How does
network controllability
depend on topology?
Soumya Jyoti Banerjee
Senior Research Fellow
Bose Institute, Kolkata, India
Key to Network Controllability
Soumya Jyoti Banerjee and Soumen Roy
arXiv: 1209.3737v1 (2012)
Structural controllability of networks:
Identify minimum set of nodes that, if driven by different signals, is
sufficient to fully control the network’s dynamics.
●
These nodes are the 'driver nodes'
Algorithm to identify a minimum set of driver nodes:
Matching Links: The maximum set of links of a network that do not share
starting or ending nodes.
Matched Nodes: The end nodes of all the matching links.
Unmatched Nodes: The nodes, which are not matched.
●
Actually these unmatched nodes are the driver nodes.
Liu YY, Slotine JJ & Barabási AL, Controllability of complex networks,
Nature 473, 167-173 (2011).
Is controllability completely decided by network's
in-out degree distribution P(Kin, Kout) ?
Consider, for example, above
two directed N = 4 node
networks: a chain graph, G1,
and a rather densely connected
graph, G2.
Above graphs have very different P(kin,kout) and degree correlations.
So, G1 and G2 are expected to have different number of driver nodes
(ND), according to Liu et al.
●
●
However, ND = 1, for both G1 and G2.
Are distance based network metrics important for controllability
rather than in-out degree ?
Liu YY, Slotine JJ & Barabási AL, Controllability of complex networks,
Nature 473, 167-173 (2011).
Banerjee and Roy, arXiv: 1209.3737v1 (2012)
Structural controllability is defined as: nD = ND/N
where, ND = Number of driver nodes (Calculated using maximum matching
algorithm) and N = Total number of nodes.
●
We observe that controllability of real networks changes
with change of distance based metrics :
●
Closeness (C)
●
Betweenness (B)
Herein, we will examine the dependence of nD on X,
where X is a very simple function:
<C> and <B> are
average closeness and
betweenness centrality
of networks.
Since, <C> ≠ 0 for all
connected networks,
X(C,B) is well defined.
Liu, Y., Slotine, J. and Barabasi, A.L., Controllability of complex network, Nature: 473, 167-173 (2011)
Banerjee and Roy, arXiv: 1209.3737v1 (2012)
FIG. 1. Change in controllability, X = < C > + < B > / < C > versus nD for 32 different
real world networks with different degree distributions.
Fig.1(a) presents a reasonably coherent picture of controllability.
●
nD decreases (i.e controllability increases) with increase of X(C,B).
Enhancement of Controllability within
Same In-Out Distribution:
Figs.(b-d): Change in nD
versus NS [number of X
increasing in-out degree
preserving swaps for little rock
food web (LRFW) and
C. elegans metabolic and
neuronal networks respectively.
Figs.(e-g): Elucidates
interaction between
< C >(red) and < B >(green) in
deciding nD for the same three
networks.
LRFW showed maximum
deviation between nDreal and
nDrand−degree
●
Only 12% of edge swaps lead
to decrease of nD by 35% for
LRFW network.
Banerjee and Roy, arXiv: 1209.3737v1 (2012)
Summary
●
●
●
●
Can controllability be decided completely by the neighborhood of
a node?
Do distant nodes have no direct role in deciding controllability?
X(C,B) is a possible measure of controllability, which captures
both local and global information of a network.
X(C,B) has potential to enhance structural controllability of real
networks merely using few edge swaps without disturbing the
in-out degree distribution P(Kin, Kout) of network.
---------------------------------------------------
Thank
You
Questions ??