Optimal Filter Design for Consensus on Random Directed Graphs

OPTIMAL FIL TER DESIGN FOR CONSENSUS ON RANDOM DIRECTED GRAPHS Stephen Kruzick and J os ´ e M. F . Moura Carnegie Mellon Uni versity , Department of Electrical Engineering 5000 Forbes A venue, Pittsb urgh, Pennsylv ania 15213 ABSTRA CT Optimal design of consensus acceleration graph filters relates closely to the eigen values of the consensus iteration matrix. This task is complicated by random networks with uncertain iteration matrix eigen values. Filter design methods based on the spectral asymptotics of consensus iteration matrices for large-scale, random undirected networks have been pre- viously dev eloped both for constant and for time-varying network topologies. This work builds upon these results by extending analysis to large-scale, constant, random dir ected networks. The proposed approach uses theorems by Girko that analytically produce deterministic approximations of the empirical spectral distribution for suitable non-Hermitian random matrices. The approximate empirical spectral dis- tribution defines filtering regions in the proposed filter opti- mization problem, which must be modified to accommodate complex-v alued eigen values. Presented numerical simula- tions demonstrate good results. Additionally , limitations of the proposed method are discussed. Index T erms — graph signal processing, filter design, dis- tributed average consensus, random graph, random matrix, spectral statistics, stochastic canonical equations 1. INTRODUCTION Distributed av erage consensus, an iterative network process in which nodes compute the av erage of data spread among the netw ork nodes through local communications, represents an important network agreement problem [1]. This task ap- pears in network-related applications such as processor load balancing [2], sensor data fusion [3], distrib uted inference [4], and formation control or flocking of autonomous agents [5]. Every network node begins with an initial scalar data element, collected into a vector x 0 in which each row corresponds to a node. They each maintain a scalar state v ariable o ver time, collected into a vector x n at time iteration n . At each iter- ation, the nodes communicate with neighboring nodes in the network and update their state according to a linear combina- tion of locally av ailable data. This implements the dynamics x n = W ( G ) x n − 1 (1) where W ( G ) is the consensus iteration weight matrix, which must respect the network graph structure G [4]. For a giv en Stephen Kruzick (skruzick@andrew .cmu.edu) and Jos ´ e M. F . Moura (moura@andrew .cmu.edu) are with the Department of Electrical and Com- puter Engineering at Carnegie Mellon University in Pittsbur gh, P A, USA. This work was supported by NSF grant # CCF 1513936. connected network topology , there are many possible itera- tion weight matrices. Consensus will be reached provided the following conditions hold, ` > W = ` > , W 1 = 1 , ρ ( W − J ` ) < 1 , J ` = 1 ` > / ` > 1 (2) where ρ is the spectral radius and J ` is the matrix that pro- duces consensus to an av erage weighted by ` , the left eigen- vector of W corresponding to eigen value λ = 1 [4]. That is, lim n →∞ x n = J ` x 0 . (3) at an exponential rate go verned by ln ( ρ ( W − J ` )) [4]. Design of consensus dynamics for fast conv ergence can be approached in sev eral w ays, including design of the weight matrix W given the netw ork topology G [6] as well as design of the network topology G under constraints gi ven a weight matrix scheme W ( G ) [7]. A third approach in volves design- ing filters that are applied to the system state at each node. Some example filter design methods proposed for various sce- narios can be found in [8–14]. For this paper , each node im- plements (1) at each iteration. Additionally , a de gree d filter will periodically be applied to the state v alue recorded at each node ev ery d iterations. For filter coefficients { a n } n = d n =0 , this may be expressed as x n := P k = d k =0 a k x n − d + k , n ≡ 0 ( mod d ) . (4) Furthermore, for constant network topology G , this can be expressed as x n = p ( W ( G )) x n − d where p is a polynomial graph filter with coefficients { a n } n = d n =0 . The filter should be designed to optimize the con vergence rate by minimizing the con vergence factor 1 d ln ( p ( W ) − J ` ) . For constant, random networks, [15 – 18] find deterministic approximations for the empirical spectral distrib ution of consensus iteration matrices for large-scale random, symmetric networks and use this for consensus acceleration filter design, while [19] handles net- works with time-v arying (switching) network topology . This paper extends pre vious work connecting spectral asymptotics to graph filter design for consensus accelera- tion by e xamining the filter design problem in the conte xt of constant (not time-varying), random networks of large-scale. Section 2 discusses a random matrix theory method useful for analyzing the asymptotics of the empirical spectral dis- tribution of non-Hermitian random matrices, demonstrating its application to a consensus iteration matrix for an e xample directed random graph model. Section 3 poses a filter design optimization problem for directed random networks that se- lects eigen value filtering regions based on the deterministic approximations to the empirical spectral distributions of the iteration matrix, as deriv ed in the preceding section. Section 4 supports the proposed design method with numerical simu- lation results comparing filtered conv ergence rates. Finally , Section 5 provides concluding analysis. 2. DIRECTED NETWORKS: SPECTRAL ST A TISTICS Consider a potentially non-Hermitian N × N random ma- trix Ξ N with potentially complex eigenv alues λ i (Ξ N ) for i = 1 , . . . , N . The empirical spectral distribution and corre- sponding empirical spectral density , functions of the real and imaginary components of a complex parameter , encapsulate the eigen value information. These functions are, respecti vely , giv en by F Ξ N ( x,y ) = 1 N i = N X i =1 χ ( x ≤ Re { λ i (Ξ N ) } , y ≤ Im { λ i (Ξ N ) } ) (5) f Ξ N ( x,y ) = 1 N i = N X i =1 δ ( x − Re { λ i (Ξ N ) } , y − Im { λ i (Ξ N ) } ) (6) where χ is an indicator function and δ is the Dirac delta func- tion. Although these are random functions, the limiting be- havior of the empirical spectral distribution sometimes admits a deterministic characterization that pro vides useful informa- tion through theorems from random matrix theory . Classic ex- amples include the Wigner semicircular law [20], Marchenko- Pastur la w [21, 22], and Girko circular law [23]. Matrices arising from random networks necessitate analy- sis methods that handle random matrix models with structure that require certain entries to be zero. While many random matrix theory tools focus on matrices with independent, iden- tically distributed entries, theorems called stochastic canon- ical equations described by Girko [23] accommodate non- identically distributed entries and, thus, zeros forced by graph structure. These methods in volv e solving a system of equa- tions that depends on the random matrix model to find a de- terministic equiv alent for the empirical spectral distribution of a large-scale matrix. For symmetric matrices, Girko’ s K1 Equation was applied to network adjacenc y matrices and con- sensus iteration matrices in [15, 16, 24], information that was then used to inform filter design optimization problems for consensus acceleration in [17 – 19]. For the non-symmetric random network models on which this paper focuses, a much more complex method shown, in abridged form, as Theorem 1 (Girko’ s K25 Equation) is required to perform analysis. Theorem 1 (Girko’ s K25 Equation (abr .) [23]) Let Ξ N be a family of complex-valued N × N random matrices with independent entries that satisfy se veral re gularity conditions. (See Theor em 25.1 of [23] for the full list.) Let Ξ N have e x- pectation B N = E [Ξ N ] and centralization H N = Ξ N − B N with entry variance σ 2 N ,ij = E[ | ( H N ) ij | 2 ] . Then lim β → 0 + lim N →∞    F Ξ N ( x, y ) − b F Ξ N ,β ( x, y )    = 0 (7) almost sur ely , where ∂ 2 b F Ξ N ,β ( t,s ) ∂ x∂ y = ( − 1 4 π R ∞ β  ∂ 2 ∂ t 2 + ∂ 2 ∂ s 2  m N ( u,t, s ) du ( t, s ) / ∈ G 0 ( t,s ) ∈ G (8) (with the r egion G defined below) and m N ( u,t,s ) = 1 N tr h C 1 ( u,s,t ) + .. . ( B N − ( t + is ) I ) C 2 ( u,s,t ) − 1 ( B N − ( t + is ) I ) ∗  − 1  (9) for u > 0 . The matrices C 1 ( u, s, t ) and C 2 ( u, s, t ) ar e diag- onal matrices wit h entries that satisfy the system of equations ( C 1 ) kk ( u,s, t ) = u + j = N X j =1 σ 2 N ,kj   C 2 ( u,s, t ) + . . . ( B N − ( t + si ) I ) ∗ C 1 ( u,s, t ) − 1 ( B N − ( t + si ) I )  − 1  j j (10) ( C 2 ) `` ( u,s, t ) = 1 + j = N X j =1 σ 2 N ,j `   C 1 ( u,s, t ) + . . . ( B N − ( t + si ) I ) C 2 ( u,s, t ) − 1 ( B N − ( t + si ) I ) ∗  − 1  j j (11) for k , ` = 1 , . . . N . Ther e exists a unique solution to this system of equations among r eal positive analytic functions in u > 0 . The re gion G is given by G = ( ( t,s )      limsup β → 0 + limsup N →∞     ∂ ∂ β m N ( β ,t,s )     < ∞ ) . (12) Thus, for large-scale random matrices that satisfy the conditions of Theorem 1, an approximation to the empirical spectral density can be achie ved in which the pointwise er- ror conv erges to zero almost surely . The methods presented in Section 3 employ this approximate distribution and the corresponding density to define regions for filter response optimization. T owards that goal, the application of Girko’ s method to consensus iteration matrices that arise from di- rected random networks must first be justified, described, and demonstrated. In order to obtain a deterministic approximation of the empirical spectral distrib ution, the system of equations (10)-(11) must be solved numerically for numerous u values ranging from a small v alue of β > 0 to a very large value such that the integral in (8) can be approximated. This must be ac- complished for all t + si for which the v alue of F Ξ N ,β ( t, s ) is required. Hence, the system must be solved for numerous points in a three dimensional re gion. This represents a signif- icant computational burden that must be accomplished of fline in advance of network deployment using foreknowledge of the network iteration matrix distribution. Note that the system can be solved via an iterative fixed point search, similar to the procedure in [15, 16], because a unique solution exists. Also note that the system has 2 N equations, where N is typically large, presenting a computational challenge. When possible, reduction of the system of equations (10)-(11) via diagonalization addresses this problem, as done in [15, 16] for symmetric matrices and Girko’ s K1 equation. The follo wing example analyzes the application of this equation for deter- ministically approximating the empirical spectral distrib ution of a consensus iteration matrix for a non-symmetric stochastic block model (briefly , proofs and deri vations omitted). These results, and the design methods produced in Section 3 were used to produce the simulation results shown in Section 4. Example 1 (Stochastic Block Model) Consider a di- rected stochastic block model network [24] with M popula- tions of equal size S such that there are N = M S nodes. Let each pair of distinct nodes in populations 1 ≤ i, j ≤ M connect with probability Θ ij = Θ j i . Note that the con- nection probability is symmetric, but the outcomes for each link direction are independent. Furthermore, for each pair of populations 1 ≤ i, j ≤ M , let there be some automorphism on the populations that preserves Θ and maps population i to population j . This produces node transitivity on the distribution (b ut not outcome) of the random graph. For filter design in this paper , an estimate of the spectral distribution for the consensus iteration matrix (based on the directed, row normalized Laplacian through W N = I − α b L R ) is required, which can be accomplished through a scaled adja- cency matrix Ξ N = 1 /γ A N ( G ) with γ = ρ (E [ A N ]) under certain conditions. The approximate distrib ution for W N will be deriv ed from that found for Ξ N through b f W N ,β ( x, y ) = 1 α 2 b f Ξ N ,β  ( x − 1) α + 1 , y α  . (13) The approach for solving Girko’ s equation for this model is briefly described, with full details omitted for space. For any random iteration matrix model arising node-transiti ve random network, the diagonal matrices C 1 , C 2 from Theorem 1 must become scalar matrices C 1 = c 1 I , C 2 = c 2 I and the vari- ance ro w sums and column sums must be equal. This allo ws simplified computation via the trace by summing both sides of (10)-(11). In the right side expressions, a trace emerges that then can be written in terms of the mean matrix eigen values. Further reduction occurs because B N is real and symmetric for this case. The resulting equations appear below . c 1 = 1 N tr( C 1 ) = u +  1 N k = N X k =1 σ 2 N ,kj  × . . . i = N X i =1  c 2 +1 /c 1  λ i ( B N ) 2 − 2 tλ i ( B N )+ | t + is | 2  − 1 (14) c 2 = 1 N tr( C 2 ) = 1+  1 N ` = N X ` =1 σ 2 N ,j `  × . . . i = N X i =1  c 1 +1 /c 2  λ i ( B N ) 2 − 2 tλ i ( B N )+ | t + is | 2  − 1 (15) These equations can then be approximately solved at every necessary v alue of ( u, t, s ) via an iterativ e fixed point search as done for Girko’ s K1 equation in [15, 16]. Numerical inte- gration is then conducted to find the density . 3. DIRECTED NETWORKS: FIL TER DESIGN This section considers consensus acceleration filter design for a random network with constant (non-time-v arying) topol- ogy described by a random graph G under a fixed scheme for determining the consensus iteration matrix from the ran- dom graph topology . Thus, the consensus iteration matrix is drawn once from a random matrix distribution and used for all time iterations. This section presents filter design criteria for this scenario. Assume that a deterministic equiv alent for the empirical spectral distribution of the corresponding ran- dom iteration matrix is av ailable, such as through the method in Section 2. Recall that there are many potential choices for consensus iteration matrices that satisfy the consensus con- ver gence conditions (2). In particular , this paper uses iteration matrix scheme W ( G ) = I − α b L R ( G ) (16) based on the directed, row normalized Laplacian matrix b L R ( G ) = I − D ( G ) − 1 A ( G ) (17) where A ( G ) is the network graph adjacency matrix, D ( G ) is the diagonal matrix of node outdegrees, and α is chosen to satisfy the spectral radius condition in (2). While this produces a weighted average (unlike unnormalized Lapla- cian based weights for symmetric graphs), this choice is easier to analyze through Girko’ s methods and can hav e con- ver gence rate advantages [17, 18]. Unlike the undirected context in which the weighting could be corrected through pre-multiplication if each node knows its degree [17], this weighted average must be accepted as the left-eigenv ector ` is not easily computable from basic degree information. Consensus acceleration filters seek to minimize the fol- lowing expression for the conv ergence rate, where W is the random, constant (non-time-varying) iteration matrix. lim k →∞    p ( W ) k − J `    1 /k 2 = lim k →∞    ( p ( W ) − J ` ) k    1 /k 2 (18) By Gelfand’ s formula [25], this reduces to the spectral radius ρ ( p ( W ) − J ` ) . Note that the eigen values of p ( W ) are pre- cisely p ( λ i ( W )) for each eigen v alue λ i ( W ) of W by the spectral mapping theorem [26]. Let the eigen values of W be λ 1 ( W ) , . . . , λ N ( W ) , where λ N ( W ) = 1 , and be ordered such that | λ i ( W ) | ≤ | λ j ( W ) | for 1 ≤ i < j ≤ N . Subtract- ing J ` remov es the consensus eigen value λ N ( W ) = 1 . Thus ρ ( p ( W ) − J ` ) is the maximum absolute v alue of p ( λ i ( W )) for 1 ≤ i ≤ N − 1 . Therefore, the worst case consensus con ver gence rate with periodic filtering can be approximately minimized through the follo wing minimax optimization problem, where P d is the space of (real coefficient) polynomials of degree at most d . min p ∈ P d max λ ∈ Λ κ,τ | p ( λ ) | s.t. p (1) = 1 Λ κ,τ = n | λ − 1 | > κ    b f W N ,β (Re { λ } , Im { λ } ) > τ o (19) The filtering region defined by the set Λ κ,τ is determined by a deterministic approximation for the empirical spectral den- sity , which can be obtained as described in Section 2. This substitutes for knowledge of the true set of eigen values from the random iteration matrix. Because computation of b f W N ,β in volves numerical computation of integrals and limits, it is necessary to threshold the result to define the filtering re gion. The parameter τ (small v alue chosen) fills this role, while the parameter κ (small value chosen) pro vides a transition region around the equality constraint. Note that some computation Fig. 1 : Exp. empirical spectral density E [ f W N ] of W N ( 1000 Monte-Carlo trials) with outline of Λ κ,τ (blue) dervied from b f W N ,β (see Figure 2) Fig. 2 : Deterministic approx. den- sity b f W N ,β computed from b f Ξ N ,β via Girko’ s theorem as described in Sec- tion 2 along with Λ κ,τ outline (blue) Fig. 3 : W orst case filter response (log scale, per iteration) compared for sev- eral filters (proposed design in blue), determines the con vergence rate could be sav ed by simply transforming the complement of the region G from Theorem 1, but the above formulation is more directly analogous to that from [17, 18]. By introducing ε to bound the maximum filter response magnitude squared and examining the response only at sam- ple points Λ S ⊆ Λ κ,τ , an approximate solution to (19) can be found by solving the following problem. min p ∈ P d ,ε ε s.t. p (1) = 1 | p ( λ i ) | 2 < ε for all λ i ∈ Λ S (20) The precise scheme to determine Λ S is of little importance, but it should be sufficient to approximately capture the struc- ture of Λ κ,τ in discretized form. Collecting the filter coeffi- cients { a n } n = d n =0 into a vector a , the optimization problem (20) can be recast as min a ∈ R d +1 ,ε ε s.t. 1 > a = 1 a > Q ( λ i ) a < ε for all λ i ∈ Λ S (21) where Q ( λ i ) is the real, positiv e semidefinite matrix Q ( λ i ) = 1 2  V ( λ i ) ∗ V ( λ i ) + V  λ i  ∗ V  λ i   (22) and V ( λ i ) is the V andermonde ro w vector V ( λ i ) =  λ 0 i , . . . , λ d i  . (23) This optimization problem has linear objecti ve function with positiv e semidefinite quadratic contraints for each sample point (QCLP), and, thus, is con vex [27]. This approach mir- rors that from [17, 18] for symmetric matrices, where the real-valued eigen values produce linear inequality constraints. Section 4 shows numerical simulation results that demon- strate improv ed filters computed through this method. 4. DIRECTED NETWORKS: SIMULA TIONS This section sho ws example results, displayed in Figures 1-2, for a directed stochastic block model with N = 600 nodes divided into M = 6 populations each with S = 100 nodes. For this simulation, the independent link probabilities Θ ij be- tween two disinct nodes in each ordered pair of populations i, j are Θ ij = 0 . 05 for i = j and Θ ij = 0 . 01 for i 6 = j . The simulation uses W = I − α b L R for the consensus iteration matrix ( α = 1 ). Figure 1 shows the expected empirical spectral distribu- tion, averaged over 1000 Monte-Carlo trials (independently drawn random networks), in heat map form along with the outline of the region Λ κ,τ ( κ = 10 − 2 , τ = 10 − 4 ) isolated from the approximate density function b f W N ,β ( β = 10 − 6 ). The heatmap for the approximate density function b f W N ,β ap- pears in Figure 2, also with the outline of Λ κ,τ . Figure 3 plots the expected worst case exponential con- ver gence rate per iteration 1 d ln ( ρ ( p ( W ) − J ` )) of the fil- tered consensus system, av eraged over 1000 Monte-Carlo tri- als (independently drawn random netw orks) for filter degrees d = 1 , . . . , 6 . The proposed filter design method is compared against the tri vial filter (no filtering), a filter designed to mini- mize response at the eigen values of the mean iteration matrix [9] (only for d ≤ K − 1 where K is the number of distinct mean matrix eigen values, for this simulation K = 3 ), and a filter designed with oracle knowledge of the true eigen val- ues (optimal). The proposed method (blue) performs nearly as well as the optimal filter (green), pro viding strong support. Note that attempting to optimize at only the eiven values of the mean iteration matrix (black) performs poorly in this case because the eigen values spread o ver a wide re gion. 5. CONCLUSION This paper considered graph filter design for accelerated consensus via periodic filtering on lar ge-scale, directed ran- dom networks that hav e random non-symmetric consensus iteration matrices with tractable spectral asymptotics. Sim- ilar to the approach pre viously taken for undirected random network models with symmetric random iteration matrices, this work first examined tools from random matrix theory to compute deterministic approximations for the empirical spectral distribution of the random consensus iteration ma- trix. Subsequently , filter design criteria were proposed that employ this information to define filtering regions, resulting in an optimization problem to approximately minimize the con vergence rate of the filtered consensus dynamics. Nu- merical simulations demonstrated that filters designed via the proposed method achieve con ver gence rates close to the optimal con vergence rate with full kno wledge of the iteration matrix eigen values. This approach has limitations, including complex numerical computations in Girko’ s K25 equation, restrictions on the random matrix models to which Girko’ s K25 equation can be efficiently applied, and a dif ficult to cor - rect weighted av erage consensus. Continuing work focuses on e xtending analysis to additional directed random network models and on handling time-varying random netw orks. 6. REFERENCES [1] R. Olfati-Saber , A. Fax, and R. M. Murray , “Consen- sus and cooperation in network ed multi-agent systems, ” Pr oceedings of the IEEE , vol. 98, no. 7, pp. 1354–1355, June 2010. [2] G. Cybenko, “Dynamic load balancing for distributed memory multiprocessors, ” Journal of P arallel and Distributed Computing , vol. 7, no. 2, pp. 279–301, Oct. 1989. [3] L. Xiao, S. Boyd, and S. Lall, “ A scheme for robust distributed sensor fusion based on av erage consensus, ” Pr oceedings of the 4th International Symposium on In- formation Pr ocessing in Sensor Networks (IPSN 2005) , pp. 63–70, April 2005. [4] S. Kar and J. M. F . Moura, “Consensus+innov ations distributed inference over networks: Cooperation and sensing in networked systems, ” IEEE Signal Pr ocessing Magazine , v ol. 30, no. 3, pp. 99–109, May 2013. [5] R. Olfati-Saber, “Flocking for multi-agent dynamic systems: Algorithms and theory , ” IEEE T ransactions on Automatic Contr ol , vol. 51, no. 3, pp. 401–420, Mar . 2006. [6] L. Xiao and S. Boyd, “Fast linear iterations for distributed av eraging, ” Systems and Contr ol Letters , vol. 53, pp. 65–78, Feb . 2004. [7] S. Kar and J. M. F . Moura, “T opology for global aver - age consensus, ” 40th Asilomar Confer ence on Signals, Systems and Computers (A CSSC 2006) , May 2007. [8] S. Sundaram and C. Hadjicostis, “Finite-time distrib uted consensus through graph filters, ” Pr oceedings of the 26th American Contr ol Confer ence (ACC 2007) , pp. 711–716, July 2007. [9] E. K okiopoulou and P . Frossard, “Polynomial filtering for fast conv ergence in distributed consensus, ” IEEE T ransactions on Signal Processing , vol. 57, pp. 342– 354, Jan. 2009. [10] A. Sandryhaila, S. Kar , and J. M. F . Moura, “Finite-time distributed consensus through graph filters, ” Pr oceed- ings of ICAASP 2014 , pp. 1080–1084, May 2014. [11] E. Montijano, J. I. Montijano, and C. Sagues, “Cheby- shev polynomials in distrib uted consensus applications, ” IEEE T ransactions on Signal Pr ocessing , vol. 61, no. 3, pp. 693–706, Feb . 2013. [12] A. Loukas, A. Simonetto, and G. Leus, “Distributed au- toregressi ve moving av erage graph filters, ” IEEE Sig- nal Pr ocessing Letters , vol. 22, no. 11, pp. 1931–1935, Nov . 2015. [13] S. Apers and A. Sarlette, “ Accelerating consensus by spectral clustering and polynomial filters, ” IEEE T rans- actions on Contr ol of Network Systems , v ol. 4, no. 3, pp. 544–554, Sept. 2016. [14] F . Gama and A. Ribeiro, “W eak law of large numbers for stationary graph processes, ” 2017 IEEE International Confer ence on Acoustics, Speech, and Signal Pr ocess- ing (ICASSP 2017) , pp. 4124–4128, March 2017. [15] S. Kruzick and J. M. F . Moura, “Spectral statistics of lattice graph percolations, ” Arxiv: https://arxiv .or g/abs/1611.02655 , Sept. 2016. [16] S. Kruzick and J. M. F . Moura, “Spectral statistics of lat- tice graph structured, non-uniform percolations, ” 42nd IEEE International Conference on Acoustics, Speech, and Signal Pr ocessing (ICASSP 2017) , pp. 5930–5934, Sept. 2016. [17] S. Kruzick and J. M. F . Moura, “Optimal filter design for signal processing on random graphs: Accelerated consensus, ” IEEE T ransactions on Signal Pr ocessing (pr eprint) , Nov . 2017. [18] S. Kruzick and J. M. F . Moura, “Graph signal process- ing: Filter design and spectral statistics, ” 2017 IEEE International W orkshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP 2017) , Dec. 2017. [19] S. Kruzick and J. M. F . Moura, “Consensus state Gram matrix estimation for stochastic switching networks from spectral distribution moments, ” 2017 Asilomar Confer ence on Signals, Systems, and Computers (A C- SSC 2017) , Oct. 2017. [20] E. W igner, “On the distrib ution of the roots of cer- tain symmetric matrices, ” The Annals of Mathematics , vol. 67, no. 2, pp. 325–327, Mar . 1958. [21] V . Mar ˇ chenko and L. Pastur , “Distribution of eigen val- ues for some sets of random matrices, ” Mat. Sb . , v ol. 1, no. 4, pp. 507–536, 1967. [22] R. Couillet and M. Debbah, Random Matrix Methods for W ir eless Communications . Cambridge Univ ersity Press, 2011. [23] V . Girko, Theory of Stochastic Canonical Equations . Springer Science+Business Media, 2001, vol. 1-2. [24] K. A vrachenkov , L. Cottatellucci, and A. Kada- vankandy , “Spectral properties of random matrices for stochastic block model, ” 4th International W orkshop on Physics-Inspir ed P aradigms in W ir eless Communi- cations and Networks , pp. 537–544, May 2015. [25] P . Lax, Functional Analysis . W iley-Interscience, 2002. [26] P . Lax, Linear Algebr a and Its Applications, second edi- tion . Wile y-Interscience, 2007. [27] S. Boyd and L. V andenberghe, Con vex Optimization . Cambridge Uni-versity Press, 2004.