Corrected MeanField Model for Random Sequential Adsorption on Random Geometric Graphs
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Abstract
A notorious problem in mathematics and physics is to create a solvable model for random sequential adsorption of nonoverlapping congruent spheres in the ddimensional Euclidean space with \(d\ge 2\). Spheres arrive sequentially at uniformly chosen locations in space and are accepted only when there is no overlap with previously deposited spheres. Due to spatial correlations, characterizing the fraction of accepted spheres remains largely intractable. We study this fraction by taking a novel approach that compares random sequential adsorption in Euclidean space to the nearestneighbor blocking on a sequence of clustered random graphs. This random network model can be thought of as a corrected meanfield model for the interaction graph between the attempted spheres. Using functional limit theorems, we characterize the fraction of accepted spheres and its fluctuations.
Keywords
Random geometric graph Random sequential adsorption Jamming fraction Functional limit theorems Meanfield analysis1 Introduction
Equivalently, we may think of the interaction network of the n chosen centers of spheres by drawing an edge between two points if they are at most 2r distance apart. This is because a deposition attempt can block another deposition attempt if and only if the centers are at most 2r distance apart. The obtained random graph is known as the random geometric graph (\(\textsc {rgg}\)) [32]. The fraction of accepted spheres can be obtained via the following greedy algorithm to find independent sets of \(\textsc {rgg}\): Given a graph G, initially, all the vertices are declared inactive. Sequentially activate uniformly chosen inactive vertices of the graph and block the neighborhood until all the inactive vertices are exhausted. We refer to the above greedy algorithm as \(\textsc {rsa}\) on the graph G. If G has the same distribution as \(\textsc {rgg}\) on n vertices, then the final set of active vertices has the same distribution as the number of accepted spheres in the continuum after n deposition attempts. Thus, we one can equivalently study \(\textsc {rsa}\) on \(\textsc {rgg}\) to obtain the fraction of accepted spheres when \(\textsc {rsa}\) is applied in continuum.
In the classical setting, one keeps adding spheres until there is no place left to add another sphere and studies the fraction of area \(N_n(d)\) covered by the accepted spheres. Detailed results about \(N_n(d)\) involving law of large numbers and Gaussian fluctuations have been obtained in [33, 38]. The settings under consideration in this paper, however, are known as fixed or finite input packing in the literature. It was shown in [34] that there exist constants J(c, d) and V(c, d) such that \(J_n(c,d) \xrightarrow {\scriptscriptstyle \mathbb {P}} J(c,d)\) and \(\sqrt{n}(J_n(c,d)  J(c,d))\xrightarrow {\scriptscriptstyle d} {\mathrm {Normal}}(0,V(c,d))\). More detailed results about the point process of locations of accepted spheres and the number of accepted spheres have been obtained in [4, 5, 6, 17, 39]. This includes both moderate and large deviation results for \(N_n(d)\). To the best of our knowledge, finding an explicit quantitative characterization of J(c, d) for dimensions \(\ge 2\) remains an open problem, although numerical estimates have been obtained through extensive simulations [3, 13, 19, 41, 44, 47, 51].
In this paper, we do not aim to analyze the rsaprocess on rgg’s directly. Rather, we introduce an approximate approach for studying J(c, d) and V(c, d) by considering \(\textsc {rsa}\) on a clustered random graph model, designed to match the local spatial properties of the \(\textsc {rgg}\) model in terms of average degree and clustering. Exact analysis of this random graph model leads to expressions for the limiting jamming fraction and its fluctuations, in turn providing approximations for J(c, d) and V(c, d). Using simulations we show that these approximations are accurate. havior of \(\textsc {rsa}\) on \(\textsc {rgg}\) and our proposed random graph model via. extensive simulation.
2 Clustered Random Graphs

Partition the n vertices into random households of size \(1+{\mathrm {Poisson}}(\alpha c)\). This can be done by sequentially selecting \(1+{\mathrm {Poisson}}(\alpha c)\) vertices uniformly at random and declaring them as a household, and repeat this procedure until at some point the next \(1+{\mathrm {Poisson}}(\alpha c)\) random variable is at most the number of remaining vertices. All the remaining vertices are then declared a household too, and the household formation process is completed.

Now that all vertices are declared members of some household, the random graph is constructed according to a local and a global rule. The local rule says that all vertices in the same household get connected by an edge, leading to complete graphs of size 1+Poisson(\(\alpha c\)). The global rule adds a connection between any two vertices belonging to two different households with probability \((1\alpha )c/n\).
Seen as the topology underlying the \(\textsc {rsa}\) problem, the \({\textsc {crg}}(c,\alpha )\) model incorporates local clusters of overlapping spheres, which occur naturally in random geometric graphs; see Fig. 1b. We can now also consider rsaon the \({\textsc {crg}}(c,\alpha )\) model, by using the greedy algorithm that constructs an independent set on the graph by sequentially selecting vertices uniformly at random, and placing them in the independent set unless they are adjacent to some vertex already chosen. The jamming fraction \(J^\star _n(c,\alpha )\) is then the size of the greedy independent set divided by the network size n. From a highlevel perspective, we will solve the rsaproblem on the \({\textsc {crg}}(c,\alpha )\) model, and translate this solution into an equivalent result for \(\textsc {rsa}\) on the \(\textsc {rgg}(c,d)\).
Our ansatz is that for large enough n, a unique relation can be established between dimension d in \(\textsc {rgg}\) and the parameter \(\alpha = \alpha _d\) in \({\textsc {crg}}\), so that the jamming fractions are comparable, i.e., \(J_n(c, d)\approx J^\star _n(c, \alpha _d),\) and virtually indistinguishable in the large network limit. In order to do so, we map the \({\textsc {crg}}(c,\alpha )\) model onto the \(\textsc {rgg}(c,d)\) model by imposing two natural conditions. The first condition matches the average degrees in both topologies, i.e., c is chosen to be equal to \(nV_d(2r)\). The second condition tunes the local clustering.
\(\alpha _d\) for dimensions 1 to 5
\(\hbox {d}\)  1  2  3  4  5 

\(\alpha _d\)  0.750000  0.586503  0.468750  0.379755  0.310547 
The main goal of these works was to find greedy independent sets (or colorings) of large random networks. All these results, however, were obtained for nongeometric random graphs, typically used as first approximations for sparse interaction networks in the absence of any known geometry.
3 Main Results
3.1 Limiting Jamming Fraction
For the \({\textsc {crg}}(c,\alpha )\) model on n vertices, recall that \(J^\star _n(c,\alpha )\) denotes the fraction of active vertices at the end of the \(\textsc {rsa}\) process. We then have the following result, which characterize the limiting fraction:
Theorem 3.1
Upon substituting \(\alpha =\alpha _d\), \(J^\star (c,\alpha _d)=\lim _{n\rightarrow \infty }J^\star _n(c,\alpha _d)\) is completely characterized by (3.1) and serves an approximation for the intractable counterpart J(c, d), the limiting jammed fraction for the \(\textsc {rgg}(c,d)\) model. The choice of \(\alpha _d\), as discussed earlier, is given by (2.1) and shown in Table 1. Figure 3 validates the meanfield limit for the crgmodel, and shows the theoretical values \(J^\star (c,\alpha _2)\) from Theorem 3.1, along with the simulated values of \(J_n(c,2)\) on the \(\textsc {rgg}(c,2)\) model for values of c ranging from 0 to 30. Figure 4 shows further comparisons between \(J^\star (c,\alpha _d)\) and \(J_n(c,d)\) for dimensions \(d=3,4,5\), and densities \(0\le c\le 30\). All simulations use \(n=1000\) vertices. The remarkable agreement of the \(J^\star (c,\alpha _d)\)curves with the simulated results across all dimensions shows that the integral equation (3.1) accurately describes the meanfield largenetwork behavior of the \(\textsc {rsa}\) process, not only for the crgmodel, but also for the \(\textsc {rgg}\) model. The following result is a direct consequence of Theorem 3.1, and gives a simple law to describe the asymptotic fraction \(J^\star (c,\alpha _d)\) in the large density (\(c\rightarrow \infty \)) regime.
Corollary 3.1
As \(c\rightarrow \infty \), \( J^\star (c,\alpha _d) \sim (1+\alpha _dc)^{1}. \)
3.2 Fluctuations of the Jamming Fraction
The next theorem characterizes the fluctuations of \(J_n^\star (c,\alpha )\) around its mean:
Theorem 3.2
Comparison between the observed mean and scaled variance \(n{\mathrm {Var}}(J_n(c,2))\) for the \(\textsc {rgg}\) model, and the theoretical mean and variance from Theorem 3.2 in dimension 2. The sample means and variances for the \(\textsc {rgg}\) model are calculated over 150 samples
rgg  crg  

n  c  \(J_n(c,2)\)  \(V_n(c,2)\)  \(J^\star (c,\alpha _2)\)  \(V^\star (c,\alpha _2)\) 
200  10  0.1618  0.0166  0.1454  0.0178 
500  10  0.1608  0.0158  
1000  10  0.1623  0.0155  
200  20  0.0887  0.0062  0.0786  0.0057 
500  20  0.0892  0.0068  
1000  20  0.0890  0.0067  
200  30  0.0619  0.0039  0.0538  0.0032 
500  30  0.0620  0.0041  
1000  30  0.0615  0.0043 
Figure 5a confirms that the asymptotic analytical variance given in (3.3) and (3.4) is a sharp approximation for the crgmodel with only 2000 vertices. Table 2 shows numerical values of \(V^\star (c,\alpha _d)\) and compares the analytically obtained values of \(J^\star (c,\alpha _d)\) and \(V^\star (c,\alpha _d)\), and simulated mean and variance for the random geometric graph ensemble. The agreement again confirms the appropriateness of the \({\textsc {crg}}(c,\alpha _d)\) model for modeling the continuum \(\textsc {rsa}\). Furthermore, \(V^\star (c,\alpha _d)\) serves as an approximation for the value of V(c, d), the asymptotic variance of J(c, d) (suitably rescaled). Figure 5b shows the density function of the random variable based on the Gaussian approximation in Theorem 3.2. We observe that both the mean and the fluctuations around the mean decrease with c. Indeed, the variancetomean ratio has been typically observed to be smaller than one for rsain the continuum, and it is generally believed that the jamming fractions are typically of subPoissonian nature with fluctuations that are not as large as for a Poisson distribution; see for instance the Mandel Q parameter in quantum physics [36]. So, while a closedform expression remains out of reach (as for the Mandel Q parameter [36]), our solvable model gives a way to describe approximately the variancetomean ratio as \(V^\star (c,\alpha _d)/J^\star (c,\alpha _d)\).
4 Proof of Theorem 3.1
In this section we analyze several asymptotic properties of rsaon the \({\textsc {crg}}(c,\alpha )\) model. In particular, we will prove Theorem 3.1. We first introduce an algorithm that sequentially activates the vertices while obeying the hardcore exclusion constraint, and then analyze the exploration algorithm (see [8, 9, 16] for similar analyses in various other contexts). The idea is to keep track of the number of vertices that are not neighbors of already actives vertices (termed unexplored vertices), so that when this number becomes zero, no vertex can be activated further. The number of unexplored vertices can then be decomposed into a drift part which converges to a deterministic function and a fluctuation or martingale part which becomes asymptotically negligible in the meanfiled case (Theorem 3.1) but gives rise to the a system of SDEs with variance (3.3). The proof crucially relies on the Functional Laws of Large Numbers (FLLN) and the Functional Central Limit Theorem (FCLT). The key challenge here is that the process that keeps track of the number of unexplored vertices while the exploration algorithm is running does not yield a Markov process, so we have to introduce another process to make the system Markovian and analyze this twodimensional system.
For each vertex, the neighboring vertices inside and outside its own household will be referred to as ‘household neighbors’ and ‘distant neighbors’, respectively. If H denotes the size of the households, then \(H\sim 1+{\mathrm {Poisson}}(\alpha c)\). Therefore, \(\mathbb {E}\left( H\right) =1+\alpha c\), and \({\mathrm {Var}}(H)=\alpha c\). Furthermore, any two vertices belonging to two different households are connected by an edge with probability \(p_n=(1\alpha )c/n\), so the number of distant neighbors is a Bin\((nH1,p_n)\) random variable, Poisson\(((1\alpha )c)\) in the large n limit. As mentioned earlier, the total number of neighbors, is then asymptotically given by a Poisson(c) random variable. In this section we fix \(c>0\) and \(\alpha \in [0,1]\), and simply write \(J^\star _n\) and \(J^\star \) for \(J^\star _n(c,\alpha )\) and \(J^\star (c,\alpha )\) respectively.
4.1 The Exploration Algorithm
Instead of fixing a particular realization of the random graph and then studying rsaon that given graph, we introduce an algorithm which sequentially activates the vertices onebyone, explores the neighborhood of the activated vertices, and simultaneously builds the random graph topology on the activated and explored vertices. The joint distribution of the random graph and active vertices obtained this way is same as those obtained by first fixing the random graph and then studying rsa. The idea of exploring in the above fashion simplifies the whole analysis, since the evolution of the system can be described recursively in terms of the previous states, as described below in detail.

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 some vertex in A(t) as a distant neighbor.
4.2 State Description and Martingale Decomposition

\(X_n(t)\) decreases by one, when a new vertex v becomes active.

The household neighbors of v are selected from \(Y_n(t1)\) vertices, and \(X_n(t)\) decreases by an amount of the number of such vertices which are in \(\mathrm {U}(t1)\).

\(X_n(t)\) decreases by the number of distant neighbors of the newly active vertex that belong to \(\mathrm {U}(t1)\) (since they are transferred to \(\mathrm {BO}(t)\)).
4.3 Quadratic Variation and Covariation
Claim 1
Proof
The proof is immediate by observing that the random variable denoting the household size has variance \(\sigma ^2\). \(\square \)
Claim 2
Proof
Claim 3
Proof
Based on the quadratic variation and covariation results above, the following lemma shows that the martingales when scaled by n, converge to the zeroprocess.
Lemma 4.1
Proof
4.4 Convergence of the Scaled Exploration Process
The claim below establishes that \(J^{\star }<1/\mu \).
Claim 4
\(J^\star <1/\mu \).
Proof
We now complete the proof of Corollary 3.1.
Proof of Corollary 3.1
5 Proof of Theorem 3.2
Lemma 5.1
Proof
Having proved the above convergence of martingales, we now establish weak convergence of the scaled exploration process to a suitable diffusion process.
Proposition 5.2
Proof
Proof of Theorem 3.2
Lemma 5.3
6 Clustering Coefficient of Random Geometric Graphs
Proposition 6.1
Proof
Lemma 6.2
7 Discussion
We introduced a clustered random graph model with tunable local clustering and a sparse superimposed structure. The level of clustering was set to suitably match the local clustering in the topology generated by the random geometric graph. This resulted in a unique parameter \(\alpha _d\) that for each dimension d creates a onetoone mapping between the tractable random network model and the intractable random geometric graph. In this way, we offer a new perspective for understanding \(\textsc {rsa}\) on the continuum space in terms of rsaon random networks with local clustering. Analysis of the random network model resulted in precise characterizations of the limiting jamming fraction and its fluctuation. The precise results then served, using the onetoone mapping, as predictions for the fraction of covered volume for rsain the Euclidean space. Based on extensive simulations we then showed that these prediction were remarkably accurate, irrespective of density or dimension.
In our analysis the random network model serves as a topology generator that replaces the topology generated by the random geometric graph. While the latter is directly connected with the metric in the Euclidian space, the only spatial information in the topologies generated by the random network model is contained in the matched average degree and clustering. One could be inclined to think that random topology generators such as the \({\textsc {crg}}(c,\alpha _d)\) model may not be good enough. Indeed, this random network model reduces all possible interactions among pairs of vertices to only two principal components: the local interactions due to the clustering, and a meanfield distant interaction. There is, however, building evidence that such randomized topologies can approximate rigid spatial topologies when the local interactions in both topologies are matched. Apart from this paper, the strongest evidence to date for this line of reasoning is [25], where it was shown that the typical ensembles from the latentspace geometric graph model can be modeled by an inhomogeneous random graph model that matches with the original graph in terms of the average degree and a measure of clustering. We should mention that [25] is restricted to onedimensional models and does not deal with \(\textsc {rsa}\), but it shares with this paper the perspective that matching degrees and local clustering can be sufficient for describing spatial settings.
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