This post was written by Innaxis researcher, Massimiliano Zanin.
In the last few years, researchers have realized that interactions between the constituting elements of complex systems seldom develop on a single channel. Let’s take the case of a social network: information exchange may happen orally, electronically, or even indirectly; additionally, people interact according to different types of relationships, like friendship and co-working. This is important because the type of information shared may significantly depend on the channel and on the type of relation: you would probably not say the same to a co-worker in an email as you would to a significant other face to face. Due to this, it may be necessary to include different types, or layers of links, in order to obtain a meaningful representation of the system under study. Neglecting such multi-layer structure, or in other words working with the projected network, may alter our perception of the topology and dynamics, leading to a wrong understanding of the properties of the system.
Since a couple of years, I’ve been interested in the multi-layer structure of the air transport system, see for instance Refs. [1, 2]. Clearly, not all connections are the same: it is straightforward to identify that a clear multi-layer structure is created by airlines and airline alliances, which allow an easy movement of passengers between them, but difficult inter-layer movements. Last year we have published a huge monograph on multi-layer networks, which includes all aspects: from defining topological metrics, analysing dynamical process, up to a review of applications. You can find it in Ref. . (But please, do not print it before checking the number of pages!)
More recently, I’ve started asking myself: “what about multi-layer functional networks?” Let’s take one step back, and see what functional networks are.
In the early stages of complex network theory, such paradigm was mainly used to analyze systems whose structure, either physical or virtual, could be directly mapped into a network. Once again, this is the case of the air transport system, as links (direct flights between pairs of cities) have a physical nature and are easily accessible. It was soon clear that in certain cases this is not possible, as the only information obtainable from the system itself was the evolution through time of some observables. Such measurable variables reflect the behavior of the interacting elements constituting the system, and as such, the value of every observable is expected to be a “function” of the values of other peers. When the structure of such interactions is inferred from the dynamics of the observables, the result is then called a functional network.
Many examples are available of functional networks, but probably the most famous is the study of brain dynamics. First of all, it has to be noticed that physical connections between brain regions do exist, but they are quite difficult to assess… especially if you don’t want to damage the brain! Also, physical connections are interesting, but much more important are the connections that actually activate when the brain is performing some kind of task. A functional network representation can be the perfect solution. By considering the magnetic or electric field generated by spiking neurons, links are established whenever some kind of synchronisation is detected between the recorded time series, usually by means of metrics like Pearson’s linear correlation, Synchronization Likelihood, or Granger Causality. When two regions are synchronised, they are (probably, indeed this point can be discussed!) exchanging some kind of information, and thus participating in a specific computation: functional networks thus represent these collaborative processes.
Now, what about the multi-layer structure of the brain? It is well known that the human cortex has a six-layer structure, in which each layer is responsible for a different level of information abstraction and integration. This structure is nevertheless neglected, due to the limited spatial resolution of magnetic and electric sensors, and the analysed time series just correspond to the global activity of the top-most layers. We are thus projecting the multi-layer network into a single layer. Are we confident that the resulting network is still representative of the original brain activity? Notice that the non-linear nature of the projection process can foster the appearance of constructive or destructive interferences: a link may appear in the projection even if no relationship is present in any layer; or links in two layers can interfere, and disappear from the projection.
How can we validate this hypothesis? It cannot be done with brain data, as we still cannot solve the spatial resolution problem – let’s see how technology will evolve in the next decade. I found a solution by moving back to aviation. Specifically, we can create functional networks of delays: nodes are airports, pairwise connected when there is a correlation (or causality) between the time series representing their average hourly delay. Airports are thus connected if a delay propagation process is detected between them. The concept is not new: see for instance the work performed in the POEM WP-E project ; the advantage is that delay propagation can be assessed without any modelling process, and without having to gather information about aircraft turn-arounds, crews, etc. Moreover, the availability of high-resolution real data allows the reconstruction of a complete multi-layer picture, in which each layer corresponds to a different airline. But we can also collapse the dynamics, in order to simulate the creation of a single-layer representation, and compare the structures of the single- and multi-layer representations.
This is exactly what I’ve done in a paper recently published in Physica A . Results are quite startling! First, the most central nodes in the projections do not correspond to the nodes of high centrality in each layer; therefore, the former analysis give biased estimations, which cannot reliably be used to detect the most critical elements in the system. If you then try to use a single-layer model to allocate resources, you would probably end up giving money to the wrong airport! Furthermore, when a simple dynamical model is executed, the magnitude of the error yielded by considering a single layer projection is as big as the results themselves, thus indicating that any estimate obtained with this simplification is meaningless.
So, what does this mean in terms of complex systems modelling? Can we neglect the multi-layer structure? The answer is clearly NO.
Let’s consider the problem of modeling and forecasting the dynamics of the air transport network. First, results obtained imply that any simulation performed to understand the dynamics of the system may yield misleading results when the multi-layer structure created by airlines is neglected. In spite of this, most of the recent research works in this fields fail to include this essential ingredient, both in the analysis of delay propagation and of the network robustness to disruption and attacks. Second, it has to be noticed that the air transport system is created by the interactions between a large number of agents, which may create different layers along different dimensions. For instance, multiple flights do not just share the airline, but they may also be connected by the crew operating them. Disregarding these different layer dimensions, like crews, aircraft types or flight type (cargo or passengers), may further bias our understanding of the system. If the most important airports, in term of delay propagation, cannot reliably be detected with a projected functional network, the identification of functional hubs in the brain dynamics may be confused by the fact that the multi-layer structure of the cortex is neglected. Therefore, global hubs may not correspond to the most important nodes in each layer: the single layer analysis may then be misinforming about the real structure created by information flows. Or, in other words, it is possible that all results obtained in neuroscience by means of functional networks may be biased… quite a big problem!
Summing up: complex networks, and their functional version, are very powerful tools to understand the hidden dynamics behind real complex systems. Yet, one has to remember this: one layer does not fit all!
P.S.: One last comment: if you want to play with the data of Ref. , you can find an interactive version of the paper here.
 Cardillo, A., Gómez-Gardeñes, J., Zanin, M., Romance, M., Papo, D., Del Pozo, F., & Boccaletti, S. (2013). Emergence of network features from multiplexity. Scientific reports, 3. Freely accessible here.
 Cardillo, A., Zanin, M., Gómez-Gardeñes, J., Romance, M., del Amo, A. J. G., & Boccaletti, S. (2012). Modeling the multi-layer nature of the European Air Transport Network: Resilience and passengers re-scheduling under random failures. arXiv preprint arXiv:1211.6839. Preprint available here.
 Boccaletti, S., Bianconi, G., Criado, R., Del Genio, C. I., Gómez-Gardeñes, J., Romance, M., … & Zanin, M. (2014). The structure and dynamics of multilayer networks. Physics Reports, 544(1), 1-122. Preprint available here.
 Cook, A., Tanner, G., Cristóbal, S., & Zanin, M. (2013). New perspectives for air transport performance. Third SESAR Innovation Days, 26th – 28th November 2013. PDF available here.
 Zanin, M. (2015). Can we neglect the multi-layer structure of functional networks?. Physica A: Statistical Mechanics and its Applications. Preprint available here.