This project's goal was to develop analytical tools and results for spreading processes over multilayer and interconnected networks.

Multilayer networks are an abstract representation of interconnection among nodes representing individuals, where the interconnection has multiple nature (see figure).

Interconnected networks are abstract representation where two or more simple networks, possibly with different and separate dynamics upon them, are interconnected to each other.

Studying multilayer and interconnected networks is an important topic in network science, with numerous potential applications to realistic social, biological, and technological networked systems.

Among the numerous results obtained during the project,  in the following, we summarize the most important ones:

1) We developed an algorithm --GEMFsim-- for exact, continuous-time numerical simulation of GEMF-based processes. The generalized epidemic modeling framework (GEMF) develops a class of epidemic models as stochastic processes at the node/ individual level, where nodes interact through different layers -- multilayer networks. The implementation of this algorithm, GEMFsim, is publicly available in our webpage in popular scientific programming platforms such as MATLAB, R, Python, and C. GEMFsim facilitates simulating stochastic spreading models that fit in the GEMF framework. We have used GEMFsim to study the transmission of several infectious diseases such as Zika, Ebola, syphilis, and Japanese encephalitis.

2) We studied the epidemic threshold of a model where each node has a default contact neighborhood set and switches to a different contact set once she becomes alert about infection among her default contacts. Since each agent can adopt either of two possible neighborhood sets, the overall contact network switches among an exponentially-large number of possible configurations. Notably, a multilayer network representation with two layers can fully model the underlying adaptive, state-dependent contact network. The epidemic threshold for the presented adaptive contact network belongs to the solution of a nonlinear Perron-Frobenius (NPF) problem, which does not depend on the contact adaptation rate monotonically. Furthermore, the network adaptation model predicted a counter-intuitive scenario where adaptively changing contacts may adversely lead to lower network robustness against epidemic spreading if the contact adaptation is not fast enough. This original result for a class of NPF problems facilitate the analytical developments of this work and can be useful to solve many other similar problems.

3) We found the exact coupling threshold of interconnected networks uncovering network topologies with unexpected behaviors. An interconnected network features a structural transition between two regimes: one where the network components are structurally distinguishable and one where the interconnected network functions as a whole. As a consequence, an emergent phenomenon appears concerning the vulnerability of interdependent networks: below the coupling threshold, failures stay localized in the originating network layer, while beyond this threshold, failures will invade across layers and lead to whole-scale catastrophic failure. We were able to determine the exact value of this critical threshold. Furthermore, we showed conditions under which superdiffusion can occur despite the network components functioning distinctly. Moreover, we find that components of certain interconnected network topologies are indistinguishable despite very weak coupling between them.

4) We solved the problem of finding an optimal weight distribution for one-to-one interlayer links of interconnected networks under a budget constraint. We showed that for the special multiplex configurations with identical layers, the uniform weight distribution is always optimal. On the other hand, when the two layers are arbitrary, increasing the budget reveals the existence of two different regimes. Up to a certain threshold budget, the second eigenvalue of the supra-Laplacian is simple, the optimal weight distribution is uniform, and the Fiedler vector is constant on each layer. Increasing the budget past the threshold, the optimal weight distribution can be nonuniform. The interesting consequence of this result is that there is no need to solve the optimization problem when the available budget is less than the threshold, which can be easily found analytically.

5) We developed a framework to infer missing links among nodes of an epidemic network. In our approach, we use the state of nodes through time as the input. Moreover, since we are using a Bayesian inference method, we can incorporate different prior information about the links in the inference problem. This is particularly useful when the trace of nodes' state does not include enough information. Although we only showed the application of our approach for the susceptible-infected-susceptible epidemic model, our approach can be easily applied to more complicated models that may involve multilayer networks.

In terms of broader impacts this project provided the opportunity to 1) train one postdoc, one Ph.D. student, and two MS students 2) collaborate with Piet van Mieghem from TU Delft and Jose Marzo from University of Girona, and 3) integrate project results on the graduate course ECE 841 Network Science at Kansas State University, where GEMFsim is used by the students in their final project.

Last Modified: 07/23/2018
Modified by: Caterina Scoglio