Silvio C. Ferreira

Multigraphs are graphs in which multiple links between pairs of nodes are allowed, whereas they are forbidden in simple graphs, the latter being widely used in network science. Simple graphs generated by the configuration model have served as a benchmark for validating theoretical approaches to dynamical processes on networks. However, generating large scale-free networks with degree exponent γ < 3 introduces uncontrolled disassortative correlations and severe computational limitations due to the prohibition of reconnecting hubs. These constraints do not exist in multigraphs. We investigate how multiple connections affect epidemic spreading by comparing several epidemic models exhibiting an active steady state on simple graphs and multigraphs sharing the same degree sequence and natural upper cutoff. By analyzing epidemic thresholds, finite-size scaling, and localization, we show that differences between simple graphs and multigraphs emerge only when epidemic activity can persist on isolated hubs (star subgraphs) for times exponentially long in the hub degree. Our results remove a methodological barrier to the study of dynamical processes on large scale-free networks.

Bridging nodes, which connect critical components of a network, play an important role in maintaining structural integrity and facilitating communication within the network, representing indirect yet relevant connections. Epidemic triggering mechanisms in networks often involve long-range mutual activation of hubs, mediated by paths composed of low-degree nodes. While low-degree nodes are abundant in networks, their role in bridging central nodes in epidemic activation mechanisms has not been thoroughly analyzed. Starting with a backbone network with a power-law degree distribution, we investigate the role of adding degree-2 bridging nodes that are preferentially attached to hubs. Our findings reveal that bridging nodes can mediate an indirect feedback interaction between hubs that modifies the epidemic localization and activation mechanisms of the epidemic processes with recurrent infections. In particular, the collective activation observed in the presence of waning immunity, which produces a finite epidemic threshold in power-law networks with degree exponent γ>3, is altered to a localized activation with a vanishing threshold. Our numerical results are analytically supported by the nonbacktracking matrix properties that emerge in the recurrent dynamical message-passing theory.

Network models adeptly capture heterogeneities in individual interactions, making them well-suited for describing a wide range of real-world and virtual connections, including information diffusion, behavioural tendencies, and disease dynamic fluctuations. However, there is a notable methodological gap in existing studies examining the interplay between physical and virtual interactions and the impact of information dissemination and behavioural responses on disease propagation. We constructed a three-layer (information, cognition, and epidemic) network model to investigate the adoption of protective behaviours, such as wearing masks or practising social distancing, influenced by the diffusion and correction of misinformation. We examined five key events influencing the rate of information spread: (i) rumour transmission, (ii) information suppression, (iii) renewed interest in spreading misinformation, (iv) correction of misinformation, and (v) relapse to a stifler state after correction. We found that adopting information-based protection behaviours is more effective in mitigating disease spread than protection adoption induced by neighbourhood interactions. Specifically, our results show that warning and educating individuals to counter misinformation within the information network is a more effective strategy for curbing disease spread than suspending gossip spreaders from the network. Our study has practical implications for developing strategies to mitigate the impact of misinformation and enhance protective behavioural responses during disease outbreaks.

We analyze the critical time series of the order parameter generated with active to inactive phase transitions of spreading dynamics running on the top of heterogeneous networks. Different activation mechanisms that govern the dynamics near the critical point were investigated. The time series were analyzed using the visibility graph (VG) method where a disassortative degree correlation of the VG is a signature of criticality. In contrast, assortative correlation is associated with off-critical dynamics. The signature of criticality given by the VG is confirmed for collective activation phenomena, as in the case of homogeneous networks. Similarly, for a localized activation driven by a densely connected set of hubs, identified by a maximum k-core decomposition, critical times series were also successfully identified by the VG method. However, in the case of activation driven by sparsely distributed hubs, the time series criticality is blurred, being observable only for very large systems. In the case of strong structural localization induced by the presence of rare regions, an assortative VG degree correlation, typical of off-critical series, is observed. We conclude that while macroscopic times series remain good proxies for the analysis of criticality for collective or maximum k-core activation, systems under spatial localization can postpone the signatures of or, in case of extreme localization, lead to false negatives for criticality of the time series.

Infectious diseases are marked by recovering time distributions which can be far from the exponential one associated with Markovian/Poisson processes, broadly applied in epidemic models. In the present work, we tackled this problem by investigating a susceptible-infected-recovered-susceptible model on networks with η independent infectious compartments (SI η RS), each one with a Markovian dynamics, that leads to a Gamma-distributed recovering time. We analytically develop a theory for the epidemic lifespan on star graphs with a center and K leaves, which mimic hubs on networks, showing that the epidemic lifespan scales with a non-universal power-law. Compared with standard susceptible-infected-recovered-susceptible dynamics, the epidemic lifespan on star graphs is severely reduced as the number of stages increases. In particular, the case η → ∞ leads to a finite lifespan. Numerical simulations support the approximated analytical calculations. We investigated the SI η RS dynamics on random power-law networks. When the epidemic processes are ruled by a maximum k-core activation, either the epidemic threshold or the epidemic localization pattern are unaltered. When hub mutual activation is at work, the localization is reduced but not sufficiently to alter the threshold scaling with the network size. Therefore, the activation mechanisms remain the same as in the case of Markovian healing.