In contrast to the conventional wisdom that scale-free networks are prone to epidemic propagation, in the paper we present that disease spreading is inhibited in fractal scale-free networks. We first propose a novel network model and show that it simultaneously has the following rich topological properties: scale-free degree distribution, tunable clustering coefficient, "large-world" behavior, and fractal scaling. Existing network models do not display these characteristics. Then, we investigate the susceptible-infected-removed (SIR) model of the propagation of diseases in our fractal scale-free networks by mapping it to bond percolation process. We find an existence of nonzero tunable epidemic thresholds by making use of the renormalization group technique, which implies that power-law degree distribution does not suffice to characterize the epidemic dynamics on top of scale-free networks. We argue that the epidemic dynamics are determined by the topological properties, especially the fractality and its accompanying "large-world" behavior.
In recent years, there has been much interest in the study of the structure and dynamics of complex networks [1,2,3,4]. One aspect that has received considerable attention is the epidemic spreading taking place on top of networks [5], which is relevant to computer virus diffusion, information and rumor spreading, and so on. In the study of epidemic spreading, the notion of thresholds is a crucial problem since it finds an intermediate practical application in disease eradication and vaccination programs [6,7]. In homogeneous networks, there is an existence of nonzero infection threshold, if the spreading rate is above the threshold, the infection spreads and becomes endemic, otherwise the infection dies outs quickly. However, recent studies demonstrate that the threshold is absent in heterogeneous scale-free networks [8,9,10,11]. Thus, it is important to identify what characteristics of network structure determine the presence or not of epidemic thresholds.
To date the influences of most structural properties on disease dynamics have been studied, which include degree distribution [9,10,11], clustering coefficient [12], and degree correlations [13]. However, these features do not suffice to characterize the architecture of a network [14]. Very recently, by introducing and applying box-covering (renormalization) technique, Song, Havlin and Makse found the presence of fractal scaling in a variety of real networks [15,16]. Examples of fractal networks include the WWW, actor collaboration network, metabolic network, and yeast protein interaction network [20]. The fractal topology is often characterized through two quantities: fractal dimension d B and degree exponent of the boxes d k , both of which can be calculated by the box-counting algorithm [17,18]. The scaling of the minimum possible number of boxes N B of linear size ℓ B required to cover the network defines the fractal dimension d B , namely
Similarly, the degree exponent of the boxes
where k B (ℓ B ) is the degree of a box in the renormalized network, and k hub the degree of the most-connected node inside the corresponding box. Interestingly, for fractal scale-free networks with degree distribution P (k) ∼ k -γ , the two exponents, d B and d k , are related to each other through the following universal relation:
Fractality is now acknowledged as a fundamental property of a complex network [14]. It relates to a lot of aspects of network structure and dynamics running on the network. For example, in fractal networks the correlation between degree and betweenness centrality of nodes is much weaker than that in non-fractal networks [19]. In addition, several studies uncovered that fractal networks are not assortative [16,20,21]. The peculiar structural nature of fractal networks make them exhibit distinct dynamics. It is known that fractal scale-free networks are more robust than non-fractal ones against malicious attacks on hub nodes [16,21]. On the other hand, fractal networks and their non-fractal counterparts also display disparate phenomena of other dynamics, such as cooperation [23,22], synchronization [21], transport [24], and first-passage time [26,25]. Despite of the ubiquity of fractal feature and its important impacts on dynamical processes, the dynamics of disease outbreaks in fractal networks has been far less investigated.
In this current paper, we focus on the effects of fractality on the dynamics of disease in fractal scale-free networks. Firstly, we propose an algorithm to create a class of fractal scale-free graphs by introducing a control parameter q. Secondly, we give in detail a scrutiny of the network architecture. The analysis results show that this class of networks have unique topologies. They are simultaneously scale-free, fractal, ’large-world’, and have tunable clustering coefficient. Thirdly, we study a paradigmatic epidemiological model [6,7], namely the susceptible-infected-removed (SIR) model on the proposed fractal graphs. By mapping the SIR model to a bond percolation problem and using the renormalization-group theory, we find the existence of non-zero epidemic thresholds as a function of q. We also provide an explanation for our findings.
This section is devoted to the construction and the relevant structural properties of the networks under consideration, such as degree distribution, clustering coefficient, average path length (APL), and fractality.
The proposed fractal networks have two categories of bonds (links or edges): iterative bonds and noniterated bonds which are depicted as solid and dashed lines, respectively. The networks are constructed in an iterative way as shown in Fig. 1. Let F t (t ≥ 0) denote the networks after t iterations. Then the networks are built in the following way: For t = 0, F 0 is two nodes (vertices) connected by an iterative edge. For t ≥ 1, F t is obtained from F t-1 . We replace each existing iterative bond in F t-1 either by a connected cluster of links on the top middle of Fig.
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