Synchronization in complex networks of phase oscillators: A survey <s> Clock Synchronization in Decentralized Networks <s> Synchronization is considered a particularly difficult task in wireless sensor networks due to its decentralized structure. Interestingly, synchrony has often been observed in networks of biological agents (e.g., synchronously flashing fireflies, or spiking of neurons). In this paper, we propose a bio-inspired network synchronization protocol for large scale sensor networks that emulates the simple strategies adopted by the biological agents. The strategy synchronizes pulsing devices that are led to emit their pulses periodically and simultaneously. The convergence to synchrony of our strategy follows from the theory of Mirollo and Strogatz, 1990, while the scalability is evident from the many examples existing in the natural world. When the nodes are within a single broadcast range, our key observation is that the dependence of the synchronization time on the number of nodes N is subject to a phase transition: for values of N beyond a specific threshold, the synchronization is nearly immediate; while for smaller N, the synchronization time decreases smoothly with respect to N. Interestingly, a tradeoff is observed between the total energy consumption and the time necessary to reach synchrony. We obtain an optimum operating point at the local minimum of the energy consumption curve that is associated to the phase transition phenomenon mentioned before. The proposed synchronization protocol is directly applied to the cooperative reach-back communications problem. The main advantages of the proposed method are its scalability and low complexity. <s> BIB001 </s> Synchronization in complex networks of phase oscillators: A survey <s> Clock Synchronization in Decentralized Networks <s> This article has explored history, recent advances, and challenges in distributed synchronization for distributed wireless systems. It is focused on synchronization schemes based on exchange of signals at the physical layer and corresponding baseband processing, wherein analysis and design can be performed using known tools from signal processing. Emphasis has also been given on the synergy between distributed synchronization and distributed estimation/detection problems. Finally, we have touched upon synchronization of nonperiodic (chaotic) signals. Overall, we hope to have conveyed the relevance of the subject and to have provided insight on the open issues and available analytical tools that could inspire further research within the signal processing community. <s> BIB002 </s> Synchronization in complex networks of phase oscillators: A survey <s> Clock Synchronization in Decentralized Networks <s> This paper studies the problem of realizing a common software clock among a large set of nodes without an external time reference (i.e., internal clock synchronization), any centralized control, and where nodes can join and leave the distributed system at their will. The paper proposes an internal clock synchronization algorithm which combines the gossip-based paradigm with a nature-inspired approach, coming from the coupled oscillators phenomenon, to cope with scale and churn. The algorithm works on the top of an overlay network and uses a uniform peer sampling service to fulfill each node's local view. Therefore, differently from clock synchronization protocols for small scale and static distributed systems, here, each node synchronizes regularly with only the neighbors in its local view and not with the whole system. An evaluation of the convergence speed and the synchronization error of the coupled-based internal clock synchronization algorithm has been carried out, showing how convergence time and the synchronization error depends on the coupling factor and the local view size. Moreover, the variation of the synchronization error with respect to churn and the impact of a sudden variation of the number of nodes have been analyzed to show the stability of the algorithm. In all these contexts, the algorithm shows nice performance and very good self-organizing properties. Finally, we showed how the assumption on the existence of a uniform peer-sampling service is instrumental for the good behavior of the algorithm and how, in system models where network delays are unbounded, a mean-based convergence function reaches a lower synchronization error than median-based convergence functions exploiting the number of averaged clock values. <s> BIB003 </s> Synchronization in complex networks of phase oscillators: A survey <s> Clock Synchronization in Decentralized Networks <s> Distributed synchronization has gradually gained importance over the last two decades. The ad-hoc nature of new applications has increased the need for robust and scalable distributed algorithms that are capable of generating high precision timing information. However, current solutions usually produce phase errors when the frequencies are heterogeneous. This paper proposes a distributed synchronization procedure that can achieve consensus in both frequency and phase. The algorithm uses only local information and is robust to frequency heterogeneity and network topology. A sufficient condition for global convergence is shown by leveraging recent results on coupled oscillators. We further characterize an invariant constant of the algorithm that relates the limiting frequency ω* with the harmonic mean of the clocks' natural frequencies. Simulations are provided to illustrate and verify these properties. <s> BIB004 </s> Synchronization in complex networks of phase oscillators: A survey <s> Clock Synchronization in Decentralized Networks <s> Synchronization is crucial to wireless sensor networks. Recently a pulse-coupled synchronization strategy that emulates biological pulse-coupled agents has been used to achieve this goal. We propose to optimize the phase response function such that synchronization rate is maximized. Since the synchronization rate is increased independently of transmission power, energy consumption is reduced, hence extending the life of battery-powered sensor networks. Comparison with existing phase response functions confirms the effectiveness of the method. <s> BIB005
Another emerging technological application of oscillator networks is clock synchronization in decentralized computing networks, such as wireless and distributed software networks. A natural approach to clock synchronization is to treat each clock as an oscillator and follow a diffusion-based (or pulse-coupling) protocol to synchronize them, see the surveys  BIB002 and the interesting recent results BIB001 BIB003 BIB004 BIB005 . For illustration, consider a set of distributed processors V = {1, . . . , n} connected by a (possibly directed) communication network. Each processor is equipped with an internal clock. These clocks need to be synchronized for distributed computing and network routing tasks. As discussed in the surveys by  and BIB002 , we consider only analog clocks with continuous coupling since digital clocks are essentially discretized analog clocks, and pulse-coupled clocks can be modeled continuously after a phase reduction and averaging analysis. For our purposes, the clock of processor i is a voltage-controlled oscillator (VCO) generating a harmonic waveform s i (t) = sin(θ i (t)), where θ i (t) is the accumulated instantaneous phase. For uncoupled clocks, each phase θ i (t) evolves according to where T nom > 0 is the nominal period, T i ∈ R is an offset (or skew), and θ i (0) ∈ S 1 is the initial phase. To synchronize their internal clocks, the processors follow a diffusion-based protocol. In a first step, neighboring oscillators communicate their respective waveforms s i (t) to another. Second, through a phase detector (PD) each node measures a convex combination of phase differences where f : S 1 → R is an odd 2π-periodic function, and a ij ≥ 0 are detector-specific convex weights satisfying n j=1 a ij = 1. Finally, cvx i (θ(t)) is fed to a phase-locked loop filter (PLL) whose output drives the local phase. A first-order and constant PLL with gain K results iṅ (11) The diffusion-based synchronization protocol (11) is illustrated in Fig. 4 , and its objective is to synchronize the frequenciesθ i (t) and possibly also the phases θ i (t) in the processor network. For an undirected communication protocol with symmetric weights a ij = a ji and a sinusoidal coupling function f (·) = sin(·), the protocol (11) reduces to the coupled oscillator model (1). The tools developed in the next section will enable us to state conditions when the protocol (11) successfully achieves phase or frequency synchronization. The protocol (11) is merely a starting point, more sophisticated higher-order PLLs can be constructed to enhance steady-state deviations from phase synchrony, communication and phase noise as well as non-constant timedelays can be considered in the design, and the phase coupling functions f can be optimized to increase the synchronization rate or minimize energy consumption.