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We study some nonlinear gossip algorithms for wireless sensor networks. Firstly, two types of nonlinear single gossip algorithms are proposed. By using Lyapunov theory, Lagrange mean value theorem, and stochastic Lasalle’s invariance principle, we prove that the nonlinear single gossip algorithms can converge to the average of initial states with probability one. Secondly, two types of nonlinear multigossip algorithms are also presented and the convergence is proved by the same methods. Finally, computer simulation is also given to show the validity of the theoretical results.

Wireless sensor networks that are composed of a large number of unreliable cheap sensors have drawn much attention from academia to industry in the past decade [

Recently, there emerge lots of studies on gossip algorithms in the field of wireless communication [

The remainder of this paper is organized as follows. In Section

Suppose there is a connected wireless sensor network which consists of

The gossip algorithm can be described as the following: without loss of generality, we assume in discrete time slot

This is the traditional gossip algorithm, which is called linear gossip algorithm. Specially, if

In gossip algorithms, different gossip functions denote different gossip rules. If

In this subsection, we consider the following nonlinear gossip algorithm:

In (

Suppose the gossip function

First we will prove nonlinear gossip algorithm (

Without loss of generality, in

According to Lagrange mean value theorem

if

if

if

Hence, from (i), (ii), and (iii), we have

Invoking the stochastic version of LaSalle’s Theorem [

Now we will prove that all the gossip variables converge to the average of their initial values. Consider

Equation (

We consider the following nonlinear gossip algorithm:

In (

Suppose the gossip function

Similar to the proof of Theorem

According to Lagrange mean value theorem

if

if

if

Hence from (i), (ii), and (iii), we have

By using the stochastic version of LaSalle’s Theorem, we have

Consider

Equation (

In aforementioned gossip algorithms, each sensor is allowed to gossip with at most one of its neighbors in a fixed time slot, which leads to a low average convergence rate. For example when sensor

It is similar to nonlinear single gossip algorithm that we present two types of nonlinear multigossip algorithms.

Assume in discrete time slot

In multigossip algorithm, if nonlinear function

In

Similar to the proof of Theorem

Therefore we have

Hence, we get

Consider

Equation (

Assume in discrete time slot

In multigossip algorithm, if nonlinear function

The analysis is similar to the proofs of Theorems

In this section, we give several computer simulation examples for the presented nonlinear gossip algorithms. A connected wireless sensor network is considered which is composed of

The topology of wireless sensor networks.

Simulations of the single gossip algorithm for type-1 and type-2.

Simulations of the multigossip algorithm for type-1 and type-2.

Figure

Nonlinear gossip algorithms for wireless sensor networks are considered. It is proved that the proposed algorithms can converge to the average of the initial values with probability one. The proposed algorithm is a general approach to the gossip algorithm, while the traditional linear gossip algorithm can be viewed as a special case of the nonlinear gossip algorithm. In the simulation, we find that different gossip functions bring about different convergence rates. How to determine and accelerate the convergence rate of nonlinear gossip algorithms may worth further study.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (nos. 61371107, 61261017, and 61304160), by the Fundamental Research Funds for the Central Universities (Grant no. JB140406), and by Guangxi Natural Science Foundation (2013GXNSFAA019334).