16 in Ref. [24]. We reproduce this solution below for the sake of convenience. The master equation describing the time evolution at a mean-field level of the average density of infected individuals ρ ( t ): where k is the degree of all nodes in the regular graph. The first term on the right hand side of Eq. (7) accounts for infected nodes becoming healthy. The second term on the right hand side of Eq. (7) accounts for healthy nodes becoming infected: a randomly chosen node is healthy with probability 1 − ρ ( t ), this healthy node can be infected by either of its k neighbor nodes with total probability of βkρ ( t ). The stationary endemic state is reached when dρ ( t ) /dt = 0 which leads to indicating the existence of a nonzero epidemic threshold of β = 1 /k . The innermost core of a network consisting only of nodes with degrees k ≥ k$_{S}$$_{max}$ will have epidemic threshold The above inequality holds for all considered networks. Moreover, this inequality becomes an equality for CNI and collaboration networks where nearly all nodes in the innermost cores have degree k ≈ k$_{S}$$_{max}$ . β$_{c}$ ≤ 1 /k$_{S}$$_{max}$. (9) ρ = 1 − 1 βk , (8) dρ ( t ) dt = − ρ ( t ) + βkρ ( t )(1 − ρ ( t )) , (7)