We live surrounded by networks: traffic, infrastructures, WWW, etc. We also live embedded into networks, such as those formed by our familial and friendship ties, collaborations, social media. But more importantly, we are conformed by networks, such as our muscle-bone networks, organ-organ communication networks, vascular webs, brain networks, among others. All these networks exist to allow the navigation of items between their nodes at different time scales. How is it that we are not constantly suffering from the collapse of these networks due to such intensive navigational use? Our hearts are submitted to a heavy blood flow, particularly when we perform exercise training. However, our hearts create “collateral vessels” or “natural bypasses”, formed by very small, hair-like vessels (capillaries) that interconnect coronary arteries and their service areas. These natural bypasses facilitate the blood flow in situations of high demand, such as during exercise training or when occlusions are present in the main arteries.
IFISC (UIB-CSIC) researchers Ernesto Estrada and Lucas Lacasa, together with Jesús Gómez-Gardeñes (Universidad de Zaragoza), have published a study in Proceedings of the National Academy of Sciences of the USA which shows how the formation and evolution of networks has a hidden mechanism for naturally creating bypasses that, they show, facilitate network navigability. This mechanism increases the efficiency of the communication between pairs of entities (nodes) in a network as the system complexifies, by saving some energy in the process that the system can use later for other functions. In a network there are always paths that connect two nodes with a minimum number of steps. These are called shortest (geodesic) paths between the corresponding nodes. However, to navigate a network using shortest paths one necessary needs to have a map of its global structure, something which is an impossible task in many real-world situations like in the brain. Then, most of the navigation in these networks occurs through diffusive processes, such as when coffee is dropped in a glass of milk. The authors discovered that during network evolution, some randomly-created paths emerge as an alternative to the shortest ones. While these alternative paths connect the same pair of nodes using routes slightly longer than the shortest path, these paths in turn traverse less connected regions of the network.
The authors developed a mathematical theory that showed that these alternative paths allow a navigator without a map of the whole network to arrive at their destination with less chances of getting lost than when using the shortest path. To quantify whether a “blind” navigator would prefer to go through the shortest or the alternative path they defined a way to quantify the amount of energy saved by going through the alternative route. Sometimes, navigating through these alternative routes compensate the navigator to use a slightly longer path. Researchers called it network “bypasses”. Bypasses cannot be created without a certain cost to the network. Such cost is paid in terms of disorder, i.e., increasing the entropy of the network. However, they found that a little increase in the disorder of a network is enough to generate useful bypasses.
Estrada, Gómez-Gardeñes and Lacasa applied their findings to study a wide range of real-world networks ranging from social, technological, infrastructural, biological, and informational ones. In general, they found that most of the networks contain bypasses, but some contain significantly more than others. At the bottom of the ranking, they found networks that had not been designed to be navigated in a random way, such as power grids or electronic circuits. The lack of effective bypasses in these networks is an indication of their vulnerability, as we have seen during blackouts –a power grid failure cascade--, the authors argue. In these cases, the amount of energy saved by using the existing bypasses is indeed relatively low.
Conversely, the authors found networks --such as the human brain-- where using bypasses saves a lot of energy and time in the communication process. For instance, the analysis of a human brain network displaying regions that coactivate together identifies that there are pairs of regions whose communication via bypasses represents more than 200% of time saved in comparison with the navigation via the shortest path, which points out the importance of these results to understand the plasticity of complex systems.
Image: A network in which the nodes (yellow and green balls) are connected with links. The blue path linking the two green nodes is the shortest, but it turns out that the pink "brachistochronous" path is more advantageous, as it avoids congestion and is less resistive. This pink path constitutes a bypass.
Estrada, Ernesto, et al. Network Bypasses Sustain Complexity. Proceedings of the National Academy of Sciences (2023). https://doi.org/10.1073/pnas.2305001120.