Controllability and maximum matchings of complex networks

Previously, the controllability problem of a linear time-invariant dynamical system was mapped to the maximum matching (MM) problem on the bipartite representation of the underlying directed graph, and the sizes of MMs on random bipartite graphs were calculated analytically with the cavity method at zero temperature limit. Here we present an alternative theory to estimate MM sizes based on the core percolation theory and the perfect matching of cores. Our theory is much more simplified and easily interpreted, and can estimate MM sizes on random graphs with or without symmetry between out- and in-degree distributions. Our result helps to illuminate the fundamental connection between the controllability problem and the underlying structure of complex systems.

Figure 1

An example of a directed graph, its bipartite representation, its MM and MDNS, and the Karp-Sipser algorithm on it. (a) A small directed graph with five vertices and six arcs. (b) The undirected bipartite representation of (a). (c) A MM of three edges in red dashed lines on the undirected graph in (b). Correspondingly, the in-vertices in dashed circles are unmatched. (d) A configuration of MDNS, in which the driver nodes or the actuators in dashed circles are simply the unmatched in-vertices in (c). (e) A bipartite representation of a hypothetical directed graph with six vertices and 12 arcs. (f) A MM of five edges in red dashed lines from the Karp-Sipser algorithm on the undirected graph in (e). The first stage of the GLR procedure consists of three elementary steps, after which the three red filled circles outside the trapezoid are roots and the five vertices and the six edges enclosed in the trapezoid form the core. The three dashed edges adjacent to these roots are matched edges constructed from the GLR procedure. In the core, a randomized edge removal procedure is applied as an edge $(4^{+},6^{-})$ is removed and correspondingly matched. On the residual graph after the randomized edge removal procedure, the second stage of the GLR procedure involves only one elementary step, in which the root $5^{-}$ is removed. Correspondingly, an edge $(6^{+},5^{-})$ [chosen randomly from $(5^{+},5^{-})$ and $(6^{+},5^{-})$] is matched. Till now the bipartite graph (f) leaves no edge and the set of dashed edges constitutes a matching for (e). The in-vertex $1^{-}$ in a dashed circle is the only unmatched vertex in the in-vertex set.

Figure 2

The relative sizes of MM $y$, the fractions of out- and in-vertices in cores $n^{\pm }$, and the fraction of roots $w$ on random directed graphs. (a)–(c) Results on the directed random regular (RR) graphs, directed Erdös-Rényi (ER) random graphs, and directed asymptotical scale-free networks generated with the static model (SM) with degree exponents ${\gamma }_{+}={\gamma }_{-}=3.0$, which are listed from left to right, respectively. (d)–(f) Results on the random directed graphs with Poissonian out-degree distribution and power-law in-degree distribution with an exponent ${\gamma }_{-}=3.0$ generated with the static model (ERSM) and the directed asymptotical scale-free networks generated with the static model (SM) with ${\gamma }_{+}=3.0$ and ${\gamma }_{-}=2.7,2.5,2.4$, which are listed from left to right, respectively. (a), (d) Results of $y$. (b), (e) Results of $n^{\pm }$. (c), (f) Results of $w$. Each sign is for the simulation result on a single graph instance with a vertex size $N={10}^5$. Each solid line is for the analytical result on infinitely large random graphs. Each dotted line in (e) and (f) is for a discontinuity in the analytical result.

Figure 3

The differences of the MM sizes $\mathrm{\Delta }y$, the core asymmetries $\mathrm{\Delta }n$, and the root sizes $\mathrm{\Delta }w$ between simulation and analytical theory on 24 real-world network instances. $\mathrm{\Delta }y,\mathrm{\Delta }n$, and $\mathrm{\Delta }w$ are shown against the degree asymmetry $\mathrm{\Delta }c$ in (a), (b), and (c), respectively.