The Effect of Network Topology on the Spread of Epidemics

A. Ganesh'. L. Massoulie", D. Towsley2 'Microsoft Research 7 J.J. Thomson Avenue CB3 OF3 Cambridge, United Kingdom USA { ajg, lmassoul } @microsoft.com

2Dept. of Computer Science University of Massachusetts Amherst, MA 01003

towsley@cs.umass.edu

Absfract-Many network phenomena are well modeled as spreads of epidemics through a network. Prominent examples include the spread of worms and email viruses, and, more generally, faults. Many types of information dissemination can also be modeled as spreads of epidemics. In this paper we address the question of what makes an epidemic either weak or potent, More precisely, we identify topological properties of the graph that determine the persistence of epidemics. In particular, we show that if the ratio of cure to infection rates is larger than the spectral radius of the graph, then the mean epidemic lifetime is of order Iogn, where 7) is the number of nodes. Conversely, if this ratio is smaller than a generalization of the isoperimetric constant of the graph, then the mean epidemic lifetime is of order en=, for a positive constant a. We apply these results to several network topologies including the hypercube, which is a representative con- nectivity graph for a distributed hash table, the complete graph, which h an important connectivity graph for BGP, and the power law graph, of which the AS-level Inkmet graph is a prime example. We also study the star topology and the Erd6s-Rhyi graph as their epidemic spreading behaviors determine the spreading behavior of power law graphs.

iadex Terms- Graph theory, Stochastic processes

I . INTRODUCTION Many network phenomena are well modeled as

spreads of epidemics through a network. Prominent examples include the spread of worms and email viruses, and, more generally, faults. The spread of informa- tion can also often be modeled as the spread of an epidemic. An epidemic spreads along the underlying network topology from an initial set of irgected nodes to susceptible nodes. In many cases infected nodes can be cured, reverting back to being susceptible and possibly

D. Towsley was a visitor at the Cambridge Microsoft Research Lab at the t ime this work was completed.

being reinfected. It is imperative, then to understand how the topology affects the spread of an epidemic, in other words how he topology either impedes or facilitates its spread and maintenance. Such an understanding can lead to better techniques for preventing and fighting worms and viruses, avoiding cascading failures, and for improving the dissemination of information.

In this paper we take a step towards developing an understanding of how the topology affects epidemic spread. We model the spread of an epidemic as a contact process over a finite undirected graph (with n nodes) where a node can be infected by its infected neighbors at a rate that is proportional to their number, and a node can be cured when it is infected. For such a system, given any set of initially infected nodes, the epidemic dies out in a finite amount of time. We develop general, topology dependent conditions under which the epidemic dies out either quickly or slowly. More precisely, if T

denotes the length 'of time that the epidemic is active (i.e., at least one node is infected), then a sufficient condition for quick die out, defined as E[T] = O(logn), is that the ratio of cure rate to infection rate be larger than the spectral radius of the adjacency matrix of the underlying topology graph. A sufficient condition for slow die out, defined as E171 = R ( e n Q ) for some a > 0, is that the ratio of cure rate to infection rate be smaller than the isoperimeuic constant associated with the graph. These conditions are not necessary. For some topologies, such as a hypercube, clique, and ErdBs-Rdnyi graph, the conditions are quite close. In others, such as a star and a power law graph, the conditions are farther apart. We narrow the gap through supplementary analysis for these latter cases. We focus on real world topologies, nameIy the hypercube, clique and power law graph. We also extensively study the star topology and the Erdds- Rknyi graph since their properties play prominent roles in the analysis of the power law graph. In all of these

1455 0-7803-8%8-9/05/320.00 (C)ZOOS IEEE

applications. the condition for fast die out based on the spectral radius appears to’ be tight.

Several papers have touched on different aspects of the problem studied here. Much of the work has focused on infinite scale-free graphs, estabhhing conditions under whch epidemics either die out or sustain themselves forever; see. e.g., [15]. Soinc work has been done on finite graphs, using primarily heuristic arguments to ob- tain conditions under which epidemics spread quickly or not. For example, [16] uses a mean field approximation for establishing conditions for quick and slow die out in scale-free graphs and [12] provides an approximate analysis for the case of an Erdgs-Rknyi graph. We rigorously establish similar conditions in Section V. An exception is the work of Durrett and Liu [SI, which presents a rigorous anaIysis Ieading to conditions for fast and slow die out on a finite one dimensional linear network. Also relevant is the work of Coffman et al., [7 ] , which models cascading BGP failures on a fully connected topology. They identify regimes where BGP routers recover quickly (corresponding to fast die out)and others where they go through long periods of repeated failures (corresponding, to slow die out). We treat this in a rigorous manner in Section V. Last, Wang et al. [18] show, through simulation and approximate analysis. that the condition for fast die out relates to the spectral radius of the adjacency matrix of the underlying graph.

Our model is an example of what is termed an SIS (Susceptible-Infective-Susceptible) model in the epi- demic literature. Related work has been concemed with the so-called SIR (Susceptible-Infective-Removed) model; s e Ball et al. 121 for recent references as well as a study of vaccination strategies in that context.

The .paper is structured as follows. We introduce the epidemic spreading model in Section 11. Sufficient conditions for the fast die out and slow die out of an epidemic are derived in Sections I11 and IV respectively. Applications of these results to the star, hypercube, clique, ErdBs-Rinyi graph, and power law graph are found in Section V. Section VI summarizes the paper and describes further directions to pursue.

TI. MODEL

Consider the following continuous time version of the epidemic spread model in Wang et al. [18]. We represent the system by a connected graph G = (V, E) . Here V is a set of ~t sites, E C V 2 and ( i , j ) E E represents a link between the pair of sites i , j E IT. The state at time t is represenled by a vector X ( t ) ; site i is infected (respectively, healthy) at time t iff X i ( t ) = 1 (respectively, Xi(&) = 0). ‘Assume that infected nodes

contaminate neighbours at rate ,L? and recover at rate 6. This defines a Markov process with transition rates:

Xi : 1 + 0 at rate 5.

Henceforth. without loss of generality, we set S 1 1. Denote by A the adjacency matrix of the graph structure on the set of sites. Denote by p(A) the spectral radius of A, namely, its largest eigenvalue. Observe that the Markov process X is such that one can reach the absorbing state 0 starting from any state z . Thus, it is the case that epidemics always die out. Even more is true: the probability that they have not d i d out by time f will decay exponentially with t , This fact follows from standard theory of Markov processes with absorbing states, reviewed for instance in [4]. The question of interest is then: how quickly do the epidemics die out, or how quickly does the system recover from the epidemic? More precisely, define T to be the time until the epidemic dies out provided there is at least one infected node initially. We are interested in determining the behavior of E[T] as a function of the system size n, whether it dies out quickly. i.e., E[T] = O(logn), or slowly, i s , ,

log E[T] = n(na) for some a > 0. Although parameters of interest such as /3 may depend on the system size n, we do not make this dependency explicit in ow notation.

111. A SUFFICIENT CONDITION FOR FAST RECOVERY

We show that the following condition,

implies that the epidemic dies out fast, and the charac- teristic time before extinction is related to the difference between the two terms in (1). Note that the threshold condition (1) is the one proposed in [18].

More precisely, we have the following result. Theorem 3.1: Suppose condition 1 holds. Then, the

probability that the epidemic has not died out by time t , given the initial condition X ( 0 ) E (0 , admits the following upper bound:

where IlX(O)ll1 = cy=l Xi(0). In addition, under the condition (l), the time to extinction T verifies

for any initial condition, X(0).

(3)

1456

Pruot Consider the continuous time Markov pro- 17, and transition cess Y = {x}iElr, with values in

rates Y, : I; --+ k + 1 Yi : k i k - 1 at rate I.;.

at rate j3 C(i, j)EE 5,

Standard coupling arguments yield X ( t ) I s t Y ( t ) for all t 2 0 when starting from the same initial conditions; here, X Sst T.’ denotes that Y stochastically dominates X . This implies that

P(EXZ(t) = 0) 2 P(C YJt) = 0). i i

Moreover, it holds that:

i i

However. the transition rates for process Y are such that d dt --E(Y(t)) = (PA - 1) E ( Y ( t ) ) ,

where 1 denotes the identity matrix, Hence,

E(Y(t)) = exp( tM)Y(O) ,

where M = ,!?A - 1. Note that exp(tA4) is a symmetric matrix (since A is) with spectral radius Hence,

IIE(Y(t)) 112 I e(Bp(A)-’)t(IE(Y(0))112, where IIE(Y(d))ll2 = Jcy=l E(X(’z(t))*. But, by the Cauchy-Schwarz inequality,

I n

Equation (2) follows, upon replacing yZ(0) by Xi (0 ) ; note that X ~ ( O ) ~ = &(o) since x;(o) takes values in (01 1).

Now, write

E(T) = L u P ( ~ > t ) d t = P(X(t) # 0)dt. J-lm Replacing P(X(t) # 0) by the trivial upper bound 1 on [O,(log(n)/(l - &(A))] , and by the upper bound n e x p ( ( p p ( A ) - Ijt) on the interval [(log(n)/(l - Pp(A)) , CO), in h s expression yields the claimed re- suit (3).

In other words, under condition (l), the probability that the epidemic hasn’t. died out by time a +t decays exponentially in t.

However, Condition (1) may not be necessary for en- suring exponentially fast death of epidemics, with small characteristic time before extinction. We wilI observe this in the case of a star-shaped network in Section V-A.

Iv. SUFFICIENT CONDITION FOR L4STING INFECTION

We now provide a condition under which an epidemic will survive for a long time. To this end, we introduce the so-called gemraked isoperinzcrric coiis?anf of the graph G, defined as:

In the above, E ( S , z ) denotes the number of edges connecting the set of vertices S to the complementary set, 3. When m = Ln/2j, is corresponds to the standard isoperimetric constant q(G). Henceforth we omit the reference to G. Define now a Markov process 2 on (0:. . . m}, with transition rate:

z --+ z + 1 : z --+ z - 1 :

at rate q(m.)pz, at rate 2 .

Now, standard coupling arguments yield ( Z ( t ) } I s t

{ziEITXi(t)} due to the fact that (i) Z( t ) can never exceed m, (ii) their downward transition rates are iden- tical, and (iii) the upwards transition rate for {Z(t)} is q(m),Ba, that for {EiEV X ; ( t ) } is E(S, s),O where S = {i E V : X i ( t ) = l}, and we have E(S,T) 2 q(m)z whenever 15’1 = z 2 m.

This is useful for establishing the following result. Theorem 4.1: Assume that the following inequality

holds:

Then for any initial condition X ( 0 ) with 0, it holds that,

X;(Oj >

I-?- P (T > ‘?-;:’’) > e (1 + O ( T ~ ) ) . (6)

Proufi First consider the embedded discrete time Markov chain associated with process Z, which tracks the successive states visited by 2. We denote it by Zt, where now t E , Its transition matrix has as non- negative terms:

- m ( m ) ’

p ( i , i - 1) = - flo(m)+l> l i = l > . . * ! m l P ( 0 , O ) = 1, p(m,m- 1) = 1.

1457

Let q ( k ) denote the probability that, starting from state k , the Markov chain 2, will enter state ni before entering state 0. The evaluation of q ( k ) is a standard problem, known as the gambler's ruin problem. It has the following solution:

where T = 1/(,8q(m)). Let T denote the number of steps before X t is absorbed at 0. The solution ot the gambler's ruin problem implies that:

Indeed. T is going to be larger than t if 2 pays at least t distinct visits to m before being absorbed. The prob- ability of the first visit taking place before absorption is the first term in the above expression; after a visit to na, the probability of having a subsequent visit prior to absorption is given by (1 - P-')/(l - P) as can be seen by considering what happens in the step just after the visit to m, where the process is necessarily in state M - 1. The formula above follows. Taking t = LT-"+'~ in expression (7), the right-hand side is then at least (1 - r ) / e (1 + O(?)). Conditioned on the event that there are indeed t visits to m before absorption at zero, the time before absorption is then larger than the sum of t exponentially distributed random variables with parameter m, describing the sojourn times in state m. When the number of such visits is large, the total sojourn time is then at least t / ( Z m ) , with a probability 1 - E where the error e is exponentially small in t , and the result (6) follows.

We have the following corollary. CoroELay 4.1: Consider a sequence of graphs indexed

by n. Suppose there is an IL > 0 and a sequence m. = @(nu) such that T < 1 unifordy in n, for T defined as in (5 ) . Then log (E~T]) = il (na).

Proof: We have 00

E[T] = P(T > z)dz, I f l . - ~ - L l , ( 2 m ) J O

P

jT-m+l ] 1 - 7- 2 -( 2m e

Upon taking logarithms, this yields

log (E[T]) 2 O(na) - G

which establishes the corollary.

7 > X)dX,

+ O ( P ) ) .

We now use Theorem 4.1 to provide sufficient con- ditions for exponentially long survival of the epidemics

stated in terms of the spectral struclure of the underlying graph G. This will allow comparison with the former sufficient condition for fast extinction, (l),

The Laplacian matrix L of the graph G is by definition D - A, where D is the diagonal matrix with as entries the degrees of the vertices in G, and A is the adjacency matrix of G. We shall denote the eigenvalues of L by XI(L) 5 X a ( L ) 5 . . . 5 X,(L). We now have:

CoroZlaq> 4.2: Let

Then (6) is still valid, with r replaced by T' and 7n =

The proof follows from Corollary 3.8 in [14], which states that for any graph G, the following inequality

1 4 2 1 ,

holds:

Corollary 4.3: Assume that the graph G is regular, i.e. all its vertices have the same degree, say d. Denote by Xl(A) 5 , . .&(A) the eigenvalues of its adjacency matrix A. Then a sufficient condition for exponentially long survival of the epidemics is that:

2 < P(d - A n d = P ( p ( A ) - L l ) . (9)

, Indeed, Condition (9) coincides with (8) in the case of a d-regular graph. Ignoring the factor 2 in (91, this condition amounts to comparing the ratio 1/p not to p ( A ) , as is done in Condition (l), but rather to the gap between the two largest eigenvalues of A, namely p(A) - A,-1.

v. SOME EXAMPLES

In this section we apply the previous results to several important classes of topologies: star-shaped networks, hypercubes, cliques, Erdds-Rhyi random graphs, and random power law graphs.

A. Star-shaped network The star-shaped network is of interest for several

reasons. First, it illustrates that neither of the preceding conditions are sufficient. We are able to get much tighter conditions because the star is simple enough to study exhaustively. Second, the spreading behavior of a large class of power law graphs is determined by the spreading behavior of stars embedded within them, as we will observe in Section V-E.

Consider a star-shaped network, with n + 1 nodes, where the only edges are (0; i), i = 1, . . . ~ n. Let us first

1458

identify the spectral radius of the corresponding matrix A . An eigenvec.tor x associated with eigenvalue X must

n satisfy

a=1

The only solutions for A are 0 and &fi, and'thus p ( A ) = fi. The results from the previous two sections tell us that

1) epidemics die out quickly provided lhat j3 <

2) epidemics die out slowly provided that p > 1. These results are clearly quite loose. In this section we derive tighter thresholds for fast and slow die out.

We describe the state of the epidemic on a star-shaped network by a vector ( X ( t ) , N ( t ) ) of length two. Here X(t) is the indicator that the hub is infected, and N ( t ) is the number of infected leaves, at time f , The painvise infection rate is ,O, while each infected node recovers at rate 1 . Thus, the process ( X ( t ) , N ( t ) ) is Markovian, with transition rates

l/fi, and

(0 , i ) + ( l , i ) , rate pi, 11, i ) + (O.z ) , rate 1, ( 0 , i ) + ( 0 , i - I), rate i (1 , i ) + ( 1 7 i - l), rate i ,

all other transition rates equal zero. Standard coupling arguments establish ( X ( i ) , N ( t ) ) to be stochastically nondecreasing in ( X ( 0 ) . N ( 0 ) ) . During each interval over which the state of the hub is fixed at 1, the leaves evolve as independent Markov chains on ( 0 , l ) with transition rate ,O from 0 to 1, and 1 from 1 to 0. Denote the transition rate matrix by Q1. A straightforward calculation yields that the transition probability matrix is

( 1 , i ) + (1,i + I), rate /3(n - x ) ;

~ ' ( t ) = e Q't

)U01 1 1 + Pe-(l+@)t ~ ( 1 - e-( l+W) - -4 1 + p 1 - ,++PP p + , - ( l tP) t

Define the process fl(t) to be the second component of the Markov process ( X ( t ) , N ( t ) ) , conditioned on X I S ) = 1 for all s E ( 0 , ~ ) . It is easy to see by a coupling argument that, for my t > 0, ~ ( t ) 5, N ( t ) provided N(0) ss fi(0); for random variables X, Y , we write X 5, Y to mean that X is stochastically dominated by Y . But conditioning on X ( s ) 1 ensures that the transition probabilities for each leaf are given by (10). Hence, if we take N ( 0 ) = n, then N ( t ) is binomial with parameters n and (~+e- ( l+P)" ) l (~ + p ) + Since the latter is a decreasing function oft, we also note that N ( s ) stochastica~ly dominates N ( t ) for any s 5 t . In fact, if T 2 s is a random time which is independent of the N ( . ) process, then N ( s ) stochastically dominates N(T). The

following theorem shows that (1) is not sufficient for fast die out.

i7ieoreni 5.1: Suppose p = C/&i for a fixed G > 0. Then, E[T] = O(1ogn) when either X(0) = 1 and/or

Proof As ( X ( t ) > N ( t ) ) is nondecreasing in the size of the initial population of infectives, it suffices to consider

so that

N ( 0 ) 2 1.

the initial state (1,n). Fix > 0, and take t = -w, lodPe)

p+,-(l+b)t = w. Define 1 +P

: X(s) = o} (11)

to be the first time after t that the central hub becomes uninfected in the original Markov process (X(t), N ( t ) ) . By the arguments above, N(T1) is stochastically domi- nated by fijt), Note that TI - t is stochastically dom- inated by the time for h e hub to become uninfected (assuming that it is infected at time t) , which is an exponentially distributed random variable with mean 1, denoted U1. Since = C/&. we have

(12) Next, observe that, conditioned on N(T1) = I;, the

probability that the infection dies out before the hub becomes infected again is 1/(1 + /?Ik. This is because each leaf becomes uninfected before infecting the hub with probability 1/(1 + ,B), and the infection attempts of different leaves are independent. LRt SI = inf{t > TI : X ( t ) = l}, with the convention that SI = CO if the infection dies out before the centre is re-infected. Then,

Recall that t = -* and that "(t) is binomid with

parameters n and w, Hence, using the generating function for the binomid, we have

Substituting ,O = C/fi, we have for all n sufficiently large, that

1 2

q := P(S1 = CO) 2 -exp(-(1 + e)C2). (13)

Next, we bound the time by which either the hub becomes re-infected or the infection dies out. Associate with each leaf i an exponential random variable of unit mean, Z;, denoting the time at which this node would become uninfected if it was initially infected. Define

1459

SI = TI + Zi. Then, either the infection has died out before 5'1. which happens with probability at least q (as given by (13)), or the hub was re-infected at some time in (TI! SI). On the latter event, the evolution of the process after SI can be stochastically dominated by the evolution which starts at 31 in the state (1,n). But this is the state we considered at time 0, so is a regeneration time for the dominating process. We have thus shown that the time to extinction is stochastically dominated by the sum of 1 + V independent copies of SI, where V is geometric with mean (1 - q ) / q .

2,. We have Now consider Sl - 571 =

In other words, SI - TI - logn is stochastically domi- nated by an exponential random variable with mean 1, denoted U2. We now have from (12) that

3

3

31 5 5 log n - Iog(cC) + UI -t- ~ 2 ,

(14) E[Si] 5 5 logn - log(&') + 2, where U1 and U2 are independent Exp(1) random variables defined on the same probability space as SI. Finally, the time to extinction, 7, satisfies

1 +v 7 5 cs j , P ( V = j ) = q ( l - q ) j : j = 0 , 1 , 2 ,...

j = 1

(15) where S j are iid and independent of V. Here q is given by (13), and is bounded below by a constant that doesn't depend on 71.. It is now immediate from (14) and (15) that E [ T ] = O(1ogn). This completes the proof of the theorem. w

Next, suppose ,O = na-i, for some a E ( 0 , i ) . Define K = and let Bn

R

c = U ((0, k ) U (Wh b=h'

be the set of states in which K or more leaves are infected. We shall show that, starting from any state in C, the Markov chain ( X ( t ) , N ( t ) ) returns to the set C before hitting (0,O) with high probability.

Theorern 5.2: Suppose 0 = n"-f? for some cy E (0,;) and C is defined as above. Then: 1) There is a constant IC > 0 such that, for all t i

sufficiently large and all x E C,

Pz( ( X ( t ) , N ( t ) ) hits (0,O) before returning to C ) 5 exp(-m").

Here, P, denotes probabilities for the Markov chain ( X ( t ) , N ( t ) ) , conditioned on ( X ( O ) , N ( 0 ) ) = IE. 2) For all z # (0; 0), log E,[T] = f2(na).

The proofs of I) and 2) reIy on the following three lemmas, whose proofs can be found in the Appendix.

The first lemma states that, for the Markov chain started in (O,k) , with high probability the huh becomes infected before the number of infected leaves decreases by fi or more.

Lernmu 5.1: Suppose ( X ( O ) , N ( 0 ) ) = (O! k ) . with k > ,/E. k t (7 = inf{t > 0 : X(t) = 1). Then.

7P

1 + P P(N(u) 5 k - 6) 5 ..p(---->.

Recall that K = a. Fix k < K and consider the Markov chain ( X ( t ) , N ( t ) ) started in the state ( 1 , k ) . The following lemma states that, with high probability, either X ( t ) = 0 or N ( t ) = K before N ( t ) = k - fi+ Note that. if N ( t ) = K , then ( X ( t ) , Ai(t)) E C.

Lemnia 5.2: Suppose (X(O) , N ( 0 ) ) = (1, k ) , with k 5 K . Let cr = inf{t > 0 : ( X ( t ) , N ( t ) ) E C or X ( t ) = O}. Then,

- P ( N ( ~ ) I - ,/E for some t E [o, 01 I 4-6.

The last lemma states that the Markov chain ( X ( t ) , N ( i ) ) , starting in state ( 1 , k ) for any k , has probability at least one third of hitting the set C before the hub becomes uninfected.

Lemma 5.3: Suppose ( X ( O ) , N(O)) = (I, k), with k < K . Let U = inf{t > 0 : X ( t ) = 0). Then,

1 P ( ( X ( 4 , N ( 4 ) E C ) 2 S?

for all n sufficiently large. Proof of 7?ieorem 5.2: Let x be an arbitrary state in the set C. Define TC to be the return time to the set C; TC = 0 if the first transition out of state x takes the chain to another state in C. We want to show that P(T 5 TC) 5 exp( -ma).

We recursively define the times Ti and Si as follows: TO = inf{t > 0 : X(t) = 0}, which could possibly be zero; for i 2 1, Si inf(t > Tz-l : X ( t ) = 1) and Ti = inf(t > Si : X ( t ) = 0). Note that the state (0,O) is absorbing. Hence, if N(Ti) = 0 for some a, then we define Si and Ti to be 03 for all j > i. Now, by Lemmas: 5.1 and 5.2, we have

1460

and, for all IC 1 1,

2 1 - exp(-filog4).

N ~ ~ , p so - -ZL > for a11 12 sufficiently large. Thus, we obhm by the unmn bound that

4(l+P) 2 f i 7 8( l+B) - .9

Note that, on this event, T > Tnap Moreover, on each interval [Sk,Tk], k I na/Y, there IS probability at least 4 of hitting the set C, by Lemma 5.3. Hence,

Taking K. = 1 + 9 M 3 / 2 ) for instance, we see that the first claim of the theorem is satisfied.

To establish the second claim, observe that the Markov chain started in any state of the form ( 0 , k ) with k > 0 dominates Ihe chain started in the state (0,l). Likewise, the chain started in (1, k) dominates that started in (1 I 0). Thus, we need only consider the initial conditions (0,l) and ( I , 0). Now, if the chain is started in (0, l), it hits ( 1 , l ) before (0,O) with probability /? = na-+, and subsequently dominates the chain started in (1,O). Moreover, by Lemma 5.3, the chain started in ( 1 , O ) hits the set C before hitting (O,O), with probability at least a third. Consequently, Px(rc < T) 2 n"-$/3 for all x # (0,O). Next , starting from any state y E C, we have shown that Py(. TC) 5 exp(-ma). Consequently,

the number of returns to C before hitting (0,O) stochas- tically dominates a geometric random variable with mean exp(ma). The time for each return to C dominates the time for a single transition from state y; for a11 y E C, this time dominates an exponential random variable with mean 1/77,. Consequently,

for large enough n, and suitably chosen IE' > 0; K' = ~ / 2 will suffice. Thus log E,[.] = Q(?t"). This completes the

Remark 5.1: It is interesting to note that the system behavior is greatly affected by the initid set of infected hosts in the following way. If X ( 0 ) = 0 and N ( 0 ) = ~ ( n ~ / ~ - * ) , then the probability that the hub becomes infected goes to zero as n + 03. In particular, if T =

for A > 0 and large n. Furthermore, these epidemics are of short duration, O(10g 1 1 ) . However, even though nearly all epidemics die out (with high probability), the rare occasions that they do not yield epidemics of such long durations that the average epidemic length is exponential in n. On the other hand, if either X ( 0 ) = 1 or N(0) = O ( T L ~ / ' - ~ ) , then a non-zero fraction of the epidemics do not die out quickly, even as n 4 00.

proof of the theorem. I

inf{s : X(s) = I}, then P(T < m) < Ae- N(O)jn'/2--a

B. Hyperciibes

The hypercube is of interest because of the widespread and growing interest in distributed hash tables and ap- plications, such as file sharing [17], being built on top of them. AIready woms and viruses have appeared in some applications, [l 11. As many DHT structures are hypercubic in nature, it is important to understand the spreading behavior of such worms on a hypercube. Here we represent a hypercube as a graph G with vertex set (0, 1 I E for some .f E , and where the edge ( U , w) is present if and only if the Hamming distance d ~ ( v , w ) equals 1. As a hypercube is a regular graph, its spectral radius is p(G) = log,n. Hence, from Section I11 we know that the epidemic dies out provided that /3 < 1/ log, n. We now determine conditions for the epidemic to die out slowly. In particuIar, we estdblish that the epidemic dies out slowly provided /3 > (l-n)llog ,, for any 0 < a < 1. Thus, unlike the star topology, here is essentially no gap for the hypercube.

We first establish the following result.

1461

nieortm 5.3: Let m. = P , for some ,k E (0, . . . , k' - 1). Assume that the following inequality holds:

1 p ( e - k ) l-

r :=

Assume further that rm goes to zero as n. -+ m, Then the time to extinction satisfies identity (61, with our current choices for and m.

The result follows by a direct application of Theo- rem 4.1. In order to prove the theorem, we need the following result established by Harper [lo] (see also [l] for background and recent extensions}.

Lerima 5.4: (Harper [lo]) Let S be a set of m vertices of the hypercube (0, l}'. Then the edge-boundary size E ( S , T ) is larger than or equal to E ( S * , S ' ) , where S* denotes the set of the n7. smallest vertices according to lexicographic order.

root (of Theorem 5.3) Let m = 2'. for some b < e. Let us first establish the following inequality:

IW Indeed, in view of the lemma it is the case that the set S* achieving the minimum isoperimetric ratio q(m) consists of all points @-j+'4-', where 4-l spans the whole set { O , l ) j - l , plus some points of the type O'-jlz;-', for some j 6 k. It turns out that any point &'z{, where

pp = 1 and O&-jzf belongs to S*, contributes exactly one edge to E(S*,s '* ) . Hence the lower bound

q(mJ 1 e - b.

from which (18) directly follows. The result follows by applying Theorem 4.1. Now reverting to the notation n and m, condition (17)

1 log, n - log, m.

can be expressed as

P >

Take m. = nn for some a such that 0 < a < 1, we obtain 1

(1 - U ) log:! 12. P >

C. Complete graph The complete graph also plays a prominent role in

networks. For example, the BGP routers belonging to the top level autonomous systems of the Internet form a completely connected component. In addition, large ISPs often organize their internal BGP (iBGP) routers into a set of route rejectors that are completely connected. Because of the importance of BGP to the Intemet, it is important to understand what effect BGP router failures can have on each ocher, At a very high level, the behavior

of routers failing and coming up can be modeled as the spread of an epidemic within a complete graph, [7].

Consider a complete graph with n vertices, namely the graph where an edge is present between each pair of nodes. The adjacency matrix of this graph is A = 1 lT - I , where 1 denotes the column vector of ones, and 1' is its tsanspose. A has the spectral radius p(A) = ? E - 1. The isoperimetric constant ~ ( m ) is easily shown to be n.-n2. Application of Theorem 3.1 and Corollary 4.1 tell us that epidemics die out quickly when ,O < 1/(n - I) and slowly when > l/(n-nz), when n7 = ?tu for some a > 0. Thus 1/71 is essentially the correct threshold and, as in the case of the hypercube, there is no gap.

D. Erdiis-Rkyi randow graphs The Erdiis-Rdnyi graph G(n., p ) with parameters 71. and

p is defined as a random graph on n nodes, where the edge between each pair of nodes is present with prob- ability p, independent of all other edges. The spreading behavior of an epidemic on an ErdBs-Rdnyi graph is of interest for a number of reasons. First, it is a graph that has received considerable attention in the past 131. Second, it is an important component of the class of power law random graphs that model the Internet AS graph. Thus if we are to understand the robustness of the Internet AS-level graph, we need to characterize the robustness of the Erd6s-Rknyi graph.

We shall consider a sequence of such graphs indexed by '1%. Denote by d the corresponding average degree, i.e. d = (n - 1)p. Note that p and d depend on n, but this is suppressed in the notation. We say that a property holds with hgh probability if its probability goes to 1

We consider the regime log(n) << d, i.e., log(n)/d --+

0 as n -3 00. In this case, it is known that the graphs are connected with high probability. Moreover, by the Perron-Frobenius theorem, the spectral radius p(A) lies between the smallest and largest node degree. Since the node degrees are binomial, B n - ~ , p , it is easy to see using the Chernoff bound that p ( A ) = (1 + o(1))np = (1 + o( l))d, with high probability.

Theorenr 5.4: Let m be such that m/n --t a as TI +

00, for a fixed a E (0,l) . Assume further that log(n) << d. Then it holds that, with hgh probability,

(19)

as n -+ 00.

v(G.m) == (1 + 0(1))(l - a)d .

Proofi Fix some b E (0 , l - a). By the union bound,

n

P(q(G,nz) < kd) 5 P ( E ( S : s ) < Mi) .

1462

Note that E ( S , s ) has a binomial distribution with parameters .i(n - i> and p . Denote by Bi!n-ijs a generic Binomial random variable with this distribution. We thus have

Since k < 1 - cr and m = ( 1 + o( l ) )an , we have for 11

large enough that (71 - i ) p > kd for all I E {I. . . . . ni} and that 5 1/( 1 - a). We now apply the following Chernoff bound,

P ( B < (1 - b)E(B) ) < e-E(B)d2/2. (20)

vatid for any Binomial random variable B , [13. Theorem 4-21, to B,(lz-z),p in the above expression. Taking 6 = 1 - k z 2 1 - A, this yields

> 0. We thus have the upper bound

i=l

where we have also used the upper bound ( y ) 5 ni/z! to obtain the first inequality. By assumption, log(.) << d = (n - l ) p , and t is a positive constant that doesn't depend, on n. Thus, for any constant K > 0, it holds that, for n large enough,

m .

This suffices to conclude that for any k < 1 - a, with high probability q(G, m) 2 kd.

In order to obtain an inequality in the opposite di- rection, choose any set 5 of cardinality m,. Then one certainlv has

Let I; > (1 - a ) be fixed. One then has that

P(v (G,m) > k 4 I w%7-&(n-m),p > w n - 1 ) P ) .

It is readily seen using a Chernoff bound that the right- hand side goes to zero as n + m. This concludes the proof of the theorem.

Theorern 5.5: Consider ErdGs-Rknyi random graphs G(71.,p) with log(n) << d = np The following claims

hold with high probability: an epidemic on G(7e ,p ) dies out quickly, E[T] = O(logn), provided ,O < (1 - u ) / d for 0 < t~ < 1. On the other hand, the epidemic dies out slowly, log E[T] = Sr(n,) provided that /3 > (1 + v)/d for w > 0.

Proof: The theorem follows from Theorem 3.1 and Corollary 4.1, the expression p(A) = (I 4- o( 1))d for he

I Notice that there is very little gap between the two

thresholds as U and 'U can be chosen to be arbitrarily close to zero.

spectral radius, and Theorem 5.4.

E. Power law graphs

There has been considerable interest in power law graphs since it was first noticed that the Internet AS- level graph exhibits a power law degree distribution, [Y]. Briefly a power law graph is one where the number of nodes with degree I; is proportional to k-r for some y > 1. For the mean degree to be finite, we need y > 2 and this is the range we shall consider. The Internet AS- level graph is characterized by y M 2.1

In this section we consider a class of random power law graphs first intxoducd in 161. Let tu1 WZ, . . . , W n

denote the expected degrees of the nodes in the graph. An edge is assigned to a pair of vertices with probability wiwjl WI;. I x t d denote the average degree and m the maximum degree. If wi = ~ ( i o + a > ?--I (1 I i I n> where c = number of nodes with degree k is proportional to k-r. Theorem 4 from 161 states under mild conditions that, with high probability, the spectral radius of the graph is

. --

i o = 12 ( - 47-21 )7-1 then he Y-1 mir-1)

We have the following result on the fadslow die out of epidemics on the above class of power law graphs. Here we will take RZ to be the folIowing increasing function of n, m = nx where 0 < X < 1.

Theorem 5.6: 1) For y 2 2.5,

{a)

Y-1

the infection dies out quickly, E(T] = Ojlogn), provided that ,Ll < (1 - u ) / J m for some 1 > U > 0,

(b) the infection dies out slowly, l o g E [ ~ ] = C2(nxa), provided that > m0-1/2 for some cy f @,I).

2) For y E (2,2.5),

1463

(a) the infection dies out quickly, E[T] = O(logsi.), provided that pp < (1 - U ) where

p = d

for some 0 < 'IL < 1. (b) the infection dies out slowly, logE[T] =

Q(+Nr-1) ) provided that (i) ,9q > (1 + U )

where

for some U > 0. and (ii) the following condition holds:

q >> log(i0). (23) Prooj The first part of each of the claims follows

from Theorem 3.1 and (22). Consider claim 1.b. We can bound the time to die out of an epidemic that starts in either the maximum degree node or one of its neighbors by the time to die out on a star of size m +- 1. Now consider the case that some other randomly chosen node is initially infected. Theorem 4 in [ 5 ] states that, with high probability, the diameter of the power law graph is O(1ogn) when y > 2, Hence the probability that an epidemic starting from a randomly infected node spreads to either the maximum degree node or its immediate neighbors is O(l/nalogn) for some a > 0 independent of n. This combined with arguments used to establish Theorem 5.2 establishes the claim.

We focus on claim 2.b. We shall now identify a subgraph of the original power-law random graph that is an Erdiis-Rhyi random graph. Consider the first N nodes, where N < n is to be chosen, with respective weights wio,. . . , w,,+N. Then the probability of an edge being present between any two of these nodes is at least as large as

N + i o

The original graph thus contains an Erdiis-Rknyi graph with parameters N and p ~ . Survival of the epidemic on the original graph will be at least as long as on the Erdds-Rknyi subgraph provided that at least one node within the ErdBs-Rhyi component is initially infected. The average degree of the subgraph is then given by

Let us now take N = € i o , for some positive E . The average degree of the induced ErdBs-Rinyi graph is then equivalent to

( N - 1)PN i3-7)/(Y--l)

Simple calculus shows that this is maximized by setting E = (y - 1)/(3 - T), for which choice we arrive at

( N - 1 ) P N - The die out result is a direct application of Theo-

rem 5.4 and Corollary 4.1, which is applicable under the condition (23).

If the initially infected nodes are not within the Erdiis-Rhyi subgraph, then because the diameter of the power law graph is O(logn) with high probability, the probability that the epidemic dies out before infecting a node within the ErdBs-Rbnyi component is O( l/nu log ") for some a > 0 independent of n. This combined with arguments used to establish Theorem 5.2 establishes the claim.

Remark. Note that in the case of y € (2,2.5), the thresholds for fast arid slow recovery are almost the same, since q in the statement of Theorem 5.6 is smaller than p(A) by a factor of

Put together, our results determine the outcome of the epidemics for such graphs in the ranges &(A) < 1 (fast extinction) and P p ( A ) > [2/(3 - y)]2/(9-') (long survival). Our techniques do not cover the range in between at the present time.

VI. SUMMARY We have presented a preliminary investigation of how

topology affects the spread of an epidemic, motivated by networking phenomena such as worms and viruses, cascading failures, and dissemination of information. We have developed sufficient conditions under which epidemics either die out quickly (logarithmically in the size of the network) or slowly (exponentially in the size of the network). These conditions are tight for sev- eral network topologies, such as hypercubes, complete

1464

c graphs, and Erd6s-Rknyi random graphs. They are not tight for topologies such as stars and power law graphs. We provided a supplementary analysis of the star that significantly Lightened the condition for slow die out so that i t is close to that for fast die out. In the case of a power law graph, we presented tight conditions by drawing on results for the star and Erdiis-Rknyi random graph. In all cases, the condition for fast die out appears to be tight.

There are several interesting directions to pursue this. As we remarked within the paper, the behavior of the epidemic in the c a e of the star and power law graph is sensitive to the initial set of infected nodes. It would be useful to understand this relationship better. In addition, as pointed out in 1121, there is a notion of a mefasfable set of nodes that are infected given that one is infected in the regime of slow die out. They present an approximate analysis of the distribution of the number of nodes that belong to this set, It would be useful to pursue this in a more rigorous manner and for other classes of network topofogies.

APPENDIX PROOF OF LEMMAS 5.1, 5.2, 5.3

Proof of Lemma 5.1: When the Markov chain is in state (0, IC), the probability that it hits (0, I; - 1) before (1, A:) is k / ( k + Dk) = 1/(1 + ,$); no other transitions are possibk from (0, k). Hence, by the Markov property,

P ( N ( a ) 5 I; - A) = fi P(O,~) ( hit ( 0 , j - 1) before (1,j) )

we observe that M(t) is a martingale and that 5 is ii stopping time. Define y = P( f i (8 ) = fi(0) - fi). By the optional stopping theorem, E M ( 8 ) = EA4(0), i.e.,

Pn (4+3Ph - (4+38)K > P - Now, K = 4(1+8), so 1 2 - K = ~ 4(1+fl) - F. Hence, it is immediate from the above that

This establishes the claim of the lemma. Proof of Lemma 5,3: Clearly, the Markov chain started

in (1 ,k ) stochastically dominates the chain started in ( l , O ) , so it suffices to establish the claim of the Iemma for k = 0. By conditioning on the time U that it takes the hub to become uninfected, which is exponentially distributed with unit mean, we get

w

P ( N ( a ) > K ) = p ? P ( N ( t ) 1 KlN(0) = 0:

X(S) = 1"s E [O,t])dt. (26)

Now, conditional on N ( 0 ) = 0 and X(s) = 1 on [O: t], N ( t ) is binomial with parameters 72. and p(1 - e-(l+pIt)/(l + p). Applying the Chemoff bound (20) with 6 = 1 - 4 ( 1 - - e - ( l + P ) t ) yields for t > $

j=k-&+l ,on' ) 4 j l + 0) P ( N ( t ) <

)> p ( 3 - qe-(1+P)t)2

< exp -n

< exp -

( 32(1+ P)((l - e-(l+Blt)

( 16(1 + P >

Using the inequality log 5 5 z - 1, we obtain

as claimed. w Proof of Lenima 5.2: Observe that. while X(t) = 1,

N ( t ) evolves as a birth-death Markov chain with birth rate ,B(n - N ( t ) ) and death rate N( t ) . Now, over the time interval lO,u.), X ( t ) = 1 and N ( t ) 5 K . Hence,

Here t = minimizes the denominator and t = 00

maximizes the numerator of the exponent in the right hand expression of the first inequality thus yielding the second inequality.

P ( N ( a ) 2 K ) over [O, U ) , N(t) 'stochastically dominates the Markov chain N(t) with birth rate P(n - K ) and death rate K , having the same initial condition N ( 0 ) = b. Hence, i t e-tdt suffices to show that, with probability at least 1 - 4-&, #( t ) hits K before it hits k - fi. Defining

1 t P

(27)

for all n sufficiently large. This establishes the claim of

1 > - - 3 ?

6 = inf{i > o : fi(t) = IC or N ( O ) - 61, the lemma.

1465

REFERENCES [l] M. C. Azizoglu and 0. Ekecioglu, "Extrema1 sets minimizing

dimension-normalized boundary in Hamming graphs". SIAM J- Discrere Mnrh.. 17 (2003). 219-236.

[ 2 ] F. Ball. T. Britton and 0. Lyne, "Stochastic multitype epidemics in a community of households: estimation and form of optimal vaccination schemes", Mahenratical Biosciences, 19 1 (2004). 1940.

I31 B. Bollobiis. R a n d m Grup'hs, 2nd Edision. Cambridge Univer- sity Press, 2001.

[4] P. BrCmaud. Markov Chairrs. Gibbs fieZds. Monre Carlo sirnzr- larion. and qiieues, Springer. New York 1999.

[SI E Chung and L. Lu. "Tbe average distances in random graphs with given expected degrees", Iruernet Mafhemarics. 1 (2003).

I61 E Chung. L. Lu and V. Vu. "Eigenvalues of random power law gnphs", Annals of Cumbinaforics. 7 (2003). 21-33.

[7] E.G. Coffman Jr.. Z. Ge, V. Misra and D. Towsley. "Network tesllience: exploring cascading failures within BGP", Pmc. 40111 Annital Allerton Conference on Communications, Co~irputing and Confro/. October 2002.

[SI R. Durrett and X.-E Liu. "The contact process on a finite set," Ann. Prob.. 16 (19881, 1158-1173.

191 M. Faloutsos, P. Faloutsos and C. Faloutsos. "On power-law

91-114.

relationships of the Internet topology", Proc. ACM -Sigcoiiim, 1999.

[lo] L. Harper, "Optimal assignment of numbers to vertices": J. Soc. Ind. Appl. Math., 12 (1964). 131-135.

[ 1 11 http://www.kazaa.codus/helphown-virus.htm [12] J,O. Kephart and S.R. Whte. "Directed-graph epidemiological

models of computer viruses." Proc. 1991 IEEE Compufer Societ?, Symposiuni on Research in Securiry and Privacy (1991). 343-359.

[l?] R. Motwani and P. Raghavan. Randomized Algorithms. Cam- bridge University Press, 1995.

1141 B. Mohar, "Some npplications of Laplace eigenvalues of graphs", in Graph Symmetv: Algebraic Merhods and Applica- tiom, Eds. G. Hahn and G. Sabidussi. NATO AS1 Ser. C 497, Kluwer (1997). 225-275.

[I51 R. Pastor-Satorras and A. Vespignani. "Epidemic spreading in scale-free networks." Physical Review Lerrers, 86 (ZOOl) , 3200- 3203.

[16] R. Pastor-Satomas and A. Vespignani. "Epidemic dynamics in finite scale-free networks." Physicd Review E, 65 (2002).

[17] A. RowstKIn and P. Druschel, "Pastry: Scalable, distributed ob- ject location and routing for large-scale peer-to-peer systems", Proc. Middleware 2001, Germany. November 2001.

[ lS] Y. Wang, D. Chakrabarti, C. Wang and C. Faloutsos. "Epidemic spreading in real networks: An eigenvalue viewpoint", Proc. IEEE SRDS. 2003.

1466