平成23年度 電気関係学会四国支部連合大会

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Eﬀective Associative Memory by Small World
HNN with Adjusted Shortcut Connections
Tomoya SHIMA, Chihiro IKUTA, Yoko UWATE and Yoshifumi NISHIO(Tokushima University)

1. Introduction
Associative memory is one of the important applications of the Hopﬁeld Neural Network (HNN). Thisnetwork model usually has full and symmetric connectivity. However, biological neural systems have localand non-symmetric connectivity. Also, the HNN causes
high cost to generate the network in terms of implementation. A few years ago, Bohland et al. have proposed
Small World HNN (SWHNN) [1] model in order to resolve these problems. The SW network improves ability for transmission by introducing the shortcut connections into the sparse regular network. We focus on
the relationship between the SWHNN and the shortcut
connections. In this study, in order to conﬁrm the relationship between the SWHNN and the shortcut connections, we propose the SWHNN with Adjusted Shortcut
Connections (ASC-SWHNN).

2. HNN working as Associative Memory
We consider the associative memory by the HNN.
The HNN stores patterns by determining the weightparameter. The weight parameters w is given by,{ 1 ∑P(p) (p)(i ̸= j)p=1 xi xj
Pwij =(1)0(i = j),where P is the number of patterns and x is the patternof memory. All neurons are initialized by an unknown
pattern. Namely, the network is given the input. The
state of each neuron is determined by Eq. (2).
n∑ui (t) =wij (t)xj (t),(2)few longer-range edges than edges which connects each
unit to its k nearest neighbors.

Figure 1: Example of the one-dimensional ring model of theSW network, when n = 16 and k = 4.

4. Simulation Results
In this study, we apply three networks, which are theHNN based on the regular network {1}, the SWHNN{2} and the ASC-SWHNN {3}, to the associative memory. Also, we compare the recall rate of three networks
when k is changed. In the computer simulations, weconsider each network with 400 neurons. We prepare 3
stored binary patterns at random and produce 3 × 50
binary patterns to be input, which diﬀers from eachstored patterns by 25%. It is practically reasonable
that a pattern within 1% error is considered as being recalled. In the case of the SWHNN or the ASC-SWHNN,
we randomly compose 10 network and calculate the average of the recall rate of the networks. The simulation
results are shown in Fig. 2 and show that the ASCSWHNN provides the better recall performance thanother networks. Therefore, we assume that the propagation of the SW network improves by the strong shortcut connections.
1j=1

The HNN recalls a similar pattern to the input patternfrom the stored patterns by repeating Eq. (2) to Eq. (3).
The SWHNN is the SW network which is trained using
these normal HNN rule.

3. ASC-SWHNN ModelFirst, we consider a one-dimensional ring model ofthe SW network model. We start with a n ring regularlattice, with each unit connected to its k nearest neighbors by edges, and then rewire with probability p. The
one-dimensional ring model is shown in Fig. 1. The
ASC-SWHNN is composed based on the SW network
and is adjusted by multiplying the weight parameters of
the shortcut connections in the SWHNN by an adjusting coeﬃcient α. Here, the shortcut connections are aPowered by TCPDF (www.tcpdf.org)

recall rate

0.8where u is the internal state of the neuron and x is
the input or output. The value of u is binary. Neuron’s output is determined by a threshold function. Theequation of the determining neuron’s output is given by
Eq. (3).
{1,(ui (t) ≥ 0)xi (t + 1) =(3)−1,(ui (t) < 0).

0.6
0.4{1}{2}{3} (α=3)

0.2001020304050kFigure 2:

Process of recall in diﬀerent networks, the sparse
HNN, the SWHNN and the ASC-SWHNN, when p = 0.1.

5. Conclusions
In this study, we have proposed the ASC-SWHNN.
andapply the associative memory. The computer simulation results showed that the ASC-SWHNN provided
the better recall performance than other networks. We
assumed that the propagation of the SW network improves by the strong shortcut connections.
References[1] J.W. Bohland and A.A. Minai, “Eﬃcient Associative Memory
Using Small-World Architecture,” Neurocomputing, vol. 38-40,
pp. 489-496, 2001.