Ph.D. Thesis – Summary of results
Structure and controllability of complex
networks
M´arton P´osfai
Supervisor: Prof. G´abor Vattay, D.Sc.
Department of Physics of Complex Systems
E¨otv¨os Lor´and University, Budapest, Hungary
Graduate School of Physics
Head of School: Prof. L´aszl´o Palla, D.Sc.
Doctoral Program for Statistical Physics,
Biological Physics and Physics of Quantum Systems
Head of Program: Prof. Jen˝o K¨
urti, D.Sc.
Budapest
2014
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Introduction
Complex systems consist of many interacting elements, and the web of these interactions
are best described by complex networks. Therefore, studying such networks and exploring
the consequences of their properties is essential to understand complexity. While during
the past decade significant efforts have been devoted to understand the structure, evolution and dynamics of complex networks, only recently has attention turned to an equally
important problem: our ability to control them. Liu et al. used the framework of structured linear systems to investigate the controllability of complex networks. Particularly,
they focused on identifying the minimum set of driver nodes, such that by directly imposing appropriate external signals on only these driver nodes, the entire system can be
controlled, i.e. it can be driven from any initial state to any final state in finite time.
This provided tools that enabled the research community to apply the full arsenal of
network science to the problem, uncovering various nontrivial phenomena emerging from
the complexity of the structure of the system.
The aim of my PhD project was to investigate how various network characteristics
influence controllability. For this, first core percolation was studied on directed networks.
Core percolation is a structural phase transition not only interesting on its own right,
but it is also closely related to the controllability of networks. Indeed, the analytical
framework developed here was later employed to solve control related questions. I then
investigated how known general properties of real systems (clustering, degree correlations
and community structure) affect controllability using numerical simulations, analytical
calculations and measurements in real networks. Then, observing that the minimal driver
node set is not unique, the nodes were categorized according to role in control, uncovering
that above the core percolation transition the network can be in two drastically different
control modes. I also examined the structural controllability of systems for which the
timescale of the dynamics we control and the timescale of changes in the network are
comparable. I proved the independent path theorem which connects structural controllability to temporal network features. Relying on this theorem, I calculated the controllable
subnetwork using a single input in a temporal network both in model and real systems.
Depending on the density of interactions in the system, the emergence of a giant controllable subspace was witnessed.
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Methods
My PhD thesis relies on the framework of structured systems and structural controllability which connects linear controllability to a purely graph combinatorial problem, thus
making it possible to investigate control related problems using only information about
network structure. I heavily made use of the minimum input theorem of Liu et al., who
showed that finding the minimum set of driver nodes in a network is equivalent to finding
the maximum matching.
Numerical simulations were carried out on Erd˝os-R´enyi networks, scale-free networks
generated by the static model and on publicly available real network datasets. In Chapter 3, correlations were added to the networks using a combination of degree preserving
rewiring and simulated annealing. The maximum matching of the networks was calculated
using the Hopcroft-Karp algorithm. All algorithms were implemented in C++ by the author. The analytical solutions for uncorrelated and degree correlated model networks with
arbitrary degree distribution were derived using the generating function formalism.
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3.1
Results
Core percolation and structural controllability
Core percolation was analytically studied on complex networks with arbitrary
degree distributions. The condition for core percolation was derived, and it was found
that purely scale-free networks have no core for any degree exponents. It was shown that
for undirected networks if core percolation occurs then it is always continuous while for
directed networks it becomes discontinuous when the in- and out-degree distributions
are different. I uncovered the connection between core percolation, maximum
matching and structural controllability, allowing us to use the framework developed
here to address control related questions.
Related publications: [1] and [5].
3.2
Effect of correlations on the controllability of complex networks
I studied the impact of various network characteristics on the minimal number of driver
nodes required to control a network. I found that clustering and modularity have
no discernible impact, but the symmetries of the underlying matching problem can produce linear, quadratic or no dependence on degree correlation
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coefficients, depending on the nature of the underlying correlations. The results are supported by numerical simulations and help narrow the observed gap between
the predicted and the observed number of driver nodes in real networks.
Related publications: [4].
3.3
Emergence of bimodality in network controllability
The observation that the set of driver nodes is not unique prompts us to classify each node
in a network based on their role in control. Accordingly, a node is critical, intermittent
or redundant if it acts as a driver node in all, some or none of the control configurations.
An algorithm was developed to identify the category of each node, and we provided an
analytical solution for uncorrelated networks with arbitrary degree distributions. For
model networks, below a critical average degree the fraction of redundant
nodes increases with the average degree. Above the critical point the network
can be in two drastically different modes: in the distributed mode the fraction of
redundant nodes further increases, meaning that most nodes can be used as drivers; and
in the centralized mode, the fraction of redundant nodes drops, that is only a small subset
of nodes can be drivers. I showed that the emergence of bimodality coincides with the
core percolation and the control mode of a network is determined by the structure of the
core.
Related publications: [1] and [3].
3.4
Structural controllability of temporal networks
I examined the structural controllability of systems for which the timescale
of the dynamics we control and the timescale of changes in the network are
comparable. By mapping the time-varying network to a larger static system, I proved for
discrete time dynamics the independent path theorem linking controllability of temporal
networks to temporal network characteristics. Thereby allowing us to formulate questions
using information about the network topology only. I proved that the minimum input
node problem is NP-complete. And I provided a polynomial time algorithm to compute
the maximum controllable subnetwork for a given set of input nodes.
Related publications: [2].
3.5
Phase transition in the controllability of temporal networks
I investigated the average controllable subnetwork using a single input in a temporal
network. I developed an analytical solution for a generic class of model networks. A
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phase transition was witnessed depending on the density of the interactions: for temporal
networks with hki < 1 in each time step, only finite number of nodes can be controlled;
hki ≥ 1 a giant controllable subspace emerges. I showed the existence of the two phases in
real-world networks. Using randomization procedures, we found that the overall activity
and the degree distribution of the underlying network are the main features influencing
controllability.
Related publications: [2].
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4
Publications
4.1
Publications used in thesis
1. T. Jia, and M. P´osfai. Connecting Core Percolation and Controllability of Complex
Networks, Scientific Reports 4:5379 (2014).
2. M. P´osfai and P. H¨ovel. Structural controllability of temporal networks, arXiv preprint
arXiv:1312.7595 (2013) – under review
3. T. Jia, Y.-Y. Liu, E. Cs´oka, M. P´osfai, J.-J. Slotine and A.-L. Barab´asi. Emergence
of bimodality in controlling complex networks, Nature Communications, 4:2002
(2013)
4. M. P´osfai, Y.-Y. Liu, J.-J. Slotine and A.-L. Barab´asi. Effect of correlations on
network controllability, Scientific Reports, 3:1067, (2013)
5. Y.-Y. Liu, E. Cs´oka, H. Zhou and M. P´osfai.* Core percolation on complex networks,
Phys. Rev. Lett., 109:205703, (2012)
*All authors contributed equally
4.2
Other publications
1. D. Kondor, I. Csabai, J. Sz¨
ule, M. P´osfai and G. Vattay. Inferring the interplay of
network structure and market effects in Bitcoin – under review
2. D. Kondor, M. P´osfai, I. Csabai and G. Vattay. Do the rich get richer? An empirical
analysis of the Bitcoin transaction network, PLoS ONE, 9(2):e86197 (2014)
3. B. T´oth, R. Boha, M. P´osfai, Z. A. Ga´al, A. K´onya, C. J. Stam and M. Moln´ar.
EEG synchronization characteristics of functional connectivity and complex network
properties of memory maintenance in the delta and theta frequency bands, Int J
Psychophysiol., 83(3):399-402, (2012)
4. M. P´osfai, A. Fekete, and G. Vattay. Shortest path sampling of dense homogeneous
networks, EPL 89:18007, (2010)
5. A. Fekete, G. Vattay and M. P´osfai. Shortest path discovery of complex networks,
Phys. Rev. E, 79:065101(R), (2009)
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