Modelling dynamical processes in complex socio-technical systems

Journal name:
Nature Physics
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Published online


In recent years the increasing availability of computer power and informatics tools has enabled the gathering of reliable data quantifying the complexity of socio-technical systems. Data-driven computational models have emerged as appropriate tools to tackle the study of dynamical phenomena as diverse as epidemic outbreaks, information spreading and Internet packet routing. These models aim at providing a rationale for understanding the emerging tipping points and nonlinear properties that often underpin the most interesting characteristics of socio-technical systems. Here, using diffusion and contagion phenomena as prototypical examples, we review some of the recent progress in modelling dynamical processes that integrates the complex features and heterogeneities of real-world systems.

At a glance


  1. Phase diagram of epidemic models.
    Figure 1: Phase diagram of epidemic models.

    Illustration of the behaviour of the prevalence for the SIS and SIR model in a heterogeneous network (solid line) as a function of the spreading rate β/μ, compared with the theoretical prediction for a homogeneous network (dashed line). The figure clearly shows the difference between homogeneous and heterogeneous networks, where the epidemic threshold is shifted to very small values. For scale-free networks with degree distribution exponent γ≤3, however, the associated prevalence is extremely small over a large range of values of β/μ. In other words, as noted since the first work on epidemic spreading in complex networks, the bad news about the suppression (or very small value) of the epidemic threshold is balanced by the very low prevalence attained by the epidemic46.

  2. Progression of an epidemic process.
    Figure 2: Progression of an epidemic process.

    The progression of a susceptible–infected (SI) epidemic in a heavy-tailed network at three snapshots of the process, corresponding to time t=5, 10 and 20, measured in unitary time integration steps of the model. The SI model assumes that infected nodes will spread the infection indefinitely to neighbours with rate α. In this case we know that the system is eventually completely infected, whatever the spreading rate of the infection. However, we can highlight the effect of topological fluctuations on the spreading hierarchy. Susceptible nodes are coloured blue and infected nodes are coloured from yellow to red according to the time of infection (red corresponding to later times). The size of a node is proportional to the node degree. In general, the first nodes to be infected are the large hubs with high degree, then the epidemic progresses in time by a dynamical cascade through degree classes, finally affecting low-degree nodes.

  3. Illustration of the global threshold in reaction-diffusion processes.
    Figure 3: Illustration of the global threshold in reaction–diffusion processes.

    a, Schematic of the simplified modelling framework based on the particle–network scheme. At the macroscopic level the system is composed of a heterogeneous network of subpopulations. The contagion process in one subpopulation (marked in red) can spread to other subpopulations as particles diffuse across subpopulations. b, At the microscopic level, each subpopulation contains a population of individuals. The dynamical process, for instance a contagion phenomena, is described by a simple compartmentalization (compartments are indicated by different coloured dots). Within each subpopulation, individuals can mix homogeneously, or according to a subnetwork, and can diffuse with rate d from one subpopulation to another, following the edges of the network. c, A critical value dc of the diffusion strength for individuals or particles identifies a phase transition between a regime in which the contagion affects a large fraction of the system and one in which only a small fraction is affected (see the discussion in the text). Panels a and b reproduced from ref. 118.

  4. Visualization of the dynamical network generated by Twitter interactions.
    Figure 4: Visualization of the dynamical network generated by Twitter interactions.

    Twitter is a microblogging tool that allows users to post and relay (’re-tweet’) short messages. The topic of the message is signalled by short identifiers (@mentions, #hash-tags and urls). This feature allows one to trace the spreading of specific discussion topics (also called memes). The figure shows the diffusion network for the tag #gop. Each node corresponds to an individual user. Blue edges represent re-tweets and orange edges represent mentions. Two communities are clearly visible, corresponding to politically left- and right-leaning users113. Communications between the two communities take place primarily through the use of mentions, while within a group communication occurs through re-tweets. The figure, obtained using the Truthy infrastructure114, clearly exemplifies the co-evolution of the communication network with the spreading process.


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  1. Department of Physics, College of Computer and Information Sciences, Bouvé College of Health Sciences, Northeastern University, Boston, Massachusetts 02115, USA

    • Alessandro Vespignani
  2. Institute for Scientific Interchange (ISI), Torino, 10133, Italy

    • Alessandro Vespignani

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