The viral conductance of a network

Computer Communications 35 (2012) 1494–1506

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The viral conductance of a network Piet Van Mieghem ⇑ Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science, P.O. Box 5031, 2600 GA Delft, The Netherlands

a r t i c l e

i n f o

Article history: Received 9 April 2012 Received in revised form 15 April 2012 Accepted 18 April 2012 Available online 2 May 2012 Keywords: SIS epidemics Network robustness Viral conductance Epidemic threshold

a b s t r a c t Besides the epidemic threshold, the recently proposed viral conductance w by Kooij et al. [11] may be regarded as an additional characterizer of the viral robustness of a network, that measures the overall ease in which viruses can spread in a particular network. Motivated to explain observed features of the viral conductance w in simulations [29], we have analysed this metric in depth using the N-intertwined SIS epidemic model, that upper bounds the real infection probability in any network and, hence, provides safe-side bounds on which network protection can be based. Our study here derives a few exact results for w, a number of different lower and upper bounds for w with variable accuracy. We also extend the theory of the N-intertwined SIS epidemic model, by deducing formal series expansions of the steadystate fraction of infected nodes for any graph and any effective infection rate, that result in a series for the viral conductance w. Though approximate, we illustrate here that the N-intertwined SIS epidemic model is so far the only SIS model on networks that is analytically tractable, and valuable to provide first order estimates of the epidemic impact in networks. Finally, inspired by the analogy between virus spread and synchronization of coupled oscillators in a network, we propose the synchronizability as the analogue of the viral conductance. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction We investigate the influence of the network topology on the spread of viruses, digital as well as biological ones, whose dynamics is modeled by a susceptible-infected-susceptible (SIS) type of process [2]. Digital viruses (of all kind, and generally called malware) are living in cyberspace and use mainly the Internet as the transport media, while biological viruses contaminate other living beings on earth and use ‘‘contacts’’ among their victims as their propagation networks. The increasing threats from cybercrime and the expected outbreak of new lethal, biological viruses justify studies on virus spread in graphs. In particular, network operators are interested to know (a) how vulnerable their network is for epidemics and (b) how to protect or modify their infrastructure, most often, at minimal cost (see e.g. [10]). In contrast to the negative connotation of a threat, information spread (news, rumors, etc.) in the new types of on-line social networks like Facebook, Twitter, Digg [20], etc. resembles epidemic diffusion. While this analogy still needs to be verified and evaluated via extensive measurements, epidemic theory is expected to be fruitful in assessing the spreading speed and depth (i.e. how many users are reached) of news in social networks. A network is represented by an undirected graph GðN; LÞ with N nodes and L links. The network topology is described by a ⇑ Tel.: +31 152782397. E-mail address: [email protected] 0140-3664/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.comcom.2012.04.015

symmetric adjacency matrix A, in which the element aij ¼ aji ¼ 1 if there is a link between nodes i and j, otherwise aij ¼ 0. In the sequel, we confine ourselves and make the following simplifying assumptions. The state of a node i is specified by a Bernoulli random variable X i 2 f0; 1g: X i ¼ 0 for a healthy node and X i ¼ 1 for an infected node. A node i at time t can be in one of the two states: infected, with probability v i ðtÞ ¼ Pr½X i ðtÞ ¼ 1 or healthy, with probability 1  v i ðtÞ. We assume that the curing process per node i is a Poisson process with rate d, and that the infection rate per link is a Poisson process with rate b. All involved Poisson processes are independent. The effective infection rate is defined as s ¼ bd. This is the general description of the simplest type of a SIS virus spread model in a network and the challenge is to determine the virus infection probability v i ðtÞ for each node i in the graph G. This SIS model can be expressed exactly in terms of a continuous-time Markov model with 2N states as shown in [27]. Unfortunately, the exponentially increasing state space with N prevents the determination of the set of fv i ðtÞg16i6N in realistic networks, which has triggered a spur of research to find good approximate solutions. For an overview of SIS heuristics and numerous extensions, we refer to [3,14,29]. Our continuous-time mean-field approximation of the exact SIS model for the spreading of a virus in a network, called the N-intertwined model [27], was earlier considered by Ganesh et al. [9] and by Wang et al. [28] in discrete-time, whose paper was later improved in [6]. A remarkable property of the SIS model is the appearance of a phase-transition [4] when the effective infection

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rate s approaches the epidemic threshold sc . Below the epidemic threshold sc , the network is virus-free in the meta-stable state,1 while above sc , there is always a fraction of nodes that remains infected. The determination of the precise epidemic threshold sc in any network is a long-standing open, difficult problem and a major contribution of the mean-field N-intertwined model is the lower bound sc ¼ k11 6 sc , where k1 is the largest eigenvalue of the adjacency matrix A, also called the spectral radius. Here, we will complement the analysis of the N-intertwined SIS model inspired by a new metric, the viral conductance w, introduced in [11,29] and reviewed in Section 3. The viral conductance w, precisely defined in (17), measures the integrated effect over all possible viral infection strengths on the network’s infectious vulnerability. This paper is part of a larger study on network robustness [25], in which a framework is proposed to assess the level of robustness of a given network, based on network metrics. As argued in Section 3, the vulnerability of a network against epidemics should be measured by a set of metrics including the epidemic threshold sc ¼ k11 and the viral conductance w, rather than only by the epidemic threshold, because w complements sc naturally. Other possible viral metrics can be taken into account as well, since any proposal of a metric can be motivated or debated. However, as pointed out in [25], a relevant and desirable set of network metrics should be small and consisting of as independent2 as possible metrics. Finally, we point here to another dynamic process on a network, that bears resemblance to virus spread. The synchronization of coupled oscillators in a network features a surprisingly similar phase transition: the onset of oscillator coupling occurs [17] at a critical coupling strength g c ¼ gk10 and the behavior of the phase transition around g c is mathematically similar [24] to that of the epidemic threshold sc . Synchronization [19] plays a role in sensor networks, human body (hart beat, brain, epilepsy), light emission of fire-flies, etc. In Section 3, we propose the analogue of the viral conductance, namely the synchronizability of a network, that measures the ease in which coupled oscillators in a certain network synchronize for all possible coupling strengths. The inspiring analogy between virus spread and synchronization in networks may open new avenues: we speculate that results here obtained for the viral conductance may translate to the setting of coupled oscillators in a network. The outline of the paper is as follows. First, we review the Nintertwined SIS virus spread model in Section 2 and derive series expansion around the epidemic threshold sc and around s ! 1 in Section 4 that will be used to express the viral conductance w as a series in Section 5. Section 6 gives analytic bounds for w, while Section 7 explains theoretically an observation from simulations in [29]: the steady-state fraction y1 of infected nodes in the steadystate around s ’ NL is about y1 ’ 12, where N is the number of nodes and L is the number of links in the network. Proofs are placed in the Appendix. Apart from the overview in Section 2, the presented analytic results in the subsequent sections are new and add to the general theory of the N-intertwined model in [27].

2. The N-intertwined SIS model in brief

1 Since the exact SIS Markov process has an absorbing state, namely the overall healthy state, the steady-state equals this overall healthy state in which the virus has disappeared from the network. Unfortunately, already for reasonably small networks (N P 100), the time to reach this absorbing state is huge on average (see [8, p. 99]) so that the exact steady-state is hardly ever reached in real networks, while the metastable state reflects the observed viral behavior fairly well.The steady-state in the Nintertwined virus spread model refers to the meta-stable state, which is reached exponentially rapidly and which reflects real epidemics more closely. 2 Since topological metrics (such as the hopcount, clustering coefficient, degree, betweenness, centrality, etc.) are derived from the adjacency matrix A of the network, most of them are correlated [13].

In contrast to all published SIS-type of models, the N-intertwined model, proposed and investigated in depth in [27] and reviewed in [23], only makes 1 (mean-field) approximation in the exact SIS model and is applicable to all graphs. In [15,5], we show that the mean-field approximation upper bounds the exact Pr½X i ðtÞ ¼ 1 (which is useful to guarantee epidemic safety bounds in real networks) and that it implies that the random variables X j and X i are implicitly assumed to be independent. Since this basic assumption is increasingly good for large N, we expect that the deductions from the N-intertwined model in sufficiently dense networks (i.e. with average degree increasing with N) are asymptotically (for N ! 1) nearly exact. The homogenous N-intertwined model, where the infection and curing rate is the same for each link and node in the network, has been extended to a heterogeneous setting in [26], where each node i has its own infection rate bi and curing rate di . Furthermore, the method and analysis of the N-intertwined model has been transferred to the SIR model in [30], while in [7], the N-intertwined model is extented to an SAIS infection, where the Alert (A) state is introduced next to the S ðX i ¼ 0Þ and I ðX i ¼ 1Þ state. A very general extension of the N-intertwined model is proposed in [18], while second order mean-field improvements are studied in [5]. The N-intertwined model is shown in [12] to be generally superior to another widely used mean-field model, the heterogeneous mean-field model of Pastor-Satorras and Vespignani [16]. 2.1. Governing equations We briefly summarize the key aspects of the N-intertwined model. The governing differential equation in the N-intertwined model for a node i is N X dv i ðtÞ ¼ bð1  v i ðtÞÞ aij v j ðtÞ  dv i ðtÞ dt j¼1

ð1Þ

In words, the time-derivative of the infection probability of a node i consists of two competing processes: (1) while healthy with probability ð1  v i ðtÞÞ, all infected neighbors, with an average number of PN j¼1 aij v j ðtÞ, try to infect the node i with rate b and (2) while infected with probability v i ðtÞ, the node i is cured at rate d. Defining the vector VðtÞ ¼ ½ v 1 ðtÞ v 2 ðtÞ    v N ðtÞ T , the matrix representation based on (1) becomes

dVðtÞ ¼ ðbA  dIÞVðtÞ  b diagðv i ðtÞÞAVðtÞ dt

ð2Þ

where diagðv i ðtÞÞ is the diagonal matrix with elements v 1 ðtÞ; v 2 ðtÞ; . . . ; v N ðtÞ. In the sequel, we focus on the steady-state where v i1 ¼ limt!1 v i ðtÞ and limt!1 dvdti ðtÞ ¼ 0. From (1), we obtain

v i1 ¼

P b Nj¼1 aij v j1 1 ¼1 P PN 1 þ s Nj¼1 aij v j1 b j¼1 aij v j1 þ d

ð3Þ

Beside the trivial solution v i1 ¼ 0, (3) illustrates that there is another positive solution reflecting the meta-stable state in which we are interested here. For regular graphs, where each node has degree d, symmetry in the steady-state implies that v i1 ¼ v 1 for all nodes i and it follows from (3) with the definition of the degree P di ¼ Nj¼1 aij that

v 1; regular ¼ y1; regular ðsÞ ¼ 1 

1

sd

ð4Þ

P where y1 ¼ N1 Ni¼1 v i1 is the fraction of the average number of infected nodes in the steady-state. We now provide a couple of new analytic relations that will be used in Section 7. Summing (1) over all i is equivalent to right mul-

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tiplication of VðtÞ by the all one vector PN T i¼1 v i ðtÞ ¼ u VðtÞ. Then, we find from (2) that

uT

because

where

¼ bðu  VðtÞÞT AVðtÞ  duT VðtÞ Hence, we obtain a relation for y1 2 ½0; 1 in terms of the vector V 1 : T

Ny1 ¼ uT V 1 ¼ sðu  V 1 Þ AV 1

ð5Þ T

Since the degree vector D satisfies uT A ¼ DT because A ¼ A , we can write (5) as

N

DT V 1  V T1 AV 1



  Ny1 ¼ s uT DV 1 þ V T1 DV 1  V T1 DV 1  V T1 AV 1   ¼ s ðu  V 1 ÞT DV 1 þ V T1 ðD  AÞV 1 Introducing the Laplacian Q ¼ D  A of the graph G, the steady-state fraction of infected nodes y1 is expressed as a quadratic form in terms of the Laplacian [22],

y1 ¼

N

T

ðu  V 1 Þ DV 1 þ

V T1 QV 1



After left-multiplication of the steady-state version of (2) by the  k1   k  T k k vector  V T1 diag v i1 ¼ v 11 v 21    v N1 , which we denote k by V 1 , we obtain the scalar equation

ð8Þ

j¼1

For k ¼ 0 in (8), and introducing the all one vector u ¼ limk!0 V k1 , we obtain (5) again. For k ¼ 1 in (8), the norm P kV 1 k22 ¼ V T1 V 1 ¼ Nj¼1 v 2j1 obeys

  V T1 V 1 ¼ s V T1 AV 1  V T1 diagðv i1 ÞAV 1

ð9Þ

When

summing (8) over all k from m P 0 to infinity and taking

v j1 < 1 into account the telescoping nature of the right-hand side leads to 1  N T X X V k1 V 1 ¼

v mþ1  T j1 ¼ s Vm 1 AV 1 1  v j1 j¼1

k¼m

ð10Þ

When m ¼ 0, we have that V m 1 ¼ u and we obtain, with the degree vector uT A ¼ DT , N X v j1 ¼ DT V 1 ¼ dj v j1 s j¼1 1  v j1 j¼1

N 1X

ð11Þ

As shown earlier in [27], the characteristic structure (10) of the Nintertwined model follows more elegantly from the governing Eq. (2) in the steady-state

V 1 ¼ s diagð1  v i1 ÞAV 1 for finite

s such that v i1 < 1. Indeed, after left-multiplying both  

sides by ðdiagð1  v i1 ÞÞ

1

s

diag

or



ð12Þ



1

¼ diag

1 V 1 ¼ AV 1 1  v i1

1 1v i1

, we have

V1 1V 1

T

Since the eigenvectors x1 ; x2 ; . . . ; xN belonging to the eigenvalues k1 P k2 P    P kN of the adjacency matrix A span the N-dimensional vector space, we can write the steady-state infection probability vector V 1 ðsÞ as a linear combination of the eigenvectors of A,

V 1 ð sÞ ¼

N X

ck ðsÞxk

ð14Þ

k¼1

where the coefficient ck ðsÞ ¼ xTk V 1 ðsÞ is the scalar product of V 1 ðsÞ and the eigenvector xk and where the eigenvector xk obeys the normalization xTk xk ¼ 1. Physically, (14) maps the dynamics V 1 ðsÞ of the process onto the eigenstructure of the network, where ck ðsÞ determines the importance of the process in a certain eigendirection of the graph. The definition y1 ðsÞ ¼ N1 uT V 1 ðsÞ shows that

y1 ðsÞ ¼

ð7Þ

  N  T  T X T V k1 V 1 ¼ v kþ1 V k1 AV 1  V kþ1 AV 1 1 j1 ¼ s

ð13Þ

2.2. An eigenvector approach

ð6Þ

Finally, we write the degree vector D as D ¼ Du, where D ¼ diagðd1 ; d2 ; . . . ; dN Þ, so that

s



¼ AV 1

h i v v 21 v N1 ¼ 111 v 11 1v 21    1v N1 . By left-multiplication  T of (13) by V m 1 , we obtain (10) again.

T

s

V1

s 1  V1

du VðtÞ ¼ uT ðdiagð1  v i ðtÞÞbA  dIÞVðtÞ dt

y1 ðsÞ ¼

1

N 1X c ðsÞuT xk N k¼1 k

ð15Þ

Substitution of (14) into (5) yields

y1 ðsÞ ¼

N sX

N k¼1

  kk ck ðsÞ uT xk  ck ðsÞ

ð16Þ

For irregular graphs, generally, cm ðsÞ ¼ xTm V 1 ðsÞ – 0 for m > 1 and all eigenvalues and eigenvectors in (16) play a role. Moreover, P cm ðsÞ can be negative, as well as km , while Nk¼1 kk ¼ 0 (see [22, p. 30]). The larger the spectral gap k1  k2 and the smaller jkN j, the more y1 is determined by the dominant k ¼ 1 term in (16), and the more its viral behavior approaches that of a regular graph. Graphs with large spectral gap possess strong topological robustness [22], in the sense that it is difficult to tear that network apart. 3. The viral conductance The viral conductance w of a virus spreading process in a graph was first proposed in [11] as a new graph metric and then elaborated in more detail in [29]. The viral conductance w is defined as

w¼

Z 0

k1

y1 ðsÞds

ð17Þ

where s ¼ 1s. Below the epidemic threshold sc , the network is virusfree in the steady-state. Hence, v i1 ðsÞ ¼ 0 for s < sc , and equivalently, v i1 ðsÞ ¼ 0 for s > s1c ¼ k1 . Since the function y1 ðsÞ versus s is not integrable over all s, Kooij et al. [11] have proposed to con sider y1 1s versus s ¼ 1s. Fig. 1 illustrates the typical behavior of y1 ðsÞ versus s ¼ 1s and y1 ðsÞ versus s in the insert for a regular and irregular graph. Theoretically – though debatable – one might argue that the effective curing rate s ¼ bd ¼ 1s is more natural than the effective infection rate s ¼ bd, because a Taylor expansion of y1 ðsÞ around s ¼ 0 can be deduced, while the corresponding one for y1 ðsÞ is a Laurent series in 1s around s ! 1 (see Lemma 2). In most published work so far, network G1 was considered to be more robust against virus spread than network G2 if the epidemic threshold sc ðG1 Þ > sc ðG2 Þ. For example, in Fig. 1, the regular graph with the same number N of nodes and nearly the same number L of links possesses a higher epidemic threshold than the star, and, thus, according to the above robustness criterion, the regular graph is more robust against virus propagation than the star. However,

P. Van Mieghem / Computer Communications 35 (2012) 1494–1506

1497

where u ¼ 1g and r is the order parameter [17] instead of the fraction of coupled or phase-locked oscillators. Thus, the analogue of the effective infection rate s is the coupling strength g, the epidemic threshold sc ¼ k11 corresponds to the onset of synchronization g c ¼ gk10 , while the steady-state fraction of infected nodes y1 translates to the order parameter r 2 . Similar to w, the synchronizability S measures the ease in which a set of oscillators in a given network synchronize for all possible coupling strengths g: the higher S, the easier oscillator nodes in a network synchronize. The synchronizability S plays a same role as the viral conductance and can be used to distinguish between networks. In particular in medical sciences, a high synchronizability S of the functional brain network would indicate that the patient is vulnerably for epileptic seizures. 3.1. Some exact expressions for w Fig. 1. The steady-state fraction y1 of infected nodes versus s ¼ 1=s (and versus s in the insert) for a regular graph with degree d ¼ 2 and a star with N ¼ 100 nodes. Both graphs have almost the same average degree.

For regular graphs with degree d, the basic relation (4) for y1;regular shows that the viral conductance equals

wregular ¼

Z

d



1

0





when the effective infection rate s > 1 ¼ 2sc Gregular , we observe from Fig. 1 that the percentage of infected nodes in the star is smaller than in the regular graph. The extent of the virus-free region is one aspect of the network’s resilience against viruses, but once that barrier, the epidemic threshold, is crossed, the virus may conduct differently in networks with high and low epidemic threshold. This observation has led Kooij et al. [11] to propose the viral conductance as an additional metric. As mentioned in Section 1, robustness metrics should be as complementary as possible. The sensitivity or dependence of the viral conductance w on the epidemic threshold sc could be assessed by

dw ¼ dk1

Z

k1

0

@y1 ðsÞ ds @k1

which is difficult to compute or by correlation simulations. Such } s–Rényi and small-world graphs simulations on scale-free, Erdo indicate the relatively low correlation. The viral conductance w is a graph metric that represents the overall conductance of the virus for all possible effective infection rates s: when w is high for the graph G, the virus can spread easily in G. Thus, instead of grading graphs only based on their epidemic threshold sc from virus vulnerable, where sc is small (high spectral radius k1 ) to virus robust (where sc is large), the viral conductance complements this classification with an average infection notion because

1 y1 ¼ k1

Z 0

Z 0

k1

r 2 ðuÞdu

mn  s12  sþm n

i 2 Nn

ð18Þ

mn  s12  sþn m

j 2 Nm

ð19Þ

v i1 ¼ 1 and

v j1 ¼ 1

Since y1 ðsÞ ¼

nv i1 þmv j1 , nþm

we obtain

  mn  s2 1 1 þ y1 ðsÞ ¼ N sþm sþn

ð20Þ

pffiffiffiffiffiffiffi and, with k1 ðK m;n Þ ¼ mn (see [22, p. 130]), the viral conductance of the complete bipartite graph K n;m equals

wK n;m

1 ¼ N

Z 0

pffiffiffiffiffi mn



mn  s2 mn  s2 þ ds sþm sþn

  2 Using ds ¼ A  m2 log ðs þ mÞ  s2 þ ms þ c, where c is a constant, we obtain R

y1 ðsÞds < 1

so that w ¼ ys1c < s1c ¼ k1 . Graphs with small epidemic threshold may possess a small average fraction of infected nodes y1 so that the viral conductance can be equal to graphs with large epidemic threshold and large y1 . The concept of viral conductance can be transferred to the synchronization of coupled oscillators in a network [17]. The dynamic coupling of oscillators (nodes) is comparable to virus spread in networks: there is a critical coupling strength g c ¼ gk10 , where g 0 is a constant and the behavior of the phase transition around g c is mathematically similar [24]. The direct translation of the viral conductance to the synchronizability S of a network is defined as

S¼

which is independent of the size N of the network! For the complete bipartite graph K m;n , where there are two partitions Nm with m nodes and Nn with n nodes so that N ¼ m þ n and L ¼ mn, we have [27] that

wK n;m ¼

k1

s d L ds ¼ ¼ d 2 N

As2 sþm

 pffiffiffim pffiffiffiffiffiffiffi mn ðn  mÞ 1 þ mn mn  þ log  pffiffiffin N N 1 þ mn

ð21Þ

Notice the symmetry in (21), namely that wK n;m ¼ wK m;n . Also, if m ¼ n, we again obtain a regular graph, for which wK n;n ¼ 2n. The other extreme occurs for a star K 1;n , where m ¼ 1 and n ¼ N  1 in (21). After some tedious manipulations, we obtain

wstar ¼

 log N 1 1  þ O pffiffiffiffi 2 2 N

ð22Þ

The cycle, which is a regular graph with degree d ¼ 2, only has wcycle ¼ 1, irrespective of the number N of nodes. Given a fixed number N of nodes, the viral conductance wK n;Nn (as well as the number L ¼ nðN  nÞ of links) increases with n from n ¼ 1 (star) up to n ¼ N2 (a regular graph). When both N and the average degree E½D ¼ 2L are N fixed, the following lemma is proved in Appendix E: Lemma 1. For a given N and L, the viral conductance of the complete bipartite graph K m;n is minimal when it is regular (m ¼ n).

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P. Van Mieghem / Computer Communications 35 (2012) 1494–1506

From (28), we have

4. Derivatives of y‘ ðsÞ at s ¼ 0 and s ! k1 Extremely high effective infection rates ðs ! 1Þ correspond to s ¼ 0 at which y1 ðsÞ ¼ 1 and below the epidemic threshold, s < sc or s > sc ¼ k1 ; y1 ðsÞ ¼ 0. Besides these two points, the ðsÞ 1 ðsÞ derivative at lims!0 dy1 and lims!k1 dyds can be computed exactly, ds as shown here for the first time. This section provides new series expansion for the viral steady-state probabilities v i1 , which will be used in Section 5 to bound the viral conductance w. The following Lemma 2 is proved in Appendix C.

PN

dy1 ðsÞ

1 j¼1 ðx1 Þj ¼  P ds s¼k1 k1 N Nj¼1 ðx1 Þ3j which is shown in [24] to be bounded as



1 dy1

1 6 6 <0 k1 ds s"k1 k1 Nmaxðx1 Þj

Theorem 3 suggests, for all 1 6 k 6 N, the existence of the power series

v i1 ðsÞ ¼ 1 þ

gm ðiÞs

ð23Þ

m¼1

possesses the coefficients

1 g1 ðiÞ ¼  di

ð24Þ

1 2 di

1

N X j¼1

aij dj

! ð25Þ

and for m P 2, the coefficients obey the recursion N X 1 aij gmþ1 ðiÞ ¼  gm ðiÞ 1  di d j¼1 j

For

regular

graphs

!

with

degree

d,

we

observe

ð26Þ that

with (4). From the Laurent series (23) in Lemma 2 after the transðsÞ form s ¼ 1s, the derivative at lims!0 dy1 can be computed exactly as ds

dy ðsÞ 1 lim 1 ¼  s!0 ds N

j¼1

1 1 ¼ E dj D

ð31Þ

ð32Þ

valid for s P k11 . Similarly, from the eigenvector expansion (14), all steady-state infection probabilities v i1 ðsÞ are expanded as 1 X N X   1 j cj ðkÞðxk Þi s1 c s

ð33Þ

We extend the above analysis and determine all coefficients cj ðkÞ in a recursive way as specified in Lemma 4, which is proved in Appendix D. Lemma 4. Defining N X

ðxm Þq ðxl Þq ðxk Þq

q¼1

the coefficients cj ðmÞ in (31) obey, for m > 1 and j > 2, the recursion

cj ðmÞ ¼

cj1 ðmÞ f1  c1 ð1Þðk1 þ km ÞX ðm; m; 1Þg k1  km N c 1 ð 1Þ X ðk1 þ kk Þcj1 ðkÞX ðm; k; 1Þ  k1  km k¼1;k–m N X N XX 1 cjn ðlÞcn ðkÞkk X ðm; l; kÞ k1  km n¼2 l¼1 k¼1 j2

ðsÞ with equality for the regular graph. Hence, the slope lims!0 dy1 is ds the least steep for the regular graph as also illustrated in Fig. 1. We also mention that (27) has been inferred in [27] based on simulations, but not rigorously proved as here. While the right-derivative4 lims#k1 dy1dsðsÞ ¼ 0, the left-derivative at lims"k1 dy1dsðsÞ can be computed explicitly from the following theorem, proved in [24]:

Theorem 3. For any graph with spectral radius k1 and corresponding P 2 eigenvector x1 normalized such that xT1 x1 ¼ N j¼1 ðx1 Þj ¼ 1, the steady-state fraction of infected nodes y1 obeys

when

1 X N   1X 1 j cj ðkÞuT xk s1 c s N j¼1 k¼1

ð27Þ

N 1 1X 1 1 1 P ¼ P D N j¼1 dj E½D k1

PN    1 j¼1 ðx1 Þj  1 1 2 y1 ðsÞ ¼ s  s1 þ O s1 P c s k1 N Nj¼1 ðx1 Þ3j c

j

ticular, from the definition (15), we obtain the series expansion of 1 y1 ðsÞ around s1 as c s

X ðm; l; kÞ ¼

which is minus the harmonic mean of the degrees in the graph. The harmonic, geometric and arithmetic mean inequality (see e.g. [22, p. 138]) shows3 that

E

1 s1 c s

j¼1 k¼1

g1 ðiÞ ¼ g1 ¼  1d and gm ðiÞ ¼ 0 for all m > 1, which is in agreement

N X



 P 1 where c1 ðkÞ ¼ 0 for 2 6 k 6 N and c1 ð1Þ ¼ k1 Nj¼1 ðx1 Þ3j . In par-

v i1 ðsÞ ¼

m N X 1X  gmþ1k ðiÞ aij gk ðjÞ di k¼2 j¼1

cj ðkÞ

j¼1

y1 ðsÞ ¼

and

g2 ðiÞ ¼

1 X

ck ðsÞ ¼ m

ð30Þ

16j6N

Lemma 2. The Laurent series of the steady-state infection probability 1 X

ð29Þ

ð28Þ

s approaches the epidemic threshold sc from above.

3 Since f ðxÞ ¼ 1x is convex for x > 0, Jensen’s inequality [21], f ðE½XÞ 6 E½f ðXÞ, applies as well. 4 The notation limx#x0 f ðxÞ is the limit towards x0 when x approaches x0 from above, i.e. x > x0 .



ð34Þ

while, for j ¼ 2 and m > 1,

c2 ðmÞ ¼ 

1 X ðm; 1; 1Þ k1  km k1 X 2 ð1; 1; 1Þ

ð35Þ

 P 1 and c1 ðmÞ ¼ 0. For m ¼ 1, there holds that c1 ð1Þ ¼ k1 Nj¼1 ðx1 Þ3j and for j > 1, the coefficients cj ð1Þ satisfy the recursion

cj ð1Þ ¼  

N X 1 ðk1 þ kk Þcj ðkÞX ð1; 1; kÞ ð Þ k1 X 1; 1; 1 k¼2 j1 X N X N X

cjþ1n ðlÞcn ðkÞkk X ð1; l; kÞ

ð36Þ

n¼2 l¼1 k¼1

The radius of convergence of the Laurent series (23) and of the series (31) is, in general, unknown and still an open problem. In addition to Theorem 3, we present an orthogonality condition, proved in Appendix B, for the derivative of the vector V 1 defined h iT T as dVds1 ¼ dvds11 dvds21    dvdsN1 at s ¼ k1 :



is orthogonal to the vector Lemma 5. In any graph, the vector dVds1

s¼k1 D  k1 u.

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P. Van Mieghem / Computer Communications 35 (2012) 1494–1506

Expression (28) illustrates that, among all graphs with N nodes has the largest epiand L links, the regular graph with degree d ¼ 2L N demic threshold sc ¼ 1d, but also the largest (in absolute value)



derivative dyds1

¼  k11 ¼  1d because equality in (30) is reached. s!k1

This means that a higher effective infection rate s is needed to cause a non-zero steady-state fraction of nodes in the regular graph to be permanently infected, but that, slightly above that critical rate sc , a higher relative fraction of nodes is infected than in other graphs. In other words, the change in virus conductivity at s ¼ sc þ  is highest in regular graphs. Thus, when we are asked to classify networks according to their proneness to viral infections, the epidemic threshold alone is insufficient as metric alone. Our analysis based on Theorem 3 supports the viral conductance w as an additional candidate metric that complements the epidemic threshold sc ¼ k11 . 5. Analysis of the viral conductance w As in [29], one may device heuristics to approximate the steadystate fraction of infected nodes y1 ðsÞ in the s-interval ½0; k1  based on the two endpoints, their derivative and on the convexity5 of y1 ðsÞ. Strict convexity (for irregular graphs) also means that the derivative dyds1 decreases from s ¼ 0 up to s ¼ k1 consistent with both Lemma 2 and Theorem 3. Here, we present an analysis of the viral conductance w using the knowledge of y1 ðk1 Þ ¼ 0; y1 ð0Þ ¼ 1 and the corresponding derivatives, y01 ðk1 Þ and y01 ð0Þ specified in Section 4, together with the convexity of y1 ðsÞ. Since y1 ðsÞ 6 1, we obtain the trivial upper bound w 6 k1 for the viral conductance (17). Convexity of y1 ðsÞ implies that y1 ðsÞ 6 1  ks1 from which the sharper upper bound

w6

k1 2

Z 0

1 þ k1 y01 ðk1 Þ  y01 ðk1 Þ

y01 ð0Þ

generates a maximum

 2 1 0 1 1 þ k1 y01 ðk1 Þ 2 f ðfÞ ¼  y1 ðk1 Þk1  2 2 y01 ð0Þ  y01 ðk1 Þ Thus, for irregular graphs, our optimized lower bound for w is

wP

  2  1 1  k1 y01 ðk1 Þ k21  0



y1 ðk1 Þ þ 2 y01 ð0Þ  y01 ðk1 Þ

2

Explicitly, with (27) and (29),

! k1 ð1  Z Þ2

wP Z þ

 1  2 k1 E D  Z

k1

y1 ðsÞds ¼ x 

Z

k1

0

ðs  xÞy01 ðsÞds

ð38Þ

PN ðx1 Þj where Z ¼ N1 PNj¼1 3 and f ¼ k j¼1

1Z ½ Z

1 1E D

ðx1 Þj

  6 1 because k1 E D1 P Ek½D1  P 1.

Equality in (38) occurs for regular graphs where Z ¼ 1 and k1 ¼ d. With the bounds for irregular graphs

1
Z

k1

0

ðs  xÞ2 00 1 y1 ðsÞds > min y001 ðsÞ 2 s2½0;k1  2 ¼

is immediate. After partial integration, the definition of the viral conductance (17) becomes, for any real x,

w¼

f¼

k1

ðs  xÞ2 ds

0

k1 k2 min y001 ðsÞ 1  k1 x þ x2 2 s2½0;k1  3

evaluated at x ¼ f < 1 because summary, we arrive at

k1 wP 2

Z

k21 3

( Z þ fð1  Z Þ þ min

s2½0;k1 

y001 ðsÞ

!

 k1 f þ f2 > 0 when k1 P 3. In

k21  k1 f þ f2 3

!) ð39Þ

Again partially integrating yields

1 1 1 w ¼ x þ x2 y01 ð0Þ  ðk1  xÞ2 y01 ðk1 Þ þ 2 2 2 Z k1 2 00  ðs  xÞ y1 ðsÞds

Similarly, a same type of upper bound follows directly as

ð37Þ

k1 w6 2

( Z þ fð1  Z Þ þ

max y001 ðsÞ s2½0;k1 

k21  k1 f þ f2 3

!) ð40Þ

0

Convexity, i.e. y001 ðsÞ P 0 with equality only for regular graphs, implies that

1 1 w P f ðxÞ ¼ x þ x2 y01 ð0Þ  ðk1  xÞ2 y01 ðk1 Þ 2 2   1   1 ¼ 1 þ y01 ðk1 Þk1 x þ x2 y01 ð0Þ  y01 ðk1 Þ  y01 ðk1 Þk21 2 2 Furthermore,

f 0 ðxÞ ¼ 1 þ xy01 ð0Þ þ ðk1  xÞy01 ðk1 Þ while

f 00 ðxÞ ¼ y01 ð0Þ  y01 ðk1 Þ < 0 demonstrates that the solution of f 0 ðfÞ ¼ 0, i.e. 5 A long the same lines as in [26], we can prove that v j1 ðsÞ is (strict) convex in s, for s < sc .

For the complete bipartite graph K n;m , the second derivative of (20) equals

y001 ðsÞ and

¼

  2ðm  nÞ2 m2 n  s3 þ mnðn þ 3sÞ ðm þ nÞðm þ sÞ3 ðn þ sÞ3

mins2½0;k1  y001 ðsÞ ¼ y001 ðk1 Þ ¼ 2

2ðmnÞ2 pffiffiffiffiffi pffiffiffi pffiffi 4 ðmþnÞ mnð mþ nÞ

Þ y001 ðsÞ ¼ y001 ð0Þ ¼ 2ðmmn 2 n2 . The ratio

y001 ð0Þ y001 ðk1 Þ



and

pffiffiffin  n

¼ 1þm

m

1þ

maxs2½0;k1  pm ffiffiffi4 n

in-

creases with increasing irregularity, i.e. with increasing ratio mn . The class of complete bipartite graphs can range from regularity m ¼ n to extreme irregularity in the star, where m ¼ 1 and n ¼ N  1. Therefore, that class of K n;m is our benchmark. If the right-hand side in (39) can be shown to exceed 12 E½D ¼ NL , then it would imply that the viral conductance of the regular graph is the smallest among all graphs with N nodes and L links. Lemma 1 states that the claim holds for the class of complete bipartite graphs and simulations in [29] suggest the truth of the claim for other types of graphs as well.

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P. Van Mieghem / Computer Communications 35 (2012) 1494–1506

5.1. Series expansions for the viral conductance w

Hence, the viral conductance is bounded by

When using the Laurent expansion in (23), the steady-state fraction of infected nodes can be formally written as

w6

y1 ðsÞ ¼ 1 þ

! N 1X 1 ds N i¼1 1 þ di 0 s  N 1X k1 ¼ di ln 1 þ N i¼1 di Z

1 X N 1X g ðiÞsm N m¼1 i¼1 m

while (32) yields

k1

1

ð41Þ

For regular graphs, this upper bound (41) translates to VC regular 6 d ln 2 ¼ 0:693d. Also, we deduce from (41) that

1 X N 1X y1 ðsÞ ¼ cj ðkÞuT xk ðk1  sÞj N j¼1 k¼1

  N k1 1 X k1 w < ln 1 þ di ¼ E½D ln 1 þ dmin N i¼1 dmin

We write the integral in (17) as

w¼

Z

k1 2

0

y1 ðsÞds þ

Z

where E½D denotes the average degree. The second convergent of the continued fraction expansion (50) is

k1

y1 ðsÞds

k1 2

and use the expansion around s ¼ 0 in the first and the other expansion in the second integral to obtain

w¼

1 X N   k1 1 X kmþ1 gm ðkÞ þ cm ðkÞuT xk mþ1 1 þ 2 N m¼1 k¼1 2 ðm þ 1Þ

This series for w is valid provided the convergence radius of both series for y1 ðsÞ is at least k21 . Executing the first term yields

v i1 6 1 

1 1 þ sdi  s

PN

aij j¼1 1þsdj

and the viral conductance is bounded by

Z

0 k1

w6 0

B @1 

1

N 1X

N

i¼1

1 PN

1 þ dsi  1s

aij

j¼1

dj

C Ads

1þ s

0 1 N 2 k1 1 @X 1 uT x1 A k1 þ R2 w¼   XN 2 N k¼1 dk k ðx1 Þ3j 8 1 j¼1    k1 1 1 k1 E Z þ R2 ¼ 1 4 D 2

But this integral cannot be evaluated formally. Instead, we approximate the integral by

where the remainder R2 is

Z

R2 ¼

1 X N  1X

N m¼2 k¼1

gm ðkÞ þ cm ðkÞuT xk



mþ1

6. Bounds for the viral conductance In addition to the upper (40) and lower bound (39) derived in Section 5 and based on derivatives of y1 ðsÞ, we here present other types of upper and lower bounds that lead to relatively simple expressions in terms of graph properties. 6.1. Upper bounds for the viral conductance By exploiting Theorem 9, proved in [27] and alternatively in Appendix A, considerably sharper bounds can be deduced. Each convergent of the continued fraction expansion (50) provides an upper bound for v i1 . The first convergent of (50),

1 1 þ sdi

k1

0

ðk1 Þ

2mþ1 ðm þ 1Þ

Subject to the above convergence condition, both expansions can be numerically computed up to any desired accuracy when the adjacency matrix A is given, from which the spectral expression A ¼ X KX T can be computed [22].

v i1 6 1 

1 ¼ k1  N

N Z k1 X i¼1

0

sds P a s þ di  Nj¼1 sþdij

sds P a 6 s þ di  Nj¼1 sþdij j

Z

k1

0

j

sds PN

s þ di 

aij j¼1 dj

and obtain

1 ! 0 N N X 1X aij k 1 ln @1 þ w6 di  P aA N i¼1 d di  Nj¼1 ij j¼1 j

P a When di  Nj¼1 dijj ¼ 0, this i-th term does not contribute to the viral conductance (because limx!0 x ln ð1 þ c=xÞ ¼ 0). For a regular graph, this bound simplifies to

 d w 6 ðd  1Þ ln 1 þ d1   ¼ ðd  1Þ ln 2 þ ln 1 þ ¼ ðd  1Þ ln 2 þ

1 2ðd  1Þ

1 X ð1Þk1



1

k

k1 k2 ðd  1Þ k¼1 !  1 1 1 1  þO ¼ d ln 2  ln 2  2 2 8 d1 ð d  1Þ

which is better than the previous bound w 6 d ln 2. Using

leads to

di 

N X aij

s

P min di 

N X aij

!

16i6N dj d j¼1 j  di P min di  ¼ dmin  1 16i6N dmin j¼1

N 1X 1 y1 ðsÞ 6 1  N i¼1 1 þ di

ð42Þ

dj

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Likely, the star possesses the largest viral conductance among all graphs with N nodes and L links.

in (42), we deduce, for dmin > 1, that

) ( X N X N k1 1 N 1X aij di  N i¼1 N i¼1 j¼1 dj dmin  1 )  ( N N X k1 1 1X aij E½D  < ln 1 þ N j¼1 dj i¼1 dmin  1  k1 ¼ ln 1 þ fE½D  1g dmin  1

 w 6 ln 1 þ

7. Behaviour of y‘ ðsÞ around s ¼ E½2D

  pffiffiffiffiffiffiffiffiffiffiffiffiffi The viral conductance of the star, where k1 AK 1;N1 ¼ N  1 and E½D ¼ 2  N2 , shows that the logarithmic prefactor can be significant and that w can be larger than E½D. In particular, the bound (42) for the star K 1;N1 is

 log N 1  2 ln 2 þ O pffiffiffiffi 2 N

w/

6.2. Lower bound for the viral conductance Using the lower bound derived in [27], valid for s 6 dmin ,

1 1  ddi þ sdi

Lemma 6. For any graph, it holds that

y1 ðsÞ 6

 1 1 E½D E½D for s ¼ þ 1 2 2 k1 2

1 s P dmin or

6 v i1

Proof. Convexity of y1 ðsÞ for s 2 ½0; k1  implies that y1 ðqa þ ð1  qÞbÞ 6 qy1 ðaÞ þ ð1  qÞy1 ðbÞ for any a; b 2 ½0; k1 . By choosing a ¼ 0; b ¼ k1 and ð1  qÞb ¼ E½2D, we find (44). Equality only holds for the regular graph since only then y001 ðsÞ ¼ 0 as verified from (4). h Lemma 7. For any graph, it holds that

min

y1 ðsÞ 6 s

the viral conductance is lower bounded as

w¼

Z

dmin

y1 ðsÞds þ

0

P

1 N

N Z X j¼1

dmin

Z

dmin

dmin

y1 ðsÞds

v i1 ðsÞds P ¼

Proof. Consider the Laplacian expression (7) for steady-state fraction of infected nodes y1 . The Rayleigh principle [22] states that lN ¼ 0 6 V T1 QV 1 6 l1 V T1 V 1 , where l1 and lN are the largest and respectively smallest eigenvalue of the Laplacian Q. The eigenvector u belongs to the smallest eigenvalue lN ¼ 0 of Q. Thus, only for regular graphs where each node has degree di ¼ d and V 1 ¼ v 1 u, we observe that V T1 QV 1 ¼ 0. The first quadratic form in (7)

Z

dmin

1

1 1  ddi þ dsi

! ds

min

   di dmin dmin dmin 1þ ln di  dmin di  dmin di

 9 8 N ln ddi = dmin X 1 < min wP 1 d i N i¼1 1  dmin : 1; di

ð43Þ

dmin

v i1 ðsÞds P

d 0

 s d 1  ds ¼ d 2

illustrating that the lower bound (43) is the best possible among all   ln x graphs. Since the function 1=ð1  1=xÞ 1  x1 is strict increasing from 1=2 for x ¼ 1 to 1 when x ! 1, we conclude from (43) that the viral conductance is always larger than the minimum degree dmin for non-regular graphs. 2 In summary, the upper bound (42) and the lower bound (43) provide us with the elegant bounds

 dmin k1 6 w < ln 1 þ fE½D  1g 2 dmin  1 pffiffiffiffi The largest possible ratio dk1 ¼ O N occurs in the star, which min illustrates that the viral conductance is bounded by 12 E½D log N.

is maximal when, at each node k; v k1 ¼ 12, resulting in P ðu  V 1 ÞT DV 1 6 14 Nj¼1 dj ¼ 2L . Due to symmetry, this maximum is only reached in the regular graph and (45) is proved. h Lemma 7 shows for the regular graph that the ‘‘tangent’’ line through the origin ðs ¼ 0Þ lies above y1;regular and only touches y1;regular at the point not larger than we find

y1 ðsÞ 6

By taking the limit di ! dmin in (43), we find the same result (after invoking de l’Hospital’s rule twice).

E½D 4

s ¼ E½2D ¼ 2d. For any other graph, the slope is T

þ V 1 NQV 1 and, after transforming s ¼ 1s in (45),

1 2V T1 QV 1 þ 2 NE½D

for s ¼

E½D 2

that complements (44). The correction correction

2V T1 QV 1 NE½D

ð46Þ 1 2

  1  Ek½D1  in (44) and the

in (46) are positive and small, but nevertheless

show that y1 ðsÞ can be larger than 12 at s ¼ E½2D, as numerically found in [29]. Lemma 8. For any graph, it holds that

y1 ðsÞ

s 6

 1  v j1 dj v j1

j¼1

Only for regular graphs where di ¼ dmin ¼ d, equality in the lower bound (43) is achieved, because6

Z

N X 

ðu  V 1 ÞT DV 1 ¼

we arrive at the lower bound

0

ð45Þ

v i1 ðsÞds

0

dmin

!

and equality is only possible for the regular graph for which V T1 QV 1 ¼ 0.

0

0

Z

E½D V T1 QV 1 þ 4 N

k1

With

Z

ð44Þ

and equality is only possible for the regular graph.

which is correct to first order for large N as verified from (22).

1

Via extensive simulations, Youssef et al. [29] have observed that y1 ’ 12 around s ¼ E½2D. In this section, we aim to explain these observations. In [15], we have shown, for any graph, that y1 6 12 for s 6 E½1D. Here, sharper general bounds are derived.

P rðsÞ ¼

N   1X v j1 dj  k1 v j1 N j¼1

where the lower bound obeys, for any s,

ð47Þ

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P. Van Mieghem / Computer Communications 35 (2012) 1494–1506

rðsÞ 6

N2 4Nk1

Ignoring the integer nature   of m, the maximizer of the right-hand side occurs at m ¼ 2k1 E D1 , resulting in

and where N k ¼ uT Ak u denotes [22] the number of walks of length k. Proof. The Rayleigh principle [22] and the fact that all involved elements of A and V 1 are non-negative imply that 0 6 V T1 AV 1 6 k1 V T1 V 1 , with equality in the upper bound only for the regular graph, where the eigenvalue 2L ¼ k1 corresponds to N the eigenvector u and V 1 ¼ v 1 u. Hence, (6) is bounded by

y1 ðsÞ P

s N

DT V 1  k1 V T1 V 1



ð48Þ

with equality for the regular graph, from which (47) is immediate. The lower bound rðsÞ in (47) has an extremum at s ¼ n > sc that obeys N   1X v 0j1 ðnÞ dj  2k1 v j1 ðnÞ ¼ 0 N j¼1

There are three possible cases: (a) neither all v 0j1 ðnÞ ¼ 0 nor all v j1 ðnÞ ¼ 2kdj1 , which leads to a complicated, not analytically manageable relation defining n; (b) v 0j1 ðnÞ ¼ 0 for all 1 6 j 6 N, in which case v j1 ðnÞ ¼ 1, n ! 1 and y1nðnÞ ! 0. This case corresponds to a minimum and the lower bound is then negative and, hence, useless; d (c) v j1 ðnÞ ¼ 2kj1 for each 1 6 j 6 N in which case the maximum lower bound is attained

rðsÞ 6 rðnÞ ¼

N 1 X N2 2 d ¼ 4Nk1 j¼1 j 4Nk1

This demonstrates the lemma.

ð49Þ h

N2 We will now estimate the value n for which rðnÞ ¼ 4Nk in irreg1 ular graphs where Var½D > 0. The proof of Lemma 8 shows that rðnÞ attains the maximum bound (49), provided that there exists d a value of s ¼ n for which v j1 ðnÞ ¼ 2kj1 for each 1 6 j 6 N, in which case

y1 ðnÞ ¼

N 1X dj 1 2L 1 E½D 1 ¼ ¼ < N j¼1 2k1 2k1 N 2 k1 2

 y1

2k1 1 2 E ½D    ’ E½D 4k1 E D1

Notice that

2k1 1 E½D 4k1 E D1

2k1 1

6 E½Dk1 < 12.

½ 4E½D In summary, both last lower bound arguments illustrate, together with the upper bounds in Lemmas 6 and 7 that, for values of s approaching E½2D, the steady-state fraction of infected nodes y1 ðsÞ is close to 12 in any graph, in agreement with simulations [29]. 8. Conclusion We have reviewed and justified the viral conductance w as an additional virus-robustness metric to classify networks: the viral conductance w reflects the ease of virus spread in a network under all possible effective infection rates. In addition, we have derived easy to use analytic bounds on the viral conductance of different type: the general lower bounds (39) and (43) and upper bounds (40) and (42). Through a set of Lemmas 6–8, we have theoretically explained the observation that y1 ðsÞ is close to 12 for s ¼ E½2D, made by simulations in [29]. We show that the claim ‘‘among all graphs with N nodes and L links, the viral conductance is minimal for the regular graph’’ is correct, at least for the class of complete bipartite graphs. In a similar vein, we believe that the star has the largest viral conductance among all graphs with N nodes and L links. Rigorous proofs of both claims would be desirable. Finally, we have complemented the theory of N-intertwined model by deriving two series expansions (23) and (33) around s ¼ 0 and s ¼ k1 ¼ s1c , respectively. Two dynamic processes on graphs, virus spread and the coupling of oscillators, are argued to be mathematically related, which has inspired us to propose the synchronizability S as the analogue of the viral conductance w. An analysis as here still needs to be performed for the synchronizability, but we speculate that properties of w may translate to S for a same network. Acknowledgement

nN 2 Lemma 8 then states that y1 ðnÞ P 4Nk or 1

We thank Mina Youssef and Caterina Scoglio for pointing us to the upper bound w 6 k21 . This research was supported by Next Generation Infrastructures (Bsik) and the EU FP7 project ResumeNet (project No. 224619).

1 L nN2 P k1 N 4Nk1 from which

n6

4L E½D 2 ¼2 < N2 ðE½DÞ2 þ Var½D E½D

Appendix A. The continued fraction expansion

In conclusion, there may exist a value n such that sc < n < E½2D for which y1 ðnÞ ¼ 12 Ek½D1  < 12. We end by deducing, approximately though, another type of lower bound. Concavity of y1 ðsÞ for s P sc ¼ k11 similarly leads to y1 ðqsc þ ð1  qÞmsc Þ P ð1  qÞy1 ðmsc Þ. For sufficiently large m, the Laurent series (23) up to first order leads to

y1 ðmsc Þ P 1 

1 msc N

N X 1 j¼1



1

ðmsc Þ2



! 2 N  sc 2 1 X 1 E½D P 1 E½D msc N j¼1 dj ðm  1Þsc     k1 E D1 2k1 1 1 1 > m m E½D



P 1 þ sdi  s Nj¼1

is positive due to

g2 ðiÞ > 0 in (25). Choosing qsc þ ð1  qÞmsc ¼ E½2D provides us with y1

Theorem 9. For any effective spreading rate s ¼ bd P 0, the non-zero steady-state infection probability of any node i in the N-intertwined model can be expressed as a continued fraction

v i1 ¼ 1 

dj

because the second order term of O

We present a slightly different derivation of the continued fraction in Theorem 9 proved in [27]:

where di ¼

PN

j¼1 aij

1 1þsdj s

k¼1

1þsdk s

is the degree of node i.

Proof of Theorem 9. We rewrite (3) as

1  v i1 ¼

ð50Þ

aij

PN

1   P 1 þ sdi  s Nj¼1 aij 1  v j1

ajk N

P

q¼1

akq . 1þsdq  . .

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P. Van Mieghem / Computer Communications 35 (2012) 1494–1506

and define the kth convergent as

wi ðkÞ ¼

1 þ sdi  s

1 PN

j¼1 aij wj ðk

Appendix C

ð51Þ

 1Þ

Proof of Lemma 2. By introduction of the Laurent series (23) into (3)

with starting value wi ð0Þ ¼ 0 and v i1 ¼ 1  limk!1 wi ðkÞ. The difference hi ðkÞ ¼ wi ðkÞ  wi ðk  1Þ, satisfying

hi ðkÞ ¼ wi ðkÞwi ðk  1Þs

N X 1 þ s aij v j1 j¼1

N X aij hi ðk  1Þ

we obtain, with g0 ðiÞ ¼ 1,

j¼1

hi ðk Þ P

1¼

PN

j¼1 aij hi ðk  1Þ

ð1 þ sdi Þ

2

!

1 X

1

with starting value hi ð1Þ ¼ ð1 þ sdi Þ , is bounded by

s

!

gj ðiÞs

1

j¼0

P0 ¼1 

wi ð0Þ 6 wi ð1Þ 6    6 wi ðk  1Þ 6 wi ðkÞ 6   

j¼1

k¼0 j¼1

( 1 X m X

) N X gmk ðiÞ aij gk ðjÞ s1m j¼1

Rewritten,

0¼

Appendix B



1 X

1 X N X aij gmþ1 ðjÞsm

m¼0

m¼0 j¼1

gm ðiÞsm þ

( 1 mþ1 X X

N X aij gk ðjÞ

N X v j1 ¼ 1  v j1 j¼1 j¼1

dj  

s 1  v j1

! 2

sm

j¼1

Equating corresponding powers in

gm ðiÞ þ

dv j1 ds

)

gmþ1k ðiÞ

m¼0 k¼0

Proof of Lemma 5. After transforming (11) by s ¼ 1s, differentiation with respect to s yields

k¼0

1 X N X gj ðiÞsj þ aij gk ðjÞs1k

m¼0 k¼0

is non-decreasing and leads, in the limit for k ! 1, to the partial fraction (50). h

!

1 X j¼0

which shows that the sequence of convergents

N X

N 1 X X 1 þ s aij gk ðjÞsk

j

sm yields, for m P 0

N m þ1 N X X X aij gmþ1 ðjÞ  gmþ1k ðiÞ aij gk ðjÞ ¼ 0 j¼1

j¼1

k¼0

which is equivalent to

Evaluation at the critical threshold s ¼ k1 , where 1 6 j 6 N leads to

v j1 ¼ 0

for all

gm ðiÞ 

m X

N X aij gk ðjÞ ¼ 0

k¼0

j¼1

gmþ1k ðiÞ

For m ¼ 0 and m ¼ 1, we find (24) and (25), while after rewriting the above relation, we arrive at the recursion (26) from which all powers can be determined in terms of the adjacency matrix elements. h

N N X X dv j1 dv j1 dy

dj ¼ k1 ¼ k1 N 1

ds ds ds s¼k1 j¼1 j¼1 such that



N dy1

1 X dv j1

¼ d j ds s¼k1 k1 N j¼1 ds s¼k1

1 Appendix D. A power series expansion of y‘ ðsÞ around s1 c s

By definition of y1 , we also have that

Proof of Lemma 4. We substitute the definition (14) of V 1 in terms of eigenvectors of the adjacency matrix A into the general steady-state Eq. (12), rewritten as



N dy1

1X dv j1

¼ ds s¼k1 N j¼1 ds s¼k1

ð52Þ

AV 1  s1 V 1 ¼ diagðv i1 ÞAV 1 and find

Combining both yields, in vector notation,

N X 

 T 1 dV 1

u D ¼0 k1 ds s¼k1

k¼1

This equality implies that the vector dV 1

ds

the vector D  k1 u.

h

! N N X X  kk  s1 ck ðsÞxk ¼ diag cl ðsÞðxl Þi ck ðsÞkk xk

s¼k1

should be orthogonal to

l¼1

k¼1

After left-multiplication by xTm and exploiting the orthogonality between eigenvectors, we obtain



km  s

1



cm ðsÞ ¼

For a regular graph, dvdsk1 ¼  1d and D  k1 u ¼ 0. For any other non-regular graph, dmax > k1 > E½D so that D  k1 u has at least one



are nonpositive component, while all components of dVds1

¼

positive. Such condition on the vector components is necessary for orthogonality.

¼

N X

xTm diag

!

cl ðsÞðxl Þi

ck ðsÞkk xk

l¼1 N X

ðxm Þq

q¼1

s¼k1

N X l¼1

N X k¼1

N X

N X

l¼1

k¼1

cl ðsÞðxl Þq

N X

cl ðsÞ

k¼1

ck ðsÞkk

ck ðsÞkk ðxk Þq

N X ðxm Þq ðxl Þq ðxk Þq q¼1

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P. Van Mieghem / Computer Communications 35 (2012) 1494–1506

and



Permutation of the indices leaves X ðm; l; kÞ unchanged,

km  s

1



cm ðsÞ ¼

N X N X

cl ðsÞck ðsÞkk X ðm; l; kÞ

ð53Þ

l¼1 k¼1

P where X ðm; l; kÞ ¼ Nq¼1 ðxm Þq ðxl Þq ðxk Þq and X ðm; l; kÞ is unchanged by any permutation of the indices m; l and k. Introducing the power series (31) of ck ðsÞ into (53) and executing the Cauchy product for cl ðsÞck ðsÞ yields



km  s1

1 X   1 j cj ðmÞ s1 c s

¼

j¼0

Using



N N X X cj1 ðlÞX ðm; l; 1Þ þ c1 ð1Þ cj1 ðkÞkk X ðm; 1; kÞ l¼1

k¼1

N X ¼ c1 ð1Þ ðk1 þ kk Þcj1 ðkÞX ðm; k; 1Þ k¼1

leading to

ðkm  k1 Þcj ðmÞ þ cj1 ðmÞ

j¼0

( j X 1 N X N X X

c1 ð1Þk1

cjn ðlÞcn ðkÞkk X ðm; l; kÞ

) 

1 c

s s

 1 j

¼ c1 ð1Þ

k¼1

n¼0 l¼1 k¼1

s1 c ¼ k1 , the left-hand side is rewritten as

1 X  j km  s1 cj ðmÞ k1  s1

N X ðk1 þ kk Þcj1 ðkÞX ðm; k; 1Þ

þ

j2 X N X N X

¼ c1 ð1Þðk1 þ km Þcj1 ðmÞX ðm; m; 1Þ

j¼0 1 X  j ¼ km  k1 þ k1  s1 cj ðmÞ k1  s1



¼

1 X  j ðkm  k1 Þcj ðmÞ k1  s1 þ

j¼0 1 X

j¼0

 j cj1 ðmÞ k1  s1

j¼1

 j Equating corresponding powers in k1  s1 yields, for j ¼ 0, N X N X ðkm  k1 Þc0 ðmÞ ¼ c0 ðlÞc0 ðkÞkk X ðm; l; kÞ ¼ 0

þ

j2 X N X N X

Finally, after rearrangement, we arrive at the general recursion (34) for the coefficients cj ðmÞ for m > 1 and j > 2. The recursion for cj ð1Þ and j > 2 follows as

cj1 ð1Þ ¼

l¼1 k¼1

j X N X N X cjn ðlÞcn ðkÞkk X ðm; l; kÞ n¼0 l¼1 k¼1

¼

j1 X N X N X cjn ðlÞcn ðkÞkk X ðm; l; kÞ n¼1 l¼1 k¼1

Before proceeding, we confine to the case where j ¼ 1; ðkm  k1 Þ c1 ðmÞ ¼ 0, illustrating that all c1 ðmÞ ¼ 0 for m > 1, except when m ¼ 1, which is in agreement with Theorem 3. Next, the case for j ¼ 2 yields

ðkm  k1 Þc2 ðmÞ þ c1 ðmÞ ¼

N X N X c1 ðlÞc1 ðkÞkk X ðm; l; kÞ l¼1 k¼1

¼ c21 ð1Þk1 X ðm; 1; 1Þ

cjn ðlÞcn ðkÞkk X ðm; l; kÞ

n¼2 l¼1 k¼1

because c0 ðkÞ ¼ 0 for all 1 6 k 6 N and, for j > 0,

ðkm  k1 Þcj ðmÞ þ cj1 ðmÞ ¼

ðk1 þ kk Þcj1 ðkÞX ðm; k; 1Þ

k¼1;k–m

1 1 X  j X  jþ1 cj ðmÞ k1  s1 þ cj ðmÞ k1  s1 j¼0

N X

þ c1 ð1Þ

j¼0

¼ ðkm  k1 Þ

cjn ðlÞcn ðkÞkk X ðm; l; kÞ

n¼2 l¼1 k¼1

c1 ð1Þ

PN

k¼2 ðk1 þ kk Þc j1 ðkÞX ð1; 1; kÞ 1  2c1 ð1Þk1 X ð1; 1; 1Þ Pj2 PN PN cjn ðlÞcn ðkÞkk X ð1; l; kÞ þ n¼2 l¼1 k¼1 1  2c1 ð1Þk1 X ð1; 1; 1Þ

With c1 ð1Þ ¼ ðk1 X ð1; 1; 1ÞÞ1 , we arrive at (36). This proves Lemma 4. h

For example, for j ¼ 2 in (36),

c2 ð1Þ ¼ 

N X 1 ðk1 þ kk Þc2 ðkÞX ð1; 1; kÞ k1 X ð1; 1; 1Þ k¼2

Using (35),

c2 ð1Þ ¼

1

N X k1 þ kk

k21 X 3 ð1; 1; 1Þ k¼2 k1  kk

X 2 ðk; 1; 1Þ

ð54Þ

so that, for m ¼ 1, taking into account that c1 ðmÞ ¼ 0 for m > 1, After the transformation s ¼ 1s, the derivatives

c1 ð1Þ ¼ c21 ð1Þk1 X ð1; 1; 1Þ or c1 ð1Þ ¼ ðk1 X ð1; 1; 1ÞÞ1 , which is also in agreement with Theorem 3, while for m > 1, we obtain

c2 ðmÞ ¼ 

c21 ð1Þ 1 X ðm; 1; 1Þ k1 X ðm; 1; 1Þ ¼  k1  km k1  km k1 X 2 ð1; 1; 1Þ

j d y1 ðsÞ



j ds

¼ s"k1

N j! X cj ðkÞuT xk N k¼1

are immediate from (32). In particular, for j ¼ 2, we obtain

Continuing, for j > 2 and using that all c1 ðmÞ ¼ 0 for 2 6 m 6 N, except for c1 ð1Þ, we have

2 d y1 ðsÞ



2 ds

ðkm  k1 Þcj ðmÞ þ cj1 ðmÞ N X N N X N X X cj1 ðlÞc1 ðkÞkk X ðm; l; kÞ þ c1 ðlÞcj1 ðkÞkk X ðm; l; kÞ ¼

and substituting (54) and (35) yields

l¼1 k¼1 j2 X N X N X

l¼1 k¼1

cjn ðlÞcn ðkÞkk X ðm; l; kÞ

þ

n¼2 l¼1 k¼1 N X

¼ c1 ð1Þk1 þ

N X cj1 ðlÞX ðm; l; 1Þ þ c1 ð1Þ cj1 ðkÞkk X ðm; 1; kÞ

l¼1 j2 X N X N X

cjn ðlÞcn ðkÞkk X ðm; l; kÞ

n¼2 l¼1 k¼1

k¼1

2 d y1 ðsÞ



2 ds

¼ s¼k1

¼ s"k1

N 2 2X c2 ð1ÞuT x1 þ c2 ðkÞuT xk N N k¼2

N X 2uT x1 k1 2 3 Nk1 X ð1; 1; 1Þ k¼2 k1



þ kk 2 X ðk; 1; 1Þ  kk

N X uT xk X ðk; 1; 1Þ 2 k Nk1 X ð1; 1; 1Þ k¼2 1  kk

2

For example, from (20), the second derivative at s ! k1 ¼ the complete bi-partite graph is

ð55Þ pffiffiffiffiffiffiffi mn for

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P. Van Mieghem / Computer Communications 35 (2012) 1494–1506

pffiffiffiffiffi pffiffiffi 2 2 m n d y1 ðsÞ

2 pffiffiffiffiffi pffiffiffi p ffiffiffiffiffiffiffi ¼

2

mþ n N mn ds s"k

so that

NY n;m ¼

1

The general h formula i(55) requires the determination of X ðk; 1; 1Þ. u ffiffiffiffiffi up1n ffiffiffiffi , we have With xT1 ¼ p1m 2m 2n

X ðk; 1; 1Þ ¼

N X

ðxk Þq ðx1 Þ2q ¼

q¼1

m N 1 X 1 X ðxk Þq þ ðxk Þq 2m q¼1 2n q¼mþ1

rffiffiffiffiffi pffiffiffiffiffiffiffi  n mn m m log  log 1 þ ¼ þ   p ffiffiffi p ffiffiffiffi ffi 2 2 2 ð n  m Þ m n ðn  mÞ nþ m pffiffiffin r ffiffiffiffi ffi    m m m m n  log  log 1 þ ¼    pffiffiffi m m 2 n n 2 1  1þ n n NY n;m

N m N X 1 X 1 X ðxk Þq ðx1 Þq ¼ pffiffiffiffiffiffiffi ðxk Þq þ pffiffiffiffiffiffi ð xk Þ q 2m q¼1 2n q¼mþ1 q¼1

and, similarly, orthogonality of xk and xN ¼

h

u ffiffiffiffiffi p1m 2m

ffiffiffiffi  up1n 2n

ð56Þ

and

Orthogonality of xk and x1 implies that

0¼

n pffiffiffiffiffiffiffipffiffiffi pffiffiffiffiffi2 m  ðn  mÞ2 mn n  m þ ðn  mÞ log 2 m rffiffiffiffiffi  m  log 1 þ n

iT

results in

With y ¼ mn 2 ½0; 1, we have

pffiffiffi pffiffiffi y y ¼ gðyÞ ¼  pffiffiffi2  2ð1  yÞ logðyÞ  log ð1 þ yÞ 2 ðn  mÞ 1þ y NY n;m

N m N X 1 X 1 X ðxk Þq ðxN Þq ¼ pffiffiffiffiffiffiffi ðxk Þq  pffiffiffiffiffiffi ð xk Þ q 0¼ 2m q¼1 2n q¼mþ1 q¼1

Pm for 1 < k < N, we have that ¼ 0 and q¼1 ðxk Þp q ffiffiffiffiffiffiffi nm and q¼mþ1 ðxk Þq ¼ 0 so that X ðk; 1; 1Þ ¼ 0. With kN ¼ k1 ¼

Hence, PN

 1 1 1 X ðN; 1; 1Þ ¼ pffiffiffi pffiffiffiffiffi  pffiffiffi m n 2 2  1 1 1 X ð1; 1; 1Þ ¼ pffiffiffi pffiffiffiffiffi þ pffiffiffi m n 2 2

where gð0Þ ¼ 0. The function gðyÞ is monotonous increasing for pffiffi pffiffi  pffiffiffi yÞð1þ yÞ log y and  1 þ y log y > 0 6 y 6 1 because g 0 ðyÞ ¼ 4ð21 pffiffi pffiffi ð1 yÞ2 ð1þ yÞ   pffiffiffi 4 1  y . The latter inequality follows from the expansion  2k1 P 1 1z log z ¼ 2 1 , valid for ReðzÞ P 0, but z – 0 (see [1, k¼1 2k1 1þz pffiffiffi Section 4.1.27]), for z ¼ y. Hence, for 0 6 y 6 1; g 0 ðyÞ P 0 so that gðyÞ P 0 and this proves Lemma 1. h

we obtain from (55)

2 d y1 ðsÞ



2 ds

s"k1

uT xN X ðN; 1; 1Þ Nk1 X ð1; 1; 1Þ k1  kN pffiffiffiffiffi pffiffiffi 2 m n 2 ¼ pffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffi mþ n N mn

¼

References

2

2

which agrees with direct derivation of the closed form (20).

h

Appendix E Proof of Lemma 1. The viral conductance (20) of the bipartite graph K n;m is, with E½D ¼ 2nm N ,

wK n;m ¼

E½D þ Y n;m 2

where

Y n;m

 pffiffiffim pffiffiffiffiffiffiffi 2mn ðn  mÞ 1 þ mn þ log  ¼ mn  pffiffiffin N N 1 þ mn

and we need to show that Y n;m is always positive for n > m and only zero if n ¼ m. First, with

pffiffiffiffiffiffiffi 2mn mn  ¼ N

pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi mn  mn pffiffiffiffiffi pffiffiffi2 m n m þ n  2 mn ¼ N N

we have

NY n;m ¼

 pffiffiffim pffiffiffiffiffiffiffipffiffiffi pffiffiffiffiffi2 1 þ mn mn n  m þ ðn  mÞ log  pffiffiffin 1 þ mn

which shows that Y n;m ¼ 0 for n ¼ m. In addition, symmetry applies so that NY n;m ¼ NY m;n which allows us to further confine to the case that n > m. Next, assuming that n > m, we rewrite

 pffiffiffim rffiffiffiffiffi  n 1 þ mn m m log  ð n  m Þ log 1 þ log  ¼  pffiffiffi n 2 m n 1 þ mn

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