a user guide to low-pass graph signal processing and its ... · and the low-pass graph signals...

1

A User Guide to Low-Pass Graph Signal Processingand its Applications

Raksha Ramakrishna Hoi-To Wai Anna Scaglione

AbstractmdashThe notion of graph filters can be used to definegenerative models for graph data In fact the data obtainedfrom many examples of network dynamics may be viewed as theoutput of a graph filter With this interpretation classical signalprocessing tools such as frequency analysis have been successfullyapplied with analogous interpretation to graph data generatingnew insights for data science What follows is a user guide on aspecific class of graph data where the generating graph filters arelow-pass ie the filter attenuates contents in the higher graphfrequencies while retaining contents in the lower frequenciesOur choice is motivated by the prevalence of low-pass models inapplication domains such as social networks financial marketsand power systems We illustrate how to leverage properties oflow-pass graph filters to learn the graph topology or identify itscommunity structure efficiently represent graph data throughsampling recover missing measurements and de-noise graphdata the low-pass property is also used as the baseline to detectanomalies

I INTRODUCTION

A growing trend in signal processing and machine learningis to develop theories and models for analyzing data definedon irregular domains such as graphs Graphs often expressrelational ties such as social economics networks or genenetworks for which several mathematical and statistical mod-els relying on graphs have been proposed to explain trends inthe data [1] Another case is that of physical infrastructures(utility networks such as power gas water delivery systemsand transportation networks) where physical laws in additionto the connectivity define the structure in signals

For a period of time the graphical interpretation was primarilyused in statistics with the aim of making inference aboutgraphical models Meanwhile the need for processing graphdata has led to the emerging field of graph signal processing(GSP) which takes a deterministic and system theoretic ap-proach to justify the properties of graph data and to inspirethe associated signal processing algorithms A cornerstone ofGSP is the formal definition of graph filter which extends thenotions of linear time invariant (LTI) filtering of time seriessignals to data defined on a graph aka graph signals

In a similar vein as LTI filters in discrete-time signal process-ing a graph filter can be classified as either low-pass band-pass or high-pass depending on its graph frequency response

RR AS are with the School of Electrical Computer andEnergy Engineering Arizona State University AZ USA Emailsrramakr6AnnaScaglioneasuedu HTW is with theDepartment of Systems Engineering and Engineering ManagementThe Chinese University of Hong Kong Hong Kong SAR of China Emailhtwaisecuhkeduhk

Among them this article focuses on the low-pass graph filtersand the low-pass graph signals generated from them Thesegraph filters capture a smoothing operation applied to theinput graph signals which is a common property of processesobserved in many physicalsocial systems (see Section III)As a motivating example in Fig 1 we illustrate a fewreal datasets with such models from social networks powersystems and financial market and show the eigenvalue spectraof their sample covariance matrices A salient feature observedis that these sample covariance matrices are low-rank thusdisplaying an important symptom of low-pass filtered graphsignals (to be discussed in Section IV)

Previous articles such as [2] [3] have provided a comprehen-sive introduction to modeling and processing graph or networkdata using GSP favoring general abstractions over focusingon particular structures and concrete applications This userguide takes a different approach concentrating on low-passgraph filters and the corresponding low-pass graph signaloutputs Low-pass graph signals have specific properties thataffect their structure and dictate how to approach for examplesampling denoising and inference problems They are worthfocusing on because they are very common in practice Westart the article by surveying low-pass GSP properties andinsights setting the stage for the description of the concretesituations where such a model applies A set of particularexamples is then provided highlighting the fact that low-passgraph signals often appear in different application domainsIn fact resorting to existing underlying network dynamicalmodels that justify different data sets we show that low-passgraph processes are nearly ubiquitous in contexts where GSPis applicable

II BASICS OF GRAPH SIGNAL PROCESSING

Many tools introduced in this user guide involve severalfundamental concepts of GSP including a formal definitionof low-pass graph filterssignals These ideas will be brieflyreviewed in this section For more details the readers arereferred to the excellent prior overview articles such as [2][3] We denote vectors with boldfaced lowercase letters xand uppercase letters for matrices A The operation Diag(x)creates a diagonal matrix with elements from vector x

We focus on a weighted undirected graph G = (N E) withn nodes such that N = 1 n and E sube N times N is theedge set A graph signal is a function x N rarr R which canbe represented by a n-dimensional vector x = (x(i))iisinN AGraph Shift Operator (GSO) is a matrix S isin Rntimesn satisfying

arX

iv2

008

0130

5v1

[ee

ssS

P] 4

Aug

202

0

2

AAPL

ABBV

ABT

ACN

AGN

AIG

ALL

AMGN

AMZN

AXP

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BA

BIIB

BK

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BMY

CAT

CELG

CL

CMCSA

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CSCO

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CVX

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USB

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Power networkSocial network Financial network

0 10 20 30 40 50

102

103

104

index

(C

o)

Opinion dynamics sample covariance matrix Co

0 20 40 60 80 100

101

102

103

104

index

(C

s)

Stock data sample covariance matrix Cs

0 500 1000 1500 20001018

1014

1010

106

102

102

index

(C

v)

Voltage sample covariance matrix Cv

Voltage data ACTIVSg 2000 test caseOpinion dynamics data 110th US senate Daily return of SampP100 stocks (May 2018-Aug 2019)

CT

ME

MANH

RI VT

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Fig 1 Illustrating the eigenvalue spectra of sample data covariance matrix of voltage Senate rollcall and financial stock data These data admit physicalsocialmodels that can be regarded as low-pass filtered graph signals A salient feature of their low-pass nature is observed as the low-rank property of the samplecovariance matrices

[S]ij 6= 0 if and only if i = j or (i j) isin E When multipliedby a graph signal x each entry of the shifted graph signal is alinear combination of the one-hop neighborsrsquo values thereforelsquoshiftingrsquo the graph signal with respect to the graph topologyIn this article we take the Laplacian matrix as the GSOThe Laplacian matrix is defined as L = D minus A whereA is the weighted symmetric adjacency matrix of G andD = Diag(A1) is a diagonal matrix of the weighted degreesIt is also common to take the GSO as the normalized Laplacianmatrix or the adjacency matrix [4]

Having defined the GSO we discuss how to measure thesmoothness of graph signals and analyze their content in thegraph frequency domain Recall that if a signal is smooth intime the norm of its time derivative is small For a graphsignal x its graph derivative is defined as

[nablax]ij =radicAij(xi minus xj)

The squared Frobenius norm of graph derivative aka thegraph quadratic form [2] provides an idea of the smoothnessof the graph signal x

S2(x) =1

2nablax2F = xgtLx =

sumij

Aij(xi minus xj)2 (1)

Observe that if xi asymp xj for any neighboring nodes i j thenS2(x) asymp 0 As such we say that a graph signal is smooth ifS2(x)x2 is small

Let us take a closer look at the graph quadratic form S2(x)We set the eigendecomposition of the Laplacian matrix as L =UΛUgt and assume that it has eigenvalues of multiplicity oneordered as Λ = Diag(λ1 λn) with 0 = λ1 lt λ2 ltmiddot middot middot lt λn and U = (u1 u2 middot middot middot un) with ui isin Rn beingthe eigenvector for λi Observe that for any x isin Rn it holdsS2(x)x2 ge

S2(u1)u12 = λ1 and for any x orthogonal to u1 it holds

S2(x)x2 ge

S2(u2)u22 = λ2 and so on for the other eigenvectors The

observation indicates that the larger the eigenvalue the moreoscillatory the eigenvector is over the vertex set In particularthe smallest eigenvalue λ1 = 0 is associated with the flat all-ones eigenvector u1 = (1

radicn)1 as seen in Fig 2 The above

motivates the definition of graph frequencies as the eigenvaluesλ1 λn and of the Graph Fourier Transform (GFT) basisas the set of eigenvectors U [4] Therefore the ith frequencycomponent of x is defined as the inner product between uiand x

xi = ugti x i = 1 n (2)

and x = Ugtx is called the GFT of x The magnitude of theGFT vector |x| is the lsquospectrumrsquo of the graph signal x where|xi|2 represents the signal power at the λith frequency

An important concept to modelling data with GSP is the graphfiltering operation To this end a linear graph filter is describedas the linear operator

H(L) =

Pminus1sump=0

hpLp = U

( Pminus1sump=0

hpΛp)Ugt (3)

3

1 = 0

2 = 04706

10 = 52813

15 = 80818

-05

0

05

Fig 2 The GFT basis u1u2u10u15 associated to the graph Laplacian of an undirected unweighted graph with 15 nodes As the eigenvalue increasesthe eigenvectors tend to be more oscillatory

where P is the filter order (can be infinite) and hpPminus1p=0

are the filterrsquos coefficients as a convention we use L0 =I and λ00 = 00 = 1 From (3) one can see that the graphfilter has similar interpretation as an LTI filter in discrete-timesignal processing where the former replaces the time shifts bypowers of the GSO Meanwhile the second expression in (3)defines the frequency response as the diagonal matrix h(Λ) =sumPminus1p=0 hpΛ

p A graph signal y is said to be filtered by H(L)with the input excitation x when

y = H(L)x (4)

To better appreciate the effects of the graph filter note thatthe ith frequency component of y is

yi = h(λi) middot xi i = 1 n (5)

where h(λ) =sumPminus1p=0 hpλ

p is the transfer function of thegraph filter or equivalently we have y = h(λ)x It is similarto the convolution theorem in discrete-time signal processing

a) Low-pass Graph Filter and Signal Inspired by (5) wedefine the ideal low-pass graph filter with a cut-off frequencyλk through setting the transfer function as h(λ) = 1 λ le λkand 0 otherwise [5] Alternatively one can say that a graphfilter is low-pass if its frequency response is concentratedon the low graph frequencies In this article we adopt thefollowing definition from [6]

Definition 1 For any 1 le k le nminus 1 define the ratio

ηk =max|h(λk+1)| |h(λn)|

min|h(λ1)| |h(λk)| (6)

The graph filter H(L) is k-low-pass if and only if the low-pass ratio ηk satisfies ηk isin [0 1)

The integer parameter k characterizes the bandwidth or thecut-off frequency of the low-pass filter is at λk The ratioηk quantifies the lsquostrengthrsquo of the low-pass graph filter Uponpassing a graph signal through H(L) the high frequencycomponents (above λk) are attenuated by a factor of less thanor equal to ηk Using this definition the ideal k-low-pass graphfilter has the ratio ηk = 0 whose filter order has to be at leastP ge nminusk+1 and transfer function h(λ) has λk+1 λnas its roots

Finally a k-low-pass graph signal refers to a graph signal that

is the output of a k-low-pass filter subject to a lsquowell-behavedrsquoexcitation (ie does not possess strong high frequency compo-nents) which includes but is not limited to the white noise

b) The Impact of Graph Topologies From Definition 1one can observe that the low-pass ratio ηk of a graph filterdepends on the filterrsquos coefficients hpPminus1p=0 and the graphLaplacian matrixrsquos spectrum λ1 λn The condition λk λk+1 facilitates the design of a k-low-pass graph filter witha favorable ratio ηk 1 and a low filter order P As anexample the order-1 graph filter H(L) = Iminusλminus1n L is k-low-pass with the ratio ηk = λnminusλk+1

λnminusλk= 1 minus λkminusλk+1

λnminusλk where ηk

is small if λk λk+1

An example of graph topologies favoring the condition λk λk+1 is the stochastic block model (SBM) [7] for describingrandom graphs with k blockscommunities with nodes inN partitioned as N1 Nk Consider a simplified SBMwith k equal-sized blocks specified by a membership matrixZ isin 0 1ntimesk such that Zi` = 1 if and only if i isin N` anda latent model B isin [0 1]ktimesk where Bj` is the probabilityof edges between nodes in block j and ` We considerthe homogeneous planted partition model (PPM) such thatB = b11gt + aI with b a gt 0 With the above specificationthe adjacency matrix A is a symmetric binary matrix withindependent entries satisfying E[A] = ZBZgt When thegraph size grows to infinity (nrarrinfin) the Laplacian matrix ofan SBM-PPM graph converges almost surely to its expectedvalue [7 Theorem 21]

Lasminusrarr E[L] =

n(a+ kb)

kI minusZ

(b11gt + aI

)Zgt (7)

From the above it can be shown that λk+1minusλk = nak 1 for

E[L] ie a favorable graph model for k-low-pass graph filtersLastly the bottom-k eigenvectors of the expected Laplacianassociated with λ1 λk can be collected into the matrixradic

knZP where P diagonalizes the matrix B In other words

the eigenvectors corresponding to the bottom-k eigenvalues ofL will reveal the block structure

In contrast the Erdos-Renyi graphs have Laplacian matricesthat do not generally satisfy λk λk+1 In fact asymptoti-cally (n rarr infin) the empirical distribution of the eigenvaluesof Laplacian matrices tends to the free convolution of thestandard Gaussian distribution and the Wigners semi-circular

4

law [8] Such spectrum does not favor the design of a k-low-pass graph filter with ηk 1 reflecting the fact that blockstructure or communities do not emerge in Erdos Renyi graphs

c) Low-pass Graph-Temporal Filter When the excitationto a graph filter is of time-varying nature and the topology isfixed we consider a graph-temporal filter [9] with the impulseresponse

H(L t) =sumPminus1p=0 hptL

p (8)

such that the graph filterrsquos output is given by the time-domainconvolution yt =

sumts=0H(L tminuss)xs The filter is causal and

xs = 0 for s lt 0 We can apply z-transform and the GFT tothe graph signal process xttge0 to obtain the z-GFT signalX(z) given by

X(z) =suminfint=0 xtz

minust X(z) = UgtX(z) (9)

which represents xttge0 in the joint z-graph frequencydomain With that we obtain the input-output relation Y (z) =

h(z) X(z) and graph-temporal joint transfer functionH(λ z) =

suminfint=0

sumPminus1p=0 hptλ

pzminust A class of graph-temporalfilters for modeling graph signal processes is the GF-ARMA(q r) filter whose input-output relation in time domain andz-GFT domain are described below respectively

yt minusA1(L)ytminus1 middot middot middot minus Aq(L)ytminusq

= B0(L)xt + middot middot middot+ Br(L)xtminusr

a(z) Y (z) = b(z) X(z)

where a(z) = 1 minus sumqs=1 asz

minuss and b(z) =sumrs=0 bsz

minuss

are the z-transform of the graph frequency responses of thegraph filter taps As(L)qs=1 Bs(L)rs=0 for the GF-ARMA(q r) filter Note that the joint frequency response is given byH(λi z) = [b(z)]i

[a(z)]i whose poles and zeros may vary

depending on the graph frequencies λ1 λn A relevantcase is when H(L t) is a low-pass graph-temporal filterSimilar to Definition 1 we say that H(L t) is low-pass witha cutoff frequency (λk ω0) and ratio ηk if

ηk =maxλisinλk+1λnωisin(ω02π) |H(λ ejω)|

minλisinλ1λkωisin[0ω0] |H(λ ejω)| lt 1 (10)

Graph signals filtered by a low-pass graph-temporal filter arealso commonly found in applications as we will illustrate next

III MODELS OF LOW-PASS GRAPH SIGNALS

Before studying the GSP tools for low-pass graph signals anatural question is where can one find such graph signals Itturns out that many physical and social processes are natu-rally characterized by low-pass graph filters In this sectionwe present various examples and show that their generationprocesses can be represented as outputs from low-pass graphfilters

a) Diffusion Model The first case pertains to observationsfrom a diffusion process whose variants are broadly applicablein network science As an example we consider the heatdiffusion model in [10] In this example the relevant graphis a proximity graph where each node i isin N is a location(eg cities) and if locations i j are close to each other then(i j) isin E The graph is endowed with a symmetric weightedadjacency matrix encoding the distance between locations Thegraph signal yt isin Rn encodes the temperature of n locationsat time t and let x0 isin Rn be the initial heat distributionThe temperature of a location is diffused to its neighbors Letσ gt 0 be a constant we have

yt = eminustσLx0 =(I minus tσL+

(tσ)2

2L2 minus middot middot middot

)x0 (11)

where (11) is a discretization of the heat diffusion equation[10] As L1 = 0 the matrix exponential eminustσL = I minustσL + (tσ)2

2 L2 minus middot middot middot is row stochastic The temperature attime t is thus a weighted average of neighboring locationsrsquotemperatures at t = 0 ie this is a diffusion dynamicalprocess

To understand (11) under the context of low-pass filtering weobserve that yt is a filtered graph signal with the excitationx0 and the graph filter H(L) = eminustσL We verify that H(L)is k-low-pass with Definition 1 for any k lt n Note that thelow-pass ratio ηk is

eminustσλk+1

eminustσλk= eminustσ(λk+1minusλk)

As λk+1 gt λk and tσ gt 0 we see that H(L) is a k-low-passgraph filter for any k = 1 nminus 1

We have assumed that x0 is an impulse excitation affectingthe system only at the initial time In practice the excitationsignal may not be an impulse and the output graph signalyt is expressed as the convolution yt =

sumts=0 e

minus(tminuss)σLxsThis corresponds to a low-pass graph-temporal filter with thejoint transfer function H(λ z) = (1 minus eminusλσzminus1)minus1 Besidesthe diffusion process is common in network science as similarmodels arise in contagion process and product adoption toname a few

b) Opinion Dynamics This example pertains to opiniondata mined from social networks with the influence of externalexcitation [6] [11] The relevant graph G is the social networkgraph where each node i isin N is an individual and E isthe set of friendships Similar to the previous case studythis graph is endowed with a symmetric weighted adjacencymatrix A where the weights measure the trust among pairsof individuals Let α isin (0 λminus1n ) β isin (0 1) be parameters oftrust on others and susceptibility to external influence of anindividual respectively The evolution of opinions follows thatof a combination of DeGrootrsquos and Friedkin-Johnsenrsquos model[12] which is a GF-AR(1) model

yt+1 = (1minus β)(I minus αL

)yt + βxt (12)

where yt isin Rn is a graph signal of the individualsrsquo opinionsat time t and xt isin Rn is a graph signal of the external

5

opinions perceived by the social network Note that this alsocorresponds to a low-pass graph-temporal filter with the jointtransfer function H(λ z) = β[1minus (1minus β)(1minus αλ)zminus1]minus1

To discuss the steady state of (12) let us assume that xt equiv xConsidering (12) we observe that yt+1 is a convex combina-tion of x and weighted average of the neighborsrsquo opinionsat time t that is formed by taking a weighted average ofneighboring signals in yt using a diffusion operator I minus αLAs β gt 0 the recursion is stable leading to the steady state(or equilibrium) opinions

y = limtrarrinfin

yt = (I + αL)minus1x = H(L)x (13)

where we have defined α = β(1minusα)α gt 0 and y is a filteredgraph signal excited by x

The graph filter above is given by H(L) = (I + αL)minus1 Toverify that it is a k-low-pass graph filter with Definition 1 wenote that for any k lt n the low-pass ratio ηk is

1 + αλk1 + αλk+1

= 1minus αλk+1 minus λk1 + αλk

Again we observe that as λk+1 gt λk the above graph filteris k-low-pass for any k = 1 nminus 1 However we remarkthat this low-pass ratio may be undesirable with ηk asymp 1 whenα 1 Interestingly a similar generative model as (12) isfound in equilibrium problems such as quadratic games [13]

Two remarks are in order First social networks are typicallydirected and this suggests using a non-symmetric shift oper-ator as opposed to the symmetric Laplacian matrix which weused for simplicity of exposition Second many alternativemodels for social networks interactions are non-linear andlinear GSP is insufficient in those contexts

c) Finance Data Financial systems such as stock marketand hedge funds produce return reports periodically abouttheir business performances A collection of these reportscan be studied as graph signals where the relevant graph Gconsists of nodes N that are financial institutions and edgesE that are business ties between them It has been studied[14] that business performances are correlated according to thebusiness ties Moreover the returns are affected by a numberof common factors [15] Inspired by [14] [15 Ch 122] letβ isin (0 1) be the strength of external influences a reasonablemodel for the transient dynamics of the graph signal yt ofbusiness performance measures is also a GF-AR(1)

yt+1 = (1minus β)H(L)yt + βBx (14)

where H(L) is an unknown but low-pass graph filter B isinRntimesr represents the factor model affecting financial institu-tions and x isin Rr is the excitation strength The equilibriumof (14) is

y = limtrarrinfin

yt =( 1

βI minus β

βH(L)

)minus1Bx equiv H(L)Bx

where β = 1minusβ We see that Bx is the excitation signal andthe equilibrium y is the filter output Suppose that H(L) is ak-low-pass graph filter with the frequency response satisfying

h(λ) ge 0 then for H(L) we can evaluate the low-pass ratioηk as

1minus β

min`=1k h(λ`)minusmax`=k+1n h(λ`)

1minus βmax`=k+1n h(λ`)

As min`=1k h(λ`)minusmax`=k+1n h(λ`) gt 0 since H(L)

is a k-low-pass graph filter itself we observe that H(L) isagain k-low-pass according to Definition 1

For y to be a k-low-pass graph signal one has to also assumethat Bx is not high-pass (ie not orthogonal to a low-passone) This is a mild assumption as the latent factor affectingfinancial institutions are either independent of the network orare aligned with the communities Above all we remark that(14) is an idealized model where determining the exact modelis an open problem in economics see [14] [15]

d) Power Systems In the case of power systems therelevant graph G = (N E) is the electrical transmissionlines network The node (aka bus) set includes generatorbuses Ng = 1 |Ng| and non-generatorload busesN` = N Ng The edge set E refers to the transmissionlines connecting the buses The branch admittance matrixY models the effect of transmission lines and is a complexsymmetric matrix associated with G where [Y ]ij is thecomplex admittance of the branch between nodes i and jprovided that (i j) isin E The graph signals we consider are thecomplex voltage phasors denoted as vt isin Cn when measuredat time t They can be obtained using phasor measurementunits (PMU) [16] installed on each bus i isin N The graphshift operator in this case is a diagonally perturbed branchadmittance matrix

Sgrid = Y + diag([ygtg y

gt` (0)]

) (15)

where yg isin C|Ng| is the generator admittance and y`(0) isinC|N`| is the load admittance at t = 0

Note that Sgrid is a GSO on the grid graph as [Sgrid]ij = 0if (i j) isin E The complex symmetric matrix Sgrid can bedecomposed as Sgrid = UΛUgt where U is a complexorthogonal matrix satisfying UgtU = I and Λ is a diag-onal matrix with diagonal elements λ1 λn sorted as0 lt |λ1| le middot middot middot le |λn| see [17] for modeling details Letigt isin Cn be the outgoing current at each node at time tgiven by igt = [ygtg exp

(xgtt)0]gt where elements in

exp (xt) isin C|Ng| are the internal voltage phasors at thegenerator buses Applying Kirchoffrsquos current law in quasi-steady state the voltage phasors vt isin Cn are

vt = Sminus1grid igt +wt = H(Sgrid) i

gt +wt (16)

where wt isin Cn captures the slow time-varying nature of theload and other modeling approximations In other words vtis a graph signal obtained by the graph filter H(Sgrid) = Sminus1grid

and the excitation signal igt isin Cn Particularly we observethat H(Sgrid) = Sminus1grid is a low-pass graph filter Consider anyk le n the low-pass ratio ηk is

|λk||λk+1|

6

As the power grids tend to be organized as communitiesto serve different areas with high population densities thesystem admittance matrix Y is block diagonal and sparse Inparticular with k communities in the grid graph these factsindicate that the graph filter is k-low-pass satisfying ηk 1

The excitation graph signal igt itself has a low-rank structureas [igt ]i = 0 at i isin Ng The temporal dynamics of xt canbe described as an AR(2) graph filter [9] using a reducedgenerator-only shift operator Sred isin C|Ng| with the graph-temporal transfer function H(λred z)

xt =

tsums=0

H (Sred tminus s)ps

H(λred z) = σ2p

(1minus

1sump=0

ap1λpredzminus1 minus

1sump=0

ap2λpredzminus2)minus1

where ps is the stochastic power input to the system Thegraph-temporal filter is low-pass in the time domain Theoverall system in (16) has approximately the properties of alow-pass graph temporal filter according to the definition in(10)

IV USER GUIDE TO LOW-PASS GRAPH SIGNALPROCESSING

If we observe a set of low-pass graph signals such as thosefrom Section III what can we learn from these signals Canwe find efficient representations for them Can we exploit thisstructure to denoise the signal or detect anomalies To answerthese questions we begin by studying two salient features oflow-pass graph signals namely low-rank covariance matrixand smoothness as measured by the graph quadratic formThen we illustrate how these features can enable low-passGSP to sample graph signals (and therefore compress them)to infer the graph topology and to detect anomalous activitiesFurthermore when the graph topology admits a clusteredstructure we highlight how these clusters emerge in thelow-pass graph signals and provide insights on the optimalsampling patterns

We now consider a set of m low-pass graph signals thatcan be modeled as outcomes of independent and identicallydistributed random experiments given as

y` = H(L)x` +w` ` = 1 m (17)

such that H(L) is a k-low-pass graph filter defined on theLaplacian matrix x` is the excitation signal andw` is an addi-tive noise For simplicity we do not consider the more generallow-pass graph-temporal processes and assume that x` w` arezero-mean white noise with E[x`x

gt` ] = I E[w`w

gt` ] = σ2I

We remark that the following observations still hold for thegeneral setting when E[x`x

gt` ] is not white or even diagonal

The latter relaxation is important for the applications listed inSection III For instance in opinion dynamics the excitationsignals may represent external opinions that do not affect thesocial network uniformly eg they are news articles writtenin a foreign language

A Low-rank Covariance Matrix

From (17) it is straightforward to show that y`m`=1 is zero-mean with the covariance matrix

Cy = Uh(Λ)2Ugt + σ2I (18)

Recall that H(L) is a k-low-pass graph filter if ηk 1 asdefined in (6) the energy of h(Λ) will be concentrated in thetop-k diagonal elements Therefore when the noise variance issmall (σ2 asymp 0) the low-pass graph signals lie approximatelyin span(Uk) a k-dimensional subspace of Rn

a-i) Sampling Graph Signals As the k-low-pass graphsignals lie approximately in span(Uk) it is possible to mapthe graph signals almost losslessly onto k-dimensional vectorsWhile the k-dimensional representation can be obtained byprojecting on the space spanned by Uk it is not necessary todo so An alternative to generate this k-dimensional represen-tation is by decimating the graph signals

To describe the setup we let Ns = s1 sns sub N be a

sampling set with cardinality ns = |Ns| A sampled versionof y` is constructed as (omitting the subscript ` for brevity)

(i) select a subset Ns sub N (ii) set ysamp = Φy (19)

such that

[Φ]qj =

1 if j = sq

0 otherwise

and Φ isin Rnstimesn is a fat sampling matrix that compresses thegraph signal to an ns-dimensional vector To recover y weinterpolate ysamp using a matching linear transformation [18]giving y = Ψysamp with Ψ isin Rntimesns to be designed laterClearly when ns lt n it is not possible to exactly recover anarbitrary graph signal To ensure exact recovery we see thatit requires certain additional conditions on the sampling setand the graph signal

Exactly recovering y from its sampled version ysamp wouldrequire the sampled graph signal to be in the range space ofsampling matrix Φ We let y = UkU

gtk y be the projection

of y onto the (low-frequency) subspace spanned by Uk andw = yminusy be the projection error The projected graph signaly is a k-bandlimited (in fact k-low-pass) graph signal From[18 Theorem 1] a sufficient condition for exact recovery isthat if

rank(ΦUk

)= k (20)

then there exists an interpolation matrix Ψ isin Rntimesns such thatΨΦy = y In fact it is possible to recover any k-bandlimitedgraph signal from its sampled version We have

y = Ψysamp = y + ΨΦw = y + (ΨΦminus I)w (21)

As k-low-pass graph signals lie approximately in span(Uk)we have w asymp 0 provided that ηk 1 Under condition (20)the sampling-and-interpolation procedure results in a smallinterpolation error

Clearly a necessary condition to satisfy (20) is ns ge kie we require at least the same number of samples as the

7

(a)

10minus4 10minus3 10minus2 10minus1 10010minus6

10minus4

10minus2

100

|λi|maxi|λi|

GFT |v| = |Ugtv|

(b)

0 500 1000 1500 2000bus number

0

500

1000

1500

2000

bus n

umbe

r

Grid GSO S PMU position

0 500 1000 1500 2000bus number

0

500

1000

1500

2000

bus n

umbe

r

Grid GSO S PMU position (c)

1000 1100 1200 1300 1400 1500

04

06

08

bus number

Voltage

(realpart)

Actual Reconstructed

1000 1100 1200 1300 1400 1500

minus09

minus08

minus07

minus06

bus number

Voltage

(imaginarypart)

Fig 3 (a) Magnitude of Graph Fourier Transform (GFT) of voltage graph signal plotted with respect to normalized graph frequencies (b) Sampling patternoverlaid on the support of GSO Sgrid for placing synchro-phasors sensors on the synthetic 2000 bus ACTIVSg power grid system by employing a greedymethod to find k = 100 rows of Uk so that the smallest singular value of ΦUk is maximized (The first entries correspond to generator buses) This caseemulates the grid for the state of Texas (called ERCOT systems) where there are 8 areas matching the number of communities evident from the GSO (c)Reconstructed voltage graph signal using optimally placed sensors at a subset of buses (1000-1500)

bandwidth of the low-pass graph filter which produces thegraph signal y Beyond the necessary condition obtaininga sufficient condition for (20) can be difficult as it is notobvious to derive conditions on the sampling set The designof the sampling set has been the focus of work in [18]ndash[20]which propose to find Ns via a greedy method or via thegraph spectral proxies The above statements are valid for anygraph signal that has a sparse frequency support In the caseof low-pass graph signals we can obtain insights on what typeof sampling patterns are compatible with (20) Consider thespecial case of SBM-PPM graphs with k blocks discussed inSection II Note that as n rarr infin we have Uk =

radicknZP

for this model where Z is the block-membership matrixSubsequently condition (20) can be easily verified if Nscontains at least one node from each of the k blocks

In Fig 3 we consider a power system application We firstplot the magnitude of GFT of voltage graph signal with respectto normalized graph frequencies in log scale From the lineardecay it is evident that the magnitude of GFT coefficients atlower frequencies is higher confirming the signal is low-passin nature Then the sampling pattern (or optimal placement ofsensors) for graph signal reconstruction is shown in the figureThe block structure in the GSO for the electric grid guides thesampling strategy In this example the smallest singular valueof ΦUk is maximized using a greedy algorithm [20]

a-ii) Blind Community Detection Another consequence of(18) relates to learning the block or community structure whenthe graph topology is unknown When the graph topologyis known spectral clustering (SC) is often the method ofchoice The SC method computes the bottom-k eigenvectorsof Laplacian as Uk and partitions the n nodes via k-means

F = minN1Nk

F (N1 NkUk) (22)

where

F (N1 NkUk) =( ksum

q=1

sumiisinNq

∥∥∥urowi minus 1

|Nq|sumjisinNq

urowj

∥∥∥22

)12

such that urowi isin Rk is the ith row vector of Uk In fact this

is an effective method for SBM-PPM graphs where solving(22) reveals the true block membership [7]

Although only the graph signals y`m`=1 are observed weknow from (18) that the covariance matrix Cy will be dom-inated by a rank-k component spanned by Uk under thelow-pass assumption In fact this is precisely what we needfor community detection as hinted in (22) To this end [6]proposed the blind community detection (BlindCD) procedure

(i) find the top-k eigenvectors Uk isin Rntimesk

of sample covariance Cy = 1m

summ`=1 y`y

gt`

(ii) apply k-means on the rows of Uk

If we denote the detected communities as N1 Nk then

F (N1 NkUk)minus F = O(ηk + σ +mminus12) (23)

In other words the BlindCD approaches the performanceof SC if the graph filter is k-low-pass with ηk 1 theobservation noise σ is small and the number of samples mis large Notice that (23) is a general result which holds evenif E[x`x

gt` ] is non-diagonal or low-rank Moreover BlindCD

is shown to outperform a two-step approach that learns thegraph first and then apply SC see [6]

In Fig 4 we illustrate results of community detection foropinion dynamics and financial data by using data from USSenate from the 115th US Congress (2017-2019) and dailyreturn data of stocks in the SampP100 index from Feb 2013to Dec 2016 [source httpswwwkagglecomcamnugentsandp500] respectively The observed steady-state graph sig-nals y` for the opinion dynamics case are the aggregated voterecords of each state and we observe m = 502 voting roundsIn Fig 4 (a) we apply BlindCD to partition the states intoK = 2 groups where a close alignment between our resultswith the actual party memberships of this US Congress isobserved The financial dataset used contains m = 975 daysof data for n = 92 stocks In Fig 4 (b) we apply BlindCDto partition the stocks into K = 10 groups Each of the

8

(a)

Figure 1 Applying BlindCD methods on the 115th US Senate Rollcall records The states markedin redblue are found to be in dicrarrerent com s while the states marked in gray are marked as thelsquostubbornrsquo states as explained in the text (Left) Results of BlindCD (Right) Results of boostedBlindCD

1

(b)

AAPL

ABBVABT

ACN

AGN

AIGALL

AMGN

AMZN

AXP

BAC

BA

BIIB

BK

BLK

BMY

CAT

CELG

CL

CMCSACOF

COP

COST

CSCOCVS

CVX

C

DIS

DUK

EMR

EXC

FB

FDX

F

GDGE

GILD

GM

GOOGL

GOOG

GS

HAL

HD

HON

IBM

INTC

JNJ

JPM

KMI

KO

LLY

LMT

LOW

MA MCD

MDLZ

MDT

MET

MMM

MONMO

MRK

MSFT

MS

NEE

NKE

ORCL

OXY

PCLN

PEP

PFE

PG

PM

QCOM

RTN

SBUX

SLB

SO

SPG

TGT

TWX

TXN

T

UNH

UNP

UPS

USB

UTX

VZ

V

WBA

WFC

Utilities amp Real Estates

Consumer Staples

Health Care

Information Technology

Financials

Energy

Health Care amp Information Technology

Industrial

Consumer Staples amp Discretionary



Fig 4 Communities detected from opinion dynamics and stock data (a) US Senate voting records (Top) inferred membership via BlindCD (Bottom) actualparty membership of Senators taken from [httpsenwikipediaorgwiki115th United States Congress] Note that the purple color in the bottom indicates thatthe state has a Democrat Senator and a Republican Senator (b) Daily returns of SampP100 stocks Colors on the nodes represent different detected communitiesCommunities are manually labeled according to business types

community detected includes companies of the same businesstype (for instance lsquoBACrsquo (Bank of America) is with lsquoJPMrsquo(JP Morgan)) showing the effectiveness of the method

B Smooth Graph Signals

In Section II we introduced the graph quadratic form toquantify the smoothness of a graph signal Particularly ifS2(y) = ygtLy y2 the graph signal y is said to besmooth For k-low-pass graph signals we observe that

E[S2(y`)

]asymp

ksumi=1

λi |h(λi)|2 + σ2Tr(L) (24)

where we have used that H(L) is k-low-pass with ηk 1to derive the approximations In the cases when λi asymp 0i = 1 k such as large SBM-PPM graphs with parameters(a b) satisfying b 1 a asymp 1 we expect the k-low-pass graphsignal to be smooth ie E[S2(y`)] asymp 0

b-i) Graph Topology Learning The smoothness propertycan be used to learn the graph topology by fitting a Lapla-cian matrix which best smoothens the graph signals This isexemplified by the estimator

minz``=1mL

1

m

msum`=1

1

σ2z` minus y`22 + zgt` Lz`

st Tr(L) = n Lji = Lij le 0 i 6= j L1 = 0

(25)where we have used the graph quadratic form zgt` Lz` toregulate the smoothness of z` asymp y` with respect to thefitted L Dong et al [21] motivated (25) as a maximum-a-posterior (MAP) estimator for the Laplacian matrix wherey` sim N (0Ldagger + σ2I) and Ldagger is the pseudo-inverse ofthe Laplacian matrix This amounts to interpreting the dataas outcomes of a Gaussian Markov Random Field (GMRF)with precision matrix chosen as the Laplacian effectivelyconnecting statistical graphical models to GSP models Notethat methods following similar insights as (25) can be foundin [22] [23]

For graph signals that are output from low-pass graph temporalfilters a similar smoothness property to (24) can also beexploited to interpolate missing data Let Y isin Rntimesm be amatrix whose columns are yt t = 1 2 m where yt is thegraph signal at time t and Y samp be the sampled version of Ywhere some values are missing at different nodetime indicesThe key for interpolating the data is to regularize via graphquadratic form and `2 norm of the time derivative in additionto minimizing the `2 misfit between available samples Y samp

and reconstructed samples at known locations M(Y ) ie

minY isinRntimesm

M(Y )minus Y samp2F

+ γ msum`=1

ygt` Ly` +

msum`=2

y` minus y`minus122

See [24] and the references therein for a detailed discussion

C Anomalies Detection with Low-pass GSP

Consider a model consistent with (17) The fact that the low-pass graph process is dominated by low graph frequencycomponents can be considered the null-hypothesis character-ized by the low-pass properties such as low-rank covariancematrix and smoothness On the other hand many anomaliescan be modeled as an additive sparse noise signal wi ora high frequency graph signal Such noise signals arise inseveral scenarios such as a change in the graph connectivityor parameters a contingency in infrastructures the result ofmalicious activities in social networks or the sudden fall in themarket value of a financial entity High frequency noise signalsare also produced by a perturbation that is inconsistent withthe generative model For instance in infrastructure networksthis could be symptomatic of malfunctioning sensors or evena false data injection attack (FDIA) [25]

Such anomalies cause a surge in the high frequency spectralcomponents of a low-pass graph signal a fact that can beleveraged in a manner similar to the classical array processingproblem of detecting a source embedded in noise Formally

9

(a)

500 1000 1500 20000

02

04

06

graph frequency index

Magnitudeofgraphfrequen

cyresponse

FDI attack A1 No attack A0 (b)

0

05

1

Gro

und

Tru

th

0 50 100 150 200

0

05

1

Gra

ph S

igna

l

0 50 100 150 200

0

2

4

Filt

ered

Sig

nal

0 50 100 150 200Node Index

Fig 5 (a) Magnitude of graph Fourier transform of the output after ideal high-pass filtering |UTHHPF(L)y`| k = 1200 under the hypotheses of anomalyA1 and no anomaly A0 Particular example of FDIA on voltage graph signal y` from ACTIVSg2000 case is shown We see a surge in high frequencycomponents when there is an attack (b) Spatial difference filtering of the graph signal under a diffusion model with abnormal activities on 11 nodes the topfigure shows the ground truth locations of anomalies middle and bottom figures show the graph signal y` and filtered graph signal HSD(L)y` respectively

the observed signal under null and alternative hypothesis isdescribed as

y` =

H(L)x` under A0

H(L)x` +w` under A1

where w` is a high frequency graph signal Our task amountsto testing the hypothesis A0 A1 andor to estimate thelocations of non-zeros in wi when the latter is a sparse signaland under A1

Intuitively we can apply a high-pass graph filter to distinguishbetween A0 and A1 Let HHPF(L) be an ideal high-pass graphfilter with the frequency response hHPF(λ) = 1 λ ge λk+1 and0 otherwise Consider the test statistics as Γ` = HHPF(L)y`Under A0 and the k-low-pass assumption we have Γ` leHHPF(L)H(L)2x` = hHPF(λ) h(λ)infinx` =O(ηk) and thus the test statistics Γ` will be small On theother hand under A1 we obtain HHPF(L)y` asymp w` since theanomalies consist of high graph frequency components Thusthe test statistics Γ` will be large Imposing a threshold ofδ = Θ(ηk) we can consider the detector

Γ` = HHPF(L)y`A0

≶A1

δ

Furthermore if A1 holds these anomaly events can be locatedfrom the support of HHPF(L)y`

As a demonstration Fig 5 (a) shows the magnitude of GFT ofthe voltage graph signal after filtering using an ideal high-passgraph filter UgtHHPF(L)y` The voltage graph signal underthe hypothesis of no anomaly is the output of a low-pass graphfilter When there is a FDIA we observe an increase in energyof the high frequency components

To obtain a simple implementation of high-pass graph filterswe may consider HSD(L) = L = D minus A whose frequencyresponse is given by h(λ) = λ When applied on a graphsignal y` we will observe the difference between Dy` andAy` where the latter is a one-hop averaged version of y`We call this operation the spatial difference which is similarto the method proposed in [26] for anomaly detection on socialnetworks See the illustration in Fig 5 (b)

V CONCLUDING REMARKS

In this user guide we highlighted the key elements of low-passGSP in several applications like graph parameter inference andgraph signal sampling while emphasizing the intuition fromtime series analysis We also discussed several physical modelswhere low-pass GSP can be effectively used However thetools available for low-pass GSP are ever-expanding and aidthe discovery of new physical models where low-pass GSPcan be applied Additionally there are several open researchdirections as discussed below

a) Directed Graphs Throughout this article we have as-sumed that the observed data is supported on a graph topologywhich is undirected and the shift operator (Laplacian matrix)is symmetric This is clearly not a truthful model for a lotof real systems such as social and economics networks Thechallenge of extending the existing GSP tools to directedgraphs lies in defining the appropriate GFT basis For instancethe properties of a circular shift matrix is what a directed shiftoperator should emulate

Much of the prior research has focused on finding the appropri-ate GFT basis on directed graphs The definition of frequencyis again variational but based on the norm of the differencebetween and vector and the shift operator S of the correspond-ing graph does not have to be symmetric More formally theidea of smoothness is defined as xminus λminus1n Sx22 where λn isthe maximum eigenvalue of S This is the definition used in[4] for GFT on directed graphs where the GSO is set as theadjacency matrix A and the GFT is defined as y = Uminus1ysuch that U is obtained from the Jordan decomposition ofthe adjacency matrix A = UΛUminus1 Although U is a basisit is not orthogonal so the Parsevalrsquos identity does not holdsince y2 6= y2 That is not surprising since the normof y does not have the same physical interpretation of powerspectral density that applies to signals whose support is timeA potential fix is studied in [27] which searches for the GFTbasis that minimizes the directed total variation also see [28]Unfortunately the GFT basis does not admit a closed formsolution

10

The tools discussed in this article such as sampling theory[18] and anomaly detection may still work for low-passgraph signals on directed graphs with minor adjustments Thechallenge lies in the graph inferencelearning methods sincesecond-order statistics such as correlations are difficult tojustify in directed graphs where also the notion of communityis ambiguous A useful definition of community must first bestudied before tools of GSP can be applied for communityinference in directed graphs

b) Low-pass Graph Signals in the Edge Space An al-ternative form of graph signals are those that are defined onthe edges They can be defined as the function f E rarr Rand the equivalent vector f isin R|E| which are useful fordescribing flows on graphs such as traffic in transportationnetwork As suggested in [29] the shift operator can be takenas the edge Laplacian Le = BgtB where B isin Rntimes|E| isthe node-to-edge incidence matrix The null space of the edgeLaplacian Le corresponds to the cyclic flow vector which isalso the eigenvector for the lowest graph frequency λ1 = 0 Itis anticipated that a low-pass edge-graph signal whose energyis focused in the low graph frequencies will consist of mostlycyclic flows within communities An interesting direction isto develop a sampling theory for low-pass edge-graph signalsas well as the inference of edge Laplacian matrix

c) Identifying Low-pass Graph Signals So far we haverelied on domain knowledge about the data models eg theexamples in Section III to help justify various graph data aslow-pass graph signals

For graph signals taken from an unknown system one hasto be cautious before applying this low-pass GSP user guideEven though non-low-pass graph processes are rarely foundin a natural setting there is a crucial need to design tools foridentifying low-pass graph signals With known GSO it canbe done by inspecting the GFT spectrum with unknown GSOthe problem is related to the joint estimation of graph processand topology readers are referred to [30] for recent works inthis direction

Acknowledgement The authors thank the anonymous re-viewers and the guest editors for their useful feedback Thismaterial is based upon work supported in part by the US Army Research Laboratory and the U S Army ResearchOffice under contractgrant number W911NF2010153 and theNSF CCF-BSF CIF 1714672 grant Hoi-To Wairsquos work issupported by the CUHK Direct Grant 4055135

REFERENCES

[1] E D Kolaczyk and G Csardi Statistical analysis of network data withR Springer 2014 vol 65

[2] D I Shuman S K Narang P Frossard A Ortega and P Van-dergheynst ldquoThe emerging field of signal processing on graphs Ex-tending high-dimensional data analysis to networks and other irregulardomainsrdquo IEEE Signal Processing Magazine vol 30 no 3 pp 83ndash982013

[3] A Ortega P Frossard J Kovacevic J M Moura and P VandergheynstldquoGraph signal processing Overview Challenges and ApplicationsrdquoProceedings of the IEEE vol 106 no 5 pp 808ndash828 2018

[4] A Sandryhaila and J M Moura ldquoDiscrete Signal Processing onGraphsrdquo IEEE Transactions on Signal Processing vol 61 no 7 pp1644ndash1656 2013

[5] N Tremblay P Goncalves and P Borgnat ldquoDesign of Graph Filters andFilterbanksrdquo in Cooperative and Graph Signal Processing Elsevier2018 pp 299ndash324

[6] H-T Wai S Segarra A E Ozdaglar A Scaglione and A JadbabaieldquoBlind Community Detection From Low-Rank Excitations of a GraphFilterrdquo IEEE Transactions on Signal Processing vol 68 pp 436ndash4512020

[7] K Rohe S Chatterjee B Yu et al ldquoSpectral clustering and the high-dimensional stochastic blockmodelrdquo The Annals of Statistics vol 39no 4 pp 1878ndash1915 2011

[8] X Ding T Jiang et al ldquoSpectral distributions of adjacency and Lapla-cian matrices of random graphsrdquo The Annals of Applied Probabilityvol 20 no 6 pp 2086ndash2117 2010

[9] E Isufi A Loukas A Simonetto and G Leus ldquoSeparable autore-gressive moving average graph-temporal filtersrdquo in 2016 24th EuropeanSignal Processing Conference (EUSIPCO) IEEE 2016 pp 200ndash204

[10] D Thanou X Dong D Kressner and P Frossard ldquoLearning HeatDiffusion Graphsrdquo IEEE Transactions on Signal and Information Pro-cessing over Networks vol 3 no 3 pp 484ndash499 2017

[11] C Ravazzi R Tempo and F Dabbene ldquoLearning Influence Structurein Sparse Social Networksrdquo IEEE Transactions on Control of NetworkSystems vol 5 no 4 pp 1976ndash1986 2017

[12] N E Friedkin ldquoA Formal Theory of Reflected Appraisals in theEvolution of Powerrdquo Administrative Science Quarterly vol 56 no 4pp 501ndash529 2011

[13] O Candogan K Bimpikis and A Ozdaglar ldquoOptimal Pricing inNetworks with Externalitiesrdquo Operations Research vol 60 no 4 pp883ndash905 2012

[14] M Billio M Getmansky A W Lo and L Pelizzon ldquoEconometric mea-sures of connectedness and systemic risk in the finance and insurancesectorsrdquo Journal of financial economics vol 104 no 3 pp 535ndash5592012

[15] R N Mantegna and H E Stanley Introduction to econophysicscorrelations and complexity in finance Cambridge university press1999

[16] J D Glover M Sarma and T Overbye ldquoPower System Analysis andDesignrdquo Cengage Learning vol 4 2008

[17] R Ramakrishna and A Scaglione ldquoOn Modeling Voltage PhasorMeasurements as Graph Signalsrdquo in 2019 IEEE Data Science WorkshopDSW 2019 Institute of Electrical and Electronics Engineers Inc 2019pp 275ndash279

[18] S Chen R Varma A Sandryhaila and J Kovacevic ldquoDiscrete SignalProcessing on Graphs Sampling Theoryrdquo IEEE Transactions on SignalProcessing vol 63 no 24 pp 6510ndash6523 2015

[19] A Anis A Gadde and A Ortega ldquoEfficient Sampling Set Selectionfor Bandlimited Graph Signals Using Graph Spectral Proxiesrdquo IEEETransactions on Signal Processing vol 64 no 14 pp 3775ndash3789 2016

[20] M Tsitsvero S Barbarossa and P Di Lorenzo ldquoSignals on graphsUncertainty Principle and Samplingrdquo IEEE Transactions on SignalProcessing vol 64 no 18 pp 4845ndash4860 2016

[21] X Dong D Thanou P Frossard and P Vandergheynst ldquoLearningLaplacian Matrix in Smooth Graph Signal Representationsrdquo IEEETransactions on Signal Processing vol 64 no 23 pp 6160ndash6173 2016

[22] V Kalofolias ldquoHow to Learn a Graph from Smooth Signalsrdquo in ArtificialIntelligence and Statistics 2016 pp 920ndash929

[23] B Pasdeloup V Gripon G Mercier D Pastor and M G RabbatldquoCharacterization and Inference of Graph Diffusion Processes FromObservations of Stationary Signalsrdquo IEEE Transactions on Signal andInformation Processing over Networks vol 4 no 3 pp 481ndash496 2017

[24] F Grassi A Loukas N Perraudin and B Ricaud ldquoA Time-VertexSignal Processing Framework Scalable Processing and MeaningfulRepresentations for Time-Series on Graphsrdquo IEEE Transactions onSignal Processing vol 66 no 3 pp 817ndash829 2017

11

[25] R Ramakrishna and A Scaglione ldquoDetection of False Data InjectionAttack using Graph Signal Processing for the Power Gridrdquo in 2019 IEEEGlobal Conference on Signal and Information Processing (GlobalSIP)Institute of Electrical and Electronics Engineers Inc 2019 pp 1ndash5

[26] H-T Wai A E Ozdaglar and A Scaglione ldquoIdentifying SusceptibleAgents in Time Varying Opinion Dynamics Through CompressiveMeasurementsrdquo in 2018 IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP) IEEE 2018 pp 4114ndash4118

[27] S Sardellitti S Barbarossa and P Di Lorenzo ldquoOn the Graph FourierTransform for Directed Graphsrdquo IEEE Journal of Selected Topics inSignal Processing vol 11 no 6 pp 796ndash811 2017

[28] R Shafipour A Khodabakhsh G Mateos and E Nikolova ldquoA DirectedGraph Fourier Transform With Spread Frequency Componentsrdquo IEEETransactions on Signal Processing vol 67 no 4 pp 946ndash960 2018

[29] M T Schaub and S Segarra ldquoFlow smoothing and denoising graphsignal processing in the edge-spacerdquo in 2018 IEEE Global Conferenceon Signal and Information Processing (GlobalSIP) IEEE 2018 pp735ndash739

[30] V N Ioannidis Y Shen and G B Giannakis ldquoSemi-Blind Inferenceof Topologies and Dynamical Processes Over Dynamic Graphsrdquo IEEETransactions on Signal Processing vol 67 no 9 pp 2263ndash2274 2019

I Introduction

II Basics of Graph Signal Processing

III Models of Low-pass Graph Signals

IV User Guide to Low-pass Graph Signal Processing

IV-A Low-rank Covariance Matrix

IV-B Smooth Graph Signals

IV-C Anomalies Detection with Low-pass GSP

V Concluding Remarks

References

2

AAPL

ABBV

ABT

ACN

AGN

AIG

ALL

AMGN

AMZN

AXP

BAC

BA

BIIB

BK

BLK

BMY

CAT

CELG

CL

CMCSA

COFCOP

COST

CSCO

CVS

CVX

C

DIS

DUK

EMR

EXC

FBFDX

F

GD

GE

GILD

GM

GOOGL

GOOG

GS

HAL

HD

HON

IBM

INTC

JNJ

JPM

KMI

KO

LLY

LMT

LOW

MA

MCD

MDLZ

MDT

METMMM

MON

MO

MRK

MSFT

MS

NEE

NKE

ORCL

OXY

PCLN

PEP

PFE

PG

PM

QCOMRTN

SBUX

SLB

SO

SPG

TGT

TWX

TXN

T

UNH

UNP

UPS

USB

UTXVZ

V

WBA

WFC

13

Power networkSocial network Financial network

0 10 20 30 40 50

102

103

104

index

(C

o)

Opinion dynamics sample covariance matrix Co

0 20 40 60 80 100

101

102

103

104

index

(C

s)

Stock data sample covariance matrix Cs

0 500 1000 1500 20001018

1014

1010

106

102

102

index

(C

v)

Voltage sample covariance matrix Cv

Voltage data ACTIVSg 2000 test caseOpinion dynamics data 110th US senate Daily return of SampP100 stocks (May 2018-Aug 2019)

CT

ME

MANH

RI VT

DE

NJ

NY

PA

ILIN

MI

OHWI

IA

KS

MN

MO

NE

ND

SD

VA

AL

AR

FLGA

LA

MS NCSC

TX

KY

MD

OK

TN

WV

AZ

CO

ID

MTNV

NM

UT

WY

CA

OR

WA

AK

HI

Fig 1 Illustrating the eigenvalue spectra of sample data covariance matrix of voltage Senate rollcall and financial stock data These data admit physicalsocialmodels that can be regarded as low-pass filtered graph signals A salient feature of their low-pass nature is observed as the low-rank property of the samplecovariance matrices

[S]ij 6= 0 if and only if i = j or (i j) isin E When multipliedby a graph signal x each entry of the shifted graph signal is alinear combination of the one-hop neighborsrsquo values thereforelsquoshiftingrsquo the graph signal with respect to the graph topologyIn this article we take the Laplacian matrix as the GSOThe Laplacian matrix is defined as L = D minus A whereA is the weighted symmetric adjacency matrix of G andD = Diag(A1) is a diagonal matrix of the weighted degreesIt is also common to take the GSO as the normalized Laplacianmatrix or the adjacency matrix [4]

Having defined the GSO we discuss how to measure thesmoothness of graph signals and analyze their content in thegraph frequency domain Recall that if a signal is smooth intime the norm of its time derivative is small For a graphsignal x its graph derivative is defined as

[nablax]ij =radicAij(xi minus xj)

The squared Frobenius norm of graph derivative aka thegraph quadratic form [2] provides an idea of the smoothnessof the graph signal x

S2(x) =1

2nablax2F = xgtLx =

sumij

Aij(xi minus xj)2 (1)

Observe that if xi asymp xj for any neighboring nodes i j thenS2(x) asymp 0 As such we say that a graph signal is smooth ifS2(x)x2 is small

Let us take a closer look at the graph quadratic form S2(x)We set the eigendecomposition of the Laplacian matrix as L =UΛUgt and assume that it has eigenvalues of multiplicity oneordered as Λ = Diag(λ1 λn) with 0 = λ1 lt λ2 ltmiddot middot middot lt λn and U = (u1 u2 middot middot middot un) with ui isin Rn beingthe eigenvector for λi Observe that for any x isin Rn it holdsS2(x)x2 ge

S2(u1)u12 = λ1 and for any x orthogonal to u1 it holds

S2(x)x2 ge

S2(u2)u22 = λ2 and so on for the other eigenvectors The

observation indicates that the larger the eigenvalue the moreoscillatory the eigenvector is over the vertex set In particularthe smallest eigenvalue λ1 = 0 is associated with the flat all-ones eigenvector u1 = (1

radicn)1 as seen in Fig 2 The above

motivates the definition of graph frequencies as the eigenvaluesλ1 λn and of the Graph Fourier Transform (GFT) basisas the set of eigenvectors U [4] Therefore the ith frequencycomponent of x is defined as the inner product between uiand x

xi = ugti x i = 1 n (2)

and x = Ugtx is called the GFT of x The magnitude of theGFT vector |x| is the lsquospectrumrsquo of the graph signal x where|xi|2 represents the signal power at the λith frequency

An important concept to modelling data with GSP is the graphfiltering operation To this end a linear graph filter is describedas the linear operator

H(L) =

Pminus1sump=0

hpLp = U

( Pminus1sump=0

hpΛp)Ugt (3)

3

1 = 0

2 = 04706

10 = 52813

15 = 80818

-05

0

05





y = H(L)x (4)








min|h(λ1)| |h(λk)| (6)










Lasminusrarr E[L] =

n(a+ kb)

kI minusZ

(b11gt + aI

)Zgt (7)






4




p (8)








suminfint=0

sumPminus1p=0 hptλ




a(z) Y (z) = b(z) X(z)



minuss











(tσ)2


)x0 (11)




eminustσλk+1




sumts=0 e




)yt + βxt (12)


5



y = limtrarrinfin




1 + αλk1 + αλk+1







y = limtrarrinfin

yt =( 1

βI minus β

βH(L)




1minus β








gt` (0)]

) (15)






gt +wt (16)



|λk||λk+1|

6



xt =

tsums=0

H (Sred tminus s)ps

H(λred z) = σ2p

(1minus

1sump=0


1sump=0






y` = H(L)x` +w` ` = 1 m (17)


gt` ] = I E[w`w

gt` ] = σ2I












such that

[Φ]qj =

1 if j = sq

0 otherwise





rank(ΦUk

)= k (20)





7

(a)


10minus4

10minus2

100

|λi|maxi|λi|

GFT |v| = |Ugtv|

(b)

0 500 1000 1500 2000bus number

0

500

1000

1500

2000

bus n

umbe

r


0 500 1000 1500 2000bus number

0

500

1000

1500

2000

bus n

umbe

r


1000 1100 1200 1300 1400 1500

04

06

08

bus number

Voltage

(realpart)


1000 1100 1200 1300 1400 1500

minus09

minus08

minus07

minus06

bus number

Voltage

(imaginarypart)



radicknZP




F = minN1Nk

F (N1 NkUk) (22)

where

F (N1 NkUk) =( ksum

q=1

sumiisinNq


|Nq|sumjisinNq

urowj

∥∥∥22

)12






summ`=1 y`y

gt`








8

(a)


1

(b)

AAPL

ABBVABT

ACN

AGN

AIGALL

AMGN

AMZN

AXP

BAC

BA

BIIB

BK

BLK

BMY

CAT

CELG

CL

CMCSACOF

COP

COST

CSCOCVS

CVX

C

DIS

DUK

EMR

EXC

FB

FDX

F

GDGE

GILD

GM

GOOGL

GOOG

GS

HAL

HD

HON

IBM

INTC

JNJ

JPM

KMI

KO

LLY

LMT

LOW

MA MCD

MDLZ

MDT

MET

MMM

MONMO

MRK

MSFT

MS

NEE

NKE

ORCL

OXY

PCLN

PEP

PFE

PG

PM

QCOM

RTN

SBUX

SLB

SO

SPG

TGT

TWX

TXN

T

UNH

UNP

UPS

USB

UTX

VZ

V

WBA

WFC


Consumer Staples

Health Care


Financials

Energy


Industrial








E[S2(y`)

]asymp

ksumi=1

λi |h(λi)|2 + σ2Tr(L) (24)



minz``=1mL

1

m

msum`=1

1






minY isinRntimesm

M(Y )minus Y samp2F

+ γ msum`=1

ygt` Ly` +

msum`=2

y` minus y`minus122





9

(a)

500 1000 1500 20000

02

04

06



cyresponse


0

05

1

Gro

und

Tru

th

0 50 100 150 200

0

05

1

Gra

ph S

igna

l

0 50 100 150 200

0

2

4

Filt

ered

Sig

nal

0 50 100 150 200Node Index



y` =

H(L)x` under A0

H(L)x` +w` under A1



Γ` = HHPF(L)y`A0

≶A1

δ








10






REFERENCES

























11







I Introduction








References

3

1 = 0

2 = 04706

10 = 52813

15 = 80818

-05

0

05





y = H(L)x (4)








min|h(λ1)| |h(λk)| (6)










Lasminusrarr E[L] =

n(a+ kb)

kI minusZ

(b11gt + aI

)Zgt (7)






4




p (8)








suminfint=0

sumPminus1p=0 hptλ




a(z) Y (z) = b(z) X(z)



minuss











(tσ)2


)x0 (11)




eminustσλk+1




sumts=0 e




)yt + βxt (12)


5



y = limtrarrinfin




1 + αλk1 + αλk+1







y = limtrarrinfin

yt =( 1

βI minus β

βH(L)




1minus β








gt` (0)]

) (15)






gt +wt (16)



|λk||λk+1|

6



xt =

tsums=0

H (Sred tminus s)ps

H(λred z) = σ2p

(1minus

1sump=0


1sump=0






y` = H(L)x` +w` ` = 1 m (17)


gt` ] = I E[w`w

gt` ] = σ2I












such that

[Φ]qj =

1 if j = sq

0 otherwise





rank(ΦUk

)= k (20)





7

(a)


10minus4

10minus2

100

|λi|maxi|λi|

GFT |v| = |Ugtv|

(b)

0 500 1000 1500 2000bus number

0

500

1000

1500

2000

bus n

umbe

r


0 500 1000 1500 2000bus number

0

500

1000

1500

2000

bus n

umbe

r


1000 1100 1200 1300 1400 1500

04

06

08

bus number

Voltage

(realpart)


1000 1100 1200 1300 1400 1500

minus09

minus08

minus07

minus06

bus number

Voltage

(imaginarypart)



radicknZP




F = minN1Nk

F (N1 NkUk) (22)

where

F (N1 NkUk) =( ksum

q=1

sumiisinNq


|Nq|sumjisinNq

urowj

∥∥∥22

)12






summ`=1 y`y

gt`








8

(a)


1

(b)

AAPL

ABBVABT

ACN

AGN

AIGALL

AMGN

AMZN

AXP

BAC

BA

BIIB

BK

BLK

BMY

CAT

CELG

CL

CMCSACOF

COP

COST

CSCOCVS

CVX

C

DIS

DUK

EMR

EXC

FB

FDX

F

GDGE

GILD

GM

GOOGL

GOOG

GS

HAL

HD

HON

IBM

INTC

JNJ

JPM

KMI

KO

LLY

LMT

LOW

MA MCD

MDLZ

MDT

MET

MMM

MONMO

MRK

MSFT

MS

NEE

NKE

ORCL

OXY

PCLN

PEP

PFE

PG

PM

QCOM

RTN

SBUX

SLB

SO

SPG

TGT

TWX

TXN

T

UNH

UNP

UPS

USB

UTX

VZ

V

WBA

WFC


Consumer Staples

Health Care


Financials

Energy


Industrial








E[S2(y`)

]asymp

ksumi=1

λi |h(λi)|2 + σ2Tr(L) (24)



minz``=1mL

1

m

msum`=1

1






minY isinRntimesm

M(Y )minus Y samp2F

+ γ msum`=1

ygt` Ly` +

msum`=2

y` minus y`minus122





9

(a)

500 1000 1500 20000

02

04

06



cyresponse


0

05

1

Gro

und

Tru

th

0 50 100 150 200

0

05

1

Gra

ph S

igna

l

0 50 100 150 200

0

2

4

Filt

ered

Sig

nal

0 50 100 150 200Node Index



y` =

H(L)x` under A0

H(L)x` +w` under A1



Γ` = HHPF(L)y`A0

≶A1

δ








10






REFERENCES

























11







I Introduction








References

4




p (8)








suminfint=0

sumPminus1p=0 hptλ




a(z) Y (z) = b(z) X(z)



minuss











(tσ)2


)x0 (11)




eminustσλk+1




sumts=0 e




)yt + βxt (12)


5



y = limtrarrinfin




1 + αλk1 + αλk+1







y = limtrarrinfin

yt =( 1

βI minus β

βH(L)




1minus β








gt` (0)]

) (15)






gt +wt (16)



|λk||λk+1|

6



xt =

tsums=0

H (Sred tminus s)ps

H(λred z) = σ2p

(1minus

1sump=0


1sump=0






y` = H(L)x` +w` ` = 1 m (17)


gt` ] = I E[w`w

gt` ] = σ2I












such that

[Φ]qj =

1 if j = sq

0 otherwise





rank(ΦUk

)= k (20)





7

(a)


10minus4

10minus2

100

|λi|maxi|λi|

GFT |v| = |Ugtv|

(b)

0 500 1000 1500 2000bus number

0

500

1000

1500

2000

bus n

umbe

r


0 500 1000 1500 2000bus number

0

500

1000

1500

2000

bus n

umbe

r


1000 1100 1200 1300 1400 1500

04

06

08

bus number

Voltage

(realpart)


1000 1100 1200 1300 1400 1500

minus09

minus08

minus07

minus06

bus number

Voltage

(imaginarypart)



radicknZP




F = minN1Nk

F (N1 NkUk) (22)

where

F (N1 NkUk) =( ksum

q=1

sumiisinNq


|Nq|sumjisinNq

urowj

∥∥∥22

)12






summ`=1 y`y

gt`








8

(a)


1

(b)

AAPL

ABBVABT

ACN

AGN

AIGALL

AMGN

AMZN

AXP

BAC

BA

BIIB

BK

BLK

BMY

CAT

CELG

CL

CMCSACOF

COP

COST

CSCOCVS

CVX

C

DIS

DUK

EMR

EXC

FB

FDX

F

GDGE

GILD

GM

GOOGL

GOOG

GS

HAL

HD

HON

IBM

INTC

JNJ

JPM

KMI

KO

LLY

LMT

LOW

MA MCD

MDLZ

MDT

MET

MMM

MONMO

MRK

MSFT

MS

NEE

NKE

ORCL

OXY

PCLN

PEP

PFE

PG

PM

QCOM

RTN

SBUX

SLB

SO

SPG

TGT

TWX

TXN

T

UNH

UNP

UPS

USB

UTX

VZ

V

WBA

WFC


Consumer Staples

Health Care


Financials

Energy


Industrial








E[S2(y`)

]asymp

ksumi=1

λi |h(λi)|2 + σ2Tr(L) (24)



minz``=1mL

1

m

msum`=1

1






minY isinRntimesm

M(Y )minus Y samp2F

+ γ msum`=1

ygt` Ly` +

msum`=2

y` minus y`minus122





9

(a)

500 1000 1500 20000

02

04

06



cyresponse


0

05

1

Gro

und

Tru

th

0 50 100 150 200

0

05

1

Gra

ph S

igna

l

0 50 100 150 200

0

2

4

Filt

ered

Sig

nal

0 50 100 150 200Node Index



y` =

H(L)x` under A0

H(L)x` +w` under A1



Γ` = HHPF(L)y`A0

≶A1

δ








10






REFERENCES

























11







I Introduction








References

5



y = limtrarrinfin




1 + αλk1 + αλk+1







y = limtrarrinfin

yt =( 1

βI minus β

βH(L)




1minus β








gt` (0)]

) (15)






gt +wt (16)



|λk||λk+1|

6



xt =

tsums=0

H (Sred tminus s)ps

H(λred z) = σ2p

(1minus

1sump=0


1sump=0






y` = H(L)x` +w` ` = 1 m (17)


gt` ] = I E[w`w

gt` ] = σ2I












such that

[Φ]qj =

1 if j = sq

0 otherwise





rank(ΦUk

)= k (20)





7

(a)


10minus4

10minus2

100

|λi|maxi|λi|

GFT |v| = |Ugtv|

(b)

0 500 1000 1500 2000bus number

0

500

1000

1500

2000

bus n

umbe

r


0 500 1000 1500 2000bus number

0

500

1000

1500

2000

bus n

umbe

r


1000 1100 1200 1300 1400 1500

04

06

08

bus number

Voltage

(realpart)


1000 1100 1200 1300 1400 1500

minus09

minus08

minus07

minus06

bus number

Voltage

(imaginarypart)



radicknZP




F = minN1Nk

F (N1 NkUk) (22)

where

F (N1 NkUk) =( ksum

q=1

sumiisinNq


|Nq|sumjisinNq

urowj

∥∥∥22

)12






summ`=1 y`y

gt`








8

(a)


1

(b)

AAPL

ABBVABT

ACN

AGN

AIGALL

AMGN

AMZN

AXP

BAC

BA

BIIB

BK

BLK

BMY

CAT

CELG

CL

CMCSACOF

COP

COST

CSCOCVS

CVX

C

DIS

DUK

EMR

EXC

FB

FDX

F

GDGE

GILD

GM

GOOGL

GOOG

GS

HAL

HD

HON

IBM

INTC

JNJ

JPM

KMI

KO

LLY

LMT

LOW

MA MCD

MDLZ

MDT

MET

MMM

MONMO

MRK

MSFT

MS

NEE

NKE

ORCL

OXY

PCLN

PEP

PFE

PG

PM

QCOM

RTN

SBUX

SLB

SO

SPG

TGT

TWX

TXN

T

UNH

UNP

UPS

USB

UTX

VZ

V

WBA

WFC


Consumer Staples

Health Care


Financials

Energy


Industrial








E[S2(y`)

]asymp

ksumi=1

λi |h(λi)|2 + σ2Tr(L) (24)



minz``=1mL

1

m

msum`=1

1






minY isinRntimesm

M(Y )minus Y samp2F

+ γ msum`=1

ygt` Ly` +

msum`=2

y` minus y`minus122





9

(a)

500 1000 1500 20000

02

04

06



cyresponse


0

05

1

Gro

und

Tru

th

0 50 100 150 200

0

05

1

Gra

ph S

igna

l

0 50 100 150 200

0

2

4

Filt

ered

Sig

nal

0 50 100 150 200Node Index



y` =

H(L)x` under A0

H(L)x` +w` under A1



Γ` = HHPF(L)y`A0

≶A1

δ








10






REFERENCES

























11







I Introduction








References

6



xt =

tsums=0

H (Sred tminus s)ps

H(λred z) = σ2p

(1minus

1sump=0


1sump=0






y` = H(L)x` +w` ` = 1 m (17)


gt` ] = I E[w`w

gt` ] = σ2I












such that

[Φ]qj =

1 if j = sq

0 otherwise





rank(ΦUk

)= k (20)





7

(a)


10minus4

10minus2

100

|λi|maxi|λi|

GFT |v| = |Ugtv|

(b)

0 500 1000 1500 2000bus number

0

500

1000

1500

2000

bus n

umbe

r


0 500 1000 1500 2000bus number

0

500

1000

1500

2000

bus n

umbe

r


1000 1100 1200 1300 1400 1500

04

06

08

bus number

Voltage

(realpart)


1000 1100 1200 1300 1400 1500

minus09

minus08

minus07

minus06

bus number

Voltage

(imaginarypart)



radicknZP




F = minN1Nk

F (N1 NkUk) (22)

where

F (N1 NkUk) =( ksum

q=1

sumiisinNq


|Nq|sumjisinNq

urowj

∥∥∥22

)12






summ`=1 y`y

gt`








8

(a)


1

(b)

AAPL

ABBVABT

ACN

AGN

AIGALL

AMGN

AMZN

AXP

BAC

BA

BIIB

BK

BLK

BMY

CAT

CELG

CL

CMCSACOF

COP

COST

CSCOCVS

CVX

C

DIS

DUK

EMR

EXC

FB

FDX

F

GDGE

GILD

GM

GOOGL

GOOG

GS

HAL

HD

HON

IBM

INTC

JNJ

JPM

KMI

KO

LLY

LMT

LOW

MA MCD

MDLZ

MDT

MET

MMM

MONMO

MRK

MSFT

MS

NEE

NKE

ORCL

OXY

PCLN

PEP

PFE

PG

PM

QCOM

RTN

SBUX

SLB

SO

SPG

TGT

TWX

TXN

T

UNH

UNP

UPS

USB

UTX

VZ

V

WBA

WFC


Consumer Staples

Health Care


Financials

Energy


Industrial








E[S2(y`)

]asymp

ksumi=1

λi |h(λi)|2 + σ2Tr(L) (24)



minz``=1mL

1

m

msum`=1

1






minY isinRntimesm

M(Y )minus Y samp2F

+ γ msum`=1

ygt` Ly` +

msum`=2

y` minus y`minus122





9

(a)

500 1000 1500 20000

02

04

06



cyresponse


0

05

1

Gro

und

Tru

th

0 50 100 150 200

0

05

1

Gra

ph S

igna

l

0 50 100 150 200

0

2

4

Filt

ered

Sig

nal

0 50 100 150 200Node Index



y` =

H(L)x` under A0

H(L)x` +w` under A1



Γ` = HHPF(L)y`A0

≶A1

δ








10






REFERENCES

























11







I Introduction








References

7

(a)


10minus4

10minus2

100

|λi|maxi|λi|

GFT |v| = |Ugtv|

(b)

0 500 1000 1500 2000bus number

0

500

1000

1500

2000

bus n

umbe

r


0 500 1000 1500 2000bus number

0

500

1000

1500

2000

bus n

umbe

r


1000 1100 1200 1300 1400 1500

04

06

08

bus number

Voltage

(realpart)


1000 1100 1200 1300 1400 1500

minus09

minus08

minus07

minus06

bus number

Voltage

(imaginarypart)



radicknZP




F = minN1Nk

F (N1 NkUk) (22)

where

F (N1 NkUk) =( ksum

q=1

sumiisinNq


|Nq|sumjisinNq

urowj

∥∥∥22

)12






summ`=1 y`y

gt`








8

(a)


1

(b)

AAPL

ABBVABT

ACN

AGN

AIGALL

AMGN

AMZN

AXP

BAC

BA

BIIB

BK

BLK

BMY

CAT

CELG

CL

CMCSACOF

COP

COST

CSCOCVS

CVX

C

DIS

DUK

EMR

EXC

FB

FDX

F

GDGE

GILD

GM

GOOGL

GOOG

GS

HAL

HD

HON

IBM

INTC

JNJ

JPM

KMI

KO

LLY

LMT

LOW

MA MCD

MDLZ

MDT

MET

MMM

MONMO

MRK

MSFT

MS

NEE

NKE

ORCL

OXY

PCLN

PEP

PFE

PG

PM

QCOM

RTN

SBUX

SLB

SO

SPG

TGT

TWX

TXN

T

UNH

UNP

UPS

USB

UTX

VZ

V

WBA

WFC


Consumer Staples

Health Care


Financials

Energy


Industrial








E[S2(y`)

]asymp

ksumi=1

λi |h(λi)|2 + σ2Tr(L) (24)



minz``=1mL

1

m

msum`=1

1






minY isinRntimesm

M(Y )minus Y samp2F

+ γ msum`=1

ygt` Ly` +

msum`=2

y` minus y`minus122





9

(a)

500 1000 1500 20000

02

04

06



cyresponse


0

05

1

Gro

und

Tru

th

0 50 100 150 200

0

05

1

Gra

ph S

igna

l

0 50 100 150 200

0

2

4

Filt

ered

Sig

nal

0 50 100 150 200Node Index



y` =

H(L)x` under A0

H(L)x` +w` under A1



Γ` = HHPF(L)y`A0

≶A1

δ








10






REFERENCES

























11







I Introduction








References

8

(a)


1

(b)

AAPL

ABBVABT

ACN

AGN

AIGALL

AMGN

AMZN

AXP

BAC

BA

BIIB

BK

BLK

BMY

CAT

CELG

CL

CMCSACOF

COP

COST

CSCOCVS

CVX

C

DIS

DUK

EMR

EXC

FB

FDX

F

GDGE

GILD

GM

GOOGL

GOOG

GS

HAL

HD

HON

IBM

INTC

JNJ

JPM

KMI

KO

LLY

LMT

LOW

MA MCD

MDLZ

MDT

MET

MMM

MONMO

MRK

MSFT

MS

NEE

NKE

ORCL

OXY

PCLN

PEP

PFE

PG

PM

QCOM

RTN

SBUX

SLB

SO

SPG

TGT

TWX

TXN

T

UNH

UNP

UPS

USB

UTX

VZ

V

WBA

WFC


Consumer Staples

Health Care


Financials

Energy


Industrial








E[S2(y`)

]asymp

ksumi=1

λi |h(λi)|2 + σ2Tr(L) (24)



minz``=1mL

1

m

msum`=1

1






minY isinRntimesm

M(Y )minus Y samp2F

+ γ msum`=1

ygt` Ly` +

msum`=2

y` minus y`minus122





9

(a)

500 1000 1500 20000

02

04

06



cyresponse


0

05

1

Gro

und

Tru

th

0 50 100 150 200

0

05

1

Gra

ph S

igna

l

0 50 100 150 200

0

2

4

Filt

ered

Sig

nal

0 50 100 150 200Node Index



y` =

H(L)x` under A0

H(L)x` +w` under A1



Γ` = HHPF(L)y`A0

≶A1

δ








10






REFERENCES

























11







I Introduction








References

9

(a)

500 1000 1500 20000

02

04

06



cyresponse


0

05

1

Gro

und

Tru

th

0 50 100 150 200

0

05

1

Gra

ph S

igna

l

0 50 100 150 200

0

2

4

Filt

ered

Sig

nal

0 50 100 150 200Node Index



y` =

H(L)x` under A0

H(L)x` +w` under A1



Γ` = HHPF(L)y`A0

≶A1

δ








10






REFERENCES

























11







I Introduction








References

10






REFERENCES

























11







I Introduction








References

11







I Introduction








References

a user guide to low-pass graph signal processing and its ... · and the low-pass graph signals...

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