#### Hybrid phase transition in the collapse of multiplex networks

- Event time: 11:30am
- Event date: 10th July 2013
- Speaker: Gareth Baxter (University of Aveiro Portugal)
- Location: Room 2511, James Clerk Maxwell Building (JCMB) James Clerk Maxwell Building Peter Guthrie Tait Road Edinburgh EH9 3FD GB

### Event details

Networks can be a useful representation of many complex systems. However, complex interactions between different systems or sub-systems cannot always be captured by a single network. Instead, we can consider a set of networks with interconnections between them, or a multiplex network, which is a set of nodes with multiple kinds of links between them. Real world examples include interdependent infrastructure (water, electricity, communications etc.), different types of economic relationships and complex interactions in ecological systems.

Interdependencies between different networks can lead to increased fragility. Failures in one system can lead to failures in another which depends on it in order to function, which in turn propagate back to the first network. Recent theoretical work [Buldyrev et al. 2010] has shown that this interdependency can lead to a discontinuous collapse of the network, as opposed to the continuous transition found in a single network. I will show that this discontinuous transition is hybrid in nature [Baxter et al. 2012], having a discontinuity like a first order transition, but critical behaviour (on one side only) similar to a second order transition. Similar transitions also appear in other network processes, such as bootstrap percolation and the $k$-core subgraph [Baxter et al. 2011]. Avalanches of damage propagate from the removal of a single node. The statistics of these avalanches reveals the critical divergence at the phase transition point. The approach to the critical point from above is signalled by increasing mean avalanche size, which finally diverges at the critical point. Below the transition, on the other hand, there are no critical precursors. The discontinuous transition persists, albeit at a small size, even in scale-free multiplex networks whenever the mean degree of at least one of the interdependent networks does not diverge.

* Buldyrev, Parshani, Paul, Stanley and Havlin, Nature 464, 08932 (2010). * Baxter, Dorogovtsev, Goltsev and Mendes, Phys. Rev. Lett. 109, 248701 (2012). * Baxter, Dorogovtsev, Goltsev and Mendes, Phys. Rev. E 83, 051134 (2011).

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