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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 $P(h|k)=(1-r)P^U_{hk}+r\delta_{hk}$, where $P^U$ is uncorrelated and $r$ (the Newman assortativity coefficient) can be very small. The effect is uniform in the scale exponent $\gamma$, if the network size is measured by the largest degree $n$. We also prove that it is possible to construct, via the Porto-Weber method, correlation matrices which have the same $k_{nn}$ as the $P(h|k)$ above, but very different elements and spectrum, and thus lead to different epidemic diffusion and threshold. Moreover, we study a subset of the admissible transformations of the form $P(h|k) \to P(h|k)+\Phi(h,k)$ with $\Phi(h,k)$ depending on a parameter which leave $k_{nn}$ invariant. Such transformations affect in general the epidemic threshold. We find however that this does not happen when they act between networks with constant $k_{nn}$, i.e. networks in which the average neighbor degree is independent from the degree itself (a wider class than that of strictly uncorrelated networks)., Comment: 18 pages, 6 figures

We generate correlated scale-free networks in the configuration model through a new rewiring algorithm which allows to tune the Newman assortativity coefficient $r$ and the average degree of the nearest neighbors $K$ (in the range $-1\le r \le 1$, $K\ge \langle k \rangle$). At each attempted rewiring step, local variations $\Delta r$ and $\Delta K$ are computed and then the step is accepted according to a standard Metropolis probability $ \exp(\pm\Delta r/T)$, where $T$ is a variable temperature. We prove a general relation between $\Delta r$ and $\Delta K$, thus finding a connection between two variables which have very different definitions and topological meaning. We describe rewiring trajectories in the $r$-$K$ plane and explore the limits of maximally assortative and disassortative networks, including the case of small minimum degree ($k_{min} \ge 1$) which has previously not been considered. The size of the giant component and the entropy of the network are monitored in the rewiring. The average number of second neighbours in the branching approximation $\bar{z}_{2,B}$ is proven to be constant in the rewiring, and independent from the correlations for Markovian networks. As a function of the degree, however, the number of second neighbors gives useful information on the network connectivity and is also monitored., Comment: 21 pages, 7 figures

We develop and test a rewiring method (originally proposed by Newman) which allows to build random networks having pre-assigned degree distribution and two-point correlations. For the case of scale-free degree distributions, we discretize the tail of the distribution according to the general prescription by Dorogovtsev and Mendes. The application of this method to Barabasi-Albert (BA) networks is possible thanks to recent analytical results on their correlations, and allows to compare the ensemble of random networks generated in the configuration model with that of "real" networks obtained from preferential attachment. For $\beta\ge 2$ ($\beta$ is the number of parent nodes in the preferential attachment scheme) the networks obtained with the configuration model are completely connected (giant component equal to 100%). In both generation schemes a clear disassortativity of the small degree nodes is demonstrated from the computation of the function $k_{nn}$. We also develop an efficient rewiring method which produces tunable variations of the assortativity coefficient $r$, and we use it to obtain maximally disassortative networks having the same degree distribution of BA networks with given $\beta$. Possible applications of this method concern assortative social networks., Comment: 17 pages, 5 figures

Using a mean-field network formulation of the Bass innovation diffusion model and exact results by Fotouhi and Rabbat on the degree correlations of Barabasi-Albert networks, we compute the times of the diffusion peak and compare them with those on scale-free networks which have the same scale-free exponent but different assortativity properties. We compare our results with those obtained by Caldarelli et al. for the SIS epidemic model with the spectral method applied to adjacency matrices. It turns out that diffusion times on finite Barabasi-Albert networks are at a minimum. This may be due to a little-known property of these networks: although the value of the assortativity coefficient is close to zero, they look disassortative if one considers only a bounded range of degrees, including the smallest ones, and slightly assortative on the range of the higher degrees. We also find that if the trickle-down character of the diffusion process is enhanced by a larger initial stimulus on the hubs (via a inhomogeneous linear term in the Bass model), the relative difference between the diffusion times for BA networks and uncorrelated networks is even larger, reaching for instance the 34% in a typical case on a network with $10^4$ nodes., Comment: 19 pages, 8 figures. Misspelling in citations corrected

We introduce a heterogeneous network structure into the Bass diffusion model, in order to study the diffusion times of innovation or information in networks with a scale-free structure, typical of regions where diffusion is sensitive to geographic and logistic influences (like for instance Alpine regions). We consider both the diffusion peak times of the total population and of the link classes. In the familiar trickle-down processes the adoption curve of the hubs is found to anticipate the total adoption in a predictable way. In a major departure from the standard model, we model a trickle-up process by introducing heterogeneous publicity coefficients (which can also be negative for the hubs, thus turning them into stiflers) and a stochastic term which represents the erratic generation of innovation at the periphery of the network. The results confirm the robustness of the Bass model and expand considerably its range of applicability., Comment: 13 pages, 5 figures

The Bass model, which is an effective forecasting tool for innovation diffusion based on large collections of empirical data, assumes an homogeneous diffusion process. We introduce a network structure into this model and we investigate numerically the dynamics in the case of networks with link density $P(k)=c/k^\gamma$, where $k=1, \ldots , N$. The resulting curve of the total adoptions in time is qualitatively similar to the homogeneous Bass curve corresponding to a case with the same average number of connections. The peak of the adoptions, however, tends to occur earlier, particularly when $\gamma$ and $N$ are large (i.e., when there are few hubs with a large maximum number of connections). Most interestingly, the adoption curve of the hubs anticipates the total adoption curve in a predictable way, with peak times which can be, for instance when $N=100$, between 10% and 60% of the total adoptions peak. This may allow to monitor the hubs for forecasting purposes. We also consider the case of networks with assortative and disassortative correlations and a case of inhomogeneous advertising where the publicity terms are "targeted" on the hubs while maintaining their total cost constant., Comment: 23 pages, 4 figures; submitted version. Chaos, Solitons and Fractals, online 9 March 2016