Interplay of network dynamics and heterogeneity of ties on spreading dynamics

The structure of a network dramatically affects the spreading phenomena unfolding upon it. The contact distribution of the nodes has long been recognized as the key ingredient in influencing the outbreak events. However, limited knowledge is currently available on the role of the weight of the edges on the persistence of a pathogen. At the same time, recent works showed a strong influence of temporal network dynamics on disease spreading. In this work we provide an analytical understanding, corroborated by numerical simulations, about the conditions for infected stable state in weighted networks. In particular, we reveal the role of heterogeneity of edge weights and of the dynamic assignment of weights on the ties in the network in driving the spread of the epidemic. In this context we show that when weights are dynamically assigned to ties in the network, a heterogeneous distribution is able to hamper the diffusion of the disease, contrary to what happens when weights are fixed in time.

Figure 1

Degree distribution of networks varying ${\gamma }_k$ and maintaining ${\gamma }_w=0$. ${\gamma }_k$ ranges from 0, yellow (upper) curve, to 1, violet (lower) curve.

Figure 2

Weight distribution of networks varying ${\gamma }_w$ and maintaining ${\gamma }_k=0$. ${\gamma }_w$ ranges from 0, light-green (upper) curve, to 1, blue (lower) curve.

Figure 3

Strength distribution of networks varying ${\gamma }_w={\gamma }_k$. ${\gamma }_w={\gamma }_k$ ranges from 0, magenta (upper) curve, to 1, blue (lower) curve.

Figure 4

Fraction of infected nodes at endemic equilibrium reached as function of $\beta$, the transmission probability. For this figure $\mu$ is fixed to 1. (a) Results on annealed networks with stable weight pattern, $\eta =0$. (b) Results on annelead networks with dynamical assignment of weights, $\eta =1$. Recovery probability $\mu$ is fixed to 1. Vertical lines show the theoretical expectation of the thresholds. On graphs associated with $\eta =0$ the heterogeneity of both edge weights and node degrees seems to support pathogen persistence. On the other hand, on networks with dynamical assignment of weights, $\eta =1$, infection stability is more often achieved on graphs with homogeneously distributed weights.

Figure 5

Estimated epidemic threshold ${\beta }_c$ as function of ${\gamma }_k$ and ${\gamma }_w$ on annealed networks. (a)–(c) Recovery probability is $\mu =0.25$; (b)–(d) we fix $\mu =1$. (a),(b) Results obtained for graphs with stable weights pattern, $\eta =0$; (c),(d) results obtained on graphs with dynamical assignment of weights, $\eta =1$. By comparing these two rows, a clear difference of behavior could be detected: for $\eta =1$ the edge weight heterogeneity ceases to be a favorable condition for infection spreading. We further explore this behavior in Fig. 10.

Figure 6

Strength of a randomly chosen node of degree 50 in $({\gamma }_k=0,{\gamma }_w=\{0,0.8,1\})$ networks with $\eta =1$ fraction edge weights permuted at each time step.

Figure 7

Probability density of the evolution rate of edge weights for dynamic heterogeneous networks $({\gamma }_k=0,{\gamma }_w=0)$ for different $\eta$ values. $\eta$ ranges from 0, blue (lower) curve, to 1, magenta (upper) curve.

Figure 8

Critical values of transmission probability ${\beta }_c$ for different fractions of dynamically assigned weights $\eta$ in annealed graphs. Recovery probability $\mu$ is fixed to 1.

Figure 9

Critical values of transmission probability ${\beta }_c$ for different fractions of dynamically assigned weights $\eta$ in quenched graphs. Recovery probability $\mu$ is fixed to 1.

Figure 10

Slope of linear fitting of ${\beta }_c({\gamma }_k=1,{\gamma }_w)$ for various $\eta$. Recovery probability $\mu$ is fixed to 1.