Structural Analysis of Viral Spreading Processes in Social and Communication Networks Using Egonets

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1 1 Structural alysis of Viral Spreadig Processes i Social ad Commuicatio Networs Usig Egoets Victor M. Preciado, Member, IEEE, Moez Draief, ad li Jadbabaie, Seior Member, IEEE arxiv: v1 math.oc] 3 Sep 2012 bstract We study how the behavior of viral spreadig processes is iflueced by local structural properties of the etwor over which they propagate. For a wide variety of spreadig processes, the largest eigevalue of the adjacecy matrix of the etwor plays a ey role o their global dyamical behavior. For may real-world large-scale etwors, it is ufeasible to exactly retrieve the complete etwor structure to compute its largest eigevalue. Istead, oe usually have access to myopic, egocetric views of the etwor structure, also called egoets. I this paper, we propose a mathematical framewor, based o algebraic graph theory ad covex optimizatio, to study how local structural properties of the etwor costrai the iterval of possible values i which the largest eigevalue must lie. Based o this framewor, we preset a computatioally efficiet approach to fid this iterval from a collectio of egoets. Our umerical simulatios show that, for several social ad commuicatio etwors, local structural properties of the etwor strogly costrai the locatio of the largest eigevalue ad the resultig spreadig dyamics. From a practical poit of view, our results ca be used to dictate immuizatio strategies to tame the spreadig of a virus, or to desig etwor topologies that facilitate the spreadig of iformatio virally. Idex Terms Complex Networs, Virus Spreadig, lgebraic Graph Theory, Covex Optimizatio. I. INTRODUCTION Uderstadig the behavior of viral spreadig processes taig place i large complex etwors is of critical iterest i mathematical epidemiology 1], 2]. Spreadig processes are relevat i may real scearios, such as disease spreadig i huma populatios 3] 5], malware propagatio i computer etwors 6] 7], or iformatio dissemiatio i olie social etwors 8] 9]. To study viral spreadig processes, a variety of stochastic dyamical models has bee proposed i the literature 10] 14]. I these models, the steady-state ifectio of the etwor presets two differet regimes depedig o the virulece of the ifectio ad the structure of the etwor of cotacts. I oe of the regimes, a iitial ifectio dies out at a fast (usually expoetial) rate. I the other regime, a iitial ifectio becomes a epidemic. Both umerical ad aalytical results show that these two regimes are separated by a phase trasitio at a epidemic threshold determied Mauscript Received. V.M. Preciado ad. Jadbabaie are with the Departmet of Electrical ad Systems Egieerig at the Uiversity of Pesylvaia, Philadelphia, P US. ( preciado@seas.upe.edu; jadbabai@seas.upe.edu). M. Draief is with the Departmet of Electrical ad Electroic Egieerig at Imperial College, Lodo, SW7 2Z UK. ( m.draief@imperial.ac.u). This wor was supported by ONR MURI Next Geeratio Networ Sciece ad FOSR Topological d Geometric Tools For alysis Of Complex Networs. MD is supported by QNRF through grat NPRP MD holds a Leverhulme Trust Research Fellowship RF/9/RFG/2010/02/08. by both the virulece of the ifectio ad the topology of the etwor. Oe of the most fudametal questios i mathematical epidemiology is to fid the value of the epidemic threshold i terms of the virus model ad the cotact etwor. I may cases of practical iterest it is ufeasible to exactly retrieve the complete structure of a etwor of cotacts. I these cases, it is impossible to exactly compute the epidemic threshold. O the other had, i most cases oe ca easily retrieve the structure of egocetric views of the etwor, also called egoets 1. To estimate the value of the epidemic threshold, researchers have proposed a variety of radom etwor models i which they ca prescribe structural properties that ca be retrieved from these egoets, such as the degree distributio 15], 16], local correlatios 17], 18], or clusterig 19]. lthough radom etwors are the primary tool to study the impact of local structural features o the epidemic threshold 20], this approach presets a major flaw: Radom etwor models implicitly iduce may structural properties that are ot directly cotrolled but ca have a strog ifluece o the value of the epidemic threshold. For example, it is possible to fid two etwors havig the same degree distributio, but with opposite dyamical behavior 21]. Therefore, it is difficult (if ot impossible) to isolate the role of a particular structural property i the etwor performace usig radom etwor models. Furthermore, may real etwors preset weighted edges represetig, for example, badwidth i commuicatio etwors or resistace i electric etwors. Curret radom etwors fail to faithfully recover both the structure of the etwor ad the distributio of weights over the lis. I this paper, we develop a mathematical framewor, based o algebraic graph theory ad covex optimizatio, to study how the structure of local egoets costrai the iterval of possible values i which the epidemic threshold must lie. s a result of our aalysis, we preset a computatioally efficiet approach to fid this iterval from a collectio of egoets extracted from a (possibly) weighted etwor. Our umerical simulatios show that the resultig iterval is very arrow for several social ad commuicatio etwors. This illustrates the fact that, for may real etwors, local structural properties of the etwor strogly costrai the locatio of the viral epidemic threshold. The rest of this paper is orgaized as follows. I Sectio II, we review termiology ad existig results relatig the dyamical behavior of a virus model with spectral properties of the etwor of cotacts. I Sectio III, we itroduce a approach, based o algebraic graph theory ad covex optimizatio, 1 rigorous defiitio of egoet, i graph-theoretical terms, will be give i Sectio III.

2 2 to fid upper ad lower bouds o the epidemic thresholds from local egoets. I Subsectio III-, we itroduce a approach to related these egoets to the so-called spectral momets of the adjacecy matrix. I Subsectio III-B, we propose a optimizatio framewor to derive bouds o the epidemic threshold from a collectio of spectral momets. I Sectio IV, we illustrate the quality of our approach by computig bouds o the epidemic threshold for real-world social ad commuicatio etwors. II. NOTTION & PRELIMINRIES Let G = (V, E) be a udirected, uweighted graph, where V = {1,..., } deotes a set of odes ad E V V deotes a set of udirected edges liig them. If (i, j) E, we call odes i ad j adjacet (or first-eighbors), which we deote by i j. We defie the set of first-eighbors of a ode i as N i = {j V : (i, j) E}. The degree d i of a vertex i is the umber of odes adjacet to it, i.e., d i = N i. graph is weighted if there is a real umber w ij 0 associated with every edge (i, j) E. More formally, a weighted graph H ca be defied as the triad H = (V, E, W), where V ad E are the sets of odes ad edges i H, ad W = {w ij R\ {0}, for all (i, j) E} is the set of (possibly egative) weights. The adjacecy matrix of a simple graph G, deoted by G = a ij ], is a symmetric matrix defied etrywise as a ij = 1 if odes i ad j are adjacet, ad a ij = 0 otherwise. For weighted graphs, the etry a ij is equal to the weight w ij for (i, j) E; 0, otherwise. For udirected graphs, G is a symmetric matrix; thus, G has a full set of real ad orthogoal eigevectors with real eigevalues λ 1 λ 2... λ. The largest eigevalue of G, λ 1, is called the spectral radius of G. If has oegative etries ad is irreducible (i.e., G is coected), the the Perro- Frobeius theorem 22] ca be used to show that the spectral radius λ 1 is uique, real, ad positive. We also defie the -th spectral momet of G as: m (G) 1 λ i. (1) wal of legth from ode i 1 to ode i +1 is a ordered sequece of odes (i 1, i 2,..., i +1 ) such that i j i j+1 for j = 1, 2,...,. Oe says that the wal touches each of the odes that comprises it. If i 1 = i +1, the the wal is closed. closed wal with o repeated odes (with the exceptio of the first ad last odes) is called a cycle. Give a wal p = (i 1, i 2,..., i +1 ) i a weighted graph H, we defie the weight of the wal as, ω (p) = w i1i 2 w i2i 3...w i i +1.. Stochastic Modelig of Viral Spreadig wide variety of stochastic models has bee proposed i the literature to study the dyamics of virus spreadig processes. I most models, the steady-state level of ifectio i the etwor presets two differet regimes separated by a phase trasitio taig place at a epidemic threshold. This epidemic threshold is determied by both the virulece of the ifectio ad the etwor topology. series of papers study the value of this epidemic threshold as a fuctio of the etwor structure, i both radom 23] 27] ad real topologies 10] 14]. spreadig model widely cosidered i the literature is the so-called SIS (Susceptible-Ifected- Susceptible) model. I this model, each idividual i the etwor ca be i oe of two possible states: susceptible or ifected. Give a iitial set of ifected idividuals, the virus propagates through the edges of a udirected graph G at a ifectio rate β. Simultaeously, ifected odes recover at a rate δ, returig bac to the susceptible state (see 10] for a formal descriptio of this model). I 10] 12], we fid differet (ad complemetary) approaches to fid a expressio for the SIS epidemic threshold. I all of these papers, the authors are able to decouple the effect of the etwor topology from the dyamics of idividual odes. O the oe had, the effect of the ode dyamics is completely characterized by the ratio τ SIS δ/β. O the other had, the effect of the etwor topology depeds exclusively o the largest eigevalue of the etwor adjacecy matrix, λ 1 ( G ), such that if the threshold coditio λ 1 () < τ SIS = δ/β is satisfied, a small iitial ifectio dies out expoetially fast 10] 12]. May extesios to the SIS model have bee proposed to capture differet characteristics of viral processes, such as permaet or temporal immuity of a recovered idividual, or virus icubatio time 13], 14]. s show i 14], the decouplig argumet that allows to separate the role of the etwor topology from the ode dyamics i the SIS model still holds for a variety of other virus models. Similarly, a small iitial ifectio dies out expoetially fast i these models if the coditio λ 1 ( G ) < τ V M is satisfied, where the threshold τ V M measures the virulece of the ifectio (ad is idepedet of the etwor structure). s a bottom lie, all of the above results remar the ey role played by the largest eigevalue of the adjacecy matrix, λ 1 ( G ), i virus spreadig processes. I particular, the larger λ 1 ( G ), the more efficiet a etwor is to spread a disease (or a piece of iformatio) virally. B. Spectral Estimators Based o Radom Graphs Radom etwor models are curretly the primary tool to study the relatioship betwee local structural properties of a etwor ad its epidemic threshold. lthough may radom etwors have bee proposed i the literature 15] 19], oly radom etwors icludig a very limited amout of structural iformatio are curretly ameable to aalysis. The origial radom graph model is the Erdös-Réyi graph, deoted by G (, p), i which each edge i a graph with odes is idepedetly chose with probability p, 28]. I this model, the distributio of degrees i the etwor follows a Poisso distributio with expectatio Ed i ] = ( 1)p. Furthermore, the largest eigevalue of its adjacecy matrix is almost surely λ 1 = 1 + o (1)] p (assumig that p = Ω (log )). lthough very iterestig from a theoretical poit of view, the origial radom graph presets very limited modelig capabilities, sice the degree distributios of real-world etwors almost ever follow a Poisso distributio.

3 3 I order to icrease the modelig abilities of radom graphs, Chug et al. proposed i 16] a radom graph G (w) i which oe ca prescribe a desired expected sequece of degrees, w = (w 1,..., w ). I this radom graph, edges are idepedetly assiged to each pair of vertices (i, j) with probability w i w j / =1 w. Chug et al. proved i 16] that if / w2 i j=1 w j > max {w i } log, the the largest eigevalue λ 1 (G (w)) coverges almost surely λ 1 (G (w)) a.s. 1 + o (1)] w2 i j=1 w, (2) j for large. Despite its theoretical iterest, radom graphs with a give degree distributio are by far ot eough to faithfully model the structure of real complex etwors. I particular, it is well-ow that the degree distributio aloe is ot a sufficiet statistic to aalyze the performace of may etwors. For example, lderso et al. itroduce i 21] a collectio of etwors, icludig radom graphs, presetig the same degree distributio ad radically differet dyamical performace. lthough radom graph models with more elaborated structural properties ca be foud i the literature 15] 19], these models are usually hard (if ot impossible) to aalyze from a spectral poit of view. The source of this itractability is the presece of strog correlatios amog the etries of the (radom) adjacecy matrix. These strog correlatios prevet the resultig radom adjacecy matrix from beig aalytically tractable. I the ext sectio, we preset a ovel approach to aalyze the effect of local structural properties o the largest eigevalue of a etwor without maig use of radom graphs. III. SPECTRL NLYSIS FROM EGOCENTRIC SUBNETWORKS I this sectio, we study the relatioship betwee local structural properties of a etwor ad its eigevalue spectrum. I our aalysis, we assume that we do ot have access to the complete topology of the etwor, due to, for example, privacy ad/or security costrais. Istead, we assume that we are able to access local egocetric views of the etwor topology. I this settig, we propose a approach to extract global spectral iformatio from local structural properties of the etwor. This spectral iformatio will be used i Subsectio III-B to compute upper ad lower bouds o the epidemic threshold. We ow provide graph-theoretical ad algebraic elemets to characterize the iformatio cotaied i these egocetric views of the etwor. Let δ (i, j) deote the distace betwee two odes i ad j (i.e., the miimum legth of a wal from i to j). By covetio, we assume that δ (i, i) = 0. We defie the r-th order eighborhood aroud ode i, deoted by G i,r = (N i,r, E i,r ), as the subgraph G i,r G with ode-set N i,r {j V : δ (i, j) r}, ad edge-set E i,r = {(v, w) E s.t. v, w N i,r }. Notice that G i,r provides a graph-theoretical descriptio of the egocetric view of the etwor from ode i withi a radius of r hops. Motivated by this iterpretatio, we also call G i,r the egoet of radius r aroud ode i. Egoets ca be algebraically represeted via submatrices of the adjacecy matrix G, as follows. Give a set of odes K V, we deote by G (K) the submatrix of G formed by selectig the rows ad colums of G idexed by K. I particular, we defie the adjacecy submatrix i,r G (N i,r ). Notice that i,r is itself a adjacecy matrix represetig the structure of the egoet G i,r. By covetio, we associate the first row ad colum of the submatrix i,r with ode i V, which ca be doe via a simple permutatio of the rows ad colums of i,r. 2 For a weighted graph H with weighted adjacecy matrix H, we defie the weighted egoet H i,r as the weighted graph whose adjacecy matrix is i,r H (N i,r ).. Spectral Momets from Local Egoets I this subsectio, we derive expressios for the spectral momets of the adjacecy from the owledge of local egoets usig tools from algebraic graph theory. The followig lemma provides a iterestig coectio betwee the umber of closed wals i G (a combiatorial property) ad its spectral momets (a algebraic property) 29]: Lemma 3.1: Let G be a simple graph with adjacecy matrix G = a ij ]. The G ] ii = W i,, where W i, is the set of closed wals of legth startig ad fiishig at ode i. Usig the above result, oe ca prove the followig wellow result i algebraic graph theory 29]: Corollary 3.2: Let G be a simple graph. Deote by e ad the umber of edges ad triagles i G, respectively. The, m 1 ( G ) = 0, m 2 ( G ) = 2e, ad m 3( G ) = 6. We ca geeralize Lemma 3.1 to weighted graphs as follows: Lemma 3.3: Let H = (V, E, W) be a weighted graph with weighted adjacecy matrix H. The, H ]ii = p P,i ω (p), where P,i is the set of closed wals of legth from v i to itself i H. Proof: By recursively applyig the multiplicatio rule for matrices, we have the followig expasio H ]ii = w i,i2 w i2,i 3 w i,i. (3) i 2=1 i =1 Usig the graph-theoretic omeclature itroduced i Sectio II, we have that w i,i2 w i2,i 3...w i,i = ω (p), for p = (v i, v i2, v i3,..., v i, v i ). Hece, the summatios i (3) ca be writte as ] WH ii = 1 i,i 2,...,i ω (p). Fially, the set of closed wals p = (v i, v i2, v i3,..., v i, v i ) with idices 1 i, i 2,..., i is equal to the set of closed wals of legth from v i to itself i H (which we have deoted by P,i i the statemet of the Propositio). 2 Notice that permutig the rows ad colums of the adjacecy matrix does ot chage the topology of the uderlyig graph, it simply chages the labels associated to each ode.

4 4 Fig. 1. Cycles C 4 ad C 5, of legths 4 ad 5, i a eighborhood of radius 2 aroud ode i. quatities ] i,r, i = 1,...,. For a fixed, each value 11 ( i,r ]11, i = 1,...,, ca be computed i time O N i,r 3), where N i,r is the umber of odes i the local egoet H i,r. The sparse structure of most real etwors implies that N i,r (for moderate values of r). I particular, if N i,r = o ( ɛ ) for ay ɛ > 0, we ca compute the -th spectral momets i quasi-liear time (with respect to the size of the etwor) usig (4). This result provides a clear computatioal advatage compared to computig the spectral momets via a explicit eigevalue decompositio, which ca be prohibitively expesive to compute for large complex etwors. Usig Lemma 3.3, we ca exted Lemma 3.2 to higherorder momets of weighted graphs as follows: Theorem 3.4: Cosider a weighted, udirected graph H with adjacecy matrix H. Let i,r be the (weighted) adjacecy matrix of the egoet of radius r aroud ode i. The, for a give r, the spectral momets of H ca be writte as m ( H ) = 1 ], (4) i,r 11 for 2r + 1. Proof: Sice the trace of a matrix is the sum of its eigevalues, we ca expad the -th spectral momet of the adjacecy matrix as follows: m ( H ) = 1 Trace ( ) 1 H = ] ] H ii. (5) From Lemma 3.3, we have that H = ii p P,i ω (p). Notice that for a fixed value of, closed wals of legth i H startig at ode i ca oly touch odes withi a certai distace r () of i, where r () is a fuctio of. I particular, for eve (resp. odd), a closed wal of legth startig at ode i ca oly touch odes at most /2 (resp. /2 ) hops away from i (see Fig. 1). Therefore, closed wals of legth startig at i are always cotaied withi the eighborhood of radius /2. I other words, the egoet H i,r of radius r cotais all closed wals of legth up to 2r + 1 startig at ode i. We ca cout these wals by applyig Lemma 3.3 to the local adjacecy matrix i,r. I particular, p P,i ω (p) is equal to ] i,r (sice, by covetio, ode 1 i the local 11 egoet H i,r correspods to ode i i the graph H). Therefore, for 2r + 1, we have that i,r ]11 = ] H p P,i ω (p) = ii. (6) The, substitutig (6) ito (5), we obtai the statemet of our Theorem. Remar 3.1: The above theorem allows us to compute a trucated sequece of spectral momets {m ( H ), 2r + 1}, give a collectio of local egoets of radius r, {H i,r, i V}. ccordig to (4), we ca compute the -th spectral momet by simply averagig the B. SDP-Based Bouds o the Spectral Radius Usig Theorem 3.4, we ca compute a trucated sequece of the spectral momets of a etwor H, (m 1 ( H ), m 2 ( H ),..., m 2r+1 ( H )), from a set of local egoets of radius r, {H i,r, i V}. We ow preset a covex optimizatio framewor to extract iformatio about the largest eigevalue of the adjacecy matrix, λ 1 ( H ), from this sequece of momets. We ca state the problem solved i this subsectio as follows: Problem 1: Give a trucated sequece of spectral momets of a weighted, udirected graph H, m 2r+1 = (m 0, m 1,..., m 2r+1 ), fid tight upper ad lower bouds o the largest eigevalue λ 1 ( H ). Our approach is based o a probabilistic iterpretatio of the eigevalue spectrum of a give etwor. To preset our approach, we first eed to itroduce some cocepts: Defiitio 3.1: Give a weighted, udirected graph H with (real) eigevalues λ 1,..., λ, the spectral desity of H is defied as, µ H (x) 1 δ (x λ i ), (7) where δ ( ) is the Dirac delta fuctio. The spectral desity ca be iterpreted as a discrete probability desity fuctio with support 3 o the set of eigevalues {λ i, i = 1...}. Let us cosider a discrete radom variable X whose probability desity fuctio is µ H. The momets of this radom variable satisfy the followig: Lemma 3.5: The momets of a r.v. X µ H are equal to the spectral momets of H, i.e., for all 0. E µh ( X ) = m ( H ), 3 Recall that the support of a fiite Borel measure µ o R, deoted by supp (µ), is the smallest closed set B such that µ (R\B) = 0.

5 5 Proof: For all 0, we have the followig: ( E µh X ) = x µ H (x) dx = 1 = 1 R R x δ (x λ i ) dx λ i = m ( G ). We ow preset a covex optimizatio framewor that allows us to fid bouds o the edpoits of the smallest iterval a, b] cotaiig the support of a geeric radom variable X µ give a sequece of momets (M 0, M 1,..., M 2r+1 ), where M x dµ. Subsequetly, we shall apply these results to fid bouds o λ 1 ( H ). Our formulatio is based o the followig matrices: Defiitio 3.2: Give a sequece of momets M 2r+1 = (M 0, M 1,..., M 2r+1 ), let H 2r (M 2r+1 ) ad H 2r+1 (M 2r+1 ) R (r+1) (r+1) be the Hael matrices defied by 4 : H 2r ] ij M i+j 2 ad H 2r+1 ] ij M i+j 1. (8) The above matrices are called the momet matrices associated with the sequece M 2r+1. I geeral, a arbitrary sequece of umbers (N 0, N 1,..., N ) may ot have a represetig measure µ such that x r dµ = N r, for 0 r. sequece of umbers N = (N 0, N 1,..., N ) is said to be feasible i Ω R if there exists a measure µ with support cotaied i Ω whose momets match those i the sequece N. The problem of decidig whether or ot a sequece of umbers is feasible i Ω is called the classical momet problem i aalysis 30]. For uivariate distributios, ecessary ad sufficiet coditios for feasibility ca be give i terms of certai Hael matrices beig positive semidefiite 5, as follows 31]: Theorem 3.6: 31, Theorem 3.2] Let M 2r+1 = (M 0, M 1,..., M 2r+1 ) R 2r+2. The, (a) The sequece M 2r+1 correspods to a sequece of momets feasible i Ω = R if ad oly if H 2r 0. (b) The sequece M 2r+1 is feasible i Ω = a, ) if ad oly if (c) H 2r 0 ad H 2r+1 ah 2r 0. The sequece M 2r+1 is feasible i Ω = (, b] if ad oly if H 2r 0 ad bh 2r H 2r+1 0. Usig Theorem 3.6, we have the followig result: 4 For simplicity i the otatio, we shall omit the argumet M 2r+1 wheever clear from the cotext. 5 The otatio 0 meas that the matrix is positive semidefiite. Theorem 3.7: Let µ be a probability desity fuctio o R with associated sequece of momets M 2r+1 = (M 0, M 1,..., M 2r+1 ), all fiite, ad let a, b] be the smallest iterval which cotais the support of µ. The, b β (M 2r+1 ), where β r (M 2r+1 ) mi x x s.t. H 2r 0, x H 2r H 2r+1 0. Proof: Sice M 2r+1 is the momet sequece of a probability desity fuctio µ with support o a, b] (, b], we have from Theorem 3.6.(c) that M 2r+1 satisfy H 2r 0 ad bh 2r H 2r+1 0. Sice β (M 2r+1 ) is, by defiitio, the miimum value of x such that H 2r 0 ad xh 2r H 2r+1 0, we have that β (M 2r+1 ) b. Remar 3.2: Observe that, for a give sequece of momets M 2r+1, the etries of xh 2r H 2r+1 deped affiely o the variable x. The β (m 2r+1 ) is the solutios to a semidefiite program 6 (SDP) i oe variable. Hece, β (M 2r+1 ) ca be efficietly computed usig stadard optimizatio software, e.g. 33], from a trucated sequece of momets. pplyig Theorem 3.7 to the spectral desity µ H of a give graph H with spectral momets (m 0, m 1,..., m 2r+1 ), we ca fid a lower boud o its largest eigevalue, λ 1 ( H ), as follows: Theorem 3.8: Let H be a weighted, udirected graph with (real) eigevalues λ 1... λ. The, give a trucated sequece of the spectral momets of H, m 2r+1 = (m 0, m 1,..., m 2r+1 ), we have that (9) λ 1 ( H ) β r (m 2r+1 ), (10) (where β r (m 2r+1 ) is the solutio to the SDP i (9)). Proof: Let us cosider the spectral desity of H, µ H, i Defiitio 3.1. ccordig to Lemma 3.5, the desity µ H has associated momets m 2r+1. lso, the smallest iterval which cotais the support of µ H is a, b] = λ, λ 1 ]. Therefore, applyig Theorem 3.7 to µ H, we obtai that β r (m 2r+1 ) b = λ 1. Furthermore, for r = 1, we ca aalytically solve the SDP i (9) to derive a closed-form solutio for β 1 (m 3 ), as follows: Corollary 3.9: Let G be a simple graph with adjacecy matrix G. Deote by, e, ad the umber of odes, edges, ad triagles i G, respectively. The, λ 1 ( G ) e 3 /. (11) 2e Proof: I the ppedix. Usig the optimizatio framewor preseted above, we ca also compute upper bouds o the spectral radius of H from a sequece of its spectral momets, as follows. I this case, our formulatio is based o the followig set of Hael matrices: 6 semidefiite program is a covex optimizatio problem that ca be solved i time polyomial i the iput size of the problem; see e.g. 32].

6 6 Defiitio 3.3: Give a weighted, udirected graph H with odes ad spectral momets m 2r+1 = (m 0, m 1,..., m 2r+1 ), let T 2r (y; m 2r+1, ) ad T 2r+1 (y; m 2r+1, ) R (r+1) (r+1) be the Hael matrices defied by 7 : T 2r ] ij T 2r+1 ] ij 1 m i+j yi+j 2, (12) 1 m i+j yi+j 1. Give a sequece of spectral momets, we ca compute upper bouds o the largest eigevalue λ 1 ( H ) usig the followig result: Theorem 3.10: Let H be a weighted, udirected graph with (real) eigevalues λ 1... λ. The, give a trucated sequece of its spectral momets m 2r+1 = (m 0, m 1,..., m 2r+1 ), we have that where λ 1 δ r (m 2r+1, ), δ r (m 2r+1, ) max y y s.t. T 2r 0, yt 2r T 2r+1 0, T 2r+1 + yt 2r 0. (13) Proof: Let us defie the bul of the spectrum as the set of eigevalues {λ 2,..., λ }, ad the bul spectral desity as the probability desity fuctio: µ H 1 1 δ (x λ i ). i=2 We also defie the bul spectral momets as the momets of the bul spectral desity, which satisfy: m ( H ) x µ H (x) dx R = 1 1 = 1 1 i=2 R x δ (x λ i ) dx λ i 1 1 λ 1 = 1 m ( H ) 1 1 λ 1. Therefore, the momet matrices associated to the sequece of bul spectral momets m 2r+1 = ( m 0, m 1,..., m 2r+1 ), satisfy H s ( m 2r+1 ) = T s (λ 1 ; m 2r+1, ), (14) for s {2r, 2r + 1}, where H s ad T s were defied i (8) ad (12), respectively. Sice λ i λ 1 for i 2, the support of the bul spectral desity µ H is cotaied i the iterval λ 1, λ 1 ]. 7 We shall omit the argumets from T 2r ad T 2r+1 wheever clear from the cotext. Hece, accordig to Theorems 3.6.(b)-(c), the sequece of bul spectral momets m 2r+1 must satisfy: T 2r (λ 1 ; m 2r+1, ) 0, λ 1 T 2r (λ 1 ; m 2r+1, ) T 2r+1 (λ 1 ; m 2r+1, ) 0, T 2r+1 (λ 1 ; m 2r+1, ) + λ 1 T 2r (λ 1 ; m 2r+1, ) 0. Sice δ r (m 2r+1, ) is, by defiitio, the maximum value of y satisfyig the costrais i (13), we have that δ r (m 2r+1, ) λ 1 ( H ). Remar 3.3: The optimizatio program i (13) is ot a SDP, sice the etries of the matrices T 2r (y; m 2r+1, ) ad T 2r+1 (y; m 2r+1, ) are ot affie fuctios, but higher-order polyomials, i y. Nevertheless, the program ca be cast ito a covex optimizatio program, as follows. For the matrices i (13) to be positive semidefiite, all their pricipal miors must be oegative, where each mior is a polyomial i y. I other words, positive semidefiiteess of the matrices i (13) is equivalet to a collectio of polyomials i y beig oegative. Hece, we ca substitute the semidefiite costrais i (13) by a collectio of polyomials i y beig oegative. The resultig optimizatio problem is a Sum- Of-Squares (SOS) program 34], which is a type of covex program that ca be efficietly solved usig off-the-shelf software 35]. I summary, usig Theorems 3.4, 3.8, ad 3.10, we ca compute upper ad lower bouds o the largest eigevalue of a weighted, udirected etwor, λ 1 ( H ), from the set of local egoets with radius r, as follows: (1) Usig (4), compute the trucated sequece of momets (m 0, m 1,..., m 2r+1 ) from the set of egoets, { i,r, i V}, ad (2) usig Theorems 3.8 ad 3.10, compute the upper ad lower bouds, δ r (m 2r+1, ) ad β r (m 2r+1 ), respectively. IV. NUMERICL SIMULTIONS I this sectio, we aalyze real data from several social ad commuicatio etwors to umerically verify the tightess of our bouds. I our first set of simulatios, we study a regioal etwor of Faceboo that spas 63, 731 users (odes) coected by 817, 090 friedships (edges) 36]. I order to corroborate our results i differet etwor topologies, we extract multiple medium-size social subgraphs by ruig a Breath-First Search (BFS) aroud a collectio of startig odes i the Faceboo graph. Each BFS iduces a social subgraph spaig all odes 2 hops away from a startig ode. s a result, we geerate a set of 100 differet social subgraphs, G = {G i } i 100, cetered aroud 100 radomly chose odes. For each social subgraph G i G, we compute its first five spectral momets m 5 (G i ) = (m 1 (G i ),..., m 5 (G i )) ad use Theorems 3.8 ad 3.10 to compute lower ad upper bouds o the spectral radius, β 2 (G i ) = β 2 (m 5 (G i )) ad δ 2 (G i ) = δ 2 (m 5 (G i ), i ), where i is the size of G i. Sice we have access to the complete etwor topology, we ca also umerically compute the exact value of the largest eigevalue λ 1 (G i ), for compariso purposes. It is worth remarig that, i may real applicatios, we do ot have access to the complete etwor topology, due to privacy ad/or security

7 7 costrais; therefore, we would ot be able to compute the exact value of λ 1. It is i those cases whe our approach is most useful. Fig. 2 represets a scatter plot where each red circle above the dashed diagoal lie has coordiates (λ 1 (G i ), δ 2 (G i )), ad each blue circle below the dashed diagoal lie has coordiates (λ 1 (G i ), β 2 (G i )), for all G i G. We have also icluded a blac lie coectig every pair of circles associated to the same subgraph G i. This blac lie represets the iterval of possible values i which the largest eigevalue, λ 1 (G i ), must lie. (Notice how the dashed diagoal lie cut through all those segmets.) For all the social subetwors i G, the spectral radii λ 1 (G i ) are remarably close to the theoretical bouds β 2 (G i ) ad δ 2 (G i ). I other words, i our collectio of social subgraphs, local structural properties of the etwor strogly costrai the locatio of the largest eigevalue, ad cosequetly the ability of a social etwor to dissemiate iformatio virally. Our bouds are also tight for other importat social ad commuicatio etwors. I the followig, we the compare the values of β 2 ad δ 2 with the largest eigevalue λ 1 of a ad a Iteret etwor: Example 4.1 (Ero etwor): I this example we cosider a subgraph of the Ero commuicatio etwor 37]. Nodes of the etwor are addresses ad the etwor cotais a edge (i, j) if i set at least oe to j (or vice versa). The total size of the etwor is 36, 692 odes, which is too large for us to maage computatioally. I order to compare our bouds with the exact value of the largest eigevalue, we aalyze a subgraph obtaied by a BFS of depth 2 aroud a radomly chose ode. The resultig subgraph has = 3, 215 odes ad e = 36, 537 edges. We also compute the value of its largest eigevalue to be λ 1 = Usig (4), we have the followig values for the first five spectral momets of the adjacecy matrix: m 1 = 0, m 2 = 22.47, m 3 = 394.7, m 4 = 33, 491, ad m 5 = 2, 603, 200. From (9) ad (13), we obtai the followig upper ad lower bouds o the largest eigevalue: β 2 = < λ 1 < = δ 2. Notice that the umerical value of λ 1 is remarably close to the upper boud δ 2. Sice the spectral radius measures the ability of a etwor to spread iformatio virally, our umerical results idicate that the etwor spreads iformatio very efficietly give the structural costrais imposed by the local egoets. We ca also compare our bouds with the estimator i (2), correspodig to a radom etwor with the same degree distributio. The value of the estimator is equal to λ 1 = , which is looser tha our bouds. Example 4.2 (S-Sitter Iteret etwor): I this example, we cosider a subgraph of the Iteret etwor at the utoomous Systems (S) level. The etwor topology was obtaied from the Sitter data collectio i CID 38]. Our subgraph was obtaied from the complete S graph usig a BFS of depth 2 aroud a radom ode. The resultig subgraph has = 2, 248 odes, e = 20, 648 edges, ad its largest eigevalue at λ 1 = The spectral momets of its adjacecy matrix are m 1 = 0, m 2 = 18.37, m 3 = 341.1, m 4 = 40, 001, ad m 5 = 2, 777, 018. The resultig bouds Fig. 2. Scatter plot of the spectral radius, λ 1 (G i ), versus the lower boud β 2 (G i ) (blue circles) ad the upper boud δ 2 (G i ) (red circles), where each poit is associated with oe of the 100 social subgraphs cosidered i our experimets. from (9) ad (13) are β 2 = < λ 1 < = δ 2. Notice how, the largest eigevalue is agai remarably close to the upper boud, idicatig that the etwor is able to spread iformatio efficietly, give its local structural costrais. I this case, the estimator based o radom etwors produces a value of λ 1 = 219.1, which is very loose. Therefore, usig radom etwors to aalyze spreadig processes i the Iteret graph ca be misleadig. I coclusio, our umerical results validate the quality of the lower ad upper bouds, β 2 ad δ 2, o the spectral radius λ 1 i several social ad commuicatio etwors. Our bouds provide a iterval of values i which the largest eigevalue is guarateed to lie. This is i cotrast with estimators based o radom etwors, which ca be very misleadig ad preset o quality guaratees. V. CONCLUSIONS fudametal questio i the field of mathematical epidemiology is to uderstad the relatioship betwee a etwor s structural properties ad its epidemic threshold. For may virus epidemic models, the role of the etwor topology is characterized by the largest eigevalue of its adjacecy matrix, such that the larger the eigevalue, the more efficiet a etwor is to spread a disease (or a piece of iformatio) virally. I may cases of practical iterest, it is ot possible to retrieve the complete structure of a etwor of cotacts due to privacy ad/or security costrais. Thus, it is ot possible to exactly compute the largest eigevalue of the etwor. O the other had, it is usually easy to retrieve local views of a etwor, also called egoets, by extractig the structure of eighborhoods aroud a collectio of chose odes. To estimate the value of the spectral radius whe oly egoets are available, researchers usually use radom etwor models i which they prescribe local structural features that ca be extracted from the egoets, such as the degree distributio. This approach, although very commo i practice, presets a major flaw: Radom etwor models implicitly iduce may

8 8 structural properties that are ot directly cotrolled ad ca be relevat to the spreadig dyamics. I this paper, we have preseted a alterative mathematical framewor, based o algebraic graph theory ad covex optimizatio, to study how egoets costrai the iterval of possible values i which the largest eigevalue (ad, therefore, the epidemic threshold) must lie. Our approach provides a iterval of values i which the largest eigevalue is guarateed to lie ad is applicable to weighted etwors. This is i cotrast with estimators based o radom etwors, which ca be very misleadig ad preset o quality guaratees. Our umerical simulatios have show that the resultig iterval i which the largest eigevalue must lie is very arrow for several social ad commuicatio etwors. This idicates that, for a importat collectio of etwors, the viral epidemic threshold is strogly costraied by local structural properties of the etwor. PPENDIX Corollary 3.9 Let G be a simple graph with adjacecy matrix G. Deote by, e, ad the umber of odes, edges, ad triagles i G, respectively. The, λ 1 ( G ) e 3 /. 2e Proof: From Corollary 3.2, we have that the first three momets of G are m 1 ( G ) = 0, m 2 ( G ) = 2e/, ad m 3 ( G ) = 6 / (by defiitio, m 0 ( G ) = 1). Substitutig the sequece of momets, m 3 = (1, 0, 2e/, 6 /), ito (9), we have that β 1 (m 3 ) is the solutio to the followig SDP: mi s.t. x R (x) x 2e/ ] 2e/ 0. 2ex/ 6 / The characteristic polyomial of R (x) ca be writte as φ (s; x) = det (si R (x)) = s 2 s tr (R (x)) + det (R (x)). The, R (x) 0, if ad oly if both roots of R (x) are oegative. By Descartes rule, this happes if ad oly if the followig two coditios are satisfied: (1) tr (R (x)) = x (1 + 2e/) 6 / 0, which implies x 6 2e + x 1. (15) (2) det (R (x)) = 2ex 2 / 6 x/ 4e 2 / 2 0, which implies x e 3 / 2e x 2. (16) We also have that, x 2 > e = 3 e > 6 2e+ = x 1.Therefore, the miimum value of x satisfyig (15) ad (16) is equal to the right had side of (11). REFERENCES 1] M.E.J. 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