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On the exponential generating function for non-backtracking
walks
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Arrigo, F, Grindrod, P, Higham, DJ & Noferini, V 2018, 'On the exponential generating function for non-backtracking walks' Linear algebra and its applications, vol. 556, pp. 381-399. DOI:
10.1016/j.laa.2018.07.010
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Linear
Algebra
and
its
Applications
www.elsevier.com/locate/laa
On
the
exponential
generating
function
for
non-backtracking
walks
Francesca Arrigoa,∗, Peter Grindrodb, Desmond J. Highama,
Vanni Noferinic,∗
aDepartmentofMathematicsandStatistics,UniversityofStrathclyde,Glasgow,
G11XH,UK
bMathematicalInstitute,UniversityofOxford,AndrewWilesBuilding,Radcliffe
ObservatoryQuarter,WoodstockRoad,Oxford,OX26GG,UK
c
DepartmentofMathematicalSciences,UniversityofEssex,WivenhoePark, Colchester,CO43SQ,UK a r t i c l e i n f o a b s t r a c t Articlehistory: Received18September2017 Accepted6July2018 Availableonlinexxxx SubmittedbyM.Benzi MSC: 05C90 05C50 15A15 65F50 Keywords: Centrality Matrixfunction Localization Networkscience
Wederiveanexplicitformula fortheexponentialgenerating function associated with non-backtracking walks around a graph. Westudybothundirected anddirectedgraphs. Our results allow us to derive computable expressions for non-backtrackingversionsofnetworkcentralitymeasuresbasedon thematrixexponential.Wefindthateliminatingbacktracking walks in this context does not significantly increase the computationalexpense.Weshowhowthenewmeasuresmay be interpreted in terms of standard exponential centrality computationon a certain multilayer network.Insights from thisblockmatrixinterpretationalsoallowustocharacterize centrality measures arising from general matrix functions. Rigorous analysis on the star graph illustrates the effect of non-backtracking and shows that localization can be eliminatedwhen werestrict to non-backtrackingwalks. We alsoinvestigatethelocalizationissueonsyntheticnetworks.
©2018TheAuthors.PublishedbyElsevierInc.Thisisan openaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).
* Correspondingauthors.
E-mailaddresses:[email protected](F. Arrigo),[email protected](V. Noferini).
https://doi.org/10.1016/j.laa.2018.07.010
0024-3795/©2018TheAuthors.PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCC BYlicense(http://creativecommons.org/licenses/by/4.0/).
1. Introduction
1.1. Motivation
Theconcept ofawalkonagraphisverynatural—on arriving atanode,thewalker maycontinuebytraversinganyedgepointingoutofthatnode.If,atanystage,theedge along whichthewalkercontinuesisthereverse oftheedgeonwhichtheyarrived, then the walk is said to be backtracking. Non-backtracking walks have been analyzed in a numberoffields.Theyplayakeyroleinthestudyofzetafunctionsongraphs[24],with applications inspectral graphtheory [4,23], numbertheory [44], discrete mathematics [12,41], quantumchaos[39], randommatrix theory[40], stochastic analysis[3], applied linear algebra [42] and computerscience [38,45]. Within network science, constraining walks to be non-backtrackinghasbeen shownto offerbenefitsincommunity detection [26,28], centrality measurement [6,20,29,36], network comparison [30] and inthe study of relatedissuesconcerningoptimalpercolation[31,32].
A natural task across all these fields is to count the number of distinct non-backtracking walks of a given length between pairs of nodes, and to form a compact expression forthe associatedgenerating function.For networkcentrality,inthecase of standard walkcounts ithasbeen argued thatanexponential-stylepowerseries givesa usefulalternativeto thestandardresolvent-styleversion[15].Forexample,basedonan analogy with aphysicaloscillator, itmaybe argued thatmovingfromthe resolventto theexponentialtakesusfromclassicaltoquantumphysics[16].Further,thereare effec-tiveandreliabletoolsforcomputingtheactionofthematrixexponential[2,21,22].This providestheinitialmotivationforourarticle,wherewestudytheexponentialgenerating functionassociatedwithnon-backtrackingwalks.Wealsonotethatmoregeneralmatrix functionshavebeen proposedinthiswalk-countingcontext[17].Hence,wethenextend theanalysis tocovergeneratingfunctionsbasedonarbitrarypowerseries.
1.2. Background
WeletA= (aij)∈ Rn×ndenote theadjacencymatrixforanunweightedgraphwith
n nodes;thatis,aij = 1 ifthereisanedgefromnodei tonodej,andaij = 0 otherwise.
Wewillassumethegraphtobeloopless,sothataii = 0 foralli,andtohavenomultiple
edges.IfwefurtherassumethegraphrepresentedbyA tobeundirected,i.e.,ifA= AT,
thenthegraphwillbesaidtobesimple.WeuseD todenotethediagonalmatrixwhose diagonal entries are Dii = (A2)ii. So Dii counts the number of reciprocal neighbours
of node i, thatis, the numberof nodes j suchthat aij = aji = 1.If A = AT, so that
all edgesarereciprocated,then Dii reduces tothevector ofdegrees and anynode i for
whichDii= 1 willbecalled aleaf.
If A = AT the network is said to be directed and we will denote by S = (s ij) ∈
Rn×nthematrixassociatedwiththegraphobtainedbyremovingalledgesthatarenot
A walk of length r from node i to node j is characterized by a sequence of edges
i= i1 → i2,i2→ i3,. . . ,ir → ir+1= j.Note thatedges andnodes mayberevisitedin
thecourseofawalk. Itfollowsfrom thedefinitionof matrixmultiplication thatArhas
(i,j)thelementthatcountsthenumberofdistinctwalksoflengthr fromi toj;see,for example,[13,Theorem 2.2.1].Givenaparametert,theordinarygeneratingfunction [46] associatedwiththissequenceofwalkcountsmayberepresentedbyaresolventfunction:
∞
r=0
trAr= (I− tA)−1,
where I is the identity matrix. If t is interpreted as a formal algebraic variable, this expressionholdswithinthealgebraofformalpowerseries.Analytically,ift isinterpreted asareal(or complex)valuedparameter,itisvalid for|t|< 1/ρ(A),where ρ(·) denotes
thespectral radius. Similarly, theassociated exponential generating function [46] takes theformofamatrixexponential:
∞
r=0
trAr
r! = exp(tA),
analyticallyvalidforallt∈ R (or inC).
Power series expansionsthat combine walk counts of different lengths havebecome widely used inthefield ofnetwork science [17]. Here,the ideais to assign aweightto eachnodethatsummarizesits“importance”inthenetwork.Ifweinterpretimportance astheabilityofanodetoinitiatemessagesalongtheedgesinthegraph,thenweseethat (I− tA)−11,for 0< t< 1/ρ(A),and exp(tA)1,fort> 0,are candidatesformeasuring andcomparingnodes,where1∈ Rn isthevectorofones.Here,theithcomponentarises
by summing thenumber of distinct walks from node i to all nodes inthe graph, with walksoflengthr discountedbytrintheresolventcaseand bytr/r! intheexponential case.TheresolventversiondatesbacktotheworkofKatz[25] andthealternativebased on the exponential function, called the total (node) communicability, was introduced andstudiedin [8]. Wealsonotethatboth versionsarerelatedto eigenvectorcentrality [10,11],whichusesthePerron–FrobeniuseigenvectorofA whentheunderlyinggraphis stronglyconnected [9].
Inrecentwork[29],Martinetal.arguedthattraditionalcentralitymeasuresmaysuffer fromunwantedlocalization effectswhenthenetworkunderstudyhasascale-freedegree distribution(see,e.g., [7,14]).Inthistypeofnetwork,there areafewnodeswithahigh numberofneighbours,thatarethusassignedmostofthecentrality,andmanynodeswith averysmallnumberof connections,thathenceshare theremainderofthe massofthe centralityvector.Thisunbalancedpartitioningmakesitdifficulttodistinguishbetween the importance of the nodes with few connections. The authors therefore proposed a variationofeigenvectorcentrality motivatedbytheconcept ofnon-backtrackingwalks. This idea was pursued in [6,20] in the context of resolvent-based centrality. Our aim
hereistodevelopandstudytheanalogousexponentialversionofnon-backtrackingwalk centrality, andalsotoconsider generalmatrixfunctions.
Formally,wesaythatawalkisbacktracking ifitcontainsatleastonepairofsuccessive edges ofthe form i→ j,j → i;thatis, having leftsomenode i it immediately returns to thatnode. Wesay the walkis non-backtracking otherwise. Weuse theabbreviation NBTW to mean non-backtracking walk. We will denote by pr(A) ∈ Rn×n the matrix
whose (i,j)th entry counts the number of NBTWs of length r fromnode i to node j
in the graph represented by the matrix A. Our first task is therefore to obtain useful expressions forthecorrespondingexponentialgenerating function
F (t) := ∞ r=0 trpr(A) r! (1)
and centralitymeasure
b := F (t)1. (2)
We do this inSections 2and 3for thecase of undirectedand directed graphs, respec-tively.Although,ofcourse,anundirectedgraphisaspecialcaseofadirectedgraph,we treat thetwo casesseparatelyfor tworeasons. First,undirectedgraphsare commonin network science, and henceit is useful to haveresults tailoredto this case.Second, in our context itseems morestraightforwardto study undirected graphsdirectly thanto simplifythemoregeneralexpressionfordirectedgraphs.InSection4,webrieflydescribe how to computethe proposed centrality measures and discuss computational costs. In Section5,weinterpretthenewexpressionsfromtheviewpointoftraversalsaround mul-tilayer networkswithpossiblynegativeinterlayerweights.Thisblockmatrixframework also allowsus to deriveexpressions forthe casewhere theexponential isreplaced bya general matrixfunction.InSection6we giveaconcreteexample ofthebenefitof non-backtracking,and furtherillustrationsonrealisticmodelnetworksappearinSection7. Weend withashort summaryinSection8.
Thekey analyticalstepsusedhereareto deriveand solveanODE—thisis awidely usedapproachinthestudyofexponentialgeneratingfunctions[46].Tothisend,wenote from (1) that F(t) = ∞ r=0 trpr+1(A) r! and F (t) =∞ r=0 trpr+2(A) r! . (3)
We also require a lemma on matrix-valued ODEs. The following result arises from columnwise application of a standard result on vector-valued linear systems; see, for example, [21, Section 2.1]. The statementof the result makes use of the shifted
expo-nentialfunctionψ1(z)= (ez− 1)/z, whichmaybedefinedformallyviathepowerseries expansion ψ1(z) = ∞ r=0 zr (r + 1)!. (4)
Lemma1.1. LetC∈ Rm×m and P,X0∈ Rm×.Then
X(t) = tψ1(tC) (P + CX0) + X0 (5)
is the unique solution to the linear, constant coefficient, inhomogeneous, autonomous, initialvalue ODEsystem
X(t) = CX(t) + P, X(0) = X0.
Moreover,in thehomogeneouscaseP = 0,theexpression(5) simplifies to
X(t) = exp(tC)X0. (6)
Remark 1.2.We note that Lemma 1.1 does not require the Jacobian matrix C to be nonsingular.
2. Exponentialgeneratingfunctionforundirected graphs
Weassumethroughoutthissectionthatthegraphisundirected.Wealsoassumefor conveniencethatthegraphisconnected—inthedisconnectedcase,eachcomponentmay bestudiedseparately.Wenote thateachnodeisthereforeassumedtohaveat leastone neighbour.
As wementioned inSection1.2, the matrixAr counts walksof length r. Hencethe trivialrelationshipAr+1= AArgivesarecurrencebetweenwalkcountsforlengthr and
r + 1. In thecase of NBTWson undirected graphs, thefollowing two termrecurrence relateswalkcountsofsuccessivelengths.Wehavep0(A)= I,p1(A)= A,p2(A)= A2−D,
and
pr+2(A) = Apr+1(A)− (D − I)pr(A), for r≥ 1. (7)
This recursion was proved in [12] for general directedgraphs. For the undirected case consideredhere,therecurrencewasre-derived independentlyin[41].
Wenowgivethemainstatementofthissection:anexplicitformulaforthegenerating function(1) andthecorrespondingcentralitymeasure (2).
Theorem 2.1. For a simple graph with associated adjacency matrix A, the exponential generating functionF (t) in(1) and totalcommunicability vectorb in(2) satisfy
F (t) =I 0tψ1(tY ) A A2− D + I, b = I 0 tψ1(tY ) A1 (A2− D)1 + 1, where Y := 0 I I− D A ∈ R2n×2n. (8)
In thecasewhere thegraphhas no leaves,theseexpressions maybe writtenas
F (t) =I 0exp (tY ) I + (D− I)−1 A − (D − I)−1, b =I 0 exp (tY ) 1 + (D− I)−11 A1 − (D − I)−11.
Proof. To obtainthe exponential generatingfunction F (t) in(1),we multiplyby tr/r!
in(7) andthen sumfrom r = 1 to∞,toobtain
∞ r=1 trp r+2(A) r! = A ∞ r=1 trp r+1(A) r! − (D − I) ∞ r=1 trp r(A) r! .
From (3),thisshows
F(t)− p2(A) = A(F(t)− p1(A))− (D − I)(F (t) − p0(A)),
whichsimplifies to
F(t)− AF(t) + (D− I)F (t) = −I. (9) To converttoafirstorder system,weintroduce
G(t) = G1(t) G2(t) := F (t) F(t) ∈ R2n×n, so that(9) becomes G(t) = Y G(t) + 0 −I , where G(0) = I A ,
G(t) = tψ1(tY ) 0 −I + Y I A + I A . (10)
Finally,wenotethattheJacobianmatrixY in(8) isinvertibleifandonlyif(D−I)−1 exists,thatis,ifthegraphdoesnotcontainleaves,i.e.,nodesofdegreeone.Inthiscase
Y−1 = (D− I)−1A −(D − I)−1 I 0 and(10) simplifiesto G(t) = exp (tY ) I + (D− I)−1 A − (D− I)−1 0 . (11)
NotingthatF (t)=I 0G(t),weseethat(10) and(11) yieldthestatement. 2 WediscusstheseresultsfurtherinSections4and5.
3. Exponentialgeneratingfunctionfordirectedgraphs
For directedgraphs thematrices pr(A) that count NBTWs satisfy athree term
re-currence.Thiswasshownin[12],withamorelinearalgebraorientedderivationgivenin [42].Withp0(A)= I,p1(A)= A,p2(A)= A2− D,therecurrencetakestheform
pr+3(A) = Apr+2(A)− (D − I)pr+1(A)− (A − S)pr(A), for r≥ 0. (12)
RecallthatS∈ Rn×nistheadjacencymatrixofthegraphobtainedbyonlyconsidering
reciprocatedlinksintheoriginalnetwork;sosij = aijaji.Weemphasizethatunlikethe
undirectedcase(7),thisrecurrenceisvalidfrom r = 0.
HereistheanaloguefordirectedgraphsofTheorem2.1.
Theorem 3.1.Let A be the adjacency matrix of an unweighted directed graph with no self-loops nor multipleedges. The exponential generating function F (t) in (1) and total communicability vectorb in(2) satisfy
F (t) =I 0 0exp(tZ) ⎡ ⎢ ⎣ I A A2− D ⎤ ⎥ ⎦ , b =I 0 0exp(tZ) ⎡ ⎢ ⎣ 1 A1 (A2− D)1 ⎤ ⎥ ⎦ ,
where Z := ⎡ ⎢ ⎣ 0 I 0 0 0 I −(A − S) −(D − I) A ⎤ ⎥ ⎦ ∈ R3n×3n. (13)
Proof. Multiplying(12) bytr/r! andsummingfrom r = 0 to∞ wearriveat
F(t) = AF(t)− (D − I)F(t)− (A − S)F (t). (14) We note that, incontrast to (9), this ODE doesnot havean inhomogeneous term.To convertto afirstorder system,introduce
H(t) = ⎡ ⎢ ⎣ H1(t) H2(t) H3(t) ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ F (t) F(t) F(t) ⎤ ⎥ ⎦ , so that H(t) = ZH(t), where H(0) = ⎡ ⎢ ⎣ I A A2− D ⎤ ⎥ ⎦ , where Z isdefinedin(13).
Using (6) in Lemma 1.1 and noting that F (t) =
I 0 0
H(t), we arrive at the statement. 2
It isofinteresttonote thattheremovalofbacktracking walksmaybeinterpretedas the introduction ofdamping terms−(D − I)F(t) and −(A− S)F (t) in (14). Further, Theorem 3.1 shows that, ingeneral, as t increases the total communicability vector b
alignswiththefirstn componentsofthedominanteigenvectorofZ.
4. Computingthecentralityvectors
In this section we consider computational issues. First, we note that the standard expression for the total(node) communicability,etA1, requires the actionof the
expo-nential of a sparse n by n matrix on a vector in Rn. It is reasonable to assume that
a suitable approximation may be obtained iteratively, using a small number of sparse matrix-vectormultiplications[2].AssumingthatA hasO(n) nonzeros(edges),theneach sparsematrix-vectormultiplicationwillcostO(n).Hence,usingk todenotethenumber of iterations,the overallcost willbe O(nk). Inpractice,it isreasonable to regardk as
In Theorem 2.1, we see that for an undirected graph with no leaves, the cost of computing the non-backtracking total communicability vector b is dominated by the actionofetY onavectorinR2n,where the2n by2n sparsematrixY is definedin(8).
We note that Y has only 2n more nonzeros than the underlying adjacency matrixA,
and hence,under theassumptions above, ifthe samenumber ofiterations is required, thenthesameO(n) cost applies.
Inthe directedcase, the expression forb in Theorem 3.1 requires the action of etZ
onavectorinR3n,wherethe3n by3n sparsematrixZ isdefinedin(13).SinceZ also
hasO(n) nonzeros,weagainrecover theO(n) cost underourassumptions.
It remainsto consider thecase ofan undirected graphwith leaves.In Theorem2.1, theexpression forb involvestheshiftedexponential,ψ1,from (4).Wemayproceed by
reducingthekeycomputationtotheapproximationoftheactionofamatrixexponential inaslightlyhigherdimension.Let
Y := tY v 0T 0 ∈ R(2n+1)×(2n+1),
wherev∈ R2nand0∈ R2nisavectorofallzeros.Itwasshownin [37,Section2.3] that
exp( Y ) = etY ψ1(tY )v 0T 1 . So,ifwelet v = A1 (A2− D)1
thenb inTheorem2.1maybecomputedbyfirstapproximatingthevectorexp( Y )e2n+1,
where e2n+1 is the (2n+ 1)th vector of the standard basisof R2n+1. Wethen extract
thetopn entriesof theresultingvector,scaleby t andshiftby1.Thisagain willhave anO(n) costunderourassumptions.
Remark4.1.It is worthemphasizing thatthefinal stepsof scalingand shiftingcan be avoidedifwe areonlyinterested,asiscustomary innetworkscience, intherankingsof thenodesratherthantheiractualcentralityscores.
5. Blockmatrixinterpretations
Theorems2.1and3.1showthat,loosely,forexponential-basedcentralitytheoperation ofeliminatingbacktrackingwalksisequivalenttoreplacingtheoriginaladjacencymatrix
A with asuitable block matrix. Inthe undirected case, the 2by 2version Y in (8) is used, whereasthedirectedcasehasthe 3by3versionZ in (13). These blockmatrices wereconstructedindirectly,asaconsequenceofthegenerating functionapproach,[46].
Inthissection, weaddinsightbyshowinghow theseblockmatricesmaybeinterpreted directly.Wealsoshow thattheyremain relevantwhen theexponentialis replacedbya generalmatrixfunction.
We beginwith analgebraic interpretation.Thematrices Z andY arethefirst com-panionmatrixlinearization [18] ofthematrixpolynomials(A−S)+(D−I)λ−Aλ2+Iλ3
and,respectively,(D−I)−Aλ+ Iλ2.Theseareinturnthereversalsof,respectively,the
directedand undirected versionof thedeformedgraphLaplacian,amatrixpolynomial thatplaysacrucialroleinthetheoryofNBTWcombinatorics,see[6,20].Algebraically, the companion matrix is themultiplication-by-λI operator in themodule obtained by formingquotientsintheringofmatrixpolynomialsusingtheleftidealgeneratedbythe linearized matrix polynomial [34]. Linear-algebraic functions of the companion matrix alsohaveananalogousinterpretation;forexample,exp(tZ) representsmultiplicationby exp(tλ)I modulo(A−S)+ (D−I)λ−Aλ2+ Iλ3.Formoredetailsonthisalgebraicview
of companionmatricessee, e.g.,[33, Sec.2], [34,Sec.5],[35,Sec.9],and thereferences therein.
There is also agraph-theoretical interpretation that we now describe in detail. We focus first on the directed case, involving Z in (13). A useful connection is that Z
represents the adjacency matrix for a three layer weighted network. This network is node-aligned [27], so that the general ith nodes from each layer may be regarded as copiesofthesamenode.LookingatthediagonalblocksofZ,weseethatthewithin-layer connectionscorrespondtotheemptygraphforlayersoneandtwo,andtheoriginalgraph forlayerthree.Next,weconsidertheoff-diagonalblocks.Theidentitymatrixinthe(1,2) blockandzeromatrixinthe(1,3) blockshowthattheonlyedgeoutofanode inlayer one points to the corresponding node inlayer two.Similarly, we may only leave layer two bymovingtothecorresponding nodeinlayerthree.Withinlayerthree,inaddition to movingwithinthelayerusingedgesfromA, traversalsarepossibletolayerstwoand one.The (3,2) blockshowsthatforDii> 1 atraversal maymovefrom nodei inlayer
three to node i in layer two, with the negative weight1− Dii assigned to that edge.
The (3,1) blockshows thatamovefrom node i inlayer threeto node j inlayeroneis allowed ifandonlyiftheoriginalgraphhasanonreciprocaledgefromnodei tonodej,
inwhichcasetheinter-layeredgecarriesweight−1.
Theexpressionforb inTheorem3.1showsthatexponentialNBTWcentralityarises fromapost-processedversionofthe“standard”exponentialcentralityonthismultilayer network,wherenegativeinter-layerweightshavetheeffectofnegatingthecontributions of backtrackingwalksintheunderlyinggraph.
ThetwolemmasandthetheorembelowshowthatZ isalsorelevantformoregeneral matrixfunctions.
Lemma 5.1.Forallr≥ 0 wehave
ZrZ 0 0 I = p r(A) pr+1(A) pr+2(A) .
Proof. For r < 2,theidentity maybe verified directly.Forr≥ 2 the resultfollows by straightforwardinductionusingthethree termrecurrence(12). 2
Theorem5.2(Directed graph).Let{αr}∞r=0 beasequencesuchthatf0(Z)=∞r=0αrZr
converges.Then ∞r=0αrpr(A) converges, and ∞ r=0 αrpr(A) = [I 0 0]f0(Z) I A A2− D .
Moreover,∞r=0αrpr(A) isequaltothe(3,3) block inf0(Z)− f2(Z),where
fk(Z) = ∞
r=0
αr+kZr.
Proof. ByLemma5.1itfollows that
∞ r=0 αrpr(A) = [I 0 0] ∞ r=0 αrZrZ 0 0 I and,since Z 0 0 I = I A A2− D
thefirstpartofthestatementfollows.
Forthesecondpart,notefirstthatr=0∞ αr+2Zralsoconvergesundertheassumptions
inthestatement.Then
∞ r=0 αrpr(A) = α0I + α1A + ∞ r=2 αrpr(A) = α0I + α1A + [0 0 I] ∞ r=2 αrZr−2Z 0 0 I = α0I + α1A + [0 0 I] ∞ r=2 αrZr 0 0 I − [0 0 I]∞ r=2 αrZr−2 0 0 I = [0 0 I] ∞ r=0 (αr− αr+2)Zr 0 0 I , asrequired. 2
Theorem 5.2 shows that we can accumulate combinations of NBTW counts in the originalnetworkbylookingatthe(3,3) blockofanappropriatefunctionofthematrixZ.
Fromamultilayerperspective,the(3,3) blockofZ isassociatedwiththelayercontaining the adjacency matrixA of theoriginal network. Theorem 5.2 therefore concerns walks of any lengththat startand end at thatlayer. This includes notonly walks thattake placewithinlayerthree,butalsothosethatmakeoneormoreintermediatevisitstothe other layers.
Thecontributioncomingfromf0(Z) countswalksinthemultilayernetworkusingthe
original weights.Thecorrectionterminvolving f2(Z) removeswalksfrom thecount by
weightingwalksoflengthr− 2 asiftheywereoflengthr.
Inmoredetail,thetermf2(Z) isusedtocorrecttheappearanceofanidentitymatrix
inZ2 thatisthenpropagatedathigherpowersofZ.Thisidentitymatrixatlevelr = 2
originates fromtheoneappearinginthe(3,2) block.Thislatterisusedintherecursion (12) definingpr(A) whenr > 2 tocorrectthepenalizationofwalksofthetype
i → · · · → k → j → k → j of length r
which are backtracking at length(r− 1)1 and thus neednot be removedbythe factor
pr−2(A)D.Walksofthistypetakeplace entirelyinthethirdlayerandarebalancedby
theexistenceofwalksoftheform
node: i → · · · → k → j → j → j
layer: [3] → · · · → [3] → [3] → [2] → [3]
providedthatr > 2.Whenr = 2,walksoftheform
node: j −−−→ j1−dj −→ j1
layer: [3] → [2] → [3]
are improperly weighted, as the first hop contains a +1 that is used to balance the removal of walks thathave never even existed. This introduces a“delay” thatis then propagated whenlongerwalksarecounted.Tocounteractit,thematrixfunctionf2(Z)
removes walks of length r− 2 that are however weightedas if they were of length r;
thistypeofweightingpreciselytargetsthosewalksobtainedfromthepropagationofthe identity matrixintroducedatr = 2 inf0(Z).
Theorem 5.2canbe usedto recover theknowncharacterisationoftheordinary gen-erating function. To see this, we let αr = tr for all r, with |t| < ρ(Z)−1. Then we
have
1
The additional requirement of the edge k → j to be reciprocated is the one that adds the term
∞ r=0 trpr(A) = [0 0 I] (I− tZ)−1− t2(I− tZ)−1 0 0 I = (1− t2)[0 0 I](I− tZ)−1 0 0 I .
Thisisequivalentto theresultin [6,Section 3].
Similarly,setting αr= tr/r! forsomet> 0,Theorem 5.2gives ∞ r=0 trpr(A) r! = [0 0 I] etZ− t2ψ2(tZ) 0 0 I , (15) whereψ2(x)= ∞
r=0xr/(r + 2)!.This descriptionofF (t) from (1) isequivalenttothat
giveninTheorem3.1.However,(15) caststheresultentirelyintermsofthe(3,3) block, which,aswehavearguedabove,hasanaturalmultilayerinterpretation.
Wefinishthissectionbybrieflydiscussingtheundirectedcase.Here,theblockmatrix
Y in (8) may be associated with a two-layer network. In layer one, the only possible transition is to theequivalent node inlayer two.We maymovewithin layer two using any edge that exists in the original graph, or,for Dii > 1, we maymove from node i
in layer two to node i in layer one along an edge with negative weight 1− Dii. The
analogues of Lemma5.1 and Theorem 5.2 are given below, with proofs following ina similarmanner.
Lemma5.3. Forallr≥ 0 wehave YrY 0 I = pr+1(A) pr+2(A) .
Theorem5.4(Undirected graph).Let{αr}∞r=0 beasequenceofnonnegativeweightssuch
that fk(Y ):=∞r=0αr+kYr convergesfork = 0. Then ∞ r=0 αrpr(A) = [0 I] (f0(Y )− f2(Y )) 0 I .
Theorem 5.4 quantifiesin theundirected case how NBTWs onthe original network may also be counted by considering walks on the associated multilayer network that startandendinthelayercontainingA.
6. Stargraphanalysis
Inthis section, we givefurtherinsights into theeffect ofrestricting to NBTWsina totalcommunicabilitycentrality measure.Foreigenvectorcentrality,theauthorsin[29] arguedthatnon-backtrackingisameanstoavoidlocalization;thatis,theconcentration
of weight on a small number of nodes. Following [19], they characterized localization through an asymptotic limit. Given v = (vi) ∈ Rn the inverse participation ratio is
definedas S(v) = v14 2 n i=1 vi4=v 4 4 v4 2 . (16)
This quantity is of order one in the case where a small number of elements in v are
responsible for mostof the weight.Moreformally, we saythatthe measure is localized
ifS(v)= O(1) andnonlocalized if S(v)= o(1),intheasymptoticlimitn→ ∞.
We will analyze this property onthe undirected star graphwith n nodes. Here the adjacency matrixtakes theform
A = 1T n−1 1n−1 , (17)
where 1s ∈ Rs is thevector of all ones. Thus node 1 is a hubwith n− 1 connections
to nodes 2,3,. . . ,n,whichare onlyconnected tonode 1.Intuitively,theeffectof back-tracking shouldbe significantinthis example.Thehubnodehasn− 1 neighbours,and hence n− 1 walks of length 1, but cannot initiate walks of length two or more with-out backtracking.Each peripheral node, incontrast, hasa singleneighbour, butn− 2
NBTWs of lengthtwo.Hence, inthenon-backtracking regime,ift is takensufficiently large thatdegreedoesnotdominate, thenhaving onehighquality link(toahub) may be comparableinimportanceto havingmanylowqualitylinks (toleaves).
It isstraightforwardto showfor(17) that(see,forexample, [5,21])
etA= ⎡ ⎣ cosh(t √ n− 1) sinh(t√n− 1)(√n− 1)−11T n−1 sinh(t√n− 1)(√n− 1)−11n−1 cosh(t 1n−11Tn−1) ⎤ ⎦ . Hence, the standard, backtracking walk version of total communicability, x = etA1,
satisfies x = cosh(t√n− 1) +√n− 1 sinh(t√n− 1) cosh(t√n− 1)1n−1+ (√n− 1)−1sinh(t√n− 1)1n−1 .
Sothevaluesx1 forthehubnodeandx2foratypicalleafnodesatisfy
x1= cosh(t √ n− 1) +√n− 1 sinh(t√n− 1), x2= cosh(t √ n− 1) + (√n− 1)−1sinh(t√n− 1).
It follows immediatelythatforany t> 0 wehavex1> x2 when thegraphhasatleast
three nodes. Hence, thestandardtotal communicabilitymeasure always ranksthehub node abovetheleaves.Wealsofindthat,forfixedt> 0,
(x1)4= n2 16e 4t√n−1+ on2e4t√n−1, (xi)4= n2 16(n− 1)2e 4t√n−1+ oe4t√n−1, for all i = 2, . . . , n, x4 2= n2 4e 4t√n−1+ on2e4t√n−1.
Itfollowsthattheinverseparticipation ratio(16) satisfies lim
n→∞S(x) =
1 4 andhencethemeasureislocalized.
Wemayevaluatethenon-backtrackingversion, b= F (t)1, fromfirstprinciples.The hubnode has n− 1 NBTWsof lengthone and no others, and eachleaf node hasone NBTWoflengthone,n− 2 NBTWsoflengthtwoand noothers. So
b1= 1 + t(n− 1) and b2= 1 + t +
t2
2(n− 2).
Straightforward computationshows thatb1 > b2 for 0 < t< 2 and b1 < b2 for t > 2.
Therefore,weobserveatransitionthatisnotpresentwithstandardexponential central-ity,albeitinaparameterregimewhere t isgreaterthanunity.
Moretellingly,forfixed t> 0
(b1)4= t4n4+ o(n4),
(bi)4=
t8
16n
4+ o(n4), for all i = 2, . . . , n,
b4 2= t8 16n 6+ o(n6). Hence, lim n→∞S(b) = 0;
sothemeasure isnonlocalized. 7. Numericaltests
Inthissectionwestudythelocalizationeffectinarealisticsetting.Weusedarandom graphmodelratherthan afixeddata set sothatwe cancontrolthe dimension,n, and studytheasymptotics. Weusedanimplementation oftheBarabási–Albert preferential attachmentmodelforundirectednetworksintheMATLABtoolboxCONTEST [43].In thismodel,nodes areadded sequentiallyuntil thedesiredsize isreached. Eachnode is
Fig. 1. Inverse participation ratio of vectors F (t)1 and etA1 as the number of nodes, n, varies.
given, upon arrival, d links to current nodes. These target nodes are chosen indepen-dently with a probability proportional to their current degree. The resulting network will have ascale-freedegreedistribution, where afewnodes (hubs) haveahigh degree andmanynodeshaveaverysmalldegree.Inthiscase,thevectorofdegreesislocalized byconstruction.Wehaveρ(A)≤ A∞,and whenthisinequalityissharpthepresence ofahigh degreenodeforcesKatztouseasmallparameterα,producingresultscloseto degreecentrality [9].
Weusedthedefaultsetting ofthetoolbox, givingeachnoded= 2 linksuponarrival to the network, and we built networks of increasing size from n = 10 to n = 1000. For each network, wecomputed the inverseparticipation ratio (16) for the normalized total communicability vector etA1 andits nonbactrackinganalogue b= F (t)1 for t =
1 fixed. In order to compute both centrality vectors, we used the MATLAB toolbox funm_kryl2 [1], which computesthe action of amatrix function over avector using a restartedArnoldialgorithm.Werepeatedourtests1000 times.
The results are displayedin Fig.1, where the sample average of the inverse partic-ipation ratio is plotted against n in a semilogarithmic scale. The error bar is used to display the standard error around the averagevalues. The results are inline with the analyticalresultspresentedinSection6—theinverseparticipationratiodecaystowards zeroforthenon-backtrackingversion,butappearsto beboundedawayfromzerowhen backtracking walksareincluded.
8. Summary
Themain contributionsofthis workwere
1. toderiveanalyticalexpressionsforthenon-backtrackingwalkexponentialgenerating functionforbothundirectedand directedgraphs,
2
2. to show that it is feasible to compute the resulting network centrality measures: thekey matrixinvolvedinthecomputations hastwice(three times)thedimension of the analogous backtracking version for undirected (directed) graphs, and has a comparablelevel ofsparsity,
3. tointerpretthenewmeasuresinthecontextoftraditionalcentralityonamultilayer network,
4. touseinsightsfromthemultilayer/blockmatrixsettinginordertoderiveexpressions forgeneralmatrixfunction-basedcentralitymeasures,
5. to study how non-backtracking within exponential centrality reduces localization effects onastargraph,
6. togivefurtherillustrativeresultsonacontrollabletest network.
Ouroverallmessageis thatit isboth analyticallyandcomputationally attractive to study non-backtrackingwalk counts in ageneral matrixfunction setting thatincludes thematrixexponential.Wethereforehope tohavepavedtheway forfutureworkthat, forexample,considers
• physicalinterpretationsofthenewmeasures, • applicationstospecificfields ofnetworkscience,
• furtherlinks withareas of discretemathematics,stochastic processesand quantum physics,
• the development and analysis of customized numerical methodsfor approximating the action of a matrix function on the type of large, sparse unstructured matrix arising innetworkscience.
Acknowledgements
The work of FA and DJH was supported by grant EP/M00158X/1 from the EP-SRC/RCUKDigitalEconomy Programme.
EPSRCData Statement:data usedinthisworkwasgenerated from theCONTEST package[43],whichispubliclyavailable.
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