Mathematical Biosciences 253 (2014) 40–49

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Epidemics on a weighted network with tunable degree–degree correlation Fabio Marcellus Lopes ⇑ Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden

a r t i c l e

i n f o

Article history: Received 14 October 2013 Received in revised form 23 March 2014 Accepted 24 March 2014 Available online 8 April 2014 Keywords: Branching processes Configuration model Weighted graph Epidemic threshold Degree–degree correlation

a b s t r a c t We propose a weighted version of the standard configuration model which allows for a tunable degree– degree correlation. A social network is modeled by a weighted graph generated by this model, where the edge weights indicate the intensity or type of contact between the individuals. An inhomogeneous Reed– Frost epidemic model is then defined on the network, where the inhomogeneity refers to different disease transmission probabilities related to the edge weights. By tuning the model we study the impact of different correlation patterns on the network and epidemics therein. Our results suggest that the basic reproduction number R0 of the epidemic increases (decreases) when the degree–degree correlation coefficient q increases (decreases). Furthermore, we show that such effect can be amplified or mitigated depending on the relation between degree and weight distributions as well as the choice of the disease transmission probabilities. In addition, for a more general model allowing additional heterogeneity in the disease transmission probabilities we show that q can have the opposite effect on R0 . Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction Random graphs are often applied to model social networks appearing in large communities and phenomena therein; see [15]. In such modeling vertices correspond to the individuals in the population and the presence of an edge between two vertices represent some type of social interaction between the respective individuals. During the past decades large efforts have been spent on developing models for epidemic spread in this setting; see [9]. These efforts have led to models allowing several types of heterogeneities and network structures aimed at capturing the diversity in real populations and epidemics. A recent development is the inclusion of information concerning the type or the intensity of the contact between individuals, which may be important to determine the risk of disease transmissions. For instance, the transmission of respiratory diseases is frequently associated with household contacts and long duration interactions; see [10] and references therein. The social network is then modeled as an edge-weighted graph, where the weights represent the contact intensities between individuals. In [5], a weighted configuration model is proposed to generate weighted random graphs with prescribed degree and weight distributions. An inhomogeneous Reed–Frost epidemic model

⇑ Tel.: +46 0760680936. E-mail address: fl[email protected] http://dx.doi.org/10.1016/j.mbs.2014.03.013 0025-5564/Ó 2014 Elsevier Inc. All rights reserved.

(see Section 4) is analyzed on that network, where the disease transmission probabilities depend on the edge weights. The work demonstrates that if vertices with large degree tend to have large (small) weights on their edges and the transmission probability increases with the edge weight, then it is easier (harder) for the epidemic to take off compared to a randomized epidemic with the same degree and weight distribution. Furthermore, it also provides good examples of when ignoring weights and the dependencies that may exist among weights, degrees and disease transmission probabilities can lead to over/under-estimation of the epidemic threshold R0 , the so-called basic reproduction number (see definition in Section 4). The degree–degree correlation q of a network is defined as the Pearson correlation coefficient of the degrees associated with an edge selected uniformly at random. There is strong statistical evidence showing that real networks exhibit different correlation patterns. For instance, it is commonly reported that many biological and social networks tend to present positive q while technological networks tend to have negative q; see e.g., [14]. In this work, we propose a simple generalization of the weighted random graph model of [5] which allows explicit control over the degree–degree correlation q, which can be used to analyze the effect of different correlation patterns on network properties and epidemics therein. For instance, by exploring the tuning of q and different parameterizations of the model we are able to analyze such effects. In fact, our model provides a framework to

F.M. Lopes / Mathematical Biosciences 253 (2014) 40–49

generate and study graphs with arbitrary prescribed degree and weight distributions, and degree–degree correlations that may appear in real networks. Next, we give a short summary of our results. We study asymptotic properties, as the number of vertices of the network tends to infinity, for the inhomogeneous Reed–Frost epidemic model on the networks we propose in this work. We derive the epidemic threshold R0 for this epidemic model, and theoretical expressions for the degree–degree correlation coefficient q of our network model. We then analyze through examples the effects of tuning q and of different patterns for the other parameters in the model on the threshold for the epidemic model. Our main conclusion is that, when the degree and weight distributions and epidemic characteristics are kept fixed, weighted graphs with higher degree–degree correlation present a higher epidemic threshold R0 than those with lower values of degree–degree correlation. Furthermore, we propose a directed version of our model in which additional heterogeneities in the disease transmission probabilities are included. Then, we show that for certain parameterizations of this model the tuning of q can even have an opposite effect on the epidemic threshold R0 (see the model in Section 5). This last observation reinforces the importance of understanding the phenomena being modeled. Recently, other studies concerning epidemics on weighted random graphs based on the standard configuration model have appeared. In [7] the model in [5] is extended to incorporate additional heterogeneity by assigning different random infectivity and susceptibility to the individuals, and [8] is concerned with the effects of the dependence between weights and transmission probabilities on the epidemics and different vaccination strategies when degrees and weights are independent. A similar model but using degrees and weights based on real data for generating the graphs was studied in [10], which includes insightful remarks on vaccination strategies. For comparison, we refer to [2] where the effects of clustering and degree–degree correlation on epidemics on unweighted graphs based on the configuration model are studied, and [6] for epidemics and vaccination strategies for the same class of graphs but without clustering and tunable degree–degree correlation. The rest of the paper is organized as follows. In Section 2, we introduce the graph models. In Section 3, we present some asymptotic properties of the network, that is, as the number of vertices tends to infinity. First, in Sections 3.1 and 3.2, the threshold for the existence of a giant component is motivated and derived. And, in the Section 3.3, we present the asymptotic expressions for the degree–degree correlation coefficient q for the graph models. The epidemic model and its threshold R0 for the occurrence of a large epidemic outbreak with positive probability are presented in Section 4, where we also illustrate through examples the effects of tuning the degree–degree correlation q on R0 and its relation to other parameters of the model. In Section 5, we propose and analyze a directed version of our model which allows additional heterogeneity. Finally, Section 6 contains a short discussion. Basic notation: For any integer valued random variable X having probability distribution fpX ðjÞgjP1 , we write lX and r2X for its mean and variance. Let ðXn ; Pn ÞnP1 be a sequence of probability spaces, and suppose that An is a measurable set of Xn for n P 1. We say that ‘‘An occurs with high probability’’ (w.h.p) if Pn ðAn Þ ! 1 as n ! 1. 2. Models In this section, we present two models for weighted random graphs: the weighted configuration model (WCM) introduced in [5] and the tunable weighted configuration model (TWCM). These models generalize the standard configuration model (CM); see e.g.,

41

[12]. In both cases, each half-edge independently receives a certain weight with a degree dependent distribution. To obtain the graph, weighted edges are created by pairing up half-edges with the same weight. And, in the TWCM, each half-edge also receives independently a label which plays a further role in the pairing scheme. The probability distribution of the labellings depends on a certain fixed parameter a 2 ½0; 1 which controls to what extent vertices with ‘‘low’’ degree are connected to vertices with ‘‘high’’ degree, and by varying a we are able to tune the degree–degree correlation of the graphs generated by this model. First, we present the WCM algorithm for generating a weighted random graph with n vertices, prescribed degree distribution fpD ðjÞgjP1 and a family of conditional probabilities for the half-edge w2W weights fpðwjjÞgj2D , where D and W, are respectively the support of the degree and weight distributions. For d 2 D and w 2 W, the conditional probability pðwjdÞ denotes the probability that a halfedge incident to a vertex of degree d is assigned a weight w. We observe that the CM is retrieved if all half-edges receive the same weight in the WCM. Then, we present the TWCM algorithm for generating a weighted random graph with n vertices, prescribed weight and degree distributions as in the WCM, and a fixed parameter a 2 ½0; 1. 2.1. WCM algorithm (I) Generate n degrees, D1 ; . . . ; Dn independently according to the probability distribution fpD ðjÞgjP1 . (II) For i 2 f1; . . . ; ng, associate a vertex in the graph, which we refer to as vertex i, and attach Di half-edges to vertex i. Generate independently Di weights according to the conditional probability distribution fpðwjDi Þgw2W , and assign one weight to each half-edge (a half-edge with weight w 2 W is denoted a w-half-edge). (III) For each available weight w, repeat independently the following until no pair of w-half-edges can be found. Choose two w-half-edges uniformly at random among all available w-half-edges. Connect the half-edges to create an edge with weight w between the corresponding vertices. (IV) At the end of the procedure, ignore any remaining unpaired half-edges. There is at most one unpaired half-edge for each available weight. 2.2. TWCM algorithm (I-II) Repeat Steps (I) and (II) from the WCM algorithm. (III) Fix a 2 ½0; 1. Generate independently for each half-edge a label, 0 or 1, according to a Bernolli random variable with success probability 1  a. (IV) For each available weight w, make an increasing ordering of the w-half-edges with label 0 according to the degrees of the vertices they are incident to. Let Q 10w and Q 20w be the sets of w-half-edges corresponding to the lower and upper half quantiles, respectively in that ordering (note that the cardinality of these sets differ by at most one). Repeat independently the following until no pair of w-half-edges can be found. Choose uniformly at random one w-half-edge from each set, Q 10w and Q 20w . Create an edge with weight w between the vertices that these half-edges are incident to. (V) For each available weight w, make an increasing ordering of the w-half-edges with label 1 as in Step (IV). Let Q 11w and Q 21w be the sets of w-half-edges corresponding to the lower and upper half quantiles, respectively in that ordering. Repeat independently the following until no appropriate pair of w-half-edges can be found. Choose uniformly at random a pair of w-half-edges in Q 11w and another pair in Q 21w .

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Create an edge with weight w between the vertices of the half-edges in the first pair, and another one between the vertices of the second pair. (VI) At the end of the procedure, ignore any remaining unpaired half-edges. There is at most one for each available pair of weight and label. We observe that the WCM and the TWCM algorithms primarily generate multigraphs, that is, self-loops and multiple edges can occur in their constructions. However, the same methods that are used to derive results for simple graphs based on the CM can be adapted to study the weighted models. For instance, for degree distributions with finite variance, self-loops and multiple edges constitute w.h.p. only a negligible proportion of the graph and can be removed without perturbing the asymptotic degree distribution; see [12, Chapter 7] for details. Throughout this paper, we will assume that the degree distribution and all weight distributions have finite support. Under these assumptions, the threshold behavior of the epidemic model derived here can be established rigorously using standard results on multi-type branching processes with a finite number of types e.g.,[11], and adaptations of coupling arguments which have appeared before, see [1,6]. 3. Some asymptotic properties of the networks In this section, we study some asymptotic properties of graphs generated by the WCM and TWCM algorithm, as the number of vertices n tends to infinity. We may refer to this asymptotic analysis simply as being for ‘‘ large graphs’’, and we focus our attention to the TWCM since in Example 3.1 it is shown that for large graphs the TWCM with a ¼ 1=2 is equivalent to the WCM. In Section 3.1, the asymptotic analysis allows us to use a branching process approximation to derive the threshold for the existence (w.h.p.) of a giant component in the graph, that is, a connected component that asymptotically contains a positive fraction of the vertices of the graph. This threshold is obtained in terms of the spectral radius of the mean offspring matrix of a certain branching process, whose computation depends on the probability distribution of the degrees of the neighbors associated with a vertex with a given degree. This probability distribution is made explicit in Section 3.1, and its parameters are computed in Section 3.2. Finally, in Section 3.3, we present the expressions for the degree–degree correlation coefficients for graphs generated by the WCM and TWCM. 3.1. Branching process approximation It is well-known, see e.g., [6,12,13], that a random graph with a fixed degree distribution fpD ðjÞgjP1 generated by the CM contains a giant component w.h.p. if and only if

E½DðD  1Þ

lD

> 1;

ð3:1Þ

This threshold coincides with the one for the survival threshold of a branching process with progeny distribution fpD~ ðj  1ÞgjP1 , where

pD~ ðj  1Þ ¼

jpD ðjÞ

lD

; j 2 D:

ð3:2Þ

Indeed, the threshold results from a branching process approximation of the early stages of an exploration process that dynamically reveals the connected components associated with the vertices in the graph; see e.g., [1,6,12]. The appearance of the (shifted) size-biased probability distribution (3.2) comes from the CM construction, where the probability of picking a half-edge incident to

a vertex with degree j is proportional and close to jpD ðjÞ=lD in the beginning of the procedure. The threshold for weighted graphs generated by the TWCM can be derived in a similar way as the one for graphs generated by the CM. We describe informally the exploration process for the connected component of a vertex chosen uniformly at random in a graph as follows. Consider a graph with n vertices. The vertices in the graph can be in one of the following status: exposed, unexposed and removed. At time zero, all vertices are unexposed except one that is chosen uniformly at random and declared exposed. In the next step, the exposed vertices have their labels changed to removed and all their unexposed neighbors are exposed. The procedure proceeds by revealing at each step the unexposed neighbors of the vertices exposed in the previous step. The algorithm stops when there are no more exposed vertices. The connected component of the vertex first chosen uniformly at random consists of the vertices that have been exposed during the process. Now, we motivate intuitively the branching process approximation for the exploration process. A rigorous presentation follows the arguments in [1,6]. The main difference lies in the fact that here we use a multitype branching process (MTB) in the approximation instead of a single-type branching process. The fact that the WCM and TWCM algorithms create edges by pairing half-edges with the same weights, assigned by a degreedependent distribution, makes the probability distribution of the degrees of the neighbors of a given vertex highly dependent on the degree of the vertex itself. For d 2 D, we refer shortly to a vertex of degree d as a d-vertex. Let pd ðkÞ denote the probability that a given neighbor of a d-vertex is a k-vertex, and note that the fact that the weights are assigned independently for the half-edges of a vertex makes the degrees of its different neighbors independent for large graphs. Let us consider the early stages of the exploration process for large graphs. The degree of the first exposed vertex is distributed according to fpD ðjÞgjP1 . If it is a d-vertex, then w.h.p. d new vertices are exposed in the next step of the algorithm. Each one of these d vertices is a k-vertex independently with probability pd ðkÞ. In the next step, an exposed k-vertex from the previous step can add at most k  1 new exposed vertices, since at least one of its neighbors has already been removed in the previous step. Furthermore, due to the absence w.h.p. of short cycles, self-loops and multiple edges in the graph, each k-vertex will add w.h.p. precisely k  1 new exposed vertices. The degrees of these vertices are approximately distributed according to fpk ðjÞgj2D . As long as only a negligible fraction of vertices have been exposed, the process will evolve similarly since the degrees of the successively exposed vertices will be still well approximated by fpd ðkÞgd;k2D . This process resembles a MTB whose ancestor has its type d given according to fpD ðjÞgjP1 and progeny distributed as a multinomial with parameters d and fpd ðkÞgk2D . The second and subsequent generations have the mean offspring matrix H ¼ fðd  1Þpd ðkÞgd;k2D (in fact, the offspring of an individual of type d in these generations is distributed as a multinomial with parameters d  1 and fpd ðkÞgk2D ). Note that the types in the MTB refer to the degrees. Here, we assume that the support D of fpD ðjÞgjP1 is finite, therefore, its analysis rely on standard methods for MTB with finite number of types. Furthermore, we note that individuals (vertices) of type 1 do not contribute to the growth of the MTB (connected component) since they cannot have children, therefore, we can restrict the analysis of the survival of this MTB to the submatrix H0 ¼ fðd  1Þpd ðkÞgd;kP2 . From classical branching processes theory, e.g., [11], we have that, when the mean offspring matrix H0 is non-singular and positive regular, then the MTB described above survive with strictly positive probability if and only if the spectral radius of H0 is strictly greater than 1. The methods in [1,6] can be adapted to show that the threshold for the existence of a giant component coincides with

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F.M. Lopes / Mathematical Biosciences 253 (2014) 40–49

the one for the survival of the MTB. The positive regular condition of H0 means that for some power of it all entries in the resulting matrix are strictly positive. This can be translated in terms of branching processes as that for any pair of possible degrees, d and k, an individual of type d can with strictly positive probability have a descendant of type k after a finite number of generations. Note that H is not positive regular, since individuals of type 1 do not produce children. Next in Section 3.2, we show how pd ðkÞ can be calculated for graphs generated by the TWCM. We will return to the branching processes approximation in Section 4, where we discuss the threshold for a major epidemic outbreak in a graph generated by the TWCM. 3.2. Neighbor degrees for the TWCM We now derive the probability pd ðkÞ that a given neighbor of dvertex is a k-vertex. In the TWCM construction, we need to consider the events that a w-half-edge with label x belongs to the lower or upper half-quantile, respectively associated with the ordering of all w-half-edges with label x. These half-quantiles are denoted 1 and 2, respectively, and the letter q will be used to refer generically to a quantile associated with a w-half-edge. In particular, we have to compute the asymptotic probability that a w-half-edge with label x is incident to a d-vertex and that it belongs to quantile q in the TWCM construction. We denote this probability by f ðd; qjwÞ. The label x does not affect the probability since it is attributed independently of all half-edges characteristics, and it is therefore omitted from the notation. Let rðijwÞ denote the asymptotic probability that a w-half-edge picked uniformly at random among all w-halfedges is incident to an i-vertex, similarly to (3.2) we obtain that

ipðwjiÞpD ðiÞ rðijwÞ ¼ P : j jpðwjjÞpD ðjÞ

(





a1fx¼0g þ ð1  aÞ1fx¼1g pðwjdÞf ðqjw; dÞrðkjw; qC 1fx¼0g þ q1fx¼1g Þ:

Summing over all possible weights, labels and quantiles we obtain that

pd ðkÞ ¼

X X   pðwjdÞ f ðqjw; dÞ arðkjw; qC Þ þ ð1  aÞrðkjw; qÞ : w

q

ð3:4Þ For the standard CM, pd ðkÞ is given by the (non-shifted) size-biased degree distribution kpD ðkÞ=lD . The expression (3.4) is more complicated since we have to take in account all the possible weights, labels and quantiles the half-edge incident to a d-vertex can assume, and these characteristics will determine from which quantile the partner of such half-edge will be picked uniformly at random. Example 3.1. In this example, we show that, when a ¼ 1=2, our expression (3.4) for pd ðkÞ reduces to the corresponding expression for the WCM, thus, confirming that the models are asymptotically equivalent. We compute

pd ðkÞ ¼

X   1X pðwjdÞ f ðqjw; dÞ rðkjw; qC Þ þ rðkjw; qÞ 2 w q

X   1X pðwjdÞ f ðqjw; dÞ 2rðk; qC jwÞ þ 2rðk; qjwÞ 2 w q X X X ¼ pðwjdÞrðkjwÞ f ðqjw; dÞ ¼ pðwjdÞrðkjwÞ ¼

w

q

w

X kpðwjkÞpD ðkÞ pðwjdÞ X : ¼ jpðwjjÞpD ðjÞ w j

As n ! 1, the proportion of w-half-edges incident to i-vertices among all w-half-edges converges to rðijwÞ for each i 2 D. Therefore, when ordering all w-half-edges by their degrees, the asymptotic proportion of w-half-edges incident to d-vertices in the half-quantile q converges to the part of the probability mass between P P i6d rðijwÞ and i6d1 rðijwÞ which lies in that half-quantile. Hence, as n ! 1,

f ðd;qjwÞ ¼ max min

incident to a k-vertex in the appropriate quantile qC 1fx¼0g þ q1fx¼1g , where qC denotes the half-quantile opposite to q. Hence, it follows that pd ðk; w; x; qÞ is equal to

( ) ( ) ) X X q q1  max ;0 : rðijwÞ; rðijwÞ; 2 2 i6d i6d1 ð3:3Þ

Note that (3.3) is also valid for the asymptotic probability rðd; qjwÞ that a w-half-edge picked uniformly at random among all half-edges with weight w is incident to a d-vertex and belongs to the half-quantile q. From the fact that a half-quantile contains half of the half-edges as n ! 1, we obtain that the asymptotic probability rðdjq; wÞ that a w-half-edge picked uniformly at random among all w-half-edges in the half-quantile q is incident to a d-vertex is equal to 2rðd; qjwÞ. Also, since the probability that a w-half-edge incident to a d-vertex belongs to the half-quantile q is the ratio between the number of such half-edges in q and the total number of such half-edges, we obtain that it is asymptotically equal to f ðd; qjwÞ=rðdjwÞ, which we denote by f ðqjd; wÞ. Write pd ðk; w; x; qÞ for the asymptotic probability that a given neighbor of a d-vertex is a k-vertex and that the edge connecting them has weight w, label x 2 f0; 1g and the half-edge incident to the d-vertex used to create the edge belongs to q 2 f1; 2g. In fact, the probability of the latter events is simply the asymptotic probability that a given half-edge incident to a d-vertex is assigned weight w, label x, is in half-quantile q and is paired to a w-half-edge

In the second line, we have used the fact that rðkjq; wÞ equals to 2rðk; qjwÞ, and noticed that summing the two terms we obtain P rðkjwÞ. Finally, we obtain the fourth line since q f ðqjw; dÞ sums to one. Next, let us calculate pd ðkÞ under the WCM. First, suppose that the half-edge incident to the d-vertex is a w-half-edge. The probability that it is paired to a w-half-edge incident to a k-vertex is then proportional to the fraction of such half-edges among all the w-half-edges, and this fraction is asymptotically equal to P kpðwjkÞpD ðkÞ= j jpðwjjÞpD ðjÞ. Summing over all possible weights of the half-edge incident to the d-vertex we obtain the expression in the last equality above. Note also that when weights and degrees are independent, then pd ðkÞ for the TWCM with a ¼ 1=2 coincides with the (non-shifted) size-biased degree distribution as in the standard configuration model. 3.3. Degree-degree correlation coefficients In this section we present asymptotic formulas for the degree– degree correlation coefficient for the WCM and TWCM. The degree–degree correlation coefficient is defined in [14] as the Pearson correlation coefficient between the degrees of the two vertices incident to an edge chosen uniformly at random from a network. The derivation of the formulas are deferred to the Appendix A. Let mðwÞ denote the asymptotic probability that an edge picked uniformly at random from a weighted graph has weight w, that is, the edge was created by joining two w-half-edges. Since this probability is proportional to the fraction of w-half-edges among all half-edges, we have that

P mðwÞ ¼

j jpðwjjÞpD ðjÞ

P

i ipD ðiÞ

:

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F.M. Lopes / Mathematical Biosciences 253 (2014) 40–49

Table 1 Probability distribution of the weights in cases (ii) and (iii). Degrees

Case (ii)

Case (iii)

1 : 10 11 : 20 21 : 30

Binomial (5,0.1) Binomial (5,0.7) Binomial (5,0.9)

Binomial (5,0.1) Binomial (5,0.2) Binomial (5,0.9)

The degree–degree correlation coefficient qWCM for the WCM is given by

lw  lD~ Þ2 mðwÞ ; r2D~

ð3:5Þ

where lw denotes the expected degree of a w-half-edge picked uniformly at random. Clearly, qWCM is non-negative. We can see that qWCM depends on how much lw deviates from lD~ for each w 2 W. For instance, we have that weight patterns with high absolute value for the degree–weight correlation generate graphs with higher qWCM ; see Example 3.3. The degree–degree correlation coefficient qTWCM for the TWCM with parameter a 2 ½0; 1 is given by

qTWCM ¼ qWCM þ

ð1  2aÞD

r2D~

ð3:6Þ

;

P where D ¼ w 14 ðl2;w  l1;w Þ2 mðwÞ P 0 and lq;w denotes the expected degree of a w-half-edge picked uniformly at random from the quantile q 2 f1; 2g. Here, we can see that the effect of the tuning in qTWCM depends on the value of D, which increases with the absolute values of the differences between l1;w and l2;w for each w 2 W. From (3.6) it is clear that qTWCM can be tuned monotonically by varying the parameter a 2 ½0; 1. Next, in Example 3.2, we illustrate that the effect of the tuning on the threshold parameter is nonincreasing in a. Therefore, by formula (3.6) the threshold parameter is non-decreasing in q.

4. Epidemic model In this section, we define an inhomogeneous Reed–Frost epidemic model and derive some expressions for it when the

threshold parameter

12 11

10

10

threshold parameter

13

14

14

Example 3.2. This example gives some support for the intuition that graphs with higher degree–degree correlation have a higher threshold parameter for the existence of giant component. Indeed, the vertices with high degree would then form a connected component themselves containing asymptotically a positive fraction of the vertices in the graph. Although we have not been able to demonstrate this analytically, we believe that this is the case for graphs generated by the TWCM. To illustrate this we have considered weighted graphs with Poisson distributed degrees with

Example 3.3. In this example, we give some evidence indicating that large absolute values of the degree–weight correlation do not appear together with small values of q. Suppose that the degrees are binomially distributed with parameters 10 and 0.5 and conditioned on being in [1:10]. We work with a two-point family of weight distributions, say w1 ¼ 1 and w2 ¼ 2, and have simulated 1500 realizations for which pðw1 jdÞ ¼ U d ¼ 1  pðw2 jdÞ where the U d ’s are i.i.d. uniform random variables in [0,1]. In Fig. 2, we plot q against weight–degree correlation for several values of a. For a values far from zero, it can be noticed that the scattered points tend to form a pattern similar to a smile suggesting upper and lower boundaries determining the possible combinations of weight–degree and degree–degree correlations. Note that the smallest bandwidth for the upper and lower boundary occurs at a equal to 0.5. We have also performed simulations allowing a larger number of weights by taking the conditional distributions for weights given a certain degree as the interval lengths generated by K  1 independent and uniformly distributed points in ½0; 1. We have tested values of K up to 10 (samples with 40,000 points). The evidence becomes weaker for larger values of K, but for all tested values the suggested lower boundary was to some extent present and the case with a equal to 0.5 presented the best evidence.

13

w2W ð

12

P

11

qWCM ¼

mean 10 conditioned on being in [1:30], and with three different rules for assigning weights to the half-edges, referred as cases (i), (ii), and (iii). In case (i), all half-edges have the same weight. For the other two cases, the half-edges were independently assigned weights from 1 to 5 according to the Binomial distributions conditioned on being in [1,5], see Table 1. We can see in Fig. 1 that the effect of a on the threshold parameter is monotone and non-increasing. And, using (3.6) we can see that the threshold parameter is non-decreasing in q (correlation degree–degree in Fig. 1). Furthermore, we note that different patterns in the weight distributions can change the level of the threshold and q. For instance, the weight pattern in case (iii) yields the highest threshold parameter values since most of the half-edges with high weights are incident to vertices of high degree and therefore such vertices have more chances to be matched among themselves than in cases (i) and (ii).

0.0

0.2

0.4

0.6

α

0.8

1.0

−0.6 −0.4

−0.2

0.0

0.2

0.4

0.6

0.8

corr. degree−degree

Fig. 1. Example 3.2. The leftmost plot shows the effect of tuning a on the threshold parameter for the three degree distributions in Table 1: (i) solid line, (ii) dashed line, and (iii) dotted line. The rightmost plot presents the same information but using (3.6) to express the tuning in terms of the degree–degree correlation q.

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F.M. Lopes / Mathematical Biosciences 253 (2014) 40–49

0.0

0.2

0.4

−0.6

Corr. degree−weight

−0.4

−0.2

0.0

0.45 0.35 0.25

0.6

−0.6

−0.4

−0.2

0.2

0.4

0.6

Corr. degree−weight

Corr. degree−degree

0.2 0.0

−0.4

−0.2

0.0

0.2

0.4

0.6

0.4

0.6

α=1

−0.2 −0.6

0.0

Corr. degree−weight

−0.4

Corr. degree−degree

0.3 0.2

Corr. degree−degree

0.1

−0.2

0.4

α = 0.75

0.0

−0.4

0.2

Corr. degree−weight

α = 0.5

−0.6

0.0

0.15

Corr. degree−degree

0.55 0.45 0.35

0.6

0.2

0.4

0.6

0.0

−0.2

−0.4

−0.4

α = 0.33

−0.8

−0.6

α = 0.25

0.25

Corr. degree−degree

0.8 0.7 0.6 0.5

Corr. degree−degree

α=0

−0.6

Corr. degree−weight

−0.4

−0.2

0.0

0.2

Corr. degree−weight

Fig. 2. Example 3.3. For a 2 f0; 0:25; 0:33; 0:5; 0:75; 1g each plot has scattered 1500 points. The coordinates of each point represent the weight–degree correlation and degree–degree correlation q for a simulated family of weight distributions introduced in Example 3.3.

epidemics takes place in a large graph (the number of vertices tends to infinity) generated by the TWCM. The inhomogeneity in the model refers to the fact that the disease transmission probability between two neighboring vertices in the network depends on the weight of the edge between them, which can represent the intensity or type of contact between these individuals in the population. 4.1. An inhomogeneous Reed–Frost epidemic model Suppose that the n vertices of our graph represent the individuals in a closed population. At time zero all vertices are susceptible. Then at time t ¼ 1, we a start an epidemic by choosing a vertex uniformly at random and declaring it to be infected. The infection spreads until there are no more infectives. At time t (t ¼ 1; 2; . . .) all infectious vertices infect each one of its susceptible neighbors independently with a probability that depends on the weight of the edge connecting them, that is, if the edge has weight w, then the infection probability is pðwÞ 2 ½0; 1. At time t þ 1, all vertices that were infectious at time t become immune and play no further role in the epidemic spread, while the susceptibles who were infected at time t make up the infectious vertices at t þ 1. The infectious vertices at time t are referred to as generation t in the epidemic. Note that the process is very similar to the exploration process in Section 3, if susceptible, infective and immune, are respectively replaced by unexposed, exposed and removed. A large outbreak in the epidemic is said to occur if a strictly positive proportion of the population is infected asymptotically. The basic reproduction number, R0 , is a function of the parameters of the model that determines whether a large outbreak is possible or not. Loosely speaking, R0 represents the average number of new infectives that an average infectious individual in the beginning of the epidemic generates. As usual for epidemic models, a large outbreak is possible if and only if R0 > 1, see e.g., [4]. We motivate this below. First, we note that the epidemic model can be seen as a thinning of the original weighted graph, where edges with weight w are removed independently with probability 1  pðwÞ, for w 2 W. That is, the thinned graph consist of those edges for which the disease

will be transmitted in case one of its endpoints gets infected. Therefore, the epidemic threshold can be derived as the threshold for the existence of a giant component in a thinned weighted graph as seen in Section 3. For d; k 2 D, we need to calculate, for the thinned graph, the probability that a given neighbor of a d-vertex is a k-vertex. To distinguish it from the expression (3.4), we denote this probability by pId ðkÞ. Once we have the set of probabilities fpId ðkÞgd;k2D ; R0 is computed as in Section 3 as the spectral radius of the offspring matrix H0 ¼ fðd  1ÞpId ðkÞgd;kP2 of the MTB which approximates the exploration of vertices with degree greater or equal to 2 in the early stages of the process. Therefore, we have a strictly positive probability of a large outbreak if and only if R0 > 1. The expression for pId ðkÞ is obtained as the one for pd ðkÞ in (3.4), but when summing over all possible weights for the edge, we multiply each term in the sum by pðwÞ, which is the probability that the edge created still belongs to the thinned graph. Hence, we have that

pId ðkÞ ¼

X

X

w

q

pðwÞpðwjdÞ

  f ðqjw; dÞ arðkjw; qC Þ þ ð1  aÞrðkjw; qÞ : ð4:1Þ

We also observe that it also follows from the branching process approximation that the asymptotic probability of a large outbreak of the epidemic starting with an infected chosen uniformly at random in the population coincides with the survival probability of the following MTB. Let B be a MTB whose ancestor is given type d according to fpD ðdÞgdP1 in which case its offspring are the first jDj-coordinates of a ðjDj þ 1Þ dimensional random vector fId ð1Þ; . . . ; Id jðDjÞ; Zg distributed according to a multinomial with   P parameters d and pId ð1Þ; . . . ; pId ðjDjÞ; 1  k pId ðkÞ . Here Id ðiÞ denotes the number of neighbors of type i 2 D which were infected by the infectious d-vertex. The d-individuals in the subsequent generations have a similar offspring distribution as the ancestor, but with parameter d  1 instead of d in the multinomial distribution. The computation of the survival probability for such MTB is standard, see e.g., [11], and is described also in [5, Section 5], where the same inhomogeneous epidemic model is studied for graphs generated by the WCM. We have not extended our analysis of

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F.M. Lopes / Mathematical Biosciences 253 (2014) 40–49

the effects of tuning q to the probability of a large outbreak since it has been shown, through examples in [2], that such effect may not be monotone in q, in contrast to the effect on R0 . The outbreak probability seems to depend in a more complicated way on characteristics of the graphs and the epidemic. Example 4.1. Here, we illustrate the effect on R0 of tuning a (and thus q) when the degree distribution is Binomial (20,0.1) conditioned on being strictly positive and also for a heavy-tailed degree 1:5 distribution pD ðdÞ / d for d 2 ½5; 30. We considered two-point weight distributions with weights w1 and w2 . For each case we simulated 20 realizations of the weight distribution as

pðw1 jdÞ ¼ U d ¼ 1  pðw2 jdÞ; where each U d is uniform on ½0; 1. For the Binomial case we set

pðw1 Þ ¼ 0:5 and pðw2 Þ ¼ 0:9, and for the heavy-tailed case we set pðw1 Þ ¼ 0:01 and pðw2 Þ ¼ 0:1. Fig. 3 reveals that in all cases R0 decreases monotonically as a increases. An important quantity for epidemics on weighted graphs is the so-called expected total infection pressure of a vertex, which is quantified for a d-vertex by

tðdÞ :¼ ðd  1Þ

X pðwjdÞpðwÞ ¼ ðd  1ÞEðpðWÞjD ¼ dÞ; w

where W denote the random variable associated with the weight. That is, tðdÞ is the expected number of neighbors that a d-vertex infects during the early stages of the epidemic. This determines the role of each type of vertex in the epidemics. Our intuition behind the increasing effect of q on R0 in our model is that when a decreases (so that q increases), a vertex with high expected total infection pressure has greater chances of being matched to vertices of the same type.

5. Inhomogeneous Reed–Frost epidemic model with directed edges In this section, we make a modification in the epidemic model to allow further heterogeneity in the infection transmission probabilities that may be useful to understand real epidemics. For this we propose a directed version of the epidemic model intended to capture the following additional features. The disease transmission probabilities may depend on if the contact is between people of similar or different network features, and on whom is the infectious individual. The motivation for such directed networks and epidemics comes e.g., from the modeling of the flow of information on certain online social networks, where there may exist asymmetries in the communication. For example, users with many contacts (e.g., public personalities) typically do not interact evenly with their contacts, for instance they often interact less frequently with users with few contacts to whom they are not closely related in real life. Thus, it is natural to consider models where information flows with high probability from an individual with many contacts towards one of its neighbors with few contacts, but not vice versa. Suppose G is a weighted graph generated using the TWCM. Let G! be its directed version obtained by turning each weighted edge fa; bg in G into two directed weighted edges fa ! bg and fb ! ag both with the same edge weight as fa; bg. We consider an inhomogeneous Reed–Frost epidemic model on that graph with the dynamics defined in Section 3 but now the infection probabilities between two neighboring vertices may depend on further characteristics than the weights of the half-edges used to generate the edges in G. For any pair a and b of vertices forming an edge in G, suppose that at time t 2 f0; 1; . . .g the vertex a is infectious and b is susceptible. Then a tries to infect b through the directed edge fa ! bg and succeeds with probability pðfa ! bgÞ 2 ½0; 1 which is a function of the weight, label and half quantile of the half-edge

1.4 0.6

0.8

1.0

1.0

1.2

1.5

R0

R0

1.6

2.0

1.8

2.0

2.5

Example 4.2. In this example, we explore the dependence between weights and degrees. Consider the TWCM with i.i.d. degrees distributed as a Poisson random variable with mean 4 conditioned on being in ½1; 200, and a two-point weight distribution with weights w1 and w2 (w1 < w2 ). For each d 2 D, take s pðw2 jdÞ ¼ d (and pðw2 jdÞ ¼ 1  pðw1 jdÞ) for s > 0, that is, the probability that an edge incident to a d-vertex has the larger weight w2 decays with d at rate s. Set pðw1 Þ ¼ 0:1 and pðw2 Þ ¼ 0:7. Fig. 4 shows a plot of R0 against s for a equal to 1 (dashed line), 0.5 (solid line) and 0 (dotted line). A similar example is considered in [5, Example 3.3], and since the TWCM with a ¼ 0:5 is equivalent to

the WCM our example is an extension of that one. For the three values of a in Fig. 4, it can be seen that R0 decreases with the degree–weight correlation when we vary s. This is explained by the fact that individuals with higher degrees tend to have lower weights which make their contacts less infectious as s increases. s Fig. 5 shows a similar example, but now pðw2 jdÞ ¼ 1  d , that is, we have a positive degree–weight correlation that increases with s. Consequently, the vertices with higher degrees tend to have higher weights which make R0 increase with s.

0.0

0.2

0.4

0.6

α

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

α

Fig. 3. Example 4.1. The effect on R0 of the tuning a. In each plot we present 20 different weight distributions. The leftmost plot has degree distribution Binomial (20,0.1), while the rightmost has a heavy-tailed degree distribution.

47

−0.10 −0.15 −0.20 −0.30

−0.25

1.5 0.0

0.5

1.0

R0

2.0

correlation degree−weight

2.5

−0.05

F.M. Lopes / Mathematical Biosciences 253 (2014) 40–49

0

1

2

3

0

1

τ

2

3

τ s

0.25 0.20 0.15

0

0.05

0.10

2 1

R0

3

correlation degree−weight

0.30

Fig. 4. Example 4.2: Basic reproduction numbers and degree–weight correlations are plotted against the parameter s, with pðw2 jdÞ ¼ d ; s > 0. The solid line represents the TWCM with a ¼ 0:5 (equivalent to the WCM), the dashed line a ¼ 1 and the dotted line a ¼ 0.

0

1

2

3

0

1

τ

2

3

τ s

Fig. 5. Example 4.2: Similar to Fig. 1, but now with pðw2 jdÞ ¼ 1  d ; s > 0. The solid line represents TWCM with a ¼ 0:5, the dashed line a ¼ 1 and the dotted line a ¼ 0.

incident to a used to create the edge fa; bg in G. If fa ! bg has weight w, and the half-edge incident to a used to create the edge fa; bg in G has label x 2 f0; 1g and is in the half quantile q 2 f1; 2g the transmission probability is defined as

pðfa ! bgÞ ¼ p1 ðwÞ1fx¼0;q¼1g þ p2 ðwÞ1fx¼0;q¼2g þ p3 ðwÞ1fx¼1;q¼1g þ p4 ðwÞ1fx¼1;q¼2g ;

ð5:1Þ

where p1 ðwÞ; p2 ðwÞ; p3 ðwÞ; p4 ðwÞ 2 ½0; 1 for all w 2 W. Note that (5.1) allow us to differentiate the transmissions using edges formed by half-edges in the same half-quantile to those with opposite half-quantiles and whether the disease flows from the lower half-quantile to the upper half-quantile or vice versa. These characteristics capture to some extent the features we aimed to include in our model since the half-edges in the upper half-quantiles are usually incident to vertices with larger degrees. For instance, by letting p1 ðwÞ be much smaller than p2 ðwÞ for all w, we facilitate the propagation of the disease from the vertices with high degrees to those with small degrees. Following the steps leading to (3.4) we obtain the following directed version of pId ðkÞ for the infectious d-vertices in the second generation and onwards,

!

pId ðkÞ ¼

X pðwjdÞfa½p1 ðwÞf ð1jw; dÞrðkjw; 2Þ w

þ p2 ðwÞf ð2jw; dÞrðkjw; 1Þ þ ð1  aÞ  ½p3 ðwÞf ð1jw; dÞrðkjw; 1Þ þ p4 ðwÞf ð2jw; dÞrðkjw; 2Þg; where we have used the short notation introduced in Section 3. We arrange these probabilities in a matrix H! ¼ fðd  1Þ ! pId ðkÞ gd;kP2 , and by the same argument as in Section 3 we have that the basic reproduction number R0 of the model is given by the spectral radius of H! . Example 5.1. In all the examples we have considered in the previous section, R0 is non-decreasing in q (non-increasing in a). Although we have not been able to prove this result, we believe that this holds for the epidemic model studied in Section 4; see Example 4.1. Here, we show that in the directed epidemic model, it is easy to give an example where R0 is non-increasing in q (nondecreasing in a), which illustrates that additional heterogeneity in the epidemic model may yield a different effect of the degree– degree correlation on R0 . The reason for this is that the infection probabilities in the directed model depend not only on the weights but also on the labels and half-quantiles of the half-edges. Therefore, by considering a set of infection probabilities with

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F.M. Lopes / Mathematical Biosciences 253 (2014) 40–49

sufficiently ‘‘low’’ infectivity for the edges that are constructed using half-edges which belong to the same half-quantiles (halfedges with label 1) and sufficiently‘‘high’’ infectivity for the edges that are constructed using half-egdes in opposite half-quantiles (half-edges with label 0). And, since the proportion of edges created using half-edges with label 1 is approximatelly 1  a, if we increase a the proportion of these edges with ‘‘low’’ infectivity decreases, therefore, we would expect R0 to increase. This can be verified through the following example. Let the degrees be distributed as Poisson random variables with parameter 8 conditioned on being in ½1; 50, and assign weights to the half-edges according to the following family of two-point distributions: 0:5

pðw2 jdÞ ¼ d

¼ 1  pðw1 jdÞ; d 2 D;

w2 > w1 :

Then let the disease transmission probabilities in (5.1) be given by

p1 ðw1 Þ ¼ p2 ðw1 Þ ¼ 0:5; p1 ðw2 Þ ¼ p2 ðw2 Þ ¼ 0:8; p3 ðw1 Þ ¼ p4 ðw1 Þ ¼ 0:15 and p3 ðw2 Þ ¼ p4 ðw2 Þ ¼ 0:3. In Fig. 6 we can see that R0 increases when the degree–degree correlation coefficient q is reduced by increasing the value of a.

6. Discussion

R0

3.5

4.0

4.5

In this work we have defined a weighted network with tunable degree–degree correlation q and studied an inhomogeneous epidemic on it, where the inhomogeneity refers to the fact that the infection probabilities are affected by the edge weights. The model is a generalization of the weighted configuration model in [5]. We note that in both models the weights have an active role in the construction of the network which by itself seems to induce some degree–degree correlation in the network. We have focused our analysis on how q affects the basic reproduction number R0 , which determines whether or not a large epidemic outbreak is possible. Our analysis suggests that, when the degree and weight distributions and epidemic characteristics are kept fixed, weighted graphs with higher degree–degree correlation present a higher basic reproduction number than those with lower values of degree– degree correlation. And, that this effect can be amplified or reduced depending on the relation between weight and degree distribution as well as on the choice of the disease transmission probabilities. By following the recipe in [5], the probability of a large outbreak can be computed, and coincides with the final proportion of infected individuals; see [2,5]. However, the effect on such quantities of tuning q is more complicated and presumably not necessarily monotone as the effect on R0 .

We have also analyzed a directed version of our model which allows additional heterogeneity in the disease transmission probabilities. Depending on its parametrization this model can either emphasize or reduce the effect of the tuning in the original model. The directed model may be realistic for modeling e.g., computer networks where directed edges are natural and the flow of information between the nodes may depend on their network characteristics. As for future work, it would be desirable to prove analytically in some generality that R0 is non-decreasing in q for the undirected model. Another relevant extension would be to incorporate clustering in the analysis since this feature appears in many real networks. We believe that this can be done by adapting ideas that have appeared before in the study of clustering in the unweighted context e.g., [2,3]. This extension is interesting because contrary to what has been observed for unweighted graphs, we believe that in the weighted case clustering can increase R0 and the final size of the epidemics. For instance, if the transmission probabilities associated to ‘‘clustered edges’’ (e.g., inside the households) are much larger than those associated with ‘‘global contacts’’. A third extension that may be interesting is the study of vaccination strategies for weighted models, and a subsequent analysis of the effect of the degree–degree correlation. Here, it is interesting to note that networks with negative degree–degree correlation coefficient tend to be more resistant to random and targeted attacks than those with positive degree–degree correlation; see [14]. Hence, we believe that q may have an impact on the efficiency of vaccination strategies; see [6,8]. Appendix A Here, we derive asymptotic formulas for the degree–degree correlation coefficients in Section 3.3. As mentioned before the degree–degree correlation coefficient associated to a network is the correlation coefficient between the degrees of the vertices incident to an edge chosen uniformly at random. Suppose the edge ~ a and ða; bÞ is picked uniformly at random from the network, let D ~ Db represent the degrees associated with the two endpoints of the edge and denote by W its edge weight. We begin by deriving (3.5). To this end, first note that

h i  h i h i ~ a; D ~ b Þ ¼ E cov ðD ~ a; D ~ b jWÞ þ cov E D ~ a jW ; E D ~ b jW ; cov ðD

where cov ðX; YÞ denotes the covariance between the random vari~a ables X and Y. Also, observe that given W the random variables D ~ b are asymptotically independent, implying that and D ~ a; D ~ b jWÞ ¼ 0. We obtain (3.5) by dividing the remaining part cov ðD ~ and using the in (6.1) by the variance r2D~ of the random variable D, ~ a and D ~ b both follow the size-biased degree distribution. fact that D In order to derive (3.6) we introduce some notation. Again, we ~ a and X ~ b be the labels (0 or 1) of consider the edge ða; bÞ. Let X the half-edges incident to the vertices a and b, respectively, and similarly, let Q a (or Q b ) denote the half quantile associated with the half-edge incident to a (or b) in the ordering of the half-edges. We start by noting that

2.0

2.5

3.0

h i ~ a; D ~ b Þ ¼ E cov ðD ~ a; D ~ b jX ~ a ; Q a ; WÞ cov ðD  h i h i ~ a jX ~ a; Q a; W ; E D ~ b jX ~ a; Q a; W : þ cov E D

0.0

0.2

0.4

0.6

0.8

1.0

α Fig. 6. See Example 5.1. In the model allowing more heterogeneity in the disease transmission probabilities we can obtain examples where R0 increases when the degree–degree correlation q is reduced by increasing the value of a.

ð6:1Þ

ð6:2Þ

~ a and D ~ b are indepenAgain, by letting n ! 1, we have that D ~ ~ a and Q a dent given ðX a ; Q a ; WÞ. Note that when the values of X are given, then Q is also known. Hence, we have that b i h ~ a; D ~ b jX ~ a ; Q a ; WÞ ¼ 0, and thus E cov ðD

qTWCM ¼

~ a; D ~ bÞ cov ðD

r2D~

¼

 h i h i ~ a jX ~a; Q a; W ; E D ~ b jX ~ a; Q a; W cov E D

r2D~

:

49

F.M. Lopes / Mathematical Biosciences 253 (2014) 40–49

Let lax;q;w ðaÞ and lax;q;w ðbÞ denote the expected degree of the endpoints a and b when the half-edge incident to vertex a used to form the edge ða; bÞ has label x, weight w and is in the half-quantile q, respectively. Thus using the simplified notation from Section 3 we have that qTWCM can be written as

E

h

i

h

i h

i

lax;q;w ðaÞlax;q;w ðbÞ  E lax;q;w ðaÞ E lax;q;w ðbÞ r2D~ PP P P a 2 ~ ~ k q w x klx;q;w ðbÞpðk; x; q; wÞ  lD ¼ ; 2 rD~



1 2

And, combining these facts we obtain that qTWCM can be written as w

q

x



w

P





l1;w l2;w mðwÞ þ ð1  aÞ w 12 l21;w þ l22;w mðwÞ  l2D~ : r2D~

References

a1fx¼0g þ ð1  aÞ1fx¼1g rðkjq; wÞ mðwÞ:

P P P

P

a

lax;q;w ðbÞ ¼ lq;w . We arrive at

This can be rewritten as (3.6).

~ðk; x; q; wÞ denotes the probability that the edge chosen where p uniformly at random is incident to a k-vertex, and the half-edge incident to the k-vertex used to create such edge has weight w, label x and belongs to the half-quantile q. Recall that each half-edge receives independently the label 0 or 1 with probabilities a and 1  a, respectively, and that the assignment is also independent of the weights, degrees and quantiles. Also, we remind the fact that the probability of choosing a w-half-edge in the half-quantile q con~ðk; x; q; wÞ using the verges to 12 as n ! 1. Hence, we can rewrite p notation from Sections 3 and 4 as



~ a ¼ 1 and Q a ¼ q, then we have that X the following expression for qTWCM :



lq;w lax;q;w ðbÞ a1fx¼0g þ ð1  aÞ1fx¼1g 12 mðwÞ  l2D~ ; r2D~

where lq;w denotes the expected degree of a w-half-edge picked uniformly at random in quantile q (Section 3.3). Recall that when ~ a ¼ 0 then Q a and Q b are different, and when X ~ a ¼ 1 they are equal. X ~ a ¼ 0 and Q a ¼ q, then la ðbÞ ¼ l C . Similarly, if Note that if X q ;w x;q;w

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