^{1,a)}

### Abstract

In a recent paper, it has been suggested that the controllability of a diffusively coupled complex network, subject to localized feedback loops at some of its vertices, can be assessed by means of a Master Stability Function approach, where the network controllability is defined in terms of the spectral properties of an appropriate Laplacian matrix. Following that approach, a comparison study is reported here among different network topologies in terms of their controllability. The effects of heterogeneity in the degree distribution, as well as of degree correlation and community structure, are discussed.

In recent years, synchronization of complex networks of coupled dynamical systems has become a research subject of increasing attention within the scientific community. This is partly motivated by the frequent observation of synchronization phenomena in a wide range of different contexts, ranging from biology to medical and social sciences. On the other hand, this is because synchronization is considered a paradigmatic example of phase transitions that under certain circumstances may occur when ensembles of dynamical systems are coupled together. Generally speaking, several different processes may lead to the onset of synchronization. Namely, it may be the result of a self-organized process occurring when several coupled dynamical systems, starting from different initial conditions, converge in the same dynamical evolution (which is in general not known

**); or it can be provoked by some feedback loops driving the set of all the dynamical systems toward a desired predetermined reference evolution. In this paper, we focus on synchronization of diffusively coupled networks. Moreover, we consider that a subset of nodes are selected to be controlled. This assumption is motivated by real-world network observation, where a decentralized control action is often applied only to part of the nodes. For instance, pacemaker cells have been observed to regulate several functions in living organisms; other examples are present in human networks, where particular individuals, called**

*a priori***, have been observed to be capable to influence the network collective dynamics. Here, the controllability of a given complex network, i.e., its propensity to being controlled onto a given reference evolution by means of a decentralized control action, is defined as the width of the range of the coupling strength term among the oscillators, which stabilizes the reference evolution. A detailed comparison among different complex networks in terms of their controllability, characterized by different degree distributions, degree correlation properties, as well as community structure, will be reported.**

*leaders*The author acknowledges help in the writing of the paper from Guanrong Chen. He thanks Franco Garofalo, Mario di Bernardo, Guanrong Chen, Zhengping Fan, Mark Avrum Gubrud, and Zeynep Tufekci for useful comments and discussions.

I. INTRODUCTION

II. A STRUCTURAL MEASURE OF NETWORK CONTROLLABILITY

III. EFFECTS OF HETEROGENEITY IN THE DEGREE DISTRIBUTION

IV. EFFECTS OF DEGREE CORRELATION

V. LINEAR AND SQUARE LATTICES

VI. COMMUNITY STRUCTURE

VII. CONCLUSIONS

### Key Topics

- Networks
- 49.0
- Network topology
- 18.0
- Oscillators
- 9.0
- Eigenvalues
- 7.0
- Statistical properties
- 6.0

## Figures

Networks of nodes, edges, . and are plotted vs in networks characterized by different degree distribution . The legend is as follows: , (stars); , (diamonds); , (plus); , (circles); , (squares).

Networks of nodes, edges, . and are plotted vs in networks characterized by different degree distribution . The legend is as follows: , (stars); , (diamonds); , (plus); , (circles); , (squares).

(Color online) A scale-free network of nodes with degree distribution exponent is considered. The figure shows the eigenratio as varying both the control gain with and the number of controlled nodes (in terms of the probability ).

(Color online) A scale-free network of nodes with degree distribution exponent is considered. The figure shows the eigenratio as varying both the control gain with and the number of controlled nodes (in terms of the probability ).

(Color online) Control error at regime as a function of the control gain with and the probability under the same conditions as in Fig. 2 .

(Color online) Control error at regime as a function of the control gain with and the probability under the same conditions as in Fig. 2 .

(Color online) A scale-free network of nodes with degree distribution exponent is considered. The figure shows the eigenratio as varying both the control gain with and the number of controlled nodes (in terms of the probability ).

(Color online) Control error at regime as a function of the control gain with and the probability under the same conditions as in Fig. 4 .

(Color online) Control error at regime as a function of the control gain with and the probability under the same conditions as in Fig. 4 .

(Color online) Networks of nodes. The controllability is plotted vs the number of controlled nodes for networks characterized by different degree distribution exponents: in (a), in (b), in (c), in (d). Black (red) is used for random (selective) pinning. Different lines represent different values of the control gain , ranging from to .

(Color online) Networks of nodes. The controllability is plotted vs the number of controlled nodes for networks characterized by different degree distribution exponents: in (a), in (b), in (c), in (d). Black (red) is used for random (selective) pinning. Different lines represent different values of the control gain , ranging from to .

(Color online) Random pinning of networks characterized by different degree correlation properties, , , . Here we consider a scale-free heterogeneous network with degree distribution exponent . The controllability is reported vs the number of controlled nodes , by having in each picture a different value of the control gain : (a) , (b) , (c) , (d) , (e) , (f) . Different degree correlation properties are indicated by different symbols: Triangles are used in the case of assortative networks , squares in the case of disassortative networks , and circles in the case of networks that do not display degree correlation .

(Color online) Random pinning of networks characterized by different degree correlation properties, , , . Here we consider a scale-free heterogeneous network with degree distribution exponent . The controllability is reported vs the number of controlled nodes , by having in each picture a different value of the control gain : (a) , (b) , (c) , (d) , (e) , (f) . Different degree correlation properties are indicated by different symbols: Triangles are used in the case of assortative networks , squares in the case of disassortative networks , and circles in the case of networks that do not display degree correlation .

(Color online) Selective pinning of networks characterized by different degree correlation properties, , , . Here we consider a scale-free heterogeneous network with degree distribution exponent . The controllability is reported vs the number of controlled nodes , by having in each picture a different value of the control gain : (a) , (b) , (c) , (d) , (e) , (f) . Different degree correlation properties are indicated by different symbols: Triangles are used in the case of assortative networks , squares in the case of disassortative networks , and circles in the case of networks that do not display degree correlation .

(Color online) Selective pinning of networks characterized by different degree correlation properties, , , . Here we consider a scale-free heterogeneous network with degree distribution exponent . The controllability is reported vs the number of controlled nodes , by having in each picture a different value of the control gain : (a) , (b) , (c) , (d) , (e) , (f) . Different degree correlation properties are indicated by different symbols: Triangles are used in the case of assortative networks , squares in the case of disassortative networks , and circles in the case of networks that do not display degree correlation .

A monodimensional lattice of nodes is considered. The controllability indices and are plotted as functions of the control gain . The legend is as follows: (circles), (triangles), (stars), (diamonds), and (plus).

A monodimensional lattice of nodes is considered. The controllability indices and are plotted as functions of the control gain . The legend is as follows: (circles), (triangles), (stars), (diamonds), and (plus).

A monodimensional lattice of nodes is considered. Assuming that the controlled nodes are node 1 and node , the controllability indices and are plotted vs . Different lines indicate different control gains : (asterisks), (circles), (no marker), and (dots).

A monodimensional lattice of nodes is considered. Assuming that the controlled nodes are node 1 and node , the controllability indices and are plotted vs . Different lines indicate different control gains : (asterisks), (circles), (no marker), and (dots).

A periodic square lattice of nodes is considered. The controllability indices and are plotted as functions of the control gain . The legend is as follows: (circles), (triangles), (stars), (diamonds), and (plus).

A periodic square lattice of nodes is considered. The controllability indices and are plotted as functions of the control gain . The legend is as follows: (circles), (triangles), (stars), (diamonds), and (plus).

A square lattice of nodes is considered. Assuming that two of its nodes have been chosen to be controlled, the controllability indices and are plotted vs their distance . Different lines indicate different control gains : (asterisks), (circles), (no marker), and (dots).

A square lattice of nodes is considered. Assuming that two of its nodes have been chosen to be controlled, the controllability indices and are plotted vs their distance . Different lines indicate different control gains : (asterisks), (circles), (no marker), and (dots).

(Color online) A random network of nodes composed of two communities is considered. The first one, , includes 125 nodes and the second one, , includes the remaining (375) ones. The network controllability is plotted vs in (a) and (b) [(c) and (d)] as is varied between 1 and 0, according to the technique described in the text. The pinning probability is assumed to be . Different lines indicate several values of the control gain : (points), (no marker), (squares), (diamonds), (times), and (asterisks). The left plots (a) and (c) show the cases in which the controllers are evenly distributed among the two communities. The right plots (b) and (d) show a situation in which 95% of the controllers belong to and the rest to . Observe the logarithmic scale on the axis.

(Color online) A random network of nodes composed of two communities is considered. The first one, , includes 125 nodes and the second one, , includes the remaining (375) ones. The network controllability is plotted vs in (a) and (b) [(c) and (d)] as is varied between 1 and 0, according to the technique described in the text. The pinning probability is assumed to be . Different lines indicate several values of the control gain : (points), (no marker), (squares), (diamonds), (times), and (asterisks). The left plots (a) and (c) show the cases in which the controllers are evenly distributed among the two communities. The right plots (b) and (d) show a situation in which 95% of the controllers belong to and the rest to . Observe the logarithmic scale on the axis.

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