Effect of correlations on network controllability (1203.5161v2)

Abstract: A dynamical system is controllable if by imposing appropriate external signals on a subset of its nodes, it can be driven from any initial state to any desired state in finite time. Here we study the impact of various network characteristics on the minimal number of driver nodes required to control a network. We find 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 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.

• The paper demonstrates that while degree distribution provides an initial estimate for network controllability, degree correlations significantly influence the number of driver nodes required.
• Specific correlation types show distinct impacts on driver node density ($n_D$): out-in correlations have a linear effect (positive reduces $n_D$, negative increases), while out-out and in-in correlations show a quadratic effect, and in-out correlations have minimal impact.
• The study uses an analytical framework based on the cavity method, numerical simulations, and real network data analysis to validate how correlations affect controllability, showing varied results across different network types.

Effect of Correlations on Network Controllability

Understanding the controllability of complex networks is crucial for various scientific and engineering applications, from biological systems to technological and social networks. This paper provides a detailed examination of how specific network characteristics impact the controllability of these networks, particularly focusing on degree correlations. The paper approaches network controllability using analytical, numerical, and empirical datasets to understand which topological properties influence the minimal number of driver nodes necessary for control.

Key Findings

• Degree Distribution and Network Predictions: The paper outlines that degree distribution, $P(k_{\text{in}}, k_{\text{out}})$, can predict the order of magnitude of the number of driver nodes ($N_D$) required for controllability. However, $N_D$ is affected by degree correlations, which cannot be captured solely by degree distributions.
• Influence of Clustering and Modularity: Simulation results show that clustering coefficients and modularity do not significantly impact the density of driver nodes ($n_D$). Clustering and communities offer negligible influence, contradicting prior assumptions regarding their relevance in network control.
• Degree Correlations: It is highlighted that degree correlations have a discernible effect on network controllability. This effect varies based on the nature of the correlation:
  • Out-In Correlations: Demonstrate a linear relationship with $n_D$. Negative correlations increase the number of driver nodes needed, while positive correlations reduce it.
  • Out-Out and In-In Correlations: Both demonstrate quadratic dependence, meaning correlations among the neighbors affect $$n_D symmetrically, regardless of whether correlations are positive or negative.</li> <li><strong>In-Out Correlations:</strong> Found to have minimal effect on network controllability.</li> </ul></li> </ul> <h3 class='paper-heading'>Analytic Framework</h3> <p>The paper utilizes the cavity method to analytically determine$$n_D$$for networks with specific degree distributions and correlation profiles. The analytical results reveal the linear, quadratic, or nil dependence of$$n_D$$on degree correlation coefficients, aligning well with numerical simulations.</p> <h3 class='paper-heading'>Applicability to Real Networks</h3> <p>The authors test their predictions against real network data, leading to several group categorizations based on correlations and observed deviation$$\Delta$$:</p> <ul> <li><strong>Group A:</strong> Networks like p2p Internet show negligible correlation effects, aligning predicted and observed$$n_D$$values.</li> <li><strong>Group B:</strong> Networks including electric circuits and metabolic maps display positive$$\Delta$values. Superimposed correlations increase$n_D$$, as predicted.</li> <li><strong>Group C:</strong> Networks such as social interaction maps show negative$$\Delta$. Predictions suggest less$n_D$ due to positive out-in correlations.
  • Group D and E: Additional networks varied in degree correlation profiles, showcasing the complexity in real-world applicability of correlation effects.

  Implications and Future Research

  This paper advances our understanding of network controllability by elucidating the nuanced impact of network characteristics. The detailed examination of degree correlations suggests new directions for designing optimal network structures with minimal control input. Future research could explore robust control configurations and further verify the balance of correlations in larger, more diverse network systems.

  The systematic approach presented in this paper offers a valuable framework for researchers seeking to manipulate or optimize control strategies in complex networked systems, enhancing both theoretical comprehension and practical applications.