Extreme events in stochastic transport on networks

Abstract

Extreme events are emergent phenomena in multi-particle transport processes on complex networks. In practice, such events could range from power blackouts to call drops in cellular networks to traffic congestion on roads. All the earlier studies of extreme events on complex networks had focused only on the nodal events. If random walks are used to model the transport process on a network, it is known that degree of the nodes determines the extreme event properties. In contrast, in this work, it is shown that extreme events on the edges display a distinct set of properties from that of the nodes. It is analytically shown that the probability for the occurrence of extreme events on an edge is independent of the degree of the nodes linked by the edge and is dependent only on the total number of edges on the network and the number of walkers on it. Further, it is also demonstrated that non-trivial correlations can exist between the extreme events on the nodes and the edges. These results are in agreement with the numerical simulations on synthetic and real-life networks.Extreme events often tend to be associated with natural disasters such as floods, droughts, and earthquakes. However, more generally, any event whose numerical value displays pronounced deviation from its typical average value can be regarded as an extreme event. Then, many events ranging from traffic congestion to power blackouts would be thought of as extreme events. In particular, many of these extreme events take place on the topology of a network. Hence, it is of interest to study how the network structure affects extreme event properties, and if the networks can survive the onslaught of extreme events taking place on its nodes. Earlier, extreme events on the nodes of a complex network had been studied. By modeling events as random walkers, exceedances of the number of random walkers above a prescribed threshold was identified as an extreme event. Surprisingly, it was found that extreme event occurrence probability is lower for the hubs when compared to the small degree nodes of the network. In this work, by using the same model, we study the extreme events on the edges of a network. It is shown that the extreme event probability on the edges is a constant, and is dependent only on the parameters such as the total number of edges and the number of walkers. We have obtained analytical as well as the numerical results and they agree with one another. Furthermore, the correlation between the extreme events on the edges and nodes that they link have been studied. The non-trivial correlations indicate the role played by the network structure even though the dynamics itself is that of random walkers with no memory effects.