Many real networks present a bounded scale-free behavior with a connectivity cut-off due to physical constraints or a finite network size. We study epidemic dynamics in bounded scale-free networks with soft and hard connectivity cut-offs. The finite size effects introduced by the cut-off induce an epidemic threshold that approaches zero at increasing sizes. The induced epidemic threshold is very small even at a relatively small cut-off, showing that the neglection of connectivity fluctuations in bounded scale-free networks leads to a strong over-estimation of the epidemic threshold. We provide the expression for the infection prevalence and discuss its finite size corrections. The present work shows that the highly heterogeneous nature of scale-free networks does not allow the use of homogeneous approximations even for systems of a relatively small number of nodes. PACS numbers: 87.23.Ge, 05.70.Ln In the last years it has been recognized that a large number of physical, biological, and social networks exhibits complex topological properties [1, 2] . In particular many real world networks show the small-world phenomenon, related to a very small average path length between nodes [2, 3] . More strikingly, in some cases this property is associated to a scale-free connectivity distribution, P (k) ∼ k −2−γ , with 0 < γ ≤ 1, where k is the number of links connected to a node [4] . This scale-free nature is associated to a large heterogeneity in the connectivity properties of the system. Since the second moment of the connectivity distribution k 2 is diverging when increasing the network size, the connectivity fluctuations in scale-free (SF) networks do not have an intrinsic bound and diverge in the infinite system size limit. Scale-free properties have been observed in many real systems such as the Internet [5, 6, 7] and the World-WideWeb [4, 8] , food-webs, protein, and neural networks [9] . A very important example of scale-free networks is also found in the web of human sexual contacts [10] . This is a particularly relevant case since the unambiguous definition of contacts (links) is often missing in the analysis of social networks. Since the Internet and the web of human sexual contacts appear to be scale-free, the study of epidemics and disease dynamics on SF networks is a relevant theoretical issue in the spreading of computer viruses and sexually transmittable diseases. In heterogeneous networks, it is well-known that the epidemic threshold decreases with the standard deviation of the connectivity distribution [11, 12] , and this feature is amplified in SF networks, which have diverging connectivity fluctuations in the limit of infinite network size. Indeed, it was first noted in Ref. [13] that, in infinite SF networks, epidemic processes do not possess an epidemic threshold below which diseases cannot produce a major epidemic outbreak or the inset of an endemic state. The absence of an intrinsic epidemic threshold has been found in both the susceptible-infected-susceptible (SIS) model [13] and the susceptible-infected-removed (SIR) model [14, 15] in infinite SF networks. The immunization policies are as well very much affected by the SF nature of the connectivity distribution [16, 17] . As customarily encountered in nonequilibrium statistical systems [18] , it has also been pointed out that in finite systems an epidemic threshold is induced by finite size effects [14] . Real systems are actually made up by a finite number of individuals which is far from the thermodynamic limit. This finite population introduces a maximum connectivity k c , depending on N , which has the effect of restoring a bound in the connectivity fluctuations, inducing in this way an effective nonzero threshold. More generally, we can consider a class of bounded scale-free (BSF) networks, in which the connectivity distribution has the form P (k) ∼ k −2−γ f (k/k c ), where the function f (x) decreases very rapidly for x > 1 [19] . The cut-off k c can be due to the finite size of the network or to the presence of constraints limiting the addition of new links in an otherwise infinite network [1] . In this paper we present an analytical study of the SIS model in BSF networks with a generic connectivity exponent γ (0 < γ ≤ 1), focusing on the effects introduced by a finite cut-off k c . We analyze the case of a hard cut-off, f (x) = θ(1 − x), where θ(x) is the Heaviside step function, as it happens in growing networks with a finite number of elements. We consider as well a soft exponential cut-off, f (x) = exp(−x), as often found in systems where physical constraints are at play. We derive the behavior of the epidemic threshold as a function of k c and the network size N , and find that even for relatively small networks the induced epidemic threshold is much smaller than the epidemic threshold found in homogeneous systems. This confirms that the SF nature cannot be neglected in the practical estimates of epidemic and immunization thresholds in real networks. We also provide the explicit analytic form for the epidemic prevalence (density of infected individuals) in BSF networks. The results presented here can be readily extended to the SIR case. In order to estimate the effect of k c in epidemics
Pastor-Satorras and Vespignani study epidemic dynamics in finite-size scale-free networks, and show that, even for relatively small networks, the epidemic threshold is much smaller than that of homogeneous systems REF .
Epidemic dynamics in finite size scale-free networks
Phys. Rev. E. 65, 035108(R) (2002)
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