Solvable random network model for disordered spherepackingCitation for published version (APA):Dhara, S., van Leeuwaarden, J. S. H., & Mukherjee, D. (2016). Solvable random network model for disorderedsphere packing. arXiv, (1611.05019), 1-19. https://arxiv.org/abs/1611.05019

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Solvable random network model for disordered sphere packing

Souvik Dhara, Johan S.H. van Leeuwaarden, and Debankur MukherjeeEindhoven University of Technology, Department of Mathematics and Computer Science,

P.O. Box 513, 5600 MB Eindhoven, The Netherlands

A notorious problem in physics is to create a solvable model for random sequential adsorptionof non-overlapping congruent spheres in the d-dimensional Euclidean space with d ≥ 2. Spheresarrive sequentially at uniformly chosen locations in space and are accepted only when there is nooverlap with previously deposited spheres. Due to spatial correlations, characterizing the fractionof accepted spheres or the area covered by the deposited spheres remains largely intractable. Westudy these disordered sphere packings by taking a radically novel approach that compares randomsequential adsorption in Euclidean space to the nearest-neighbor blocking on a sequence of clusteredrandom networks with a growing number of vertices. This tractable network model leads to a precisecharacterization of the coverage and elementary laws that describe the packing fraction as a functionof density and dimension. By investigating the spatial dimensions two to five over wide densityranges, we show that these laws are accurate and universal. The model also supports a previouslyconjectured lower bound on the densest packing in high dimensions.

PACS numbers: 02.50.Ga, 68.43.-h, 89.75.-k

The packing of congruent spheres in the d-dimensionalEuclidean space has been a topic of great interest acrossthe sciences, serving as basic models in condensed matterand quantum physics [1–4], nanotechnology [5, 6], infor-mation theory and optimization problems [7–9]. A hugebody of mathematical literature [10–16] is devoted to therigorous understanding of the largest possible fraction ofthe space covered by the spheres in dimensions 3, 8, 24;the problem in general dimensions remains open. Al-though the densest packings are generally believed to beordered (regular lattices or crystal structures), disorderedpackings can serve as approximations (lower bounds). Infact, there are reasons to believe that in sufficiently highdimension, the densest packing tends to become disor-dered rather than ordered, which hints at the existenceof disordered classical ground states for some continu-ous potentials [17, 18]. Having almost no analytical re-sult in high dimensions, this strongly motivates the studyof disordered packings. Disordered sphere packings alsoturn up naturally in numerous experimental and real-world settings, ranging from the deposition of nano-scaleparticles on polymer surfaces, adsorption of proteins onsolid surfaces to the creation of logic gates for quantumcomputing, and many more applications in domains asdiverse biology, ecology and sociology, see [19–21] for ex-tensive surveys.

Disordered random packings in continuum spaces arenotoriously hard to analyze, because particles displaystrong local and global spatial correlations. These cor-relations can be understood by viewing the packing asa dynamic process referred to as random sequential ad-sorption (rsa) [19, 21]. At each time epoch, a centerappears at a uniformly chosen location in space, and anattempt is made to place a sphere with the chosen pointas its center. The new sphere must either fit in the emptyarea allowed by the hard-core exclusion interaction withthe spheres deposited earlier, or its deposition attemptis discarded. Of particular interest is the proportion of

accepted spheres, or equivalently, the volume covered bythe accepted spheres. Fig. 1 illustrates three instances ofthis rsa process in 2D.

The precise setting in this paper considers packing ina finite-volume box [0, 1]d, filled with ‘small’ spheres ofradius r and volume Vd(r). We will make use of the factthat this rsa process of spheres in the continuum canequivalently be viewed as rsa of particles on the verticesof a spatial random network with nearest-neighbor block-ing. The n attempted locations in [0, 1]d of centers areregarded as the vertices of the network and two verticesare connected if they are at most 2r distance apart. Theinteraction network, thus obtained, is known as the ran-dom geometric graph (rgg). Since the volume of [0, 1]d

is 1, the probability that two randomly chosen verticesare connected is equal to the volume of a sphere of radius2r, given by

Vd(2r) =πd/2(2r)d

Γ(1 + d/2)(1)

and the average degree of a vertex is c = nVd(2r). Sincec is the average number of overlaps per sphere, withall other attempted spheres, we interchangeably use theterms density and average degree for c. To maintain aconstant density c in the large-network limit, as n→∞,the spheres need to become smaller when more verticesare added to the graph, and the radius should scale as afunction of n according to

r =1

2

[cΓ(1 + d/2)

nπd/2

]1/d. (2)

Notice that it is equivalent to consider the deposition ofspheres with fixed radii into a box of growing volume.We parameterize the rgg model by the density c andthe dimension d, and henceforth write this as rgg(c, d).A typical instance of rgg(5, 2) with n = 1000 verticesis shown in Fig. 2. Random sequential adsorption on a

arX

iv:1

611.

0501

9v1

[m

ath.

PR]

15

Nov

201

6

2

c 10 c 20 c 50

FIG. 1. Random sequential adsorption in 2D with an increasing number of deposition attempts. Dots indicate the centers ofaccepted (red) and discarded (blue) spheres. The average number of overlaps per sphere with all other attempted spheres isreferred to as the density c.

c 15

FIG. 2. rgg(15, 2) graph with 1000 vertices: two verticesshare an edge if they are less than 2r distance apart, where ris such that a vertex has on average c = 15 neighbors. Noticethe many local clusters.

graph selects vertices uniformly at random and activatesthem if none of the neighbors are active. For rsa in thecontinuum, notice that any two centers of the depositedspheres are at least 2r distance apart, and therefore thergg model with radius 2r and nearest-neighbor blockingcorresponds to the deposition of non-overlapping spheresof radius r. The number of deposited spheres after nattempts in the continuum rsa has the same law as thenumber of active vertices for rsa on rgg(c, d) with nvertices. Let Jn(c, d) be the fraction of active verticesin the rgg(c, d) model on n vertices. Since each spherecovers a region of volume Vd(r), the total volume coveredby the accepted spheres is Jn(c, d)× n× Vd(r). Further,c = nVd(2r) and Vd(2r) = 2dVd(r), so that the total

volume covered is

Jn(c, d)× c

2d. (3)

Thus the problems of determining the total volume cov-ered by the deposited spheres in [0, 1]d, and determiningthe fraction of active vertices in the rgg(c, d) model, areequivalent. That is why we will henceforth focus only onthe quantity Jn(c, d). While it was proved in [22] that thelimit Jn(c, d)→ J(c, d) exists, no quantitative character-ization of J(c, d) for dimensions ≥ 2 has been providedtill date, and so far the main methods to study this prob-lem rely on extensive simulations [23–29]. We propose anovel approach that considers rsa on a clustered randomgraph model. This random network will be designed tomatch the local spatial properties of the rgg model interms of density and clustering, but remains amenable torigorous mathematical treatment including the limitingpacking fraction.

Random graphs serve to model large networked sys-tems, but are typically unfit for capturing local cluster-ing in the form of relatively many short cycles. Thiscan be resolved by locally adding so-called householdsor small dense graphs [30–36]. Vertices in a householdhave a much denser connectivity to all (or many) otherhousehold members, which enforces local clustering. Wenow introduce a new household model, called clusteredrandom graph model (crg), designed for the purpose ofanalyzing the disordered packing problem. An arbitraryvertex in the crg model has local or short-distance con-nections with nearby vertices, and global or long-distanceconnections with the other vertices. When pairing ver-tices, the local and global connections are formed ac-cording to different statistical rules. The degree distri-bution of a typical vertex is taken to be Poisson(c) (ap-proximately) in both the rgg and crg model. In thecrg(c, α) model however, the total mass of connectivitymeasured in the density parameter c, is split into αc toaccount for direct local blocking and (1 − α)c to incor-

3

FIG. 3. Example topology generated by the crg(c, α) model.

porate the propagation of spatial correlations over longerdistances. The crg(c, α) model with n vertices is thendefined as follows (see Fig. 3):

• Partition the n vertices into random households ofsize 1 + Poisson(αc). This can be done by sequen-tially selecting 1 + Poisson(αc) vertices uniformlyat random and declaring them as a household, andrepeat this procedure until at some point the next1+Poisson(αc) random variable is at most the num-ber of remaining vertices. All the remaining ver-tices are then declared a household too, and thehousehold formation process is completed.

• Now that all vertices are declared members of somehousehold, the random graph is constructed ac-cording to a local and a global rule. The local rulesays that all vertices in the same household get con-nected by an edge, leading to complete graphs ofsize 1+Poisson(αc). The global rule adds a con-nection between any two vertices belonging to twodifferent households with probability (1− α)c/n.

This creates a class of random networks with a fixed den-sity c and tunable level of clustering via the free param-eter α. With the goal to design a solvable model for therandom packing process, the crg(c, α) model has nc/2connections to build a random structure that mimics thelocal spatial structure of the rgg(c, d) model on n ver-tices.

Seen as the topology underlying the random packingproblem, the crg(c, α) model incorporates local clustersof overlapping spheres, which occur naturally in randomgeometric graphs; see Fig. 2. We can now also considerrsa on the crg(c, α) model, by using the greedy algo-rithm that constructs an independent set on the graphby sequentially selecting vertices uniformly at random,and placing them in the independent set unless they areadjacent to some vertex already chosen. The packingfraction Jn(c, α) is then the fraction of activated verticesin the crg(c, α) model, hence the size of the independentset divided by the network size n. From a high-level per-spective, we will solve the rsa problem on the crg(c, α)

model, and translate this solution into an equivalent re-sult for rsa on the rgg(c, d). Our ansatz is that forlarge enough n, a unique relation can be established be-tween d←→ α = αd, so that the jamming constants arecomparable, i.e.,

Jn(c, d) ≈ Jn(c, αd), (4)

and virtually indistinguishable in the large network limit.In order to do so, we map the crg(c, α) model onto

the rgg(c, d) model by imposing two natural conditions.The first condition matches the average degrees in bothtopologies, i.e., c is chosen to be equal to nVd(2r). Thesecond condition tunes the local clustering. Let us firstdescribe the clustering in the rgg model. Consider twopoints chosen uniformly at random in a d-dimensionalhypersphere of radius 2r. Then what is the probabilitythat these two points are themselves at most 2r distanceapart? From the rgg perspective, this corresponds tothe probability that, conditional on two vertices u and vbeing neighbors, a uniformly chosen neighbor w of u isalso a neighbor of v, which is known as the local cluster-ing coefficient [37]. In the crg(c, α) model, on the otherhand, the relevant measure of clustering is α, the prob-ability that a randomly chosen neighbor is a neighborof one of its household members. We then choose theunique α-value that equate to the clustering coefficientof rgg. Denote this unique value by αd, to express itsdependence on the dimension d. In order to obtain anexplicit characterization for αd, let B(x, 2r) denote thed-dimensional sphere with radius 2r, centered at x ∈ Rd,and Vd(2r), its volume. Pick any point V uniformly atrandom from B(x, 2r). Write u↔ v to say that u and vshare an edge. The above reasoning gives

αd =P (u↔ v, u↔ w, v ↔ w)

P (u↔ v, u↔ w)

=1

(Vd(2r))2

∫Bd(0,2r)

P(w ∈ Bd(0, 2r) ∩Bd(v, 2r)

)dv

=1

Vd(2r)E[|Bd(0, 2r) ∩Bd(V , 2r)|]. (5)

The expected value in (5) can be calculated explicitly(see Appendix C), which gives

αd = d

∫ 1

0

xd−1I1− x24

(d+ 1

2,

1

2

)dx, (6)

where Iz(a, b) is the normalized incomplete beta integral

Iz(a, b) =

∫ z0xa−1(1− x)b−1dx∫ 1

0xa−1(1− x)b−1dx

. (7)

Table I shows the numerical values of αd in dimensions1 to 5. With the uniquely characterized αd in (6), thecrg(c, αd) model can now serve as a generator of randomtopologies for guiding the rsa process.

In contrast to the Euclidean space, the packing of thecrg(c, αd) model through rsa is analytically solvable,

4

d 1 2 3 4 5αd 0.750000 0.586503 0.468750 0.379755 0.310547

TABLE I. Numerical values for αd for dimensions 1 to 5.

even at later times when the filled space becomes moredense (c is large) and the deposition events become morecorrelated. To solve this problem, we will extend themean-field techniques recently developed for analyzingrsa on random graph models [38–41]. The main goal ofthese works was to find greedy independent sets (or col-orings) of large random networks. All these results, how-ever, were obtained for non-geometric random graphs,typically used as first approximations for sparse interac-tion networks in the absence of any known geometry.

We now consider the crg(c, α) model, and introducean algorithm that sequentially activates the vertices andexplores the graph while obeying the hard-core exclusionconstraint (see Appendix A for further details). Asymp-totic analysis of the algorithm in the large-graph limitn → ∞ then leads to explicit characterizations of thepacking fraction. The algorithm carries along three typesof vertices: active, frozen and neutral. Initially, all thevertices are neutral; at each step a neutral vertex is se-lected uniformly at random and declared active. Thisin turn causes some vertices (the neighbors which arestill neutral) to become frozen. The neighbors of thenewly active vertex are selected as follows: a numberPoisson(αc) of vertices are selected uniformly at randomfrom the set of all vertices that do not yet belong to anyhousehold, and together form the household of the acti-vated vertex. It is possible that some of these selectedvertices are already frozen due to distant connectionswith a vertex outside its household that was activatedearlier. Then the newly activated vertex connects to anyother neutral vertex with probability (1 − α)c/n, whichcreates its distant neighbors. These vertices belong toa different household, but are frozen nevertheless. Forthe crg(c, α) model on n vertices, recall that Jn(c, α)denotes the packing fraction of active vertices. We thenhave the following result.

Theorem 1. For any c > 0 and α ∈ [0, 1], as n → ∞,

Jn(c, α)P−→ J(c, α), where J(c, α) is the smallest nonneg-

ative root of the deterministic function x(t) described bythe integral equation

x(t) = 1− t−∫ t

0

(x(s)αc

1− (αc+ 1)s+ (1− α)cx(s)

)ds.

(8)In particular, as c→∞,

J(c, αd) ∼1

1 + αdc. (9)

The ODE (8) can be understood intuitively. Rescaletime by n, so that after rescaling the algorithm has toend before time t = 1 (because the network size is n).Now think of x(t) as the fraction of neutral vertices at

RGG simulation

CRG simulation

CRG mean-field

0 5 10 15 20 25 300.0

0.2

0.4

0.6

0.8

1.0

FIG. 4. Mean-field limit for crg(c, α2) and simulation with1000 vertices of rgg(c, 2) and crg(c, α2) for 0 ≤ c ≤ 30.

time t. Then clearly x(0) = 1, and the drift −t says thatone vertex activates per time unit. Upon activation, avertex blocks its αc household members and on average(1 − α)c other vertices outside its household. At time t,the fraction of vertices that are not members of any dis-covered households equals on average (1− (1 +αc)t) andall vertices which are not part of any discovered house-holds, are potential household members of the newly ac-tive vertex (irrespective of whether it is blocked or not).Since household members are uniformly selected at ran-dom, only a fraction x(t)/(1 − (1 + αc)t) of the new αchousehold members will belong to the set of neutral ver-tices. Moreover, since all x(t)n vertices are being blockedby the newly active vertex with probability (1 − α)c/n,on average (1 − α)cx(t) neutral vertices will be blockeddue to distant connections. Notice that the graph willbe maximally packed when x(t) becomes zero, i.e., thereare no neutral vertices that can become active. This ex-plains why J(c, α) should be the smallest root of (8).The simple law (9) thus describes the packing fraction asa function of density (large) and dimension, and suggeststhat the fraction of vertices in the packing as a functionof the density c scales as (1 + αdc)

−1.

A rigorous proof of Theorem 1 is presented in Ap-pendix A, and uses martingale decomposition and func-tional limit theorems. Upon substituting α = αd,J(c, αd) = limn→∞ Jn(c, αd) is completely characterizedby (8) and serves an approximation for the intractablecounterpart J(c, d), the limiting jammed fraction for thergg(c, d) model. The choice of αd, as discussed earlier, isgiven by (6) and shown in Table I. Figs. 4 and 5 validatethe mean-field limit for the crg model, and show thetheoretical values J(c, αd) from Theorem 1, along withthe simulated values of rsa on the rgg(c, d) model forvalues of c ranging from 0 to 20, d = 3, 4, 5, and 1000vertices, respectively. The remarkable agreement of theJ(c, αd)-curve with the simulated results shows that theintegral equation (8) accurately describes the mean-fieldlarge-network behavior of the packing process, not onlyfor the crg model, but also for the rgg model.

5

RGG simulation 3D

Theoretical

0 5 10 15 20 25 300.0

0.2

0.4

0.6

0.8

1.0

RGG simulation 4D

Theoretical

0 5 10 15 20 25 300.0

0.2

0.4

0.6

0.8

1.0

RGG simulation 5D

Theoretical

0 5 10 15 20 25 300.0

0.2

0.4

0.6

0.8

1.0

FIG. 5. Mean packing fraction according to the theoretical values J(c, αd) compared with simulation of the rgg(c, d) modelfor 1000 vertices, 0 ≤ c ≤ 20, d = 3, 4, 5.

Prediction by our model

Torquato's prediction

0 500 1000 1500 2000

0

500

1000

1500

-Log2 (covered volume)

FIG. 6. log2(covering volume) is plotted as a function of di-menson. The blue line is y = 0.77865x and the red dots arey = log2(2xαx) for increasing values of x.

Let us finally consider what happens in high dimen-sions. We note that αd → 0 as d → ∞. Thus, theinteraction network described by the crg(c, αd) modelbecomes almost like the mean-field Erdos-Renyi randomgraph model, which supports the widely believed con-jecture that in high dimensions the interaction networkassociated with the random geometric graph loses its lo-cal clustering property [42]. The total scaled volumecJ(c, d)/2d covered by the deposited spheres in dimensiond, serves as a lower bound for the densest packing, gener-ally believed to be ordered. Due to the accurate descrip-tion provided by the crg model, cJ(c, αd)/2

d also givesa good approximation for the disordered lower-bound.Indeed, for large c, (9) yields J(c, αd) ∼ 1/(αdc), andtherefore, in any dimension d, our model predicts thatthe lower bound takes the form

J(c, αd)×c

2d=

1

2dαd, (10)

and − log2(covered volume) = log2(2dαd). Let us com-pare this with the lower bound 2−0.77865...d conjecturedby Torquato and Stillinger [17]. Fig. 6 shows an almostunbelievable agreement between both predictions, andhence strongly supports the validity of the modeling ap-proach and the discovered laws in this paper.

In summary, the crg(c, αd) model reduces all possibleinteractions among pairs of vertices to only two principalcomponents: the local interactions due to the cluster-ing, and a mean-field long-range interaction. While thecrg(c, αd) model seems a rather crude approximation ofthe rgg(c, d) topology, it proves sufficient for the pack-ing problem, as evidenced by Figs. 4–6. More evidence isprovided in Appendix B, where the variance of the pack-ing fraction in both models is shown to match closely.There is building evidence that randomized topologiessuch as the crg model, can approximate rigid spatialtopologies when the local interactions in both topologiesare matched. Apart from this paper, the strongest evi-dence to date for this line of reasoning is [43], where it wasshown that the typical ensembles from the latent-spacegeometric graph model can be modeled by an inhomoge-neous random graph model that matches with the origi-nal graph in terms of the average degree and a measure ofclustering. We should mention that [43] is restricted toone-dimensional models and does not deal with spherepacking, but it shares with this paper the perspectivethat matching degrees and local clustering can be suffi-cient for describing spatial settings.

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Appendix A: Asymptotic analysis of the CRG model

Here we analyze the asymptotic properties of rsa on the crg(c, α) model. We introduce an algorithm thatsequentially activates the vertices while obeying the hard-core exclusion constraint, and then analyze the explorationalgorithm (see [38, 39, 44] for similar analyses in various other contexts). For each vertex, the neighbors insideand outside its own household will be referred to as ‘household neighbors’ and ‘distant neighbors’, respectively. IfH denotes the number of household neighbors of a typical vertex, then observe that H ∼ Poisson(αc). Therefore,E (H) = αc, and Var(H) = αc. Furthermore, any two vertices belonging to two different households are connected byan edge with probability pn = (1−α)c/n, so the number of distant neighbors is a Bin(n−H−1, pn) random variable,Poisson((1 − α)c) in the large n limit. Notice that the total number of neighbors, is then asymptotically given by aPoisson(c) random variable. For a crg(c, α) model on n vertices, denote the fraction of active vertices after the rsa

process reached a jammed state by Jn(c, α). In this section we fix c > 0 and α ∈ [0, 1], and simply write Jn for Jn(c, α).

Notation. Write µ = 1 + αc, λ = (1 − α)c, and σ2 = αc. Note that µ denotes the expected size of the householdcontaining a typical vertex, since a household contains a vertex and its household neighbors. In the proof, we useboldfaced letters to denote stochastic processes, and vectors. Terms like ‘vertex’ or ’node’ will be used interchangeably.

7

A sequence of random variables {Xn}n≥1 is said to be OP(f(n)), or oP(f(n)), for some function f : R → R+, if thesequence of scaled random variables {Xn/f(n)}n≥1 is tight, or converges to zero in probability, respectively. Wedenote by DE [0,∞) the set of all cadlag (right continuous left limit exists) functions from [0,∞) to a complete,

separable metric space E, endowed with the Skorohod J1 topology, and by ‘d−→’ and ‘

P−→’, convergence in distributionand in probability, respectively. In particular, if the sample paths of a stochastic process X are continuous, we write

Xn = {Xn(t)}t≥0d−→ X = {X(t)}t≥0, if for any T ≥ 0,

supt∈[0,T ]

|Xn(t)−X(t)| P−→ 0 as n→∞. (A1)

1. Main result

The following result proves the central limit theorem (CLT) for Jn.

Theorem 2 (CLT for jamming fraction). As n→∞,

√n(Jn − J)

d−→ N (0, V ),

where J is the smallest non-negative root of the deterministic function

x(t) = e−λt(1− µt)µ−1µ

(1−

∫ t

0

eλs(1− µs)−1+1µ ds

), t <

1

µ, (A2)

and V = σxx(J) with σxx(t) the unique solution of the system of differential equations

dσxx(t)

dt= 2σxx(t)f(t) + 2σxy(t)g(t) + β(t),

dσxy(t)

dt= σxy(t)f(t) + tg(t)σ2 +

√β(t)σρ(t)

(A3)

for 0 ≤ t < 1/µ, with

y(t) = 1− µt, f(t) = −µ− 1

y(t)− λ, g(t) =

(µ− 1)x(t)

y(t)2,

β(t) =

[(µ− 1)

y(t)+ λ

]x(t), ρ(t) =

σ√β(t)

x(t)

y(t).

(A4)

Theorem 2 says that for large n, the fraction of active vertices in the jammed state will be approximately given by(J · n + Z ·

√n), where Z is a Normal random variable with mean 0 and variance V . The next theorem establishes

that for large c, J(c, α) ∼ (1 + cα)−1, which is a simple law for the jamming fraction in the high-density regime.

Theorem 3 (Large density regime). As c→∞, (1 + cα)J(c, α)→ 1.

2. The exploration algorithm

Instead of fixing a particular realization of the random graph and then studying rsa on that given graph, we willsequentially activate the vertices one-by-one, explore the neighborhood of the activated vertices, and construct therandom topology of the graph. The idea of exploring this way simplifies the whole analysis, since the evolution ofthe system can be described recursively in terms of the previous states. Similar constructions have been displayed forother random graph models in the literature [38, 41]. Below we describe the specific algorithm for the crg model.

Observe that during the process of sequential activation, until the jamming state is reached, the vertices can be ineither of three states: active, blocked, and unexplored (i.e. vertices with future potential activation). Furthermore,there can be two types of blocked vertices: (i) blocked due to activation of some household neighbor, or (ii) none ofthe household neighbors is active, but there is an active distant neighbor. Therefore, at each time t ≥ 0, categorizethe vertices into four sets:

8

• A(t): set of all vertices active.

• U(t): set of all vertices that are not active and that have not been blocked by any vertex in A(t).

• BH(t): set of all vertices that belong to a household of some vertex in A(t).

• BO(t): set of all vertices that do not belong to a household yet, but are blocked due to connections with somevertex in A(t) as a distant neighbor.

Note that BH(t) ∪ BO(t) constitute the set of all blocked vertices at time t, and BH(t) ∩ BO(t) = ∅. Initially, allvertices are unexplored, so that U(0) = V (G), the set of all n vertices. At time step t, one vertex v is selected fromU(t− 1) uniformly at random and is transferred to A(t), i.e., one unexplored vertex becomes active.

We now explore the neighbors of v, which can be of two types: the household neighbors, and the distant neighbors.Further observe that v can have its household neighbors only from the set U(t−1)∪BO(t−1)\{v}, since each vertexin BH(t− 1) already belongs to some household. Define

H(t) ∼ min{

Poisson(αc), |U(t− 1) ∪ BO(t− 1) \ {v}|},

i.e., draw a Poisson(αc) random variable independently of any other process, and if it is smaller than |U(t − 1) ∪BO(t − 1) \ {v}|, then take it to be the value of H, and otherwise set H(t) = |U(t − 1) ∪ BO(t − 1) \ {v}|. SelectH(t) vertices {u1, u2, . . . , uH} at random from all vertices in U(t− 1)∪BO(t− 1) \ {v}. These H(t) vertices togetherform the household containing v, and are moved to BH(t), irrespective of the set they are selected from. To explorethe distant neighbors, select one by one, all the vertices in U(t − 1) ∪ BO(t − 1) ∪ BH(t − 1) \ {v, u1, . . . , uH}, andfor every such selected vertex u, put an edge between u and v with probability pn. Denote the newly created distantneighbors that belonged to U(t− 1) by {u1, . . . , ud}, and move these vertices to BO(t). In summary, the explorationalgorithm yields the following recursion relations:

A(t) = A(t− 1) ∪ {v},U(t) = U(t− 1) \ {v, u1, u2, . . . , uH , u1, . . . , ud},

BH(t) = BH(t− 1) ∪ {u1, u2, . . . , uH},BO(t) = BO(t− 1) ∪ {u1, . . . , ud}.

The algorithm terminates when there is no vertex left in the set U(t) (implying that all vertices are either active orblocked), and outputs the cardinality of A(t) as the number of active vertices in the jammed state.

3. State description and martingale decomposition.

Denote for t ≥ 0,

Xn(t) := |U(t)|, Yn(t) := |U(t) ∪ BO(t)|.

Observe that {(Xn(t), Yn(t))}t≥0 is a Markov chain. At each time step, one new vertex becomes active, so that|A(t)| = t, and the total number of vertices in the jammed state is given by the time step when Xn(t) hits zero, i.e.,the time step when the exploration algorithm terminates.

Dynamics of Xn. First we make the following observations:

• Xn(t) decreases by one, when a new vertex v becomes active.

• The household neighbors of v are selected from Yn(t − 1) vertices, and Xn(t) decreases by an amount of thenumber of such vertices which are in U(t− 1).

• Xn(t) decreases by the number of distant neighbors of the newly active vertex that belong to U(t − 1) (sincethey are transferred to BO(t)).

Thus,

Xn(t+ 1) = Xn(t)− ξn(t+ 1) and Xn(0) = n (A5)

9

with

ξn(t+ 1) = 1 + η1(t+ 1) + η2(t+ 1), (A6)

where conditionally on (Xn(t), Yn(t)),

η1(t+ 1) ∼ Hypergeometric(Xn(t), Yn(t), H(t)), (A7)

i.e., η1(t + 1) has a Hypergeometric distribution with favorable outcomes Xn(t), population size Yn(t), and samplesize H(t). Further, conditionally on (Xn(t), Yn(t), η1(t+ 1)),

η2(t+ 1) ∼ Bin(Xn(t)− 1− η1(t+ 1),

λ

n

). (A8)

Therefore, the drift function of the Xn process satisfies

E (ξn(t+ 1)|Xn(t), Yn(t)) = 1 +Xn(t)(µ− 1)

Yn(t)+

(Xn(t)− 1− Xn(t)(µ− 1)

Yn(t)

)λ

n

= 1 +Xn(t)(µ− 1)

Yn(t)+λXn(t)

n+OP(n−1),

(A9)

where, in the last step, we have used the fact that Xn(t) ≤ Yn(t).

Dynamics of Yn. The value of Yn does not change due to the creation of distant neighbors. At time t, it can onlydecrease due to an activation of a vertex v (since it is moved to A(t)), and the formation of a household, since all thevertices that make the household of v, were in U(t−1)∪BO(t−1), and are moved to BH(t). Thus, at each time step,Yn(t) decreases on average by an amount µ = 1 + αc, the expected household size, except at the final step when theresidual number of vertices can be smaller than the household size. But this will not affect our asymptotic results inany way, and we will ignore it. Hence,

Yn(t+ 1) = Yn(t)− ζn(t+ 1) and Yn(0) = n, (A10)

where

E (ζn(t+ 1)|Xn(t), Yn(t)) = µ. (A11)

Martingale decomposition. Using the Doob-Meyer decomposition [45, Theorem 4.10] of Xn, (A9) yields thefollowing martingale decomposition

Xn(t) = n−t∑i=1

ξn(i) = n+MX

n (t)− t−t∑i=1

[Xn(i− 1)(µ− 1)

Yn(i− 1)+λXn(i− 1)

n+OP(n−1)

],

where MX

n = {MXn (t)}t≥1 is a square-integrable martingale with respect to a suitable filtration. Let us now define

the scaled processes

xn(t) :=Xn(bntc)

nand yn(t) :=

Yn(bntc)n

.

Also define

δ(x, y) := (µ− 1)x

y+ λx, for 0 ≤ x ≤ y, y > 0. (A12)

Thus, we can write

xn(t) = 1 +MXn (bntc)n

− bntcn− 1

n

bntc∑i=1

δ

(Xn(i− 1)

n,Yn(i− 1)

n

)+OP(n−1)

= 1 +MXn (bntc)n

− t−∫ t

0

δ(xn(s), yn(s))ds+OP(n−1).

(A13)

Similar arguments yield

yn(t) = 1 +MYn (bntc)n

− µt+OP(n−1), (A14)

where MY

n = {MYn (t)}t≥1 is a square-integrable martingale with respect to a suitable filtration. We write xn and yn

to denote the processes (xn(t))t≥0 an (yn(t))t≥0 respectively.

10

4. Proof outline

The proof of Theorem 2 consists of three key steps: First we show that the processes xn and yn converges in

probability to deterministic functions. This shows that the fraction Jn converge in probability to some deterministiclimit J . Next, in Proposition 7, we show that the diffusion-scaled processes (defined in (A41)) converges to a two-

dimensional Ornstein-Uhlenbeck process. Finally, the scaled fluctuations around J of Jn are shown to have a Gaussianlimit, as stated in Theorem 2.

5. Quadratic variation and covariation

To investigate the scaling behavior of the martingales, we will now compute the quadratic variation and covari-ation terms. For convenience, denote by Pt,Et, Vart, Covt, the conditional probability, expectation, variance andcovariance, respectively, conditioned on (Xn(t), Yn(t)). Notice that, for the martingales MX

n and MY

n , the quadraticvariation and covariation terms are given by

〈MX

n 〉(bntc) =

bntc∑i=1

Vari−1(ξn(i)),

〈MY

n 〉(bntc) =

bntc∑i=1

Vari−1(ζn(i)),

〈MX

n ,MY

n 〉(bntc) =

bntc∑i=1

Covi−1(ζn(i), ξn(i)).

(A15)

Thus, the quantities of interest are Vart(ξn(t+ 1)), Vart(ζn(t+ 1)) and Covt(ξn(t+ 1), ζn(t+ 1)), which we derive inthe three successive claims.

Claim 1. For any t ≥ 1,

Vart(ζn(t+ 1)) = σ2. (A16)

Proof. The proof is immediate by observing that the random variable denoting the household size has variance σ2.

Claim 2. For any t ≥ 1,

Vart (ξn(t+ 1)) =Xn(t)(µ− 1)

Yn(t)+λXn(t)

n+OP(n−1). (A17)

Proof. From the definition of ξn in (A6), the computation of Vart(ξn(t+ 1)) requires computation of Vart(η1(t+ 1)),Covt(η1(t+ 1), η2(t+ 1)) and Vart(η2(t+ 1)). Since η1 follows a Hypergeometric distribution,

Et (η1(t+ 1)(η1(t+ 1)− 1)|H) =Xn(t)(Xn(t)− 1)(H − 1)(H − 2)

Yn(t)(Yn(t)− 1)(A18)

and

Vart (η1(t+ 1)) =Xn(t)(Xn(t)− 1)E ((H − 1)(H − 2))

Yn(t)(Yn(t)− 1)+Et (η1(t+ 1))−E2

t (η1(t+ 1))

=X2n(t)

Y 2n (t)

(σ2 + µ2 − 3µ+ 2) +Xn(t)

Yn(t)(µ− 1)− X2

n(t)

Y 2n (t)

(µ− 1)2 +OP(n−1)

=X2n(t)

Y 2n (t)

(σ2 − µ+ 1) +Xn(t)

Yn(t)(µ− 1) +OP(n−1)

=Xn(t)

Yn(t)(µ− 1) +OP(n−1),

(A19)

11

since σ2 = µ− 1 = αc. Also, we have

Et (η2(t+ 1)(η2(t+ 1)− 1))

=[(Xn(t)− 1)(Xn(t)− 2)−Et (η1(t+ 1)) [2Xn(t)− 3] +Et

(η21(t+ 1)

) ](λn

)2

=λ2X2

n(t)

n2+OP(n−1)

(A20)

and therefore

Vart(η2(t+ 1)) =λXn(t)

n+OP(n−1). (A21)

Further,

Et (η1(t+ 1)η2(t+ 1)) = Et

(η1(t+ 1)(Xn(t)− 1− η1(t+ 1))

λ

n

)=λ

n

[(Xn(t)− 1)Et(η1(t+ 1))−Et(η21(t+ 1))

]=λXn(t)

n

Xn(t)(µ− 1)

Yn(t)+OP(n−1).

(A22)

Now, from (A7), (A8),

Et(η1(t+ 1)) =λXn(t)

n+OP(n−1) and Et(η2(t+ 1)) =

Xn(t)(µ− 1)

Yn(t)+OP(n−1),

which implies that

Covt(η1(t+ 1), η2(t+ 1)) = OP(n−1). (A23)

Combining (A19), (A21) and (A23), gives (A17).

Claim 3. For any t ≥ 1,

Covt (ζn(t+ 1), ξn(t+ 1)) =Xn(t)

Yn(t)σ2 +OP(n−1). (A24)

Proof. First, observe that

Et (ζn(t+ 1)η1(t+ 1)) = Et (ζn(t+ 1)Et (η1(t+ 1)|ζn(t+ 1)))

=Xn(t)

Yn(t)Et (ζn(t+ 1)(ζn(t+ 1)− 1)) =

Xn(t)

Yn(t)(σ2 + µ2 − µ),

(A25)

and therefore,

Covt(ζn(t+ 1), η1(t+ 1)) =Xn(t)

Yn(t)σ2. (A26)

Thus,

Et (ζn(t+ 1)η2(t+ 1)) = Et (ζn(t+ 1)Et (η2(t+ 1)|η1(t+ 1), ζn(t+ 1)))

= Et

(ζn(t+ 1)(Xn(t)− 1− η1(t+ 1))

λ

n

)= λµ

Xn(t)

n+OP(n−1)

(A27)

and hence

Covt (ζn(t+ 1), η2(t+ 1)) = OP(n−1). (A28)

Combining (A26) and (A28) yields (A24).

12

6. Convergence of the exploration process on n-scale

Based on the quadratic variation and covariation results in Section A 5, the following lemma shows that the mar-tingales when scaled by n, converge to the zero-process.

Lemma 4. For any fixed T ≥ 0, as n→∞,

1

nsupt≤T|MX

n (bntc)| P−→ 0,1

nsupt≤T|MY

n(bntc)| P−→ 0. (A29)

Proof. Observe that using (A15) along with (A16) and (A17), we can claim for any T ≥ 0,

〈MX

n 〉(bnT c) = OP(n), 〈MY

n〉(bnT c) = OP(n). (A30)

Thus, from Doob’s inequality [46, Theorem 1.9.1.3], the proof follows.

Proposition 5 (Functional LLN of the exploration process). As n → ∞, the processes {xn(t)}0≤t<1/µ and{yn(t)}0≤t<1/µ converge weakly to the deterministic processes {x(t)}0≤t<1/µ and {y(t)}0≤t<1/µ, respectively, wherey(t) = 1− µt and x(t) is the unique, non-increasing, continuous function satisfying

x(t) = 1− t−∫ t

0

(µ− 1

1− µs+ λ

)x(s)ds, 0 ≤ t < 1/µ. (A31)

Proof. Recall the representations of xn, and yn from (A13) and (A14). Fix any 0 ≤ T < 1/µ. Observe that Lemma 4immediately yields

supt≤T|yn(t)− y(t)| P−→ 0. (A32)

Next note that δ(x, y), as defined in (A12), is Lipschitz continuous on [0, 1] × [ε, 1] for any ε > 0 and we can choosethis ε > 0 in such a way that y(t) ≥ ε for all t ≤ T (since T < 1/µ). Therefore, the Lipschitz continuity of δ impliesthat there exists a constant C > 0 such that

supt≤T|δ(xn(t), yn(t))− δ(x(t), y(t))| ≤ C

(supt≤T|xn(t)− x(t)|+ sup

t≤T|yn(t)− y(t)|

). (A33)

Thus,

supt≤T|xn(t)− x(t)| ≤ sup

t≤T

|MXn (bntc)|n

+

∫ T

0

supt≤u|δ(xn(t), yn(t))− δ(x(t), y(t))|du+ oP(1)

≤ supt≤T

|MXn (bntc)|n

+ C

∫ T

0

supt≤u|xn(u)− x(u)|du+ C

∫ T

0

supt≤u|yn(u)− y(u)|du+ oP(1)

≤ εn + C

∫ T

0

supt≤u|xn(t)− x(t)|du,

where, by Lemma 4 and (A32),

εn := supt≤T

|MXn (bntc)|n

+ CT supt≤T|yn(t)− y(t)|+ oP(1),

which converges in probability to zero, as n→∞. Using Gronwall’s inequality [47, Theorem 5.1], we get

supt≤T|xn(t)− x(t)| ≤ εneCT

P−→ 0, (A34)

and the proof is now complete.

13

7. Properties of the limiting function

In this subsection, we characterize x explicitly, and derive some of its properties. Notice that the integral recursionequation (A31) can be expressed as a linear differential equation with x(0) = 1, and

x′(t) = −1−(µ− 1

1− µt+ λ

)x(t)

=⇒ x′(t) +

(µ− 1

1− µt+ λ

)x(t) = −1

=⇒ r(t)x′(t) + r(t)

(µ− 1

1− µt+ λ

)x(t) = −r(t),

(A35)

where the function r(t) is such that r(0) = 1 and

r′(t) = r(t)

(µ− 1

1− µt+ λ

)=⇒ r′(t)

r(t)=

µ− 1

1− µt+ λ

=⇒ log(r(t)) = −µ− 1

µlog(1− µt) + λt+ C

=⇒ r(t) = (1− µt)−µ−1µ eλt. C = 0, as r(0) = 1.

(A36)

Thus, (A35) now implies

r(t)x(t)− r(0)x(0) = −∫ t

0

r(s)ds

=⇒ x(t) = e−λt(1− µt)µ−1µ

(1−

∫ t

0

eλs(1− µs)−1+1µ ds

), t <

1

µ.

(A37)

The smallest root of the integral equation (A31) defined a J , must be the smallest positive solution of∫ t

0

eλs(1− µs)−1+1µ ds = 1. (A38)

The claim below establishes that such a J < 1/µ exists.

Claim 4. Let J := inf{t : x(t) = 0}, where x is given by (A2). Then J < 1/µ.

Proof. The integrand in the left hand side of (A38) is positive, and tends to ∞ as t increases to 1/µ. Therefore, the

integral∫ t0eλs(1− µs)−1+1/µds tends to infinity as well. Thus, there must exist a solution of (A38) which is smaller

that 1/µ, and the proof follows.

Next we prove Theorem 3, which shows that although for fixed µ, J < 1/µ, as µ grows large, J becomes arbitrarilyclose to 1/µ.

Proof of Theorem 3. Note that in the current notation an equivalent restatement is that as µ→∞, µJ → 1. In thisproof J should be understood as a function of µ and λ. Observe that, for t < 1/µ,

1−∫ t

0

eλs(1− µs)−1+1µ ds ≥ 1− eλt(1− µt)

1µ

∫ t

0

(1− µs)−1ds

≥ 1− eλt(1− µt)1µ

[− 1

µlog(1− µs)

]t0

∼ 1− eλµ

1

µlog(1− µt),

(A39)

and

1− eλµ

1

µlog(1− µt) = 0, =⇒ t =

1

µ(1− e−µe

−λµ

) ∼ 1

µas µ→∞. (A40)

Since J ≥ (1− e−µe−λµ

), the proof follows.

14

8. Convergence of exploration process on diffusion scale

Define the diffusion-scaled processes

Xn(t) :=√n(xn(t)− x(t)), Yn(t) :=

√n(yn(t)− y(t)), (A41)

and the diffusion-scaled martingales

MX

n (t) :=MXn (bntc)√

n, MY

n (t) :=MYn (bntc)√n

.

Now observe from (A13) that

Xn(t) = MX

n (t)− (µ− 1)

[∫ t

0

Xn(s)

yn(s)ds+

∫ t

0

x(s)√n

(1

yn(s)− 1

y(s)

)ds

]− λ

∫ t

0

Xn(s)ds+OP(n−1/2)

= MX

n (t)− (µ− 1)

∫ t

0

Xn(s)

yn(s)ds+

∫ t

o

x(s)(µ− 1)

yn(s)y(s)Yn(s)ds− λ

∫ t

0

Xn(s)ds+ oP(1).

Therefore, we can write

Xn(t) = MX

n (t) +

∫ t

0

fn(s)Xn(s)ds+

∫ t

0

gn(s)Yn(s)ds+ oP(1), (A42)

where

fn(t) = − (µ− 1)

yn(t)− λ, gn(t) =

(µ− 1)x(t)

yn(t)y(t). (A43)

Furthermore, (A14) yields

Yn(t) =√n(yn(t)− y(t)) = MY

n (t) + oP(1), (A44)

Based on the quadratic variation and covariation results in Section A 5, the following lemma shows that the martingaleswhen scaled by

√n converge to a diffusion process described by an SDE.

Lemma 6 (Diffusion limit of martingales). As n → ∞, (MX

n ,MY

n)d−→ (W 1,W 2), where the process (W 1,W 2) is

described by the SDE

dW1(t) =√β(t)

[ρ(t)dB1(t) +

√1− ρ(t)2dB2(t)

], dW2(t) = σdB1(t), (A45)

where B1 and B2 are two independent standard Brownian motions.

Proof. The idea is to use the martingale functional central limit theorem (cf. [48, Theorem 8.1]), where the convergenceof the martingales are characterized by the convergence of their quadratic variation process. Using the functionalLLN obtained in Proposition 5, we compute the asymptotics of the quadratic variations and covariation of M

X

n and

MY

n . From (A13) and (A16), we obtain

〈MY

n 〉(t) =1

n

bntc∑i=1

Vari−1(ζn(i))P−→ σ2t.

Again, (A13), (A17) and Proposition 5 yields

〈MX

n 〉(t) =1

n

bntc∑i=1

Vari−1(ξn(i))P−→∫ t

0

[(µ− 1)

y(s)+ λ

]x(s)ds =

∫ t

0

β(s)ds.

Finally, from (A13), (A24) and Proposition 5 we obtain

〈MX

n , MY

n 〉(t) =1

n

bntc∑i=1

Covi−1(ζn(i), ξn(i))P−→ σ2

∫ t

0

x(s)

y(s)ds =

∫ t

0

ρ(s)× σ√β(s)ds.

15

From the martingale functional central limit theorem, we get that (MX

n ,MY

n)d−→ (W 1, W 2), where (W 1, W 2) are

Brownian motions with zero means and quadratic covariation matrix[ ∫ t0β(s)ds

∫ t0ρ(s)× σ

√β(s)ds∫ t

0ρ(s)× σ

√β(s)ds σ2t

].

The proof then follows by noting the fact that (W 1,W 2)d

== (W 1, W 2).

Having proved the above convergence of martingales, we now establish weak convergence of the scaled explorationprocess to a suitable diffusion process.

Proposition 7 (Functional CLT of the exploration process). As n→∞, (Xn, Y n)d−→ (X,Y ) where (X,Y ) is the

two-dimensional stochastic process satisfying the SDE

dX(t) =√β(t)

[ρ(t)dB1(t) +

√1− ρ(t)2dB2(t)

]+ f(t)X(t)dt+ g(t)Y (t)dt,

dY (t) = σdB1(t),(A46)

with B1, B2 being independent standard Brownian motions, and f(t), g(t) and ρ(t) as defined in (A4).

Proof. First we show that ((Xn, Y n))n≥1 is a stochastically bounded sequence of processes. Indeed stochastic bound-edness (and in fact weak convergence) of the Yn process follows from Lemma 6. Further observe that for any T < 1/µ,by Proposition 5,

supt≤T|fn(t)− f(t)| P−→ 0, sup

t≤T|gn(t)− g(t)| P−→ 0, (A47)

where f, g are defined in (A4). Therefore, for any T < 1/µ,

supt≤T|Xn(t)| ≤ sup

t≤T|MX

n (t)|+ T supt≤T|gn(t)Yn(t)|+ sup

t≤T|fn(t)|

∫ T

0

supu≤t|Xn(u)|dt,

and again using Gronwall’s inequality, it follows that

supt≤T|Xn(t)| ≤

(supt≤T|MX

n (t)|+ T supt≤T|gn(t)Yn(t)|

)× exp

(T supt≤T|fn(t)|

).

Then stochastic boundedness of (Xn)n≥1 follows from Lemma 6, (A47), and the stochastic boundedness criterion forsquare-integrable martingales given in [48, Lemma 5.8].

From stochastic boundedness of the processes we can claim that any sequence (nk)k≥1 has a further subsequence(n′k)k≥1 ⊆ (nk)k≥1 such that

(Xn′k, Y n′k

)d−→ (X ′,Y ′), (A48)

along that subsequence, where the limit (X ′,Y ′) may depend on the subsequence (nk)k≥1. However, due to theconvergence result in Lemma 6 and (A47), the continuous mapping theorem (see [49, Section 3.4]) implies that thelimit (X ′,Y ′) must satisfy (A46). Again, the solution to the SDE in (A46) is unique, and therefore the limit (X ′,Y ′)does not depend on the subsequence (nk)k≥1. Thus, the proof is complete.

Proof of Theorem 2. First observe that

√n(Jn − J)

d−→ X(J) as n→∞.

Indeed this can be seen by the application of the hitting time distribution theorem in [47, Theorem 4.1], and noting

the fact that x′(J) = −1. Now since X is a centered Gaussian process, in order to complete the proof of Theorem 2,

we only need to compute Var(X(J)). We will use the following known result [50, Theorem 8.5.5] to calculate thevariance of X(t).

16

Lemma 8 (Expectation and variance of SDE). Consider the d-dimensional SDE given by

dZ(t) = (A(t)Z(t) + a(t))dt+

d∑i=1

bi(t)dBi(t), (A49)

where Z(0) = z0 ∈ Rd, the bi’s are Rd-valued functions, and the Bi’s are independent standard Brownian motions,i = 1, . . . , d. Then given Z(0) = x0, Z(t) has a normal distribution with mean vector m(t) and covariance matrixV (t), where m(t) and V (t) satisfy the recursion relations

d

dtm(t) = A(t)m(t) + a(t),

d

dtV (t) = A(t)V (t) + V (t)AT (t) +

d∑i=1

bibi(t)T , (A50)

with initial conditions m(0) = x0 and V (0) = 0.

In our case, observe from (A46) that

A(t) =

[f(t) g(t)

0 0

], a(t) =

[00

],

b1(t) =

[ρ(t)

√β(t)

σ

], b2(t) =

[√1− ρ(t)2

√β(t)

0

].

(A51)

Denote the variance-covariance matrix of (X(t), Y (t)) by

V (t) =

[σxx(t) σxy(t)σxy(t) σyy(t)

]. (A52)

Then

d

dtV (t) =

[σxx(t)f(t) + σxy(t)g(t) σxy(t)f(t) + σyy(t)g(t)

0 0

]+

[σxx(t)f(t) + σxy(t)g(t) 0σxy(t)f(t) + σyy(t)g(t) 0

]+

[ρ(t)2β(t)

√β(t)σρ(t)√

β(t)σρ(t) σ2

]+

[(1− ρ(t)2)β(t) 0

0 0

]=

[2σxx(t)f(t) + 2σxy(t)g(t) σxy(t)f(t) + σyy(t)g(t)σxy(t)f(t) + σyy(t)g(t) 0

]+

[β(t)

√β(t)σρ(t)√

β(t)σρ(t) σ2

].

Therefore, the variance of X(t) can be obtained from the solution of the recursion equations

dσxx(t)

dt= 2(σxx(t)f(t) + σxy(t)g(t)) + β(t),

dσxy(t)

dt= σxy(t)f(t) + σyy(t)g(t) +

√β(t)σρ(t),

(A53)

and the proof is thus completed by noting that σyy(t) = σ2t.

Appendix B: Simulation and comparison with Random Geometric Graphs

Fig. 7 confirms that the asymptotic analytical variance given in (A3) and (A4) is a sharp approximation for the crg

model with only 2000 vertices. Table II shows numerical values of V (c, αd) and compares the analytically obtained

values of J(c, αd) and V (c, αd), and simulated mean and variance for the random geometric graph ensemble. Theagreement again confirms the appropriateness of the crg(c, αd) model for modeling the continuum rsa. Furthermore,

V (c, αd) serves as an approximation for the value of V (c, d), the asymptotic variance of J(c, d) (suitably rescaled).Fig. 8 shows the density function of the random variable based on the Gaussian approximation in Theorem 2. Weobserve that both the mean and the fluctuations around the mean decrease with c. Indeed, the variance-to-meanratio has been typically observed to be smaller than one for rsa in the continuum, and it is generally believed thatdisordered packings are typically of sub-Poissonian nature with fluctuations that are not as large as for a Poissondistribution; see for instance the Mandel Q parameter in quantum physics [41]. So, while a closed-form expressionremains out of reach (as for the Mandel Q parameter [41]), our solvable model gives a way to describe accurately the

variance-to-mean ratio as V (c, αd)/J(c, αd).

17

0.075 0.080 0.0850

50

100

150

FIG. 7. Fitted normal curve for 2000 repetitions of the crg(20, α2) model with 1000 vertices. The solid curve represents the

normal density with properly scaled theoretical variance V (c, α2), centered around the sample mean.

c = 30

c = 20

c = 10

-0.2 0.2 0.4

1

2

3

4

5

6

7

FIG. 8. Fitted normal curves for the crg(c, α2) model for increasing c values 10, 20 and 30. As c increases, the curve becomemore sub-Poissonian.

Appendix C: Clustering coefficient of Random Geometric Graphs

The goal of this section is to derive the (local) clustering coefficient for rgg(c, d). Consider n uniformly chosenpoints on a d-dimensional box [0, 1]d and connect two points u, v by an edge if they are at most 2r distance apart. Fixany three vertex indices u, v, and w. We write u↔ v to denote that u and v share an edge. The clustering coefficientfor rgg(c, d) on n vertices is then defined by

Cn(c, d) := P (v ↔ w|u↔ v, u↔ w) . (C1)

The following theorem explicitly characterizes the asymptotic value of Cn(c, d).

rgg crg

n c Jn(c, 2) Vn(c, 2) J(c, α2) V (c, α2)200 10 0.1618 0.0166500 10 0.1608 0.0158 0.1454 0.0178

1000 10 0.1623 0.0155200 20 0.0887 0.0062500 20 0.0892 0.0068 0.0786 0.0057

1000 20 0.0890 0.0067200 30 0.0619 0.0039500 30 0.0620 0.0041 0.0538 0.0032

1000 30 0.0615 0.0043

TABLE II. Comparison between the observed mean and scaled variance nVar(Jn(c, 2)) for the rgg model, and the theoreticalmean and variance from Theorem 2 in dimension 2. The sample means and variances for the rgg model are calculated over150 samples.

18

Theorem 9. For any fixed c > 0, and d ≥ 1, as n→∞,

Cn(c, d)→ C(d) = d

∫ 1

0

xd−1I1− x24

(d+ 1

2,

1

2

)dx.

Remark 10. In order for Theorem 9 to hold we do not need the average degree c to be some constant value. Itremains true as long as the average degree is o(n). We only need to assume that rn → 0 as n→∞, so that for largeenough n, the 2rn neighborhood of a uniformly chosen point is inside [0, 1]d, with high probability. That is why wesuppress the subscript n below, and assume r to be sufficiently small.

Remark 11. It is worthwhile to note that the clustering coefficient is only a function of the dimension, and does notdepend on the density parameter. Intuitively, this is because the asymptotic local clustering coefficient at any vertexu can be thought as the ratio of the number of triangles u is part of, and the number of wedges centered at u. Inthergg model, as the density parameter changes, both the numerator and denominator change in proportionally.

We now prove Theorem 9. Observe that the rgg model can be constructed by throwing points sequentially atuniformly chosen locations independently, and then connecting to the previous vertices that are at most 2r distanceaway. Since the locations of the vertices are chosen independently, without loss of generality we assume that inthe construction of thergg model, the locations of u, v, w are chosen in this respective order. Now, the event{u↔ v, u↔ w, v ↔ w} occurs if and only if v falls within the 2r neighborhood of u and w falls within the intersectionregion of two spheres of radius 2r, centered at u and v, respectively. Let Bd(x, 2r) denote the d-dimensional spherewith radius 2r, centered at x, and let Vd(2r) denote its volume. Since r is sufficiently small, so that Bd(x, 2r) ⊆ [0, 1]d,using translation invariance, we assume that the location of u is 0. Let v,w denote the positions in the d-dimensionalspace, of vertices v and w, respectively. Notice that, conditional on the event {v ∈ Bd(0, 2r)}, the position v isuniformly distributed over Bd(0, 2r). Let V be a point chosen uniformly from Bd(0, 2r). Then the above discussionyields

C =P (u↔ v, u↔ w, v ↔ w)

P (u↔ v, u↔ w)(C2)

=1

(Vd(2r))2

∫v∈Bd(0,2r)

P (w ∈ Bd(0, 2r) ∩Bd(v, 2r)) dv

=1

Vd(2r)E[|Bd(0, 2r) ∩Bd(V , 2r)|]. (C3)

We shall use the following lemma to compute the expectation term in (C2).

Lemma 12 ([51]). For any x with ‖x‖ = ρ, the intersection volume |Bd(0, 2r)∩Bd(x, 2r)| depends only on ρ and r,and is given by

|Bd(0, 2r) ∩Bd(x, 2r)| = Vd(2r) · I1− ρ2

16r2

(d+ 1

2,

1

2

), (C4)

where Iz(a, b) denotes the normalized incomplete beta integral, given by

Iz(a, b) =

∫ z0ya−1(1− y)b−1dy∫ 1

0ya−1(1− y)b−1dy

.

Observe that the Jacobian corresponding to the transformation from the Cartesian coordinates (x1, . . . , xd) to thePolar coordinates (ρ, θ, φ1, . . . , φd−2), is given by

Jd(ρ, θ, φ1, . . . , φd−2) = ρd−1d−2∏j=1

(sin(φj))d−1−j . (C5)

19

Thus, (C2) reduces to

C =1

(Vd(2r))2

∫x∈Bd(0,2r)

|Bd(0, 2r) ∩Bd(x, 2r)|dx

=1

Vd(2r)

∫‖x‖≤2r

I1− ‖x‖

2

16r2

(d+ 1

2,

1

2

)dx

=1

Vd(2r)

∫ 2r

0

∫ 2π

0

∫ π

0

. . .

∫ π

0

ρd−1I1− ρ2

16r2

(d+ 1

2,

1

2

) d−2∏j=1

(sin(φj))d−1−j

d−2∏j=1

dφjdθdρ

=(∫ 2r

0

ρd−1dρ)−1 ∫ 2r

0

ρd−1I1− ρ2

16r2

(d+ 1

2,

1

2

)dρ,

since

Vd(2r) = 2π

( d−2∏j=1

∫ π

0

(sin(φj))d−1−jdφj

)∫ 2r

0

ρd−1dρ.

Therefore, putting x = ρ/2r, yields

C =(∫ 1

0

xd−1dx)−1 ∫ 1

0

xd−1I1− x24

(d+ 1

2,

1

2

)dx = d

∫ 1

0

xd−1I1− x24

(d+ 1

2,

1

2

)dx. (C6)