Avoiding catastrophic failure in correlatednetworks of networks

Saulo D. S. Reis

Levich Institute and Physics Department, City College of New York

NATURE PHYSICS VOL 10, OCTOBER 2014

Paper Alert, October 19, 2015

Saulo D. S. Reis Interconnected Networks Naturally Stable Paper Alert, October 19, 2015 1 / 12

Overview

1 Choosing reasonInterested in how (why) natural networks are more stable thanartificial ones.Prediction of how structured network should be organized in order toacquire stability.

2 FindingsMany networks interact with one another by forming multilayernetworks, but these structures can lead to large cascading failures.If interconnections are provided by network hubs, and theconnections between networks are moderately convergent, the systemof networks is stable and robust to failure.Two independent experiments of functional brain networks (in taskand resting states), which show that brain networks are connectedwith a topology that maximizes stability according to the theory.

Saulo D. S. Reis Interconnected Networks Naturally Stable Paper Alert, October 19, 2015 2 / 12

Degree-degree correlations between interconnectednetworks

Consider two interconnectednetworks, each one having apower-law degreedistribution P(kin) ∼ k−γ

in ,valid up to cutoff kmax

The structure betweeninterconnected networks:

Degree of a node towardsnodes in the othernetwork kout ∼ kα

in

The average indegree ofthe nearest neighbours ofa node in other networkknnin ∼ kβ

in

Saulo D. S. Reis Interconnected Networks Naturally Stable Paper Alert, October 19, 2015 3 / 12

Interact and propagate failure modes

Conditional interaction

A node fails every time itbecomes disconnected fromthe largest component of itsown network, OR loses all itsoutgoing links.

Redundant interaction

A node fails every time itbecomes disconnected fromthe largest component of itsown network, AND loses allits outgoing links.

Measure the fraction ofnodes in the mutuallyconnected giant component

Saulo D. S. Reis Interconnected Networks Naturally Stable Paper Alert, October 19, 2015 4 / 12

Percolation theory

Single network

Two nodes of a network arerandomly linked withprobability p. For low p, thenetwork is fragmented intosubextensive components.

As p increases, there is acritical phase transition pcin which a single extensivecluster or giant componentspans the system

System of networks

Attack form: removal of afraction of 1-p nodes chosen atrandom from both networks.

Critical pc at which a cohesivemutually connected networkbreaks down into disjointsubcomponents under differentforms of attack.

Low pc are robust, high pc areindicative of a fragile network.

Saulo D. S. Reis Interconnected Networks Naturally Stable Paper Alert, October 19, 2015 5 / 12

How to calculate critical pc ?

Saulo D. S. Reis Interconnected Networks Naturally Stable Paper Alert, October 19, 2015 6 / 12

Stability phase diagram from simulation

Saulo D. S. Reis Interconnected Networks Naturally Stable Paper Alert, October 19, 2015 7 / 12

Analysis of interconnected functional brain network

Saulo D. S. Reis Interconnected Networks Naturally Stable Paper Alert, October 19, 2015 8 / 12

Stability phase diagram for brain networks from fMRI

Saulo D. S. Reis Interconnected Networks Naturally Stable Paper Alert, October 19, 2015 9 / 12

Conclusion

Saulo D. S. Reis Interconnected Networks Naturally Stable Paper Alert, October 19, 2015 10 / 12

The End

Saulo D. S. Reis Interconnected Networks Naturally Stable Paper Alert, October 19, 2015 11 / 12

Avoiding catastrophic failure in correlatednetworks of networks

Saulo D. S. Reis

Levich Institute and Physics Department, City College of New York

NATURE PHYSICS VOL 10, OCTOBER 2014

Paper Alert, October 19, 2015

Saulo D. S. Reis Interconnected Networks Naturally Stable Paper Alert, October 19, 2015 12 / 12