We study the persistent epidemic prevalence with oscillatory behavior and the extinction of computer viruses via e-mails on a contact relational network growing with new users, for which scale-free structure is estimated from real data. Typical oscillatory phenomenon is simulated in a stochastic model for the execution and detection of viruses. The conditions of extinction by random and targeted immunizations for hubs are derived through bifurcation analysis for simpler deterministic models by using a mean-field approximation without the connectivity correlations. We can qualitatively understand the mechanisms of the spread in linearly growing scale-free networks.

The short paths between any two nodes and the robustness of connectivity are advanced properties of scale-free (SF) networks; however, they may be affected by geographical constraints in realistic situations. We consider geographical networks with the SF structure based on planar triangulation for online routings, and suggest scaling relations between the average distance or number of hops on the optimal paths and the network size. We also show that the tolerance to random failures and attacks on hubs is weakened in geographical networks, and that even then it is possible for the extremely vulnerable ones to be improved by adding with the local exchange of links.

In a topological structure affected by geographical constraints on liking, the connectivity is weakened by constructing local stubs with small cycles, a something of randomness to bridge them is crucial for the robust network design. In this paper, we numerically investigate the effects of adding shortcuts on the robustness in geographical scale-free network models under a similar degree distribution to the original one. We show that a small fraction of shortcuts is highly contribute to improve the tolerance of connectivity especially for the intentional attacks on hubs. The improvement is equivalent to the effect by fully rewirings without geographical constraints on linking. Even in the realistic Internet topologies, these effects are virtually examined.

Complex network infrastructure systems for power supply, communication, and transportation support our economic and social activities; however, they are extremely vulnerable to frequently increasing large disasters or attacks. Thus, the reconstruction of a damaged network is more advisable than an empirically performed recovery of the original vulnerable one. To reconstruct a sustainable network, we focus on enhancing loops so that they are not trees, which is made possible by node removal. Although this optimization corresponds with an intractable combinatorial problem, we propose self-healing methods based on enhancing loops when applying an approximate calculation inspired by statistical physics. We show that both higher robustness and efficiency are obtained in our proposed methods by saving the resources of links and ports when compared to ones in conventional healing methods. Moreover, the reconstructed network can become more tolerant than the original when some damaged links are reusable or compensated for as an investment of resource. These results present the potential of network reconstruction using self-healing with adaptive capacity in terms of resilience.