Robustness & resilience are often thought of in terms of a system’s capacity to maintain functionality in the face of external perturbations. We see some extraordinary examples of this, ecological networks that persist despite extreme environmental changes. Communication networks like the internet can often deal with malfunction, errors, and attacks, without these local events leading to catastrophic global failures. But we also see the opposite where some small failure in say, a financial network can propagate to affect the whole system. Trying to understand how and why this happens is the study of network robustness.1 Robustness can be correlated with connectivity in that connectivity enables system integration. Without connectivity parts to the system may become disconnected and disintegrated. If blood stops flowing to some part of the body, then it will become atrophied and waste away. Or if a child stops talking to their parents, then the family unit disintegrates through lack of communications. Thus, when we are talking about robustness & resilience, we are often asking what will happen to the network’s overall connectivity and integration if we remove some components or connections, and equally, how will this failure then spread within the network system.
Failure can either be a cause of the network’s nodes, where we are interested in what will happen if we remove a certain amount – this is called node percolation – but we can also talk about failure in terms of the removal of edges; edge percolation. And another key factor here is whether the attack is random or strategic. When we are talking about robustness with respect to the nodes in the network, then a key factor is the degree of distribution between the nodes. The higher that degree distribution, meaning there will be more hubs, the more vulnerable it is to strategic attack. But if it is a random attack, then the degree distribution is not so important, as these hubs that are part of centralized networks will be particularly vulnerable to strategic attack. Thus, distributed networks will be robust to strategic attack but scale-free centralized networks are particularly susceptible. A strategic attack on large hubs will drastically reduce the number of connections within the system, increasing the average path length significantly, which is a key measure of the system’s overall integration. This has been confirmed by empirical data from the Internet and World Wide Web which show robustness to random attack but are significantly affected by strategic attack due to the presence of major hubs in the network.2
Edge percolation is when connections fail or are removed. An important factor here is the degree of betweenness to the network. Betweenness measures the number of bridging edges that represent the critical, irreplaceable connections between one cluster and another. An example of this might be the Malacca Straits, a stretch of sea between the coast of Malaysia and Indonesia that connects maritime transport in Asia with the Middle East and Europe. Approximately 40 percent of the world’s trade and 25 percent of all crude petroleum is thought to pass through this critical link in the global logistics network. This is an example of a bridging link that reduces the system’s robustness and makes it much more susceptible to a strategic attack.
Robustness does not just depend on if a node or edge will fail, but just as importantly what will happen when it fails. In other words, will the failure end there, or will it have a cascading effect as may often be the case? For example, failure in a power grid can result in other power stations becoming overloaded, allowing for the failure to propagate. When we are modeling robustness in terms of some external perturbation that propagates through the system, destroying links and nodes on its way, we want to ask how easily does it spread and what is the resistance to its spreading within the network. One method for preventing failure propagation is through buffers and redundancy. When we engineer networks, these buffers are often artificially superimposed on the network. But when we look at the robustness in ecosystems, it is built into these networks in the form of diversity. Diversity is both a buffer in that the difference between nodes will present a barrier to complete contagion, and it is a form of redundancy as components are also similar; they can to a certain extent just replace other components.
Diversity in connections often comes in the form of weak ties, where the connections in a network can be divided into strong or weak, which is really a way of defining connections in terms of their frequency of interaction.3 In a social network, a strong tie is someone you interact with on a daily basis. Thus, strong ties are often between members of a cluster or clique, whereas a weak tie might be with someone you only talk to once every few months or once a year. Whereas strong ties may dominate a network in terms of quantity, weak ties are important in that they connect nodes from different clusters, making them bridges that add some diversity to the network which can be vital to its robustness if some subset of a network becomes infected. We could cite the phenomenon of ‘group think’ here, where a small social network or clique, often quickly converges to an agreed opinion on a subject without full consideration of different options. But by say, having some external consultant join the group, they would have a weak tie to add a diverse perspective and resist ‘group think’ from prevailing.
Real-world networks are typically dynamic, where the network can adapt to some external perturbation. For example within the internet, routers have routing tables that keep track of how effective any path through the network is, and then update where it sends new packets based upon this. If we remove some set of edges, it will dynamically reroute packets to try and maintain network functionality, and this would be the same for a logistic network, a criminal network, or any other network with some kind of control system that allows it to adapt.
Connectivity & Robustness
Connections enable integration and this is a key source of robustness as it bonds the system’s components together. But connectivity can also be a pathway for disaster spreading. The most recent financial crisis might be a good example of this, where unknown linkages between complex financial instruments and institutions led to rapid contagion. Thus, we need to be aware that every link in the network has a cost in terms of robustness. If it is not contributing to the system’s integration and robustness, then it may be depleting from it as just another pathway for contagion spreading. We should be under no delusion that connectivity is always in some way a good thing. Research has shown that within certain settings connectivity can, up to a certain point, add to the system’s robustness but beyond this hyper-connectivity can just be adding pathways for failures to propagate and overall fragility.4