Spectral Analysis of the Adjacency Matrix of Random Geometric Graphs

Mounia Hamidouche?, Laura Cottatellucci†, Konstantin Avrachenkov

? Departement of Communication Systems, EURECOM, Campus SophiaTech, 06410 Biot, France † Department of Electrical, Electronics, and Communication Engineering, FAU, 51098 Erlangen, Germany Inria, 2004 Route des Lucioles, 06902 Valbonne, France

[email protected], [email protected], [email protected].

Abstract—In this article, we analyze the limiting eigen- multivariate statistics of high-dimensional data. In this case, value distribution (LED) of random geometric graphs the coordinates of the nodes can represent the attributes of (RGGs). The RGG is constructed by uniformly distribut- the data. Then, the metric imposed by the RGG depicts the ing n nodes on the d-dimensional torus Td ≡ [0, 1]d and similarity between the data. connecting two nodes if their `p-distance, p ∈ [1, ∞] is at In this work, the RGG is constructed by considering a most rn. In particular, we study the LED of the adjacency finite set Xn of n nodes, x1, ..., xn, distributed uniformly and matrix of RGGs in the connectivity regime, in which independently on the d-dimensional torus Td ≡ [0, 1]d. We the average vertex degree scales as log (n) or faster, i.e., choose a torus instead of a cube in order to avoid boundary Ω (log(n)). In the connectivity regime and under some effects. Given a geographical distance, rn > 0, we form conditions on the radius rn, we show that the LED of a graph by connecting two nodes xi, xj ∈ Xn if their `p- the adjacency matrix of RGGs converges to the LED of distance, p ∈ [1, ∞] is at most rn, i.e., kxi − xjkp ≤ rn, the adjacency matrix of a deterministic geometric graph where k.kp is the `p-metric defined as (DGG) with nodes in a grid as n goes to infinity. Then, for  1/p n finite, we use the structure of the DGG to approximate  Pd (k) (k) p  k=1 |xi − xj | p ∈ [1, ∞), the eigenvalues of the adjacency matrix of the RGG and kxi−xjkp =  (k) (k) provide an upper bound for the approximation error.  max{|xi − xj |, k ∈ [1, d]} p = ∞. Index Terms—Random geometric graphs, adjacency matrix, limiting eigenvalue distribution, Levy distance. The RGG is denoted by G(Xn, rn). Note that for the case p = 2 we obtain the Euclidean metric on Rd. Typically, the function rn is chosen such that rn → 0 when n → ∞. I. Introduction The degree of a vertex in G(Xn, rn) is the number of edges connected to it. The average vertex degree in G(Xn, rn) is In recent years, random graph theory has been applied given by [3] to model many complex real-world phenomena. A basic (d) d an = θ nrn, random graph used to model complex networks is the Erdös- Rényi (ER) graph [1], where edges between the nodes appear where θ(d) = πd/2/Γ(d/2 + 1) denotes the volume of the d- arXiv:1910.08871v1 [math.SP] 20 Oct 2019 with equal probabilities. In [2], the author introduces another dimensional unit hypersphere in Td and Γ(.) is the Gamma random graph called random geometric graph (RGG) where function. nodes have some random position in a metric space and the Different values of rn, or equivalently an, lead to different edges are determined by the position of these nodes. Since geometric structures in RGGs. In [3], different interesting then, RGG properties have been widely studied [3]. regimes are introduced: the connectivity regime in which an RGGs are very useful to model problems in which the scales as log(n) or faster, i.e., Ω(log(n))1, the thermody- geographical distance is a critical factor. For example, RGGs namic regime in which an ≡ γ, for γ > 0 and the dense have been applied to wireless communication network [4], regime, i.e., an ≡ Θ(n). sensor network [5] and to study the dynamics of a viral RGGs can be described by a variety of random matrices spreading in a specific network of interactions [6], [7]. such as adjacency matrices, transition probability matrices Another motivation for RGGs in arbitrary dimensions is and normalized Laplacian. The spectral properties of those

This research was funded by the French Government through 1The notation f(n) = Ω(g(n)) indicates that f(n) is bounded the Investments for the Future Program with Reference: Labex below by g(n) asymptotically, i.e., ∃K > 0 and no ∈ N such that UCN@Sophia-UDCBWN. ∀n > n0 f(n) ≥ Kg(n). random matrices are powerful tools to predict and analyze where the term χ[xi ∼ xj] takes the value 1 when there is a complex networks behavior. In this work, we give a special connection between nodes xi and xj in G(Xn, rn) and zero attention to the limiting eigenvalue distribution (LED) of the otherwise, represented as adjacency matrix of RGGs in the connectivity regime.  Some works analyzed the spectral properties of RGGs in  1, kxi − xjkp ≤ rn, i 6= j, p ∈ [1, ∞] different regimes. In particular, in the thermodynamic regime, χ[xi ∼ xj] =  the authors in [8], [9] show that the spectral measure of the  0, otherwise. adjacency matrix of RGGs has a limit as n → ∞. However, A similar definition holds for A(Dn) defined over due to the difficulty to compute exactly this spectral measure, G(Dn, rn). The matrices A(Xn) and A(Dn) are symmetric Bordenave in [8] proposes an approximation for it as γ → ∞. and their spectrum consists of real eigenvalues. We denote In the connectivity regime, the work in [6] provides a by {λi, i = 1, .., n} and {µi, i = 1, .., n} the sets of all real closed form expression for the asymptotic spectral moments eigenvalues of the real symmetric square matrices A(Dn) of the adjacency matrix of G(Xn, rn). Additionnaly, Bor- and A(Xn) of order n, respectively. The empirical spectral denave in [8] characterizes the spectral measure of the adja- 0 distribution functions vn(x) and vn(x) of the adjacency cency matrix normalized by n in the dense regime. However, matrices of an RGG and a DGG, respectively are defined in the connectivity regime and as n → ∞, the normalization as factor n puts to zero all the eigenvalues of the adjacency n n 1 X 1 X matrix that are finite and only the infinite eigenvalues in the v (x) = I{µ ≤ x} and v0 (x) = I{λ ≤ x}, n n i n n i adjacency matrix are nonzero in the normalized adjacency i=1 i=1 matrix. Motivated by this results, in this work we analyze the where I{B} denotes the indicator of an event B. behavior of the eigenvalues of the adjacency matrix without Let a0 be the degree of the nodes in G(D , r ). In the normalization in the connectivity regime and in a wider range n n n following Lemma 1 we provide an upper bound for a0 under of the connectivity regime. n any ` -metric. First, we propose an approximation for the actual LED of p the RGG. Then, we provide a bound on the Levy distance Lemma 1. For any chosen `p-metric with p ∈ [1, ∞] and between this approximation and the actual distribution. More d ≥ 1, we have precisely, for > 0 we show that the LEDs of the adjacency d 0 1 d 1 matrices of the RGG and the deterministic geometric graph a ≤ d p 2 a 1 + . n n 1/d (DGG) with nodes in a grid converge to the same limit when 2an √ 2 an scales as Ω(log (n) n) for d = 1 and as Ω(log (n)) for Proof. See Appendix A d ≥ 2. Then, under the `∞-metric we provide an analytical approximation for the eigenvalues of the adjacency matrix of To prove our result on the concentration of the LED of RGGs by taking the d-dimensional discrete Fourier transform RGGs and investigate its relationship with the LED of DGGs (DFT) of an n = Nd tensor of rank d obtained from the first under any `p-metric, we use the Levy distance between two block row of the adjacency matrix of the DGG. distribution functions defined as follows. A B The rest of this paper is organized as follows. In Section Definition 1. ([10], page 257) Let vn and vn be two II we describe the model, then we present our main results A B distribution functions on R. The Levy distance L(vn , vn ) on the concentration of the LED of large RGGs in the between them is the infimum of all positive such that, for connectivity regime. Numerical results are given in Section all x ∈ R III to validate the theoretical results. Finally, conclusions are A B A given in Section IV. vn (x − ) − ≤ vn (x) ≤ vn (x + ) + . Lemma 2. ([11], page 614) Let A and B be two n × n Her- II. Spectral Analysis of RGGs mitian matrices with eigenvalues λ1, ..., λn and µ1, ..., µn, To study the spectrum of G(Xn, rn) we introduce an respectively. Then auxiliary graph called the DGG. The DGG denoted by 1 L3(vA, vB) tr(A − B)2, G(Dn, rn) is formed by letting Dn be the set of n grid points n n 6 n that are at the intersections of axes parallel hyperplanes with A B −1/d 0 0 where L(vn , vn ) denotes the Levy distance between the separation n , and connecting two points xi, xj ∈ Dn A B 0 0 empirical distribution functions vn and vn of the eigenvalues if kxi − xjkp ≤ rn with p ∈ [1, ∞]. Given two nodes, we assume that there is always at most one edge between them. of A and B, respectively.

There is no edge from a vertex to itself. Moreover, we assume Let Mn be the minimum bottleneck matching distance that the edges are not directed. corresponding to the minimum length such that there exists Let A(Xn) be the adjacency matrix of G(Xn, rn), with a perfect matching of the random nodes to the grid points entries for which the distance between every pair of matched points A(Xn)ij = χ[xi ∼ xj], is at most Mn. Sharp bounds for Mn are given in [12][13][14]. We repeat Proof. The proof follows along the same lines of Proposition them in the following lemma for convenience. A.1 in [?] when extended to a unit torus and applied to i.i.d. and uniformly distributed nodes. Lemma 3. Under any `p-norm, the bottleneck matching is 1/d! We can now state the main theorem on the concentration log n • M = O , when d ≥ 3 [12]. of the adjacency matrix of G(Xn, rn). n n For d ≥ 1, p ∈ [1, ∞], a ≥ 1, M = o(r ) and  !1/2 Theorem 1. n n log3/2 n t > 0, we have • Mn = O   , when d = 2 [13]. n −a ε2 2M {L3 (v , v0 ) > t} ≤ 2n exp n 1 − n r ! P n n log −1 3 rn • M = O , with prob. ≥ 1 − , d = 1 n (d) n n n θ (rn − 2Mn)(a − 1) + 1 [14]. + ! t an(2−c) 1 + 4 d p 2d+3 Under the condition Mn = o(rn), we provide an upper a 0 2 bound for the Levy distance between vn and v in the p 2d+6 (d) (d) n d 2 θ + 2θ an following lemma. + , (2) n2t2 Lemma 4. For d ≥ 1, p ∈ [1, ∞] and Mn = o(rn), the Levy (2 − c) t 2Mn 0 where ε = 1 + − and distance between vn and v is upper bounded as p d+2 rn n d 2 an 4 d c = 1 + 1 . 1 1/d 3 0 d+1 1 X 2an L (v , v ) ≤ d p 2 N(x ) − a n n n i n i In particular, for every t > 0, a ≥ 2, > 0 and a that (1) √ n scales as Ω(log(n) n) when d = 1, as Ω(log2(n)) when 1 d+1 2 X 0 + d p 2 a − L + a , n n i n d ≥ 2, we have i 3 0 lim L (vn, vn) > t = 0. where, N(xi) denotes the degree of xi in G(Xn, rn) and n→∞ P L ∼ Bin n, θ(d) (r − 2M ). i n n Proof. See Appendix C. Proof. See Appendix B. This result is shown in the sense of convergence in probability by a straightforward application of Lemma 8 and The condition enforced on rn, i.e., Mn = o(rn) implies √ 9 on the random variable Li, then by applying Lemma 5 and that for > 0, (1) holds when an scales as Ω(log (n) n) 3 + 1+ 6 to ξn. for d = 1, as Ω(log 2 (n)) for d = 2 and as Ω(log (n)) In what follows, we provide the eigenvalues of A(D ) for d ≥ 3. n which approximates the eigenvalues of A(X ) for n suffi- In what follows, we show that the LED of the adjacency n ciently large. matrix of G(Xn, rn) concentrate around the LED of the adjacency matrix of G(Dn, rn) in the connectivity regime Lemma 7. For d ≥ 1 and using the `∞-metric, the eigen- in the sense of convergence in probability. values of A(Dn) are given by P Notice that the term N(xi)/2 in Lemma 4 counts the d msπ 0 1/d i Y sin( N (an + 1) ) λm ,...,m = − 1, (3) number of edges in G(Xn, rn). For convenience, we denote 1 d sin( msπ ) P s=1 N N(xi)/2 as ξn. To show our main result we apply the i 0 d Chebyshev inequality given in Lemma 5 on the random where, m1, ..., md ∈ {0, ...N−1}, an = (2kn +1) −1, kn = d bNrnc and n = N . The term bxc is the integer part, i.e., variable ξn. For that, we need to determine Var(ξn) the greatest integer less than or equal to x. Lemma 5. (Chebyshev Inequality) Let X be a random Proof. See Appendix D. variable with an expected value EX and a variance Var (X). Then, for any t > 0 The proof utilizes the result in [15] which shows that the eigenvalues of the adjacency matrix of a DGG in d are Var(X) T {|X − X| ≥ t} ≤ . found by taking the d-dimensional DFT of an Nd tensor of P E t2 rank d obtained from the first block row of A(Dn). Lemma 6. When x1, ..., xn are i.i.d. uniformly distributed in For > 0, Theorem 1 shows that when an scales as √ 2 the d-dimensional unit torus Td = [0, 1] Ω(log (n) n) for d = 1 and as Ω(log (n)) when d ≥ 2, the LED of the adjacency matrix of an RGG concentrate around (d) (d) Var (ξn) ≤ [θ + 2θ an]. the LED of the adjacency matrix of a DGG as n → ∞. REFERENCES [1] P. Erdos, “On random graphs,” Publicationes Mathematicae, vol. 6, pp. 290–297, 1959. [2] E. N. Gilbert, “Random plane networks,” Journal of the Society for Industrial and Applied Mathematics, vol. 9, no. 4, pp. 533–543, 1961. [3] M. Penrose, Random geometric graphs. Oxford University Press, 2003. [4] C. Bettstetter, “On the minimum node degree and connectivity of a wireless multihop network,” in Proceedings of the 3rd ACM International Symposium on Mobile ad hoc Networking & computing, 2002, pp. 80–91. [5] J. Yick, B. Mukherjee, and D. Ghosal, “Wireless sensor network survey,” Computer Networks, vol. 52, no. 12, pp. 2292–2330, 2008. [6] V. M. Preciado and A. Jadbabaie, “Spectral analysis of virus spreading in random geometric networks,” IEEE Conference on Decision and Control, 2009. log(n) [7] A. Ganesh, L. Massoulié, and D. Towsley, “The effect of (a) Connectivity regime, rn = √ , n = 2000. n network topology on the spread of epidemics,” in Proc. of Fig. 1. An illustration of the cumulative distribution IEEE Conference on Computer Communications (INFOCOM), 2005. function of the eigenvalues of an RGG. [8] C. Bordenave, “Eigenvalues of Euclidean random matrices,” Random Structures & Algorithms, vol. 33, no. 4, pp. 515–532, 2008. Therefore, for n sufficiently large, the eigenvalues of the [9] P. Blackwell, M. Edmondson-Jones, and J. Jordan, Spectra of DGG given in (3) approximate very well the eigenvalues of adjacency matrices of random geometric graphs. Unpub- lished, 2007. the DGG. [10] J. C. Taylor, An introduction to measure and probability. III. Numerical Results Springer Science & Business Media, 2012. [11] Z. D. Bai, “Methodologies in spectral analysis of large dimen- We present simulations to validate the results obtained sional random matrices, a review,” Statistica Sinica, vol. 9, pp. in Section II. More specifically, we corroborate our results 611–677, 1999. on the spectrum of the adjacency matrix of RGGs in the [12] P. W. Shor and J. E. Yukich, “Minimax grid matching and connectivity regime by comparing the simulated and the empirical measures,” The Annals of Probability, pp. 1338– 1348, 1991. analytical results. [13] F. T. Leighton and P. Shor, “Tight bounds for minimax grid Fig. 1(a) shows the cumulative distribution functions of the matching, with applications to the average case analysis of eigenvalues of the adjacency matrix of an RGG realization algorithms,” in Proc. 18th Annual ACM Symposium on Theory and the analytical spectral distribution in the connectivity of Computing, 1986, pp. 91–103. regime. We notice that for the chosen average vertex degree [14] A. Goel, S. Rai, and B. Krishnamachari, “Sharp thresholds for √ monotone properties in random geometric graphs,” in Proc. an = log(n) n and d = 1, the curves corresponding to the eigenvalues of the RGG and the DGG fit very well for a large 36th Annual ACM Symposium on Theory of Somputing, 2004, pp. 580–586. n value of . [15] A. Nyberg, “The Laplacian spectra of random geometric IV. Conclusion graphs,” Ph.D. dissertation, 2014. [16] S. Rai, “The spectrum of a random geometric graph is con- In this work, we study the spectrum of the adjacency centrated,” Journal of Theoretical Probability, vol. 20, no. 2, matrix of RGGs in the connectivity regime. Under some pp. 119–132, 2007. [17] S. Janson, T. Luczak, and A. Rucinski, Random graphs. John conditions on the average vertex degree an, we show that the LEDs of the adjacency matrices of an RGG and a DGG Wiley & Sons, 2000. [18] R. M. Gray, “Toeplitz and circulant matrices: A review,” Foun- converge to the same limit as n → ∞. Then, based on the dations and Trends R in Communications and Information regular structure of the DGG, we approximate the eigenvalues Theory, vol. 2, no. 3, pp. 155–239, 2006. of A(Xn) by the eigenvalues of A(Dn) by taking the d- dimensional DFT of an Nd tensor of rank d obtained from APPENDIX A the first block row of A(Dn). PROOFOF LEMMA 1 In this Appendix, we upper bound the vertex degree a0 V. Acknowledgement n under any `p-metric, p ∈ [1, ∞]. This research was funded by the French Government Assume that G(Xn, rn) and G(Dn, rn) are formed using 0 through the Investments for the Future Program with Ref- the `∞-metric and let an and an be their average vertex erence: Labex UCN@Sophia-UDCBWN. degree and vertex degree, respectively. In this case, for a d-dimensional DGG with n = Nd nodes, 0 the vertex degree an is given by [15] 3 0 1 2 L (vn, vn) ≤ Trace [(A(Xn) − A(Dn)] 0 d d n an = (2kn + 1) − 1, with kn = bNrnc and n = N . 1 X X 2 = χ[x x ] − χ[x0 x0 ] n i ∼ j i ∼ j Therefore, for θ(d) ≥ 2 and d ≥ 1, we have i j (a) 1 X 0 2 X 0 = N(xi) + an − N(xi, xi) d n n 2da 1 i i a0 = (2k + 1)d − 1 ≤ n 1 + n n (d) 1/d (b) θ 2n rn 1 X 0 2 X ≤ N(xi) + a − Li d n n n d 1 i i ≤ 2 an 1 + . 2a1/d 1 2 n X X 0 ≤ N(xi) − Li + an n n 0 i i Now, let bn and bn be the vertex degree and the average 1 d+1 1 X 2 X 0 vertex degree in G(Dn, rn) and G(Xn, rn), respectively ≤ d p 2 N(x ) − L + a n i n i n when using any `p-metric, p ∈ [1, ∞]. Notice that for any i i p ∈ [1, ∞], we have 1 d+1 1 X ≤ d p 2 N(x ) − a n i n kxk ≤ kxk . i ∞ p 1 2 p d+1 X 0 + d 2 an − Li + an. 0 n Then, the number of nodes an that falls in the ball of i 0 0 0 radius rn is greater or equal than bn, i.e., an ≥ bn. Hence, (a) N(x ) = P χ[x ∼ x ] d Step follows from i i j 1 j b0 ≤ a0 ≤ 2da 1 + n n n 1/d 0 P 0 0 0 2n rn and an = χ[xi ∼ xj], and by defining N(xi, xi)= j 1/p d d d 2 an 1 P 0 0 = 1 + . χ[xi ∼ xj]χ[xi ∼ xj]. Step (b) follows from noticing that 1/p 1/d d 2n rn j 0 0 when kxi − xjkp ≤ rn − 2Mn, then kxi − xjkp ≤ rn. So, all 0 points within a radius of rn−2Mn of xi map to the neighbors It remains to show the relation between bn and bn. 0 0 of xi [16]. Thus, N(xi, xi) is stochastically greater than the Assume that the RGG is formed by connecting each two random variable L Bin(n, θ(d)(r − 2M )). 1/p i ∼ n n nodes when d kxi − xjk∞ ≤ rn. This simply means that the graph is obtained using the `∞-metric with a radius equal rn an to d1/p . Then, the average vertex degree of this graph is d1/p . APPENDIX C In addition, we have PROOFOF THEOREM 1

1 kxkp ≤ d p kxk∞. We provide an upper bound on the probability that the 0 Levy distance between the distribution functions vn and vn is higher than t > 0. The following lemmas are useful for Therefore, the following studies.

1 d Lemma 8. (Chernoff Bound) Let X be a random variable. b0 ≤ d1/p2db 1 + . n n 1/d 2n rn Then, for any t > 0

F (a) {X ≥ t} ≤ , P at

APPENDIX B where F (a) is the probability generating function and a ≥ 1. PROOFOF LEMMA 4 Lemma 9. ([17], Corollary 2.3, page 27) If X ∈ Bin(n, p), 3 In this Appendix, we upper bound the Levy distance EX = np and 0 < ε ≤ 2 , we have 0 between the distribution functions vn and vn. 2 By a straightforward application of Lemma 2, we have P{|X − EX| ≥ εEX} ≤ 2 exp(−ε EX/3). Step (a) follows by applying Lemma 1 and c = d 1  P 1 + 1/d . Then,  N(xi) 2an 3 0  1 d+1 i L (v , v ) > t ≤ d p 2 − a P n n P n n B ≤  t a (2 − c)  n | L − L | > + n − 2θ(d)nM 2 P L P E i i 1 n i  d p 2d+3 4 1 d+1 i 0  +d p 2 an − + a > t t a (2 − c) n n + n L > + n  P i 1 d p 2d+3 4 ( ) X nt ≤ nP {|ELi − Li| > anε} ≤ N(x ) − na > P i n 1 t a (2 − c) d p 2d+2 n i + nP Li > 1 + , ( ) d p 2d+3 4 nt na0 X n + P nan − 2 Li > 1 − 1 . p d+2 p d+1 where i d 2 d 2 t (2 − c) 2Mn Let ε = 1 + − . d+2 d p 2 an 4 rn ( ) nt X We continue by letting A = P N(xi) − nan > 1 , p d+2 i d 2 B = {| L − L | > a ε} . and 1 P E i i n n nt na0 o X n B = P nan − 2 Li > 1 − 1 . t a (2 − c) p d+2 p d+1 n i d 2 d 2 B2 = P Li > 1 + . d p 2d+3 4 We first upper bound the term A using Lemma 5 and 6. For n sufficiently large and consequently an sufficiently 3 large, we have 1 ≤ c < 2 and 0 < ε ≤ 2 . Therefore, by ( ) applying Lemma 9, we upper bound B as nt 1 X A = P N(xi) − nan > 1 p d+2 i d 2 nt P {|ELi − Li| > anε} = P |ξn − Eξn| > 1 n o p d+3 (d) d 2 ≤ P |ELi − Li| > (an − 2nθ Mn)ε 2 2 2d+6 p 2d+6 (d) (d) d p 2 Var(ξn) d 2 θ + 2θ an 2 ≤ ≤ . −ε (d) 2 2 2 2 ≤ 2 exp a − 2nθ M . n t n t 3 n n Next, we upper bound the term B. The last term B1 is upper bounded by using the Chernoff bound in Lemma 8. ( ) The probability generating function of the binomial ran- nt na0 X n dom variable L is given by B = P nan − 2 Li > 1 − 1 i p d+2 p d+1 i d 2 d 2 0 t a n ≤ n a − 2L > − n h (d) (d) i P n i 1 1 aθ (rn − 2Mn) + 1 − θ (rn − 2Mn) . d p 2d+2 d p 2d+1 0 t an + nP 2Li − an > 1 − 1 Therefore, for n sufficiently large, 1 ≤ c < 2 and a ≥ 1, d+2 d+1 d p 2 d p 2 we have 0 t an an ≤ nP |an − Li| > 1 − 1 + d p 2d+3 d p 2d+2 2 (d) n 0 θ (rn − 2Mn)(a − 1) + 1 t an an + n L > − + B2 ≤   . P i 1 1 t a (2 − c) d p 2d+3 d p 2d+2 2 n  1 +  (a) p d+3 4 t an(2 − c) a d 2 ≤ nP |an − Li| > 1 + d p 2d+3 4 Finally, taking the upper bounds of A and B obtained t an(2 − c) from the upper bounds of B1 and B2 all together, Theorem + nP Li > 1 + . d p 2d+3 4 1 follows. d APPENDIX D Using (4) and (5), the eigenvalues of A(Dn) in T are PROOFOF LEMMA 7 given by

 N−1 X 2πimh λ = exp − In this appendix, we provide the eigenvalues of the adja- m1,...,md  N cency matrix of the DGG using the ` -metric. h1,...,hd=0 ∞  N−kn−1 When d = 1, the adjacency matrix A(Dn) of a DGG in X 2πimh 1 − exp −  − 1 T with n nodes is a circulant matrix. A well known result N appearing in [18], states that the eigenvalues of a circulant h1,...,hd=kn+1 N−k −1 matrix are given by the discrete Fourier transform (DFT) of Xn 2πimh = exp − − 1 the ﬁrst row of the matrix. When d > 1, the adjacency matrix N of a DGG is no longer circulant but it is block circulant with h1,...,hd=kn+1 d−1 d−1 −2imsπ 2imsπ N ×N circulant blocks, each of size N×N. The author d N kn N (1+kn) Y e − e in [15], pages 85-87, utilizes the result in [18], and shows = − 1 2imsπ that the eigenvalues of the adjacency matrix in Td are found s=1 −1 + e N d by taking the d-dimensional DFT of an N tensor of rank d 2imsπ −2imsπ d (1+kn) kn obtained from the ﬁrst block row of the matrix A(D ) Y e N − e N n = − 1 2imsπ s=1 −1 + e N

d N−1 msπ X 2πi Y sin( (2kn + 1)) = N − 1 λm1,...,md = ch1,...,hd exp − m.h , (4) msπ N sin( N ) h1,...,hd=0 s=1

d msπ 0 1/d Y sin( (an + 1) ) = N − 1. sin( msπ ) where m and h are vectors of elements mi and hi, re- s=1 N spectively, with m1, ..., md ∈ {0, 1, ..., N − 1} and ch1,...,hd deﬁned as [15],

  0, for k < h , ..., h ≤ N − k − 1  n 1 d n  ch1,...,hd = or h1, ...hd = 0,    1, otherwise. (5)

The eigenvalues of the block circulant matrix A(Xn) follow the spectral decomposition [15], page 86,

A = FH ΛF, where Λ is a diagonal matrix whose entries are the eigenvalues of A(Xn), and F is the d-dimensional DFT matrix. It is well known that when d = 1, the DFT of an n × n matrix is the matrix of the same size with entries

1 F = √ exp (−2πimk/n) for m, k = {0, 1, ..., n−1}. m,k n

When d > 1, the block circulant matrix A is diagonalized by N N the d-dimensional DFT matrix F = FN1 ... FNd , i.e., tensor product, where FNd is the Nd-point DFT matrix.