Matching Rules for Collective Behaviors on Complex Networks: Optimal Configurations for Vibration Frequencies of Networked Harmonic Oscillators

Abstract

The structure-dynamics-function has become one of central problems in modern sciences, and it is a great challenge to unveil the organization rules for different dynamical processes on networks. In this work, we study the vibration spectra of the classical mass spring model with different masses on complex networks, and pay our attention to how the mass spatial configuration influences the second-smallest vibrational frequency () and the largest one (). For random networks, we find that becomes maximal and becomes minimal if the node degrees are point-to-point-positively correlated with the masses. In these cases, we call it point-to-point matching. Moreover, becomes minimal under the condition that the heaviest mass is placed on the lowest-degree vertex, and is maximal as long as the lightest mass is placed on the highest-degree vertex, and in both cases all other masses can be arbitrarily settled. Correspondingly, we call it single-point matching. These findings indicate that the matchings between the node dynamics (parameter) and the node position rule the global systems dynamics, and sometimes only one node is enough to control the collective behaviors of the whole system. Therefore, the matching rules might be the common organization rules for collective behaviors on networks.

Periodic Coupling inhibits Second-order Consensus on Networks

“…This corresponds to an ensemble of n one-dimensional harmonic oscillators, coupled via linear springs with a timedependent spring constant γW ij (t). Similar harmonic oscillators with time-invariant couplings have been investigated on networks using the classical mass spring model [33]. More importantly, however, the DO model describes the linearized dynamics of non-linearly coupled oscillators, such as the second-order Kuramoto model, which was previously used to investigate synchronization phenomena on power-grids [37,38].…”

“…end we investigate a general model for coupled oscillators with Newtonian dynamics [32] in two different settings: i) linearly coupled damped oscillators (DO) [33], and ii) the classical second-order consensus model (SOC) with velocity alignment [17][18][19][22][23][24][25]. For both models, the interaction strengths are periodically modulated around well defined mean values, which are encoded in a static and symmetric backbone network.…”

Notes on Resonant and Synchronized States in Complex Networks

“…A similar approach is used in [11] where the vibration spectra of a classical mass spring model with different masses on complex networks is investigated; and in [12], where synchronization of a leader-follower version of a system of coupled harmonic oscillators connected by dampers and each attached to fixed supports by identical springs is studied in presence of random noises and time delays with an interaction topology modelled by a weighted directed graph.…”

Taba: A Tool to Analyze the Binding Affinity

“…The use of a mass‐spring approach is ubiquitous in modeling oscillating systems in nature (at least approximately). A mass‐spring approach applies for an idealized spring, and it is a reasonable approximation for a real‐live spring, homogeneous isotropic elastic materials, graphene sheets, the phenomenon of electron tunneling in transistors made from C140, and others complex networks …”

“…A massspring approach applies for an idealized spring, and it is a reasonable approximation for a real-live spring, homogeneous isotropic elastic materials, [7,8] graphene sheets, [9] the phenomenon of electron tunneling in transistors made from C140, [10] and others complex networks. [11] Let us consider arbitrary potential energy (V(x)), and we can expand this potential in a Taylor series around d 0 (the location of the minimum of potential energy). We have the following expression,…”