Effects of varying the delay distribution in random, scale-free, and small-world networks

Graph-theory-based approaches have been used with great success when analyzing abstract properties of natural and artificial networks. However, these approaches have not factored in delay, which plays an important role in real-world networks. In this paper, we (1) developed a simple yet powerful method to include delay in graphbased analysis of networks, and (2) evaluated how different classes of networks (random, scale-free, and small-world) behave under different forms of delay (peaked, unimodal, or uniform delay distribution). We compared results from synthetically generated networks using two different sets of algorithms for network construction. In the first approach (naive), we generated directed graphs following the literal definition of the three types of networks. In the second approach (modified conventional), we adapted methods by Erdös–Rényi (random), Barabasi (scale-free), and Watts– Strogatz (small-world). With these networks, we investigated the effect of adding and varying the delay distribution. As a measure of robustness to added delay, we calculated the ratio between the sum of shortest path length between every node. Our main findings show that different types of network show different levels of robustness, but the shape of the delay distribution has more influence on the overall result, where uniformly randomly distributed delay showed the most robust result. Other network parameters such as neighborhood size in small-world networks were also found to play a key role in delay tolerance. These results are expected to extend our understanding of the relationship between network structure and delay

Immunization strategies for epidemic processes in time-varying contact networks

Spreading processes represent a very efficient tool to investigate the structural properties of networks and the relative importance of their constituents, and have been widely used to this aim in static networks. Here we consider simple disease spreading processes on empirical time-varying networks of contacts between individuals, and compare the effect of several immunization strategies on these processes. An immunization strategy is defined as the choice of a set of nodes (individuals) who cannot catch nor transmit the disease. This choice is performed according to a certain ranking of the nodes of the contact network. We consider various ranking strategies, focusing in particular on the role of the training window during which the nodes' properties are measured in the time-varying network: longer training windows correspond to a larger amount of information collected and could be expected to result in better performances of the immunization strategies. We find instead an unexpected saturation in the efficiency of strategies based on nodes' characteristics when the length of the training window is increased, showing that a limited amount of information on the contact patterns is sufficient to design efficient immunization strategies. This finding is balanced by the large variations of the contact patterns, which strongly alter the importance of nodes from one period to the next and therefore significantly limit the efficiency of any strategy based on an importance ranking of nodes. We also observe that the efficiency of strategies that include an element of randomness and are based on temporally local information do not perform as well but are largely independent on the amount of information available

Stochastic Opinion Formation in Scale-Free Networks

The dynamics of opinion formation in large groups of people is a complex non-linear phenomenon whose investigation is just at the beginning. Both collective behaviour and personal view play an important role in this mechanism. In the present work we mimic the dynamics of opinion formation of a group of agents, represented by two state $\pm 1$, as a stochastic response of each of them to the opinion of his/her neighbours in the social network and to feedback from the average opinion of the whole. In the light of recent studies, a scale-free Barab\'asi-Albert network has been selected to simulate the topology of the interactions. A turbulent-like dynamics, characterized by an intermittent behaviour, is observed for a certain range of the model parameters. The problem of uncertainty in decision taking is also addressed both from a topological point of view, using random and targeted removal of agents from the network, and by implementing a three state model, where the third state, zero, is related to the information available to each agent. Finally, the results of the model are tested against the best known network of social interactions: the stock market. A time series of daily closures of the Dow Jones index has been used as an indicator of the possible applicability of our model in the financial context. Good qualitative agreement is found.Comment: 24 pages and 13 figures, Physical Review E, in pres