Controlling edge dynamics in complex networks

Journal name:
Nature Physics
Year published:
Published online


The interaction of distinct units in physical, social, biological and technological systems naturally gives rise to complex network structures. Networks have constantly been in the focus of research for the past decade, with considerable advances in the description of their structural and dynamical properties. However, much less effort has been devoted to studying the controllability of the dynamics taking place on them. Here we introduce and evaluate a dynamical process defined on the edges of a network, and demonstrate that the controllability properties of this process significantly differ from simple nodal dynamics. Evaluation of real-world networks indicates that most of them are more controllable than their randomized counterparts. We also find that transcriptional regulatory networks are particularly easy to control. Analytic calculations show that networks with scale-free degree distributions have better controllability properties than uncorrelated networks, and positively correlated in- and out-degrees enhance the controllability of the proposed dynamics.

At a glance


  1. SBD on a simple network and its mapping to the line graph.
    Figure 1: SBD on a simple network and its mapping to the line graph.

    a, An example network G with six vertices and nine edges. The SBD takes place on the edges of the network. b, The line graph L(G) corresponding to G. A linear time-invariant dynamics on the vertices of this network is equivalent to the SBD on G. Node labels refer to the endpoints of the edges in G to which they correspond. c, Applying the minimum input theorem to L(G) yields disjoint control paths, denoted by different line styles. d, The control paths in G, mapped back from L(G). Note how each path in L(G) became an edge-disjoint walk in G. Numbers represent the order in which the edges have to be traversed in the walks. The two driver nodes are a and e because each walk starts from either a or e.

  2. Expected fraction of driver nodes nD in various model networks.
    Figure 2: Expected fraction of driver nodes nD in various model networks.

    anD in Erdős–Rényi (ER) and exponential (Exp) networks as a function of the average degree left fencekright fence. bnD in scale-free networks with exponential cutoff as a function of the exponent γ of the degree distribution, for different cutoff values κ. On both panels, symbols denote the results of simulations on networks with 105 nodes, solid lines correspond to the analytical results.

  3. The dependence of the fraction of driver nodes nD on the one-point degree correlation [rho].
    Figure 3: The dependence of the fraction of driver nodes nD on the one-point degree correlation ρ.

    anD in Erdős–Rényi networks with different average degree left fencekright fence as a function of ρ. b, Fraction of driver nodes nD in scale-free networks with different exponents γ as a function of ρ. On both panels, every fifth data point is marked by a symbol. Each data point was obtained by averaging at least 20 different realizations of the network model; error bars were omitted as they were smaller than the symbols. Note that it is hard to introduce negative degree correlations in the case of scale-free networks and none of our test runs managed to decrease the correlation below −0.2.


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Author information


  1. Department of Biological Physics, Eötvös Loránd University, Pázmány Péter sétány 1/a, 1117 Budapest, Hungary

    • Tamás Nepusz &
    • Tamás Vicsek
  2. Statistical and Biological Physics Research Group of the Hungarian Academy of Sciences, Pázmány Péter sétány 1/a, 1117 Budapest, Hungary

    • Tamás Nepusz &
    • Tamás Vicsek


T.N. devised the SBD model and performed analytical calculations and simulations.T.V. initiated the research and supervised the project. T.N. and T.V. wrote the paper.

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The authors declare no competing financial interests.

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