While communication networks show the small-world property of short paths, the spreading dynamics in them turns out slow. Here, the time evolution of information propagation is followed through communication networks by using empirical data on contact sequences and the SI model. Introducing null models where event sequences are appropriately shuffled, we are able to distinguish between the contributions of different impeding effects. The slowing down of spreading is found to be caused mainly by weight-topology correlations and the bursty activity patterns of individuals. PACS numbers: 05.45.Tp Most complex physical, biological and social networks show the small-world property, where the average shortest path length is strikingly short when compared to the network size [1] . This means that there is at least one short path between any two nodes, which should give rise to rapid transmission of influence. However, dynamic phenomena on networks [2], such as spreading of pandemics, electronic viruses, and information, follow their own pathways, which are not necessarily topologically efficient [3] . Spreading on real small-world networks turns out to be surprisingly slow, e.g., new infections by a computer virus are reported years after its emergence or the introduction of an anti-virus [4]. Here we aim at resolving this puzzle. For issues such as strategies and timing of vaccinations, improvement of information diffusion, and the slow decay of prevalence of computer viruses, it is crucial to understand the role of the underlying network and temporal activity patterns in the dynamics of spreading. The dynamics of spreading is commonly studied with SI, SIR, or SIS models [5] on static lattices or in mean field, where the dynamics is defined by state changes of individuals between (S)usceptible, (I)nfectious, and (R)ecovered. These models lead to a rapid, exponential growth of prevalence at early stages of spreading, while the dynamics at later stages depend on the model and lattice. For the SI process, the prevalence grows until the whole system reachable from initial conditions is infected, with exponential slowing down towards the end. For the SIR process, competing effects set in and the spreading may remain local or percolate through the system while the SIS process has more complex dynamics. While these results capture some of the qualitative features of real-world processes, the heterogeneity of the systems limits their applicability. First, the interactions of real-world systems span networks by broad distributions of node connections and mesoscopic features in the form of communities with dense internal and sparse external connectivity. Second, interaction intensities vary and are closely coupled to network topology. Third, the daily cycle and bursty character of interaction events give rise to important temporal inhomogeneities. Some aspects of these features have already been studied. For static networks, it is known that spatial structure has an effect on epidemics (see, e.g., [6, 7] ), and community structure slows down information diffusion due to trapping in dense regions [8] [9] [10] . There is an intimate relation between inhomogeneous link weights and network topology in social and communication networks [11, 12] : Links within communities are strong, while links between them are weak. This Granovetter-type structure enhances the trapping effect of the communities, leading to additional slowing down of spreading [12] . The bursty nature of human interactions has received particular interest and it has turned out that the corresponding activity patterns are usually non-Poissonian, often power-law correlated (see [13] ). The effect of bursty dynamics on spreading has been approached using empirical data together with approximate analytical models [14, 15] . In Ref. [14] , computer worm spreading was studied using email logs and the SI model, and it was found that the non-Poissonian inter-event time distribution leads to slow spreading in the late stages of the process. Slow spreading was also observed in Ref. [15] , where an Internet viral marketing experiment was carried out and modeled as a branching process in the nonpercolating regime. It was also argued that on the contrary, in the percolating regime, broad inter-event time distributions should give rise to faster spreading. In this Letter, we study the problem of spreading dynamics in its full complexity, using time-stamped event data on human communication networks and the SI model. We apply proper null models on the event sequences and show that spreading is slowed down due to simultaneous effects of structural and temporal correlations. For the event sequences, we have used the following data: a) Mobile phone data from a European operator (national market share ∼ 20%) with ∼ 325 million timestamped voice call records over a period of 120 days. We have only retained links with bidirectional calls within
Karsai et al. studied the effects of different topological and temporal correlations on information spreading in complex communication networks REF .
Small But Slow World: How Network Topology and Burstiness Slow Down Spreading
Phys. Rev. E 83, 025102(R) (2011)
Medicine Physics Computer Science