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# Core percolation and the controllability of complex networks

Held by Márton Pósfai (Eötvös University,
Budapest)

19.03.2013, 16:00

EW 731

Abstract:

Network science has emerged as a prominent field in
complex system research, showing that complexity may arise from the
connectivity patterns that underline a system. Structural transitions
in networks were extensively studied due to their impact on numerous
dynamical processes. Here we study such a transition, the core
percolation on complex networks.

The core of a network is defined
as a spanned subgraph which remains in the network after the following
greedy leaf removal procedure: As long as the network has leaves, i.e.
nodes of degree 1, choose an arbitrary leaf and its neighbour, and
remove them together with all the adjacent links. Finally, we remove
all isolated nodes. For low mean degree the core is small (zero
asymptotically), whereas for mean degree larger than a critical value
the core covers a finite fraction of all the nodes. We analytically
solve the core percolation problem for networks with arbitrary degree
distributions. We show that for undirected networks the transition is
continuous while for directed networks it is discontinuous (and
hybrid) if the in- and out-degree distributions differ.

As an
application we use core percolation to explain phenomena related to
the controllability of networks. Recent advances in applying control
theory to complex networks have o?ered the tools to identify the
driver nodes, through which we can achieve full control of a system.
These tools predict the existence of multiple control configurations,
prompting us to classify each node in a network based on their role in
control: a node is critical if a system cannot be controlled without
it; intermittent if it acts as a driver node only in some
con?gurations; and redundant if it does not play a role in control. We
find that above the core percolation threshold networks fall into two
classes: (i) either they are in a centralized control mode, being
driven by only a tiny fraction of nodes, (ii) or a distributed control
mode, when most nodes play some role in controlling the system. We
show that the control mode of a network depends on the structure of
the emerging core.