This paper examines how interconnected networks can fail catastrophically or remain stable. The author finds that if interconnections are provided by network hubs and the connections between networks are moderately convergent, the system of networks is stable and robust to failure. Two experiments on functional brain networks show that the brain's network topology maximizes stability, as the theory predicts interconnected networks will be stable if the connections are through hubs and moderately convergent.

1. Avoiding catastrophic failure in correlated
networks 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

2. Overview
1 Choosing reason
Interested in how (why) natural networks are more stable than
artificial ones.
Prediction of how structured network should be organized in order to
acquire stability.
2 Findings
Many networks interact with one another by forming multilayer
networks, but these structures can lead to large cascading failures.
If interconnections are provided by network hubs, and the
connections between networks are moderately convergent, the system
of networks is stable and robust to failure.
Two independent experiments of functional brain networks (in task
and resting states), which show that brain networks are connected
with a topology that maximizes stability according to the theory.
Saulo D. S. Reis Interconnected Networks Naturally Stable Paper Alert, October 19, 2015 2 / 12

3. Degree-degree correlations between interconnected
networks
Consider two interconnected
networks, each one having a
power-law degree
distribution P(kin) ∼ k−γ
in ,
valid up to cutoff kmax
The structure between
interconnected networks:
Degree of a node towards
nodes in the other
network kout ∼ kα
in
The average indegree of
the nearest neighbours of
a node in other network
knn
in ∼ kβ
in
Saulo D. S. Reis Interconnected Networks Naturally Stable Paper Alert, October 19, 2015 3 / 12

4. Interact and propagate failure modes
Conditional interaction
A node fails every time it
becomes disconnected from
the largest component of its
own network, OR loses all its
outgoing links.
Redundant interaction
A node fails every time it
becomes disconnected from
the largest component of its
own network, AND loses all
its outgoing links.
Measure the fraction of
nodes in the mutually
connected giant component
Saulo D. S. Reis Interconnected Networks Naturally Stable Paper Alert, October 19, 2015 4 / 12

5. Percolation theory
Single network
Two nodes of a network are
randomly linked with
probability p. For low p, the
network is fragmented into
subextensive components.
As p increases, there is a
critical phase transition pc
in which a single extensive
cluster or giant component
spans the system
System of networks
Attack form: removal of a
fraction of 1-p nodes chosen at
random from both networks.
Critical pc at which a cohesive
mutually connected network
breaks down into disjoint
subcomponents under different
forms of attack.
Low pc are robust, high pc are
indicative of a fragile network.
Saulo D. S. Reis Interconnected Networks Naturally Stable Paper Alert, October 19, 2015 5 / 12

6. How to calculate critical pc ?
Saulo D. S. Reis Interconnected Networks Naturally Stable Paper Alert, October 19, 2015 6 / 12

7. Stability phase diagram from simulation
Saulo D. S. Reis Interconnected Networks Naturally Stable Paper Alert, October 19, 2015 7 / 12

8. Analysis of interconnected functional brain network
Saulo D. S. Reis Interconnected Networks Naturally Stable Paper Alert, October 19, 2015 8 / 12

9. Stability phase diagram for brain networks from fMRI
Saulo D. S. Reis Interconnected Networks Naturally Stable Paper Alert, October 19, 2015 9 / 12

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

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

12. Avoiding catastrophic failure in correlated
networks 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