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Hi Don,
Thanks for your message. I am working on the maths for a proof. At this
stage, however, for the hypothesis as stated simple reasoning and
building on the analyses of others seems does the job adequately enough.
The loudness power-law relation you described , like other perceptual power
law relations, is a* psychophysical* phenomena. It depends on the mechanisms
of subjective perception and judgement.
A problem has been that many physics or information systems lecturers
have taught it as if it were a physical law or that power law behaviour
derives sui generis apparently without cause from the statistics itself. If
one looks at the detail of the basis of the psychophysical phenomena then
emerges the explanation of the effects of interactions.
In fact perceptions in general are only roughly approximated to a power
law as they depends on the characteristics of the details of the human
internal psychophysical mechanisms underpinning the gross psychophysical
phenomena. This indicates the power law behaviour is driven, for example in
the case of sound, by everything from the interactions between hairs in the
inner ear to an individual's neurological and glial network systems. Many
sensory perception judgment behaviours can be approximately characterised
by power laws using a variety of exponents. See, for example, the work of
S.S. Stephens -
http://www.cis.rit.edu/people/faculty/montag/vandplite/pages/chap_6/ch6p10.h
tml (a similar analysis probably applies to Weber and Fechner's work as
you know on jnds - leading to approximations to logarithmic power-law like
relations).
In many ways you might naturally expect power law relations in these kinds
of situation due to relationship network effects at a micro scale. In many
circumstances these can be seen as characterised by relations between on one
hand a linear potential and the other a power-based energy function (e.g. of
the form mgh = 1/2 (mv^2) + K) This is found in triggerable systems such as
synapses in which the network effect of stored energy in dendrites leads to
potential triggering of a signal along the axon. The global network
behaviour follows on from a larger scale accumulation of network effects.
There are many publications that describe aspects of this, for example,
Drew, P. J and Abbot, L. F (2006) Models and Properties of Power-Law
Adaptation in Neural Systems. Journal of Neurophysiology. 96, 826-833
(http://jn.physiology.org/content/96/2/826.full.pdf+html )
In social networks, the power function and triggering effects are apparently
due primarily to information behaviour in linked clusters of relationships.
Each of the links is, in effect and reality, dependent on social
communication between people, and this appars to be the basis of different
aspects of power law behaviour (see, for example Girvan, M and Newman, M E J
(2002) Community Structure in Social and Biological Networks
(http://www.pnas.org/content/99/12/7821.short ) and Lusseau, D (2003) A
Dolphin Social Network Proc. R. Soc London. V 270 Suppl 2 S186-S188 avail
http://rspb.royalsocietypublishing.org/content/270/Suppl_2/S186.full.pdf+htm
l )
A good explanation of the generic approach of analysing social system
structure and interactions in terms of the behaviour of data is Dooley, K
J & Van de Ven, A H (1999) Explaining Complex Organisational Dynamics,
Organisational Science, v 10, no 3, pp 358-372, though they only lightly
touch on power-law behaviour through May and Oster's work on ecological
modelling and non-linear systems.
Another explanation is that one can derive power law behaviour as 'a
characteristic of the interactions of people in a complex system' when one
straightforwardly models the consequences of interactions in social groups
via agent-based modelling or system dynamics (see, for example, Farmer, JD
and Foley, D (2009) The Economy needs Agent-Based Modelling, Nature, 460, pp
685-686
The above and other publications that for brevity and ease of typing I
haven't quoted appear to provide support for at least the weak form of the
claim, which is why I posted it as an 'indicator hypothesis' rather than a
'law'.
Best regards ,
Terry
---
Dr Terence Love
PhD(UWA), BA(Hons) Engin. PGCEd, FDRS, AMIMechE, MISI
Honorary Fellow
IEED, Management School
Lancaster University, Lancaster, UK
Director,
Love Services Pty Ltd
PO Box 226, Quinns Rocks
Western Australia 6030
Tel: +61 (0)4 3497 5848
Fax:+61 (0)8 9305 7629
[log in to unmask]
--
-----Original Message-----
On Behalf Of Don Norman : Monday, 4 November 2013 10:09 PM
Terry
I don't want to turn this discussion into the usual, long arduous PhD design
debates, so I will limit myself to one response (and then shut up).
Your goal is laudatory. But I don't see the evidence. My best information is
that power-law distributions come from certain statistical properties of the
elements involved. For example, in psychological measurements, if equal
ratios lead to equal psychological judgements, then there is a power law at
play.
(In sound, where I is the intensity of the physical waveform for some signal
and L(I) is the psychological assessment of its loudness, if:
L(10*I)/L(I) = 2, independent of the value of I That is, if a sound ten
times as intense as another always seems twice as loud
this leads to a power law of loudness: L = I^n (where n = 0.3: Loudness
increases as the cube root of Intensity.)
I see no interactions here.
--
To prove your hypothesis I either need a lot of empirical data showing that
Power Law phenomena involving human social groups only occur If and Only If
(IFF) there is social interaction among the people/groups. (Or the weak
form, .. *tends* to occur where there is social interaction and *tends not
*to occur otherwise).
or
A nice mathematical formulation of why this would be the case.
----
It would be really nice if your hypothesis were true, for all the reasons
you mention. So I am not arguing that it is false: I am saying that you have
not made the case. Yet.
Enough
Don
--
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