On the Lossy Compression of Spatial Networks

In this paper, we address the lossy compression of spatial networks, namely random geometric graphs, where two nodes are connected by an edge with a probability that depends on the distance between the nodes. We carry out this study by considering the $n\mathbf{th}$ order information-distortion function, which quantifies the complexity of a random graph under a distortion criterion. Our main result is a partial characterization of the information-distortion function for a random geometric graph with the Hamming distortion measure.


I. INTRODUCTION
As graph-structured data of enormous sizes are ubiquitous nowadays, there is a need for studying the fundamental limits of graph compression.The main objectives of graph compression are to store a graph efficiently by utilizing less memory or to speed up an algorithm by running it on the compressed version.Recently, there have been some works [1]- [12] addressing various aspects of graph compression.In the case of lossless compression of a graph, the entropy of a graph governs the asymptotically optimal behavior.For instance, the works [1], [6]- [9] studied the entropy of a stochastic block graph, and the structural entropy of an Erdős-Rényi (ER) random graph and spatial networks such as random geometric graphs.
Because of storage limitations, it is also of interest to compress a large graph in a lossy way.We can allow a graph feature to be distorted within a certain level.A natural framework to study this problem is the classical rate-distortion theory.As graph sources are arbitrary and the distortion functions could be complicated, the study of lossy compression is a challenging task.For instance, [5] considered the lossy compression problem of an ER random graph with a distortion measure on the degree of the nodes.They showed lower and upper bounds on the asymptotic rate of compression.As it is difficult to extend their techniques to the other graph models such as spatial networks, which is the focus of this paper, we study the nth order information-distortion function.This is a natural analogue of entropy in the context of lossy compression.
This research was funded in whole or in part by the U. S. Army Research Laboratory and the U. S. Army Research Office (W911NF-22-1-0070), and EPSRC (EP/T02612X/1).For the purpose of Open Access, the authors have applied a CC BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission.

A. Information-Distortion Function of a Random Graph
Let G n = (V, E) be a graph with the vertex set V = [n] ≜ {1, 2, . . ., n} and the edge set E. We denote by G n the set of all graphs on n vertices.For simplicity, we restrict ourselves to undirected and simple graphs, where simple means that there are no self loops or multiple edges.For a random graph G n ∈ G n , P Gn denotes its probability distribution.
Given a distortion function ) and a distortion level D n , which specifies the cost of representing a graph with another graph, the nth order information-distortion function of a random graph G n is defined as The information-distortion function is useful for identifying the fundamental limits of lossy compression.It is well-known from the classical rate-distortion theory [13,Ch. 13] that for an i.i.d.source {X n } n≥1 and a per-letter distortion function d, the minimum rate of compression R(D) at an (average) distortion level D is given by R(D) = min The right-hand side of ( 2) is the first order informationdistortion function of the i.i.d.source {X n } n≥1 .On the other hand, the distribution P Gn of a random graph need not have the desired independence structure.For such arbitrary sources [14,Ch. 5], the nth order information-distortion function can be used to study the lossy compression.Consider the following variable-length lossy compression of random graphs.An encoder ϕ n : G n → {0, 1} * maps a graph to a codeword, which is a binary string of finite (but arbitrary) length.Here {0, 1} * denotes the set of binary strings of finite length.We also assume that the encoder satisfies the prefix condition.A decoder ψ : {0, 1} * → G n maps a codeword to a graph.The encoder and decoder pair is designed such that they satisfy the distortion constraint We can easily show that This follows from the observation that for a prefix code [13,Thm. 5.4.1] which is further lower bounded by where the reconstructed graph satisfies the average distortion condition, yielding (3).Though an upper bound on the minimum optimal length l * n (D n ) in terms of I Gn (D n ) and D n is not known for a fixed n, asymptotically optimal results for the rate are known.
Define the asymptotic rate of the compression scheme for the distortion level It was shown in [14, Thm.5.7.1 and Remark 5.7.3] that under some reasonable assumptions on the distortion function and by allowing a stochastic encoder instead of deterministic encoder.It was also noted in [14,Cor. 5.7.1] that the stochastic encoder can be implemented without loss of optimality by randomizing between two deterministic encoders.
The nth order information-distortion function is also useful in the regimes where the order of the distortion levels {D n } n≥1 is not the same as that of the source entropy.For example, in the result (6), a notion of rate is defined with respect to the scaling n 2 with D n = n 2 D. But we can consider other regimes that are of interest, say H(G n ) = Θ(n 2 ) and D n = n log nD.For such cases, where we may not be able to define a useful notion of rate, we can still work with I Gn (D n ) owing to the inequality (3).The effect of distortion on the information-distortion function can be made explicit through the Lagrangian dual formulation with the corresponding coefficient quantifying the effect of distortion on compression.With the above motivation, we study the information-distortion function of spatial networks in this paper.

II. INFORMATION-DISTORTION FUNCTION OF SPATIAL NETWORKS
Random spatial networks are those random graphs with an underlying hidden spatial embedding of the nodes.One important example of a random spatial network is a random geometric graph.In a random geometric graph, vertices are in a d-dimensional Euclidean space and the edge connectivity between a pair of nodes is governed by the distance between them.
Formally, the vertices are spatially embedded in a bounded set K ⊂ R d with strictly positive volume with respect to the Lebesgue measure.Let K ≜ sup x,y∈K ∥x − y∥ 2 be the diameter of the domain K.For a given connectivity function p : [0, K] → [0, 1] and a fixed positive integer n, a random geometric graph (RGG) is obtained as follows.We first draw n points X = (X i : 1 ≤ i ≤ n) uniformly at random from K and independently of each other.The distance between two nodes i and j is denoted by R i,j ≜ ∥X i − X j ∥ 2 .We then form a graph G n on the vertex set V = [n] by drawing an edge between a pair of vertices i and j with probability p(R i,j ) for 1 ≤ i < j ≤ n.Let P Gn denote the induced probability distribution on the space of all graphs G n with n vertices.
The presence or absence of an edge between vertices i and j in a graph is denoted by a binary variable E i,j .This allows us to view a graph G n in terms of its adjacency matrix representation (E i,j ) i,j∈ [n] , where E i,j = 1 if i and j are adjacent in G n , and E i,j = 0 otherwise.As the graphs are simple and undirected, this matrix is symmetric with zeros on the diagonal.It means that G n can equivalently be represented by {E i,j : 1 ≤ i < j ≤ n}.Hence, the probability of a random geometric graph G n is completely specified by the joint distribution of the collection {E i,j : 1 ≤ i < j ≤ n}, and vice versa.

A. Hamming Distortion Measure
While there are various distortion measures that one can work with depending on the context, in this preliminary work, we consider the Hamming distortion measure d n between two graphs G n and Ĝn that counts the number of pairs of vertices for which the values of edge random variables differ in G n and Ĝn , i.e., The Hamming distortion penalizes a reconstructed graph Ĝn that has many pairs of vertices with mismatched edge random variables from the original graph G n .

B. Main Results
In this section, we will present our result on the characterization of the nth order information-distortion function (1) of a random geometric graph with the Hamming distortion measure.For establishing the result, we first study the informationdistortion function with conditioning on the node locations X, which is defined as This quantity arises when X is available as side information to both the encoder and decoder.It is worth noting that the information-distortion functions with and without conditioning satisfy the following inequality.

Lemma 1 ([15]
).For a pair (X, G n ) and D n ≥ 0, Before we present our result, let us define a few quantities, which will simplify the notation later on.We define R ≜ ∥X − Y ∥ 2 , where X and Y are independent and uniformly distributed random variables over K.For D n ≥ 0 and a connectivity function p : [0, K] → [0, 1], define a function q * : [0, K] → [0, 1] as follows: with µ chosen to satisfy the condition and set µ = 1 2 otherwise.Theorem 1 (Characterization of I Gn|X (D n )).Let G n be a random geometric graph with X being the node locations and p : [0, K] → [0, 1] being the connectivity function.The nth order information-distortion function of G n conditioned on node locations at the Hamming distortion D n is given by where q * : [0, K] → [0, 1] is defined as in (11).
Proof.See Section IV.
The above theorem together with Lemma 1 can be used to obtain a characterization of I Gn (D n ).To state this result, we need the following concept of the largest convex lower bound.The convex envelope, where the supremum is pointwise.
Theorem 2 (Bounds on I Gn (D n )).Let G n be a random geometric graph with X being the node locations and p : [0, K] → [0, 1] being the connectivity function.The nth order information-distortion function of G n at the Hamming distortion D n is bounded as below and where q * : [0, K] → [0, 1] is defined as in (11), and the function J Gn is defined as Proof.The lower bound (13) follows readily by combining the result of Theorem 1 with the lower bound in Lemma 1.
For the upper bound (14), first note that To see this, consider, in (1), the conditional distribution ]}, we will combine the result of Theorem 1 with the upper bound in Lemma 1.Consider the mutual information I(G n ; X) term, which can be bounded using the same approach as in [8].
where (a) follows by applying the chain rule of entropy and noting that E i,j is independent of the rest of the random variables given X i and X j for all i < j, and in (b), we use the fact that for i < j, (14) follows from the definition of the convex envelope.

III. NUMERICAL EVALUATION
In this section, we will explore an example and numerically evaluate the bounds on the information-distortion function.A connectivity function that arises in spatial networks, namely wireless networks [8], is where the parameter r 0 denotes the typical connectivity range and η is called the path loss exponent.We consider two spatial domains: a unit square and a circle of unit area.In Fig. 1 and  2, we plot the upper and lower bounds ( 13) and ( 14) on the normalized information distortion function The quantities are normalized so that the effect of the connectivity function on the information-distortion function can be seen more clearly.For instance, it can be seen in Fig. 1 that if the typical connectivity range r 0 increases, the information distortion function increases up to a certain r 0 and it decreases later on.This behavior is closely related to the complexity of the random graph.On the other hand, Fig. 2  IV.PROOF OF THEOREM 1 By conditioning on the node locations, the random geometric graph G n becomes an inhomogeneous Erdős-Rényi graph, meaning that the conditional edge random variables are independent with different edge probabilities.With this observation, we can follow the approach used in [13,Thm. 10.3.1] to characterize I Gn|X (D n ).

A. A Lower Bound on
In (a), we use the conditional independence of the edge random variables.In the inequality (b), the bound for r ∈ [0, K] and i < j.The inequality (d), where we set q(r) ≜ i<j qi,j (r) for r ∈ [0, K], follows immediately from Jensen's inequality and the concavity of the binary entropy function: It immediately follows from the distortion condition that q(r) satisfies the condition, n 2 E[q(R)] ≤ D n .By minimizing the lower bound in (d) over all functions q(r) satisfying the condition n 2 E[q(R)] ≤ D n , we get The optimization problem in ( 16) is equivalent to solving the following problem: where we use the notation f (r) to denote min{f (r), 1−f (r)}, for a function f : [0, K] → [0, 1].Let us consider two separate cases to solve this optimization problem by finding an optimizer q * .In the case of n ] for every q(r) ≤ p(r).Here, q * (r) = min{p(r), µ} with µ = 1 2 .In the case of D n ≤ n 2 E[p(r)].Consider the function q * (r) = min{p(r), µ} with µ chosen such that n 2 E[q * (R)] = D n .It is possible to choose such a µ depending on the value of D n because n 2 E[min{p(r), µ}] is a continuous 1 and nondecreasing function of µ ∈ [0, 1  2 ] with the maximum value being n 2 E[p(r)], which is greater than D n .We will now show that A * = E [h 2 (q * (R))], i.e., E [h 2 (q(R))] ≤ E [h 2 (q * (R))] for any q that satisfies the constraints in the optimization problem (17).Assume that D n > 0 (which implies that µ > 0).On the other hand, if As a consequence of the concavity of the binary function, we have the following relation: for any q(r) ∈ [0, 1] and q * (r) ∈ (0, 1) with h . If q * (r) = 0 then p(r) = 0 (because µ > 0), which means that q(r) ≤ p(r) = 0.By using these observations, we can write Let A = {r : µ ≤ p(r)}.On A, q * (r) = µ, and on Using the above observations, we can bound (18) as follows: 1 As min{p(r), µ} is bounded from above by p(r), which is an integrable function, the continuity immediately follows by applying the dominated convergence theorem (DCT) for any convergent sequence µn → µ.
where (e) follows from (19), and the last inequality is due to the fact that q(r) ≤ p(r) and the fact that on the event A c , p(r) ≤ µ, which implies that log as the derivative of binary entropy function is non-increasing.This shows that A * = E [h 2 (q * (R))], with q * (r) being an optimizer.Hence we have where q * : [0, K] → [0, 1] and q * (r) is defined as above.

V. FUTURE WORK
In this work, we obtain upper and lower bounds on the nth order information-distortion function of random geometric graph under the Hamming distortion criterion.We intend to extend the current work in future by considering various distortion measure that take into account more practically relevant features such as clustering, transitivity and so on.

Fig. 1 .Fig. 2 .
Fig. 1.Upper and lower bounds on the (normalized) information-distortion function as functions of the connectivity parameter r 0 on two spatial domains: a) a unit square b) a circle of unit area.For this plot, the (normalized) distortion D = 0.005 and η = 2.