Abstract

We prove that the presence of a diagonal assortative degree correlation, even if small, has the effect of dramatically lowering the epidemic threshold of large scale-free networks. The correlation matrix considered is , where is uncorrelated and (the Newman assortativity coefficient) can be very small. The effect is uniform in the scale exponent if the network size is measured by the largest degree . We also prove that it is possible to construct, via the Porto–Weber method, correlation matrices which have the same as the above, but very different elements and spectra, and thus lead to different epidemic diffusion and threshold. Moreover, we study a subset of the admissible transformations of the form with depending on a parameter which leaves invariant. Such transformations affect in general the epidemic threshold. We find, however, that this does not happen when they act between networks with constant , i.e., networks in which the average neighbor degree is independent from the degree itself (a wider class than that of strictly uncorrelated networks).

1. Introduction

From the mathematical point of view, a network is completely characterized (up to isomorphisms corresponding to simple redenominations of the vertices) when a list of links or an adjacency matrix is given. In many applications involving large networks, however, one often summarizes the information on the network structure in a statistic-probabilistic form, by introducing the two fundamental quantities and . , called degree distribution, represents the probability that a randomly chosen vertex of the network has neighbors. , called degree correlation function, expresses the conditional probability that a vertex of degree is connected to a vertex of degree . An alternative but equivalent description involves the symmetric quantities , defined as the probabilities that a randomly chosen link connects two vertices of degree and [13].

When one considers in a purely axiomatic way a class of networks, called Markovian networks [4], which are completely defined by assigning the quantities and , one disregards higher-order correlations like , which can in general be present. It is interesting to investigate the connections between real networks and the corresponding Markovian networks. In the case of Barabasi–Albert networks, for instance, it is possible to use recipes for constructing ensembles of the two kinds (preferential attachment vs. rewiring) and compare them [5].

In any case, let us focus on and . Imagine that we know them for a certain network and we want to study some dynamical processes on the network, e.g., epidemic diffusion processes [6]. It turns out [7, 8] that several features of these processes depend on a “contracted” form of the correlations, namely, the function , called “average nearest neighbor degree” and defined as , where is the highest degree of the nodes of the network. This function of is simpler to analyse than the full matrix . Its increasing or decreasing character discriminates between assortative and disassortative networks (see [912] and refs.). We can further contract the information on the correlations into a single number, the Newman assortativity coefficient , either using the matrix or with one more summation procedure performed on .

One may wonder whether it is possible, given an admissible function (it must satisfy a normalization condition given below), to compute a full correlation matrix which returns that upon contraction on . Porto and Weber have devised a method for this purpose [13], which has been used for some applications by themselves and Silva et al. [14]. However, while the correspondence is univocal, this is not true for the opposite correspondence . One first scope of this work is to show explicitly this ambiguity in an important specific example, namely, that of a linear . To this end, we introduce in Section 2.1 the correlation matrix of Vazquez–Weigt [15], which has the simple form , where is an uncorrelated matrix. The corresponding is linear in . Then, in Section 2.2, we recall the method by Porto and Weber for building a starting from a , and in Section 2.3, we apply it to of Vazquez and Weigt. A comparison of the result with the original matrix shows remarkable differences. Also, in Section 2.4, we briefly discuss the case of real networks for which the function is only approximately linear.

While examining the differences noted in Section 2.3, we have been led to consider the eigenvalue spectra of the associated connectivity matrices . This has revealed a simple general property of the eigenvalues of , which has important consequences for the epidemic threshold in diffusion models based on this matrix (Section 3). In fact, the eigenvalues of are , where and is the largest degree in the network. It follows that the epidemic threshold (Section 3.1) is proportional to , for any fixed value of . When is small, the epidemic threshold is definitely greater than zero for small networks, but if , the threshold goes quickly to zero. The convergence is much faster than for other correlations, for which the largest eigenvalue typically grows as a root of or even as when the scale-free exponent is equal to 3 (Sections 3.2 and 3.3). The conclusion is that adding even a very small amount of assortative diagonal degree correlations to an uncorrelated network leads, in the large- limit, to a fast vanishing of the epidemic threshold. Further, numerical results for the case in which the assortativity coefficient is not small are given in Section 3.4.

In Section 4, we discuss a family of transformations of the correlation matrices which keep their functions unchanged. We show that such transformations affect in general the epidemic threshold even if this does not happen when these transformations act on networks in which the average neighbor degree is independent from the degree itself; in this case, the transformations lead to networks for which the epidemic threshold remains unchanged and which, albeit having a constant , belong to a wider class than that of strictly uncorrelated networks.

2. The Correlation Matrix of Vazquez–Weigt vs. That of Porto–Weber Reconstruction

2.1. The Vazquez–Weigt Matrix

Vazquez and Weigt [15] have defined the following assortative correlation matrix:where (more generally, below, denotes for any function ).

This Ansatz has been used in several applications [16]. It is a linear combination of a perfectly uncorrelated matrix with elements , giving a probability of connection independent from and a perfectly assortative matrix (giving a nonzero probability of connection only between nodes of the same degree). The coefficient in the linear combination can vary in the range and corresponds to the Newman assortativity coefficient.

The function for the Vazquez–Weigt correlation matrix is easily found as follows:

Since the first term is independent from , this is a linear function, with slope .

2.2. The Recipe of Porto–Weber for a Correlation Matrix Having a Predefined

In order to compute the correlation matrix starting from a given function (normalized as [12]), Porto and Weber define first the symmetric function as follows:where

In other words, is the average of the quantity with a normalized “edge” probability distribution defined as , giving the probability that a randomly chosen edge of the network is connected to a node of degree .

The conditional probability is then given by

It is immediate to check that defined in this way satisfies the normalization condition in and the network closure condition as follows:

Also, it is straightforward to replace into the definition of and obtain an identity.

2.3. Porto–Weber Recipe Applied to the Vazquez Correlation Matrix

Now, suppose we want to reconstruct starting from using the Porto–Weber recipe. Applying this recipe to the function of Vazquez–Weigt, one obtains

One can check numerically that the insertion of this function into the Porto–Weber recipe gives a correlation matrix whose coincides element by element with of Vazquez–Weigt. However, does not coincide with . The difference between the two matrices is evident looking at their dependence on and . Their traces and eigenvalues are markedly different as we shall show in the next section. A graphical representation showing the differences of the single elements is given in Figures 1 and 2.

In conclusion, with the Porto–Weber recipe, it is possible to obtain from a given function a correlation matrix which yields that , but such a correlation matrix is not the unique correlation matrix having the given as its average nearest neighbor degree function. This could and should in fact be expected since the definition of involves a summation, and thus any two matrices and , suitably normalized, such thatyield the same .

We also observe that there is no guarantee that the Porto–Weber method works for any normalized . For instance, for a linear , some (unacceptable) negative values of are obtained when is greater than a value which is approximately 0.5. For the functions of Reference [14], of the form , one obtains negative values of when is greater than a value which is approximately 0.4.

2.4. The Case of Real Networks

Summarizing, from the purely mathematical point of view, our argument above shows explicitly that fixing a function of a network is not sufficient to define its full correlations. Actually, as proven in Section 4, there exists in general an infinite set of transformations of the correlations which leave their invariant. One can say that and just represent mathematically two different Markovian networks (also having different connectivity spectra, as discussed in Section 3).

However, when it comes to the modelization of diffusion on real networks, the following question may arise: if we know that a certain real network displays a linear function, is it more appropriate to compute its diffusion properties, and in particular the epidemic threshold, using a full correlation matrix of the type or the type? This question is motivated by the fact that “empirical” functions are often used to summarize in an efficient way the information available on correlations in real networks [3]. It is also known that a linear behavior of for real assortative networks is quite frequent but only approximate. Especially for large values of the node degree , the linear behavior is not maintained in practice, and one observes instead a decreasing tail in . This happens also because in real scale-free networks, the largest hubs are present but rare, and the Markovian-probabilistic approach is stretched to its limits of applicability.

It can be shown through numerical simulations based on degree-preserving rewiring [12] that in a real assortative scale-free networks (even with high ), the large hubs cannot be connected to a sufficient number of other hubs so as to keep increasing for large simply because many hubs have in principle a nonzero probability but are missing in a random concrete realization of the network. In such cases, using the Porto–Weber method to construct the full correlations starting from a realistic function is clearly the more correct recipe.

3. Differences in the Spectrum of the Connectivity Matrix

3.1. The Connectivity Matrix and Its Relation with the Epidemic Threshold

For a Markovian network with correlation matrix , the associated “connectivity matrix” is defined as follows [7, 8, 14]:

This matrix plays an important role in studies of diffusion on networks. For instance, in the Homogeneous Mean Field approximation of the SI (Susceptible-Infected) epidemic model, the equation set which describes the behavior in time of the fraction of infected nodes with degree is as follows (see [8]):

It can be generally shown that the solutions of this equation set are characterized by an “epidemic threshold” which separates different spreading scenarios: if , the system reaches a stationary state with a finite fraction of infected population, while if , the contagion dies out exponentially fast. The threshold turns out to be equal to , where is the largest eigenvalue of the connectivity matrix .

3.2. The Epidemic Threshold for Uncorrelated Scale-Free Networks

It is therefore important to know the largest eigenvalue of the matrix, and a general result valid for scale-free networks [7, 8] states that this eigenvalue tends to when the size of the network grows. This means that for large scale-free networks, the epidemic threshold is essentially zero and the epidemics spreads and persists in the population also when the contagion probability is very small.

In the absence of degree correlations (i.e., with uncorrelated , namely, ), it has been shown that

For scale-free networks with scale exponent , is finite when the maximum degree tends to infinity, while is divergent:where is the normalization constant of the degree distribution . The divergent part of (12) is of the form ; thus, for . When , the limit is also infinite but only with slow divergence .

3.3. The Epidemic Threshold for the Assortative Networks of Vazquez–Weigt

It can be shown through general arguments that the divergence of the largest eigenvalue when holds true independently from the degree correlations (see [4, 7]; see also some special cases in [12]). However, in the case of the diagonal assortative correlation matrices introduced by Vazquez and Weigt, a simple direct proof is possible, which is not yet available in the literature. Since for these correlations the assortativity level (expressed through the Newman coefficient ) is easily tunable in the range , the proof also has interesting consequences for the epidemic threshold in general.

The connectivity matrix associated to is

In order to compute its eigenvalues , we consider the determinant of the matrix , with elements

It is immediate to note that when , the determinant of this matrix is zero because the matrix has rank 1. Thus, we immediately find the eigenvalues as follows:the largest of them being . When , this eigenvalue grows much faster than the largest eigenvalue for uncorrelated networks, especially if approaches 3. For large networks, even a small value of like is sufficient for this to occur (see Figures 46).

Reference [17] reports the results of numerical simulations which, in retrospect, can be understood as being referred to a similar case of small . On the analytical side, however, no general treatment was given but only an approximated eigenvalue expansion valid for close to 1.

The conclusion is that for large scale-free networks, the presence of a small diagonal assortative correlation guarantees a quick convergence to zero of the epidemic threshold. We recall that according to the criterion by Dorogovtsev and Mendez [18], the relation between network size (number of nodes) and maximum degree is . Therefore, the behavior of is insensitive to as a function of but not of . Still, even for , the dependence of on is , which is fast compared with the very slow increase in in an uncorrelated network .

3.4. Larger Values of

In the mathematical diffusion theory on scale-free networks, it is interesting to consider the correlation matrix in the limit of very small because this shows the existence of networks which are practically uncorrelated but have an epidemic threshold converging to zero much more quickly than for purely uncorrelated networks. From the point of view of the “statistical mechanics of networks,” the set of these almost uncorrelated networks (say, with , and small) is wider than the set of networks with exactly and can therefore be relevant in evaluations over statistical ensembles.

On the other hand, as discussed in Section 2.4, the alternative correlation matrix can improve the modeling of diffusion on real networks which display a function that is only approximately linear, especially at large . For this purpose, it is also interesting to compare the largest eigenvalues of and for greater values of the assortativity coefficient . Some results in this direction are reported in Table 1. The general trend is confirmed as follows: grows more quickly for .

4. Examples of Variations of a Matrix Which Do Not Modify Its Average Nearest Neighbor Degree Function

In this section, we explore the more general question relative to multiplicity and concrete construction of variations of a correlation matrix which do not modify the average nearest neighbor degree function . If such a variation is represented asthen the elements with are required to(1)Satisfy the Network Closure Condition (in this case, also the elements do it as one can easily check);(2)Satisfy the normalization according to which (for each , the elements give the probabilities that a vertex with degree k is connected to a vertex with degree ), which becomes in this case(3)Leave unchanged which only occurs whenholds true.

In addition, the inequalitiesmust hold true.

To obtain condition (1), we start and assume, on the trail of Porto and Weber,where is a symmetric matrix. Notice that from now on we will write for the sake of brevity. Moreover, we will assume that each (for ) is different from zero. The two conditions (2) and (3) then take the form

We will first look for matrices which satisfy (9) and only afterwards, in connection with some specific , will we check and specify when inequality (9) is satisfied too.

We narrow our search to the family of symmetric matrices , whose only nonzero elements are, together with and , those on the main diagonal and those on the first diagonal below and on the first diagonal above the main diagonal:

For any such matrix , solving system (9) amounts to solve a linear system of equations in variables (recall that is symmetric):

We rewrite this system aswhere the matrix (also denoted by ) is given bythe unknown vector is ordered as , and is the vector with all components equal to zero.

It can be seen that, if each (for ) is different from zero, the matrix has determinant zero whereas its rank is equal to . Below, we first work out the calculations for the case , which is the smallest positive integer for which the particular structure of the matrix is clearly recognisable. Then, we describe the procedure to handle the case with general .

Denote by the matrix with .

Proposition 1. The matrix has determinant equal to zero and rank equal to seven.

Proof. The matrix has the formBy substituting the 5-th row with that obtained as the difference of the 5-th row minus the first row, we get a matrix, whose determinant is easily seen to be equal towhereBy substituting the 5-th row with that obtained as the difference of the 5-th row minus 2 times the first row, we get a matrix, whose determinant is easily seen to be equal towhereWith two similar further steps (i.e., iteratively suitably substituting the 5-th row of a matrix), one easily finds thatwhereThe Laplace expansion of , iteratively applied, giveswhich in turn implies that .
In view of (31), in order to conclude that the rank of is equal to seven; it is sufficient to prove that . And this can be immediately seen because, for example, the minor corresponding to the determinant of the triangular matrixis different from zero (being for all indices by assumption).
The proof strategy can be generalised for the case of the matrix leading to the following result.

Proposition 2. For the matrix , it is and .

Proof. By performing times a procedure similar to that in the proof of the previous proposition, namely,(i)Step 1: substituting the -th row of the matrix with that obtained as the difference of the -th row minus the first row(ii)Step 2: substituting the -th row of the matrix obtained after elimination of the first row and the first column from the matrix resulting from the previous step with that obtained as the difference of the -th row minus 2 times the first row(iii)(iv)step : substituting the -th row of the matrix obtained after elimination of the first row and the first column from the matrix resulting from the previous step with that obtained as the difference of the -th row minus times the first rowone finds thatwhereThe determinant can be calculated by iteratively applying the Laplace expansion. It is not difficult to convince oneself thatTogether, (35) and (37) imply that .
It only remains to be proved that the rank of is equal to . Also here, similarly as in the proof of Proposition 1, we observe that the matrix obtained by deleting the last row and the last column in has determinant equal to . Hence, and, together with (35), this completes the proof.
Proposition 2 implies that the eigenspace of is one-dimensional, and this in turn means that equation (24) admits infinitely many solutions; precisely, there is a one-parameter family of them. By way of example, let us consider the following low-dimensional case.

Example 1. Let and let a Markovian network with degree distribution and correlation matrix be given. Assume that for . The function pertaining to is also the average nearest neighbor degree of the networks which have the same degree distribution as and correlation matrix of the form , the only nonzero elements of the symmetric matrix beingall of them expressed in terms of a unique parameter , together with those symmetrically positioned with respect to the main diagonal, provided the inequalitiesare satisfied.
We recall here that for all . Therefore, the inequalities to be checked are in fact those relative to the nonzero elements . In this example , they can be expressed asExplicit treatment of an example requires fixing both the values of the elements of the degree distribution and correlation matrix . We here consider three cases, each of them relative to a scale-free network with , where , and is constructed according to the following algorithms (see [12, 19]):(i)Case 1. We define as follows: for , , for , , andCall for any and let . Then, redefine the correlation matrix by setting the elements on the diagonal equal toand leaving the other elements unchanged: for . Finally, normalize the entire matrix by setting(ii)Case 2. Let if and with as in (41). Then, proceed as in the previous case to get elements which satisfy the normalization .(iii)Case 3. Let if and with as in (41). Again, proceed as above to get elements which satisfy the normalization .Straightforward calculations (performed with Mathematica) yield, for example, that inequalities (40) are satisfied in the following situations:In Case 1, with , provided holds trueIn Case 1, with , provided holds trueIn Case 1, with , provided holds trueIn Case 2, with , provided holds trueIn Case 2, with , provided holds trueIn Case 2, with , provided holds trueIn Case 3, with , provided holds trueIn Case 3, with , provided holds trueIn Case 3, with , provided holds true When calculating the connectivity matrix in correspondence to correlations and then also in correspondence to correlations in the Cases above, for and (various) values of compatible with the intervals just found, one observes what follows. In passing from to (namely, by taking an admissibile positive rather than ), in all Cases, 1, 2, and 3, the largest eigenvalue of increases (and, accordingly, the epidemic threshold decreases).In contrast, if one takes an uncorrelated network, by this meaning a network for which(iv)Case 4.,and considers then the matrix with elements constructed according to (20) and (22), one notices the following fact: the largest eigenvalue of the connectivity matrix remains equal to when the parameter varies in the interval which guarantees the meaningfulness of the variation . A subtle and interesting situation is taking place: on the one hand, for networks constructed considering the elements as done here, it is no more true that the conditional probability that a vertex of degree is connected to a vertex of degree which is independent of ; on the other hand, the average nearest neighbor degree function of these networks is the same of that of a strictly uncorrelated network; it is constant (and equal to , coinciding with of the original uncorrelated network).

Remark 1. Beside the choice of taking symmetric matrices as in (22), other choices can be performed. They lead both to cases in which the largest eigenvalue of the connectivity matrix in correspondence to the matrix is greater than the one obtained in correspondence to the matrix as to cases in which this eigenvalue is smaller.
In any case, one can conclude that networks with the same but different correlation matrices can exhibit different epidemic thresholds.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Open Access Publishing Fund of the Free University of Bozen-Bolzano.