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A Hopfield neural network approach to decentralized self-synchronizing sensor networks

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Abstract

Decentralized inference of a sensor network in the difficult case of a nonreciprocal nonlinear context is investigated by transforming the sensor network into a Hopfield neural network. Equilibrium states of the latter correspond to situations of global consensus in the sensor network, characterized by suitable regions (consensus regions) in the space of its parameters. The said transformation was recently proposed by the author and applied to the simple case of three sensors. The general case of more than three sensors is investigated in the present paper. A procedure is developed for determining the structure and the properties of the consensus regions.

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Abbreviations

SN:

Sensor network

N :

Number of sensors

M :

N−1

ω° :

The common value to which the frequencies of the sensors converge

CR:

Consensus region in the space of the parameters of the SN

ω i , ω :

The result of the observation carried out by the i-th sensor and the corresponding vector

θ i , θ :

The state of the i-th sensor and the corresponding vector

a ik f(.):

Takes into account the local coupling from the k-th to the i-th sensors. a ik  > 0 and f(.) is a nonlinear odd increasing function

a ij :

\( b_{ij} \sqrt {P_{j} } \) where b ij is a topological parameter of the SN

P j :

Is the power delivered by the j-th sensor

α ki :

a i+1,k /a k,i+1

B, R, W, F:

Matrices related to the topology of the SN and to its parameters a ik

THN:

The Hopfield net resulting from the proposed transformation of the SN

E :

Number of coupling links among sensors, coincident with the number of neurons of the THN, i.e. E = N(N−1)

Y, T, Ω:

Output connection matrix and input of the THN

Y 0 :

Equilibrium point of the THN

V, X:

Vectors introduced in the procedure for deriving the consensus regions from the equilibrium points of the THN

U 1 , U 2k :

Matrices of order M introduced in the procedure for deriving the consensus regions from the equilibrium points of the THN

Z i , Ω i :

Variables introduced in the procedure for deriving the consensus regions from the equilibrium points of the THN

C :

(N−1)(N−2)/2, is the number of entries of X to be arbitrarily chosen for deriving one of the consensus regions

References

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Authors and Affiliations

  1. Department of Infocom, University of Rome “La Sapienza”, Via Eudossiana 18, 00184, Rome, Italy

    Giuseppe Martinelli

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  1. Giuseppe Martinelli
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Correspondence to Giuseppe Martinelli.

Appendix: local stability of the THN

Appendix: local stability of the THN

A sufficient condition [12] for the local stability of an equilibrium point Y 0 of the asymmetric Hopfield net, described by the following system of equations

$$ {{\dot{\mathbf{Y}}}} = - {\mathbf{gY}} + {\mathbf{Tf(Y)}} + {\varvec{\Upomega}} \quad {\mathbf{g}} = {\mathbf{diag}}(g_{1} , { }g_{2 \, } \ldots g_{N} ), \, g_{i \, } > 0 $$
(33)

regards the eigenvalues of matrix \( {\mathbf{P = (T - g \, H^{ - 1} )H}} \), where H is a diagonal matrix with entries coincident with the derivatives of f(.) with respect to the components of Y, calculated in correspondence to Y 0 . The said condition requires that all the eigenvalues of P have either negative real part or are negative real. On the basis of lemma 4 of [12], this condition on the eigenvalues of P can be replaced by a similar condition on the matrix P 1  = T − gH −1, since H is a positive diagonal matrix due to the assumption that f(.) is an increasing function. Since g = 0 in the case of the THN, P 1  = T. Moreover, it is important to remark that the THN is characterized by redundant neurons. Only (N−1) of them are independent. Consequently, (N−1)2 eigenvalues of T are equal to zero. In conclusion, the local stability is assured if the remaining (N−1) eigenvalues have negative real part or are negative real.

The previous condition can be applied to matrix B t W in virtue of the cited lemma 4 of [12], since matrix F is a positive diagonal matrix. Since the eigenvalues of B t W, as it was shown in [9], are equal to

  • 0 with a multiplicity equal to (N−1)2

  • −N with a multiplicity equal to (N−1)

the local stability of the equilibrium points of the THN is guaranteed.

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Martinelli, G. A Hopfield neural network approach to decentralized self-synchronizing sensor networks. Neural Comput & Applic 19, 987–996 (2010). https://doi.org/10.1007/s00521-010-0369-5

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