Say we have a complex network made of n sub-networks and m nodes. Some of the sub-networks share some of the m nodes. Say that such complex network (aka Interdependent Network) is under attack, and say that this attack is not targeted (e.g., does not look for high degree nodes only) nor random, but spatial (both low degree and high degree nodes are being removed). Now, say that the failure cause is external, in addition to being spatial, and that it can feature many levels of spatial extent. Hence, the higher the level, the higher the number of nodes involved, the higher the disruption (theoretically). My problem relates to the failure threshold qc (the minimum size of the disrupting event that is capable of producing 0 active nodes after the attack).
My question: does the failure threshold qc depend on how nodes are connected only (e.g., the qc is an intrinsic feature of the network)? Or is it a function of how vast the spatial attack is? Or does it depend on both?
Thank you very much to all of you.
Francesco
You have not said how failure propagates, that is how failure of some nodes causes others to fail.
Have you already explored any modelling or analytical methods?
Stephen,
thank you for your answer. I have not explored any model yet as I am in the preliminary phase of my research. At this point, it would be a crucial piece of understanding if I realized what is the relationship between failure threshold and spatial extent of an attack.
I should have added that the generic interdependent network I mentioned features both directed links and undirected links. I am aware of the fact that such clues might not be enough for you to give an answer, but as I said, this is a generic network, and my concern is to figure out if the failure threshold can vary according to the spatial extent of the attacks or not, and to what degree.
¿0 active nodes? are you considered this paper http://epjti.epj.org/images/stories/news/2014/e2014-40876-y.pdf ? because it is not talking about active nodes, but nodes disconnected of the giant (or the largest) component.
And, of course, qc depend on the extent, the topology and some other measures. Cascades in interdependent networks is an active research topic nowadays and you'll find a lot of information.
Prof. Rebollo,
thank you for your contribution.
I will try to make my question more specific, after having read the paper you suggested:
- Does the spatial extent of a (spatial) attack have an influence on failure thresholds?
- What are the factors influencing failure thresholds in Interdependent Networks? (more generic)
Thank you for your time.
I wonder the qc will highly relate to the propagation mechanism. Under different failure assumption, the propagation, or, cascading failure in network, will have different behavior. The theoretical results you mention should work only to network traffic problem (by AE Motter and YC Lai)
Yes it's likely to depend on the structure of the network, as it does for a simple network. The situation you describe is quite specific, so I'm not sure the answer is known. The best way is to try to find out!
You might find this review helpful, for some basic ideas http://arxiv.org/abs/0705.0010
Addresing the threshold through a consideration of the range of tolerance for failure: the lower bound for failure is what Charles Perrow would call tightly-coupled systems and the upper bound would be loosely- coupled systems. Some subsystems may vary in their degree of coupling (tight or loose). Hence where complex networks are concern, the incidence of failure in tightly-coupled subsystems is higher, since the level of redundancy of such networks is low to zero, and the interdependency is the highest.Interactive complexity may exacerbate the whole situation, resulting in cascading effects derived from discrete failures/isolated spatial attacks.
http://www.ohio.edu/people/piccard/entropy/perrow.html
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