as implied by theLorentz distance. Our experimental work, based on extensive computer simulations, has ledus to a number of conclusions.First, we have established a clear distinction between indicators based on transitivelyclosed (Section 3) and transitively reduced (Section 4) causets. In spite of their usefulnessfor Myrheim-Meyer dimension estimation, indicators of the first type (at least those we haveconsidered in this paper) fail to discriminate between cases as different as the highly non-local sprinkled causets and highly local directed graphs such as a regular grid, thus provinginadequate for characterising ’Lorentzianity’. Furthermore we have shown that a power-lawdistribution of node degrees is not a rare property of transitively closed causets, and does notsingle out sprinkled de Sitter causets [20] as a special case of discrete spacetime, being present,for example, in percolation causets and in what we called sprinkled Minkowski cylinder (Fig-ure 2). Note that, whenever the estimation of causet node degrees is obtained by measuringlightcone areas in the embedding manifold, as done in [20], the analysis is inevitably bound toaddress only the transitively closed version of the causet, with the limitations just mentioned.Considering transitively reduced graphs proves more useful, although, unfortunately, tran-sitive reduction is computationally costly [2], being O ( | V || E | ) for a directed acyclic graph G ( V, E ) - this has been the main computational bottleneck of our investigation. We haveproved, analytically and experimentally, that the node degrees of a transitively reduced sprin-kled Minkowski interval causet exhibit O ( log ( k )) growth in 2D and O ( k / ) growth in 3D,where k is the number of sprinkled nodes, confirming what concisely stated, in slightly differ-ent terms and without proof, in [14].Still in the context of transitively reduced causets, we have introduced a new indicator: thelongest/shortest path plot. This simple but effective visual indicator is designed to directlyreflect the interplay between longest and shortest link-paths, whose length differences growvery large in causets derived from sprinkling in Lorentzian manifolds. We have shown that36he very slow growth that these plots exhibit for sprinkled Minkowski and de Sitter intervalsis not observed in the other considered causet classes, for which the growth is linear (Figures15 and 16).We found that the node degree growth and the longest/shortest path plot have analogousdiscriminative power, at least for the considered causet classes, although the latter was meantto reflect more directly a peculiar feature of the Lorentzian metrics, one related to the reversedtriangular inequality. Indeed, an attractive item for further research would be to try anddecouple these two indicators, checking whether specific causet classes exist for which theyprovide divergent responses.More work is also necessary to shed further light on the relation between longest/shortestpath plots and causet dimensionality, in light of the experimental observation that these plotstend to flatten as the latter increases.We have then introduced two new classes of deterministic, algorithmic causets, built bystateless and stateful PA (Permutation Ant) automata, and have identified a ’perfect sprinkler’- a deterministic ’ant’ able to build a causet that appears, under the lens of the longest/shortestpath length indicator, indistinguishable from a sprinkled causet. For this ’ant’ we have alsoprovided an extremely concise implementation - one line of code. A peculiarity of this approachis that, unlike in ’cumulative’, stochastic sequential approaches, the topology of the growingcauset can be modified by the ant at any location, at any step.The ability to mimic the randomness of sprinkled causets by a deterministic approach,such as our permutation-based ’ant’, is interesting, and tells something about the power ofthe automaton; but it is not too surprising, in light of the rather direct correspondence be-tween permutations and sprinkling, and of the widely known fact that many simple models ofcomputation can produce ’deterministic chaos’ (in [31] this ability is conjectured to be a cluefor computational universality). Quite oppositely, the main expected advantage of algorithmiccausets over stochastic ones is the emergence in the former of some regularity, or a mix orregularity and pseudo-randomness. The few examples we have provided here indicate thatprogress in this direction is still at an early stage, and that more work is needed for findingalgorithmic causet classes offering the mentioned mix while achieving a good performance interms of longest/shortest path plots, i.e. Lorentzian non-locality. Some promising examples,obtained very recently by a 4-state automaton similar to the one presented here, are describedin [12].When causets exhibit some regularity, preferred frames of reference appear that violateLorentz invariance, as with the discussed regular grids (the ’Lorentzian lattices’ [29] men-tioned at the end of Subsection 4.1 are an interesting exception). In this respect, node degreegrowth rate and longest/shortest path plots (Section 4) offer the advantage of providing usefulinformation on a weaker form of Lorentzianity, independent from concerns on preferred frames.The specific causets produced by deterministic Permutation Ants are always embeddablein 2D Minkowski space, like those produced by stochastic sprinkling. Of course this is not ageneral property of deterministic causets. But there is no point in insisting on direct embed-dability: quantum oscillations at ultra-low spacetime scales may conflict with this requirement,and such causets may still turn out to be embeddable in a coarse-grained form.There exists a growing body of papers on causet embeddability, manifoldlikeness and’Lorentzianity’. The objective of these efforts is to provide tools for the reconstruction, when-37ver possible, of continuum information from the discrete structure of the causet; investigatedproperties include dimensionality [21], time-like and space-like distance [22, 28], curvature [4].(Unfortunately, the relative simplicity of the definitions and tools for these properties in thecontinuum is easily lost when transposing them in the discrete setting.)The specific problem of characterising locality - local regions that are small with respect tothe curvature scale - is addressed in [18], where the reader can also find additional references forthe other mentioned properties. The problem here is that in a causet obtained by sprinkling ina Lorentzian manifold, two intervals I [ p, q ] and I [ p, r ] with the same root and the same volumemay largely differ in the depth of the neighborhood that they span, to the point that one mayfall below and the other above the scale of curvature, thus qualifying, respectively, as local andnon-local. As observed in [18], ” from the continuum perspective small, or local neighbourhoodsare essential to several geometric constructions and are key to the conception of a manifold. ”The technique elaborated in [18] is based on detecting the distribution of m -element intervalsin flat spacetime, as a function of m , and in using these profiles as a benchmark for assessingthe flatness/locality of a causet region. Subtle correlations might exist between these refinedlocality/flatness detectors and our longest/shortest path plots. However, our plots are meantto detect an important clue for Lorentzianity - the potential presence of a wide gap betweenlongest and shortest paths between the same two points - regardless of whether the inspectedregion is below or above the curvature scale (if any). Furthermore, a longest/shortest pathplot that grows very slowly can still be taken, in our opinion, as an indicator of non-locality(with possible overloading of the term), even when dealing with a flat and ’small’ region of thegraph.It would also be interesting, moving above the ground level of spacetime, to look for othercases in Nature (e.g. in the biosphere) where ’non-locality’, as revealed by our plots, plays somerole, and to study how it is implemented. For example, the evolution of plant leaf venationhas led to patterns that optimize hydraulics and tolerance to cuts, and involve long paths fornutrient transportation between points at a short distance from each other. The first author expresses his gratitude to Marco Tarini for useful discussions and experimentson shortest and longest paths in sprinkled causets, and to an anonymous referee for proposingthe ’irrational grid’ example, for suggesting improvements in the calculations of expected nodedegrees in Minkowski 2D and 3D sprinkled intervals, and for suggesting additional referenceson causet manifoldlikeness. This work was supported by CNR, Consiglio Nazionale delleRicerche/Istituto ISTI/FMT Laboratory.
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A New Kind of Science . Wolfram Media, Inc., 2002.40 ppendix A - Node degree distribution for (transitively closed)percolation causets
The node degree density of Figure 22 refers to the transitive closure of a 32k-node percolationcauset in which the probability of finding an edge between any two nodes i and j is p ( i, j ) =0 . out -degrees.) A power-law seems to be still in actionhere, up to about degree 1000, while a flat plot segment reveals a constant density for higherdegrees.Note that we are considering a full causet, not an interval, hence we cannot compare thisplot with those of Figure 1. Comparison with the plots of Figures 2 and 3, covering full causetswith rotational symmetry is perhaps more justified, although percolation causets do not sharethat symmetry.Figure 22: Node degree density for transitively closed percolation causet. Slots of geometrically increasingwidth have been used, and the node counts for each slot have been normalised accordingly.