What can a loss-of-strength gradient look like?
Competition in space and time for people’s opinions
Gordon Burt, Conflict Research Society
28 Severn Drive, Newport Pagnell, MK16 9DQ, UK
gordonjburt@gmail.com; http://www.conflictresearchsociety.org.uk/
Contents
1 Introduction
2 Spatial variation – spatial complexity
2.0 Space
2.1 Understanding spatial variation
2.2 Formulaic understanding: loss-of-power gradient
2.3 Configurational understanding: the network topography
2.4 How many centres of power are there? Scaling and complexity theory
2.5 The loss-of-strength gradient, spatial correlation and spatial regression
2.6 A fractal structure of networks
2.7 The complementary party, the combined opposition
2.8 Profiles ... party vectors in regional space ... regional vectors in party space
3 Temporal variation ... complexity dynamics
3.1 Stable configurations over the years
3.2 Constant and proportional change
3.3 The change function
3.4 The micro basis for macro change ... complexity dynamics
4 Spatial dynamics: the discourse-location-value-choice linkage
5 Economy, technology, military, cultural
6 Conclusion
7 References
Abstract
The concept of a loss-of-power gradient is central to many studies in conflict research. The simplest case is where
there is a centre and power is a decreasing function of distance from the centre. However much more complex
cases are possible. For example a geographical map with height contours can be thought of as reflecting the lossof-height gradient. Another complication is that the situation may involve the powers of many actors rather than
just one. The loss-of-power concept is often applied to violent conflict and data may be incomplete. In contrast
elections involve (sometimes) non-violent conflict and the complete election results are often available. This
suggests that an analysis of election results might provide some insight into the concept of loss-of power gradient.
To this end the results of the elections to the Scottish Parliament in 2007 and 2011 are analysed.
Parties compete for relative success (percentage vote) over space and time, a constant-sum game. (‘Success equals
power plus luck’). The space of locations may be an unstructured set or a Euclidean space or a network of nodes.
In an unstructured set, overall party strength, variation across locations and global maxima and minima are noted.
Correlations between parties indicate similar, opposing or unrelated locational profiles. In the case study, despite
substantial changes over time in overall party strength, parties’ locational profiles remain much the same (profiles
in 2007 correlated 0.9 with profiles in 2011). These macro changes derive from micro decisions by individuals and
the distribution of individuals in party value space, with complexity theory dynamics. Representing the locations
as a network of nodes, local maxima and minima are noted. Over the eight top-up regions SNP, Labour and the
Liberal Democrats have one local maximum; Conservatives and ‘others’ each have two; and the Greens have
three. The loss-of-power gradient is discussed and distinctions are made between a single-peaked function, a
central distance function and a linear or (inverse) quadratic central distance functions. The relationship between
the analysis for the eight top-up regions and a corresponding analysis for the seventy-three constituencies prompts
some speculative reflection on fractals.
1
1 Introduction
David Cameron, UK prime minister and leader of the Conservative Party, seeks to
win hearts and minds in Scotland ... and in Afghanistan. In both countries he is
confronted by other parties also seeking to win hearts and minds. What does this
competition for hearts and minds look like? In this talk I shall discuss Scotland and I
shall discuss general theory. Implicit in the general theory is the notion that it applies
not just to non-violent democracies such as Scotland, but also to violent nondemocracies – and indeed to violent democracies and non-violent non-democracies!
And of course there are gradations of violence, gradations of democracy.
Frequently events take us by surprise. We are surprised at the complexity of parties,
of religious groups or of tribes; and we are surprised at the complexity of the spatial
variation of these parties. We are also surprised at the complexity of temporal
variation - at sudden outbreaks of events and sudden fluctuations in success. We shall
find that these surprises exist in the data for Scotland and I shall suggest that
complexity theory provides one part of the explanation. Complexity theory offers a
somewhat different account of temporal variation to the account offered by standard
econometric time series analysis and a somewhat different account of spatial variation
to the account offered by spatial regression approaches.
Most people would say that my approach is quantitative rather than qualitative. That
is not how I like to think about it. What I do is mathematical social science.
Mathematics is not about numbers, it is about abstract patterns. And it is the abstract
patterns which are central to the whole paper. A more specific point is that I am
interested here in the structure of the data for what is often thought of as the
dependent variable – I do not consider any substantive explanatory variables.
I invite you to consider how these abstract patterns might apply to the particular
situations which you are investigating in your own research and in your own work and
experience.
Scotland. Back in December, The Times awarded the title ‘Briton of the Year’ to Alex
Salmond, the leader of the Scottish National Party and First Minister of Scotland. He
had ‘transformed his party and Scottish politics’, ‘a rare politician at the top of his
game’, ‘thinker, man of the people, master of all he surveys’, ‘he defied the odds; his
triumph in the elections was a remarkable political achievement’.
[The Times (2011) Briton of the Year. Thursday 27 December. pp. 1, 2, 12, 13.]
It was not always so. Support for Alex Salmond, for the Scottish National Party (SNP)
and for Scottish independence has changed over time. The percentage SNP vote in
Scottish Parliament elections was 29% in 1999, fell to 24% in 2003 and then rose to
33% in 2007 and 45% in 2011 – in other words over the past decade the percentage
vote has risen from a quarter to a third to a half. Typically, more people vote for the
SNP than want Scottish independence. However just in the last year, the percentage of
Scots wanting independence increased from 22% at the start of 2011, to 38% at the
end of 2011 and to 47% on 5th February 2012.
[The Times (2012) Hold breakaway vote in the next 18 months, PM tells Salmond. Monday January 9, p. 15.
The Sunday Times (2012) Surge in support for Scottish independence. February 5, p.17]
2
Not only does the Scottish Nationalist vote vary over time, it also varies over space.
Figure 1 below [Slide 2] displays the ‘common-border network’ for the eight Scottish
regions with lines joining each pair of regions which have a common border; and it
shows how the SNP percentage vote in May 2011 varied across these regions. The
SNP vote had two peaks, a peak of 53% in the North East and a peak of 46% in
Central; and two troughs, a trough of 40% in the city of Glasgow and a trough of 39%
in Lothian which contains the capital city of Edinburgh.
Slide 2
Figure 1 The common-border network for the eight regions and the percentage vote
for the SNP, Scottish Parliament 2011: two peaks (maxima) and two troughs
(minima)
.
.
.
.
.
.
.
.
.
Highlands
& Islands 48
|
|
|
|
|
North East 53
|
Mid &……... Fife 45 |
|
|
|
West 42 Glasgow 40 Central 46
Lothian 39
|
|
|
|
|
|
South 41
|
What I want to do in this talk is to take the Scottish Parliament elections as an
example of party competition over space and time. The parties are competing for
percentage vote - which can be taken as a measure of relative success. This is a
constant-sum game.
Barry’s dictum, ‘Success equals power plus luck’, provides a link between the
concepts of success and power. There is a distinction between power as the attribute
of an individual and power as the attribute of a relationship. And the concept of power
as an individual attribute links to the concept of strength. These conceptual
distinctions are not discussed further in this paper. The concept of loss-of-strength
gradient [see references] is central to many studies in conflict research, and the main
aim of my talk is to reflect on this central concept and how it relates to some of the
complexity theory approaches to spatial analysis. [Slide 3]
However I also want to consider temporal variation in percentage votes. Percentage
votes are a macro phenomenon and can be explained by appeal to a micro-macro
model which derives percentage votes from individual values. In turn, individual
values are formed by a spatial dynamics of social discourse, location, value and
choice – a discourse which relates to economic, technological and military issues as
well as cultural ones.
[Note: background notes are available on some of the mathematics and on the details of the Scottish Parliament
case study.]
Slide 3
2 Spatial variation – spatial complexity
3 Change over time
The micro basis for macro change – complexity dynamics
4 Spatial dynamics: the discourse-location-value-choice linkage
5 Economy, technology, military, cultural
3
2 Spatial variation – spatial complexity
Let us start then with spatial variation. My treatment will cover the following topics.
[Slide 4]
Slide 4
2 Spatial variation
2.0 Space
2.1 Understanding spatial variation
2.2 Formulaic understanding: loss-of-power gradient
2.3 Configurational understanding: the network topography
2.4 How many centres of power are there? Scaling and complexity theory
2.5 The loss-of-strength gradient, spatial correlation and spatial regression
2.6 A fractal structure of networks
2.7 The complementary party, the combined opposition
2.8 Profiles ... party vectors in regional space ... regional vectors in party space
2 Spatial variation
2.0 Space
In the real world Scotland is a continuous subspace S of three-dimensional Euclidean
space whereas in the introduction it was represented as a network of regional nodes.
Much can be said about how one moves from the continuous subspace to the network
of nodes. The first step is to construct a partition P of S into finer continuous
subspaces. Partition P is just one of many possible partitions and we can imagine the
set P* of all possible partitions of S, what might be called the partition power set. For
each n there are an infinite number of ways of partitioning S into n areas. In the
second step, given a partition P we construct a network N with the elements of P as
nodes.
Ward and Gleditsch (2008, Chapter 4.2) note that network construction can be done in
a variety of ways. Methods can be based on either topology or distance. Commonborder links may not correspond to common-node links if more than three regions
share a common node. This is not a problem in the present Scotland case study.
Topological methods are undiscriminating compared to distance-based methods.
Broadly speaking the centroids of two adjacent regions of size L are distance L apart.
The areas of Scotland are of dramatically different size. This can be illustrated by
considering cities in different areas. Edinburgh (Lothian region) is just 46 miles from
Glasgow (Glasgow region) but Lothian and Glasgow have no common border. In
contrast Inverness (Highlands & Islands region) is 114 miles from Perth (Mid & Fife)
and Highlands & Islands and Mid & Fife do share a common border.
The use here of the topological method is appropriate to the consideration of peaks
and troughs which are topological phenomena, unaffected by distance.
4
2.1 Understanding spatial variation
How are we to understand the spatial variation of party strength? I would like to
distinguish between two types of understanding: formulaic understanding and
configurational understanding.
2.2 Formulaic understanding
Formulaic understanding is extremely important and in physics has given us laws
such as Newton’s law of gravity. This specifies that gravitational force decreases with
distance according to an inverse square law. The corresponding law in peace science
is the loss-of-strength gradient. This specifies that political and military power
decrease with distance (possibly according to an inverse square law).
The mathematical specification of the law is as follows. [Slide 5] Consider a set X of
spatial locations. The power function p(x) specifies the power at each location x. The
power function may have a single global maximum (minimum) or none or many. The
set X may have an adjacency relationship defined on it and this allows the definition
of local maxima and minima of which there can be one, none or many. Furthermore
the set X may have a distance function defined on it, d(x,y) denoting the distance
between locations x and y.
How might the power function depend on the distance function? Suppose that the
power function has a single global maximum at x*. There may be additional local
maxima, but if not then the power function is referred to as single-peaked. A special
case is where the single-peaked function depends on the distance from the global
maximum, p*(x)=f(d(x,x*)), where f is a decreasing function. This ‘central distance
function’ may take specific forms such as negative linear or negative quadratic.
Slide 5
Formulaic understanding: loss-of power function
a set X of spatial locations
power function p(x)
global maxima, minima
adjacency relationship
local maxima, minima
distance function, d(x,y)
special cases:
power function has a single global maximum at x*
single-peaked (has a single ‘centre’): no local maxima
central distance function: p(x)=f(d(x,x*))
negative linear (used in regression studies), p(x)=p(x*)-md(x,x*)
negative quadratic
For example Figure 2 below [Slide 6] shows how the SNP regional percentage vote in
2011 can be approximated by a negative linear function of distance, p=0.53-0.04d.
Here the centre of power for the SNP is the North East region; and the distance of
each region from that centre is measured by the number of links in the network
between that region and the centre. The figure shows the empirical data in blue and a
crude approximating straight line in red.
5
Slide 6
Figure 2
The loss-of-power gradient, p=0.53-0.04d.
p: the percentage SNP vote; d: number of network links from SNP centre (North East)
2.3 Configurational understanding: the network topography
In this example, the power function is taken to be a negative linear function – as is
common in the literature. However there are a number of ways in which the power
function can fail to be a negative linear function. It can fail if the power function is
non-linear; or if the power function does not depend on distance; or if the power
function is not single-peaked (in other words if there are additional local maxima).
For example none of the properties of a negative linear function are to be found in a
geographical map with height contours: instead there are many local maxima; local
loss of height does not always depend on distance; and any local distance-dependence
is not always linear. So how do we use geographical maps to understand spatial
variation in height? The answer is that we develop configurational understanding.
The following mental picture may be helpful. Think of the maps of mainland Britain
and of Japan. If the sea level rose then mainland Britain would become four islands:
the Scottish Highlands; the Pennines and Cumbria; Wales; and Devon & Cornwall.
Around Japan, if the sea level fell then the four islands of Hokkaido, Honshu, Kyushu
and Shikoku would become a single mainland Japan.
This mental picture contains the following ideas. A map displays a set of ‘islands of
high ground’ and a set of ‘trenches of deep water’. [Slide 7]. However what counts as
an island depends on the level of the sea. Imagine too that what counts as a trench
depends on the level of the sea. Configurational understanding also includes other
features – for example, ridges and valleys, watersheds and saddle points.
6
Slide 7
Configurational understanding
features:
a set of ‘islands of high ground’, ‘peaks’
a set of ‘trenches of deep water’, ‘troughs’
The set of ‘islands of high ground’ depends on the level of a criterion ‘sea’.
The set of ‘trenches of deep water’ become ‘lakes’ depending on the level of a criterion ‘sea’.
Other features: ridges and valleys, watersheds and saddle points
Looking at a map of party strength in Britain in the elections of May 2011 one finds
similar features: England is a sea of Conservative blue and grey (grey corresponding
to no overall control and no elections) with an archipelago of islands of Labour red
and Liberal Democrat orange; Scotland is a yellow sea of SNP with islands of Labour
red and Conservative blue; and Wales has three deep blue Conservative trenches.
We have already seen that the SNP vote in Scotland across the eight regions shows
two peaks and two troughs. We now do a more extensive analysis using the concepts
we have developed. [Slide 8] First we consider ‘islands’: areas of high support. At the
53% level there is just the one island consisting of just the one region, the North East;
at the 48% level there is one island with two regions, North East and Highlands &
Islands; at 46% there are two islands, the new island being Central; at 45% the two
islands join to form one island linked by the region, Mid & Fife; and, finally, at lower
levels additional regions join onto the one single island.
Slide 8
Connected areas of high support at different criterion levels
level
‘islands’: areas of high support
53%
North East
48%
(North East, Highlands & Islands)
46%
(North East, Highlands & Islands); Central
45%
(North East, Highlands & Islands, Mid & Fife, Central)
42, 41, 40, 39% [West, South, Glasgow and Lothian added in turn]
Next we consider ‘trenches’: areas of low support. [Slide 9] At the 39% level there is
just the one trench consisting of just the one region, Lothian; at the 40% level there
are two trenches each with just one region, Lothian and Glasgow; at 41% there are the
same two trenches, here South joining the Lothian trench; at 42% the two trenches
join to form one trench linked by the region, West; and, finally, at higher levels
additional regions join onto the one single trench.
Slide 9
Connected areas of low support at different criterion levels
level
‘trenches’: areas of low support
45, 46, 48, 53%
North East, Highlands & Islands, Mid & Fife, Central added in turn]
42%
(Lothian, South, West, Glasgow)
41%
Glasgow; (Lothian, South)
40%
Glasgow; Lothian
39%
Lothian
The two displays for the islands and the trenches can be put together into one display.
[Slide 10]
7
Slide 10
Table 1 The ‘islands’ and ‘trenches’ for different ‘sea levels’
The high SNP vote areas and the low SNP vote areas for different criterion levels of percentage vote.
‘sea level’
.
no. of regions
X%
below
above
.
<
≥
no. of trenches/islands
below
above
<
≥
‘trenches’ of
low support
below
<
‘islands’ of
high support
above
≥
+
53
48
46
45
42
41
40
39
1
1
1
1
1
2
2
1
-
NHCMWSGL
HCMWSGL
CMWSGL
MWSGL
WSGL
(G); (SL)
(G); (L)
L
-
N
NH
(NH); C
NHCM
NHCMW
NHCMWS
NHCMWSG
NHCMWSGL
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
1
1
2
1
1
1
1
1
2.4 How many centres of power are there? Scaling, power laws and complexity
theory
Configurational understanding requires knowledge of a variety of features of the
situation, one of the most important features being the number of peaks or centres of
power. In our analysis of regions we have found that the SNP has two centres of
power. This goes against the assumption in some uses of the concept of the loss-ofpower gradient that there is just one single centre of power.
This prompts the question: how many centres of power are there? Thinking about this,
I was quite excited to realise that this was exactly the same sort of question which
Lewis Fry Richardson had been asking when he asked ‘what is the length of the
boundary between Spain and Portugal?’. Perhaps the answer to my question is the
same as the answer to his question: it depends on the scale of the analysis, on the level
of spatial analysis. At one level we can divide Scotland into eight regions. At a more
detailed level we can divide Scotland into seventy-three constituencies ...
So: how many centres of power are there – at different levels of spatial analysis? This
is the type of question which complexity theory addresses. The theory suggests that
there may be a power law, and that the centres may display a fractal structure.
One can either assume a power law and check to see whether the data supports a
power law – or one can develop a theory of the situation from which a power law may
or may not be derivable. [Slide 11] Following the latter course, let ‘n’ denote the
number of nodes and ‘c’ the number of centres of power. If the network consists of a
chain of n nodes; and if we consider all n! possible orderings of the nodes by their
power; then the expected number of centres of power E(c) equals one plus n minus 2,
times a third. E(c)=1+(n-2)/3, for n>1. [The third by the way relates to Kendall’s
simple test for the randomness of a sequence of numbers.]
8
Slide 11
How many centres of power are there in a network?
1 It depends on the scale ‘n’ of the analysis
Given a territory, divide it into ‘n’ areas
Produce the common-border network
Identify ‘c’ centres / peaks
E: expectation over all n! possible orderings
chain: E(c)=1+(n-2)/3,
n>1
E(c)=1/3+n/3
However a chain is just one type of network. Different network configurations give
different values for E(c). One way in which networks differ is in the number of links
they have. So let ‘L’ denote the number of links in the network. [Slide 12]
The number of links can vary from zero connections, L=0, to total connection,
maxL=n(n-1)/2. If L=0 then c=n and if L=maxL then c=1. Also, if the network is a
chain then L=(n-1) and E(c)=1+(n-2)/3. To my surprise I noticed that the formula
E(c)=1/3+n(n-1)/(3L) satisfied both the chain and the total connection network. Let us
refer to L*=L/maxL=2L/n(n-1) as the fractional linkage, that is the links as a
proportion of the total possible. Then the formula can be written E(c)=1/3+2/(3L*).
Note that, in the general case, this formula provides only an approximation to E(c).
We can apply the formula * in order to obtain approximations to E(C) for special
types of network. A ‘4w-grid’ is a network consisting of a grid of squares, consisting
of h horizontal lines and v vertical lines and so n=hv, and the relative proportion of
the sides of the grid is w with w=h/v. A ‘3w-grid’ is similar except that it is a grid of
triangles. The formulae for the 4w-grid and the 3w-grid are given below.
Note that all of the above refers to the number of centres. Because the argument is
based on the ordering of points, essentially the same argument and the same formulae
apply to the number of troughs, the ordering being simply reversed.
Slide 12
How many centres of power ‘c’ are there in a network?
2 It depends on the scale ‘n’ of the analysis ...
... and on L: the number of links in the network
Zero links:
L=0
c=n
Chain:
L=(n-1) E(c)=1+(n-2)/3
Total connection:
L=n(n-1)/2
c=1
E(c)=1/3+n(n-1)(/3L)
L≥(n-1) [*]
E(c)=1/3+2/(3L*)
L*=L/maxL=2L/n(n-1)
L* is the fractional linkage
chain: E(c)=1/3+n/3
4w-grid: E(c)=1/3+(n+k√n+(k2-1)(1-k/(√n-k)))/6
3w-grid: E(c)=1/3+(n+k√n-1/3+(k2-1)(A))/9
k=(w+1/w)/2; A=...
(i) Applying the formula [*] to the national regional-node network with n=8, L=14
and maxL=28, we obtain a prediction of 1.67 centres – and 1.67 troughs. Our data
covers six parties and four elections, twenty-four observations in all. The mean
observed number of peaks is 1.71 and the mean observed number of troughs is 1.50.
(ii) Applying the formula [*] to the national constituency-node network with n=73,
L=161 and maxL=2628, we obtain a prediction of 10.9 centres. The observed number
of peaks is 10 for the SNP and 13 for Labour, averaging to 11.5.
9
(iii) Applying the formula [*] to the regional constituency-node networks with n=8 to
10, maxL=28 to 45, L=8 to 16, we obtain a prediction of 2.3 centres – and 2.3
troughs. Our data covers eight regions and four elections, thirty-two observations in
all, all relating to the SNP. The mean observed number of peaks is 2.2 and the mean
observed number of troughs is 2.2.
It may have been thought that the number of peaks per region for (ii) and (iii) should
be the same since both refer to constituency node networks. However the
consideration of each region separately involves ignoring the links to adjacent regions
and also allows nodes on the border of the region to become peaks or troughs within
the region even though they are not peaks or troughs in the national network as a
whole. Fewer links and more peaks give a higher value for E(c).
Looking at the same data but this time looking at variation across the eight regions
there is a correlation between E(c) and Obs(c), r=0.85.
Slide 13
How many centres of power ‘c’ are there in a network?
3 Applying E(c)=1/3+2/(3L*) where L*=L/maxL=2L/n(n-1)
national regional-node network (24 cases)
When n=8, maxL=28, L=14,
E(c)=1.67; Obs(c)=1.71 peaks and 1.50 troughs
national constituency-node network (2 cases)
When n=73, maxL=2628, L=161,
E(c)=10.9; Obs(c)=11.5 peaks
[average, peaks per region, E(c)=1.36; Obs(c)=1.44 peaks]
regional constituency-node networks (32 cases)
When n=8to10, maxL=28to45, L=8to16
E(c)=2.3; Obs(c)=2.2 peaks and 1.9 troughs
regional constituency-node network (8 regions)
r=0.85: correlation between E(c) and Obs(c)
Finally and this has nothing to do with the Scotland data set, we can calculate the
exact values of E(c) for all the networks in the case when n=4. With n=4 and L=3,
there are three networks (chain, star and triangle & point) and E(c) = 1.67, 1.75 and 2
respectively. With n=4 and L=4, there are two networks (square and triangle & tail)
and E(c) = 1.33 and 1.42 respectively. With n=4 and L=5, there is one network
(joined triangles) and E(c) = 1.17. With n=4 and L=6, there is one network
(tetrahedron) and E(c) = 1. The correlation between the two sets of values is 0.96.
This example illustrates how there can be different types of network for the same
values for n, c and L – and that there can be correspondingly different values of E(c).
Necessarily then the formula [*] provides only an approximation to E(c) – and indeed
here always provides an underestimate. So is there a better formula which would take
into account the different types of network? A key difference between the chain, the
star and the triangle & point lies in the variation between nodes in the number of links
the node has. In the chain, the number of links varies between one link (at the end
nodes) and two links (at internal nodes). In the star, the number of links varies
between one link (at the extreme nodes) and three links (at the central node). In the
triangle & point, the number of links varies between no link (at the point) and two
links (at triangle). So the range R is 1, 2 and 2 respectively. The maximum range is n2. So a relevant measure is R*=R/(n-2). How this might be incorporated into a new
approximating equation for E(c) is unclear.
10
Slide 14
How many centres of power ‘c’ are there in a network?
4 It depends on the scale ‘n’ of the analysis ...
... and on L: the number of links in the network ...
... and on R: the range in the number of links at each node
seven abstract networks with 4 nodes
r=0.96: correlation between E(c) and Obs(c)
Consider again the chain. Here L=(n-1) and E(c)=1+(n-2)/3. The maximum number of
centres is given by max(c)=1+(n-1)/2 if n is odd and by max(c)=1+(n-2)/2 if n is even.
So E(c)-max(c)=-(n-2)/6; E(c)-min(c)=-(n-2)/3; and max(c)-min(c)=(n-2)/2 – if n is
even. Note that (E(c)-max(c))/( max(c)-min(c))=-1/3.
This suggests a measure C*=3z for the deviation of an observed value of c from the
expected value E(c): z=(c- E(c))/( max(c)-min(c))=2(c-1-(n-2)/3)/(n-2)=
2(c-1)/(n-2)-1/3. Where -1/3≤z≤1/3. So -1≤C*≤+1. Note that C*=0 when c=E(c).
So C* provides an index of centredness. It is zero when the expected number of
centres is observed; positive when there are more centres; and negative when there are
fewer centres. The argument here only applies to chains. In principle it could be
generalised, using formula [*] so as to apply to all network types, although identifying
the maximum value for E(c) is problematic.
2.5 The loss-of-strength gradient, spatial correlation and spatial regression
The index of centredness C* provides a crude measure of spatial correlation. A
standard measure of spatial autocorrelation is Moran’s I and this takes account of all
the information in the data. Spatial correlation is a key element in spatial regression.
The topic of spatial regression considers how to take account of the spatial
dependence of p on x when carrying out a regression of z on p. For example in their
book on this topic Ward and Gleditsch (2008) motivate the topic by considering the
regression of democracy on per capita GDP. The assumptions of the classical model
are violated because there is a dependence between the observations of the
independent variable. Two main types of model are considered the spatially lagged
dependent variables model and the spatial error model.
The current paper is not quite at a stage where this is applicable. Much of the paper
looks at just one variable or looks separately at each of a set of variables – and not at
the relationship between spatially-based variables. Where relationships are considered
the primary concern at this stage is with the nature of the relationship rather than on
the estimation of parameters.
The following are some tentative remarks about the link between the loss-of strength
gradient and spatial regression. The loss-of-strength gradient expresses a dependence:
the dependence of the strength at each point on the strength at the peak. It follows that
there is also a dependence of the strength at any point on the strength at any other
point:
p(x)=p(x*)-md(x,x*)
p(y)=p(x*)-md(y,x*)
11
p(y)=p(x)-md(y,x*)+md(x,x*)
p(y)=p(x)-m[d(y,x*)-d(x,x*)]
In the case of a one-dimensional space:
p(y)=p(x)-m[d(y,x)] or p(y)=p(x)-m[d(y,(x*-x)]
Suppose now that another variable z(x) depends on p(x): z(x)=a+bp(x). Then
z(x)=a+bp(x*)-md(x,x*)
z(y)=a+bp(x*)-md(y,x*)
...
2.6 A fractal structure of networks
Here are the ten constituency peaks. [Slide 15] There are two peaks in the peak region
of the North East and these are high peaks. In general each region has its peak
constituency, even regions where the overall SNP vote is low. Around each peak are
constituencies with high SNP vote together forming a ridge.
[Slide 15]
Table 2 The constituencies with peak % SNP vote
Region
constituency
% SNP vote
NE
H&I
MF
NE
C
L
G
W, S
.
MF
L
.
Banffshire coast (12)
[Outer Hebrides] (37)
Perthshire N (64)
Dundee E (25)
Falkirk W (40)
Almond (7)
Glasgow S (49)
Cunninghame N, (21)
Kilmarnock (53)
Mid Fife (56)
Edinburgh E (32)
Midlothian N (57)
67
65
64
64
55
54
54
53
52
47
This points to a fractal structure of networks. At one level of analysis we have a
network of eight regional nodes exhibiting peaks and troughs; and at another level of
analysis each regional node consists of a network of constituency nodes exhibiting
peaks and troughs. Let us look at just one regional node, namely the North East.
[Slide 16] The common-border network for the ten constituencies in the North East
region display two peaks (Banffshire & Buchan coast and Dundee East) and two
troughs (Aberdeen Central and Dundee West).
12
Slide 16
Figure 3 The common-border network for the ten constituencies in the North East
region and the percentage vote for the SNP, Scottish Parliament 2011: two peaks
(maxima) and two troughs (minima). [Inset: North East and the other regions]
Banff 67
|
|
A. Don 43
Aberdeen East 65
Aberdeen West 55
Aberdeen Central 40
|
Aberdeen South 42 |
|
|
Angus North 55
Angus South 59_|
|
Dundee West 58
Dundee East 64
.
.
.
|
Highlands
& Islands 48
|
.
|
|
|
North East 53
.
.
.
.
.
|
Mid &……... Fife 45 |
|
|
|
West 42 Glasgow 40 Central 46
Lothian 39
|
|
|
|
|
|
South 41
|
2.7 The complementary party, the combined opposition
For every party there is a notional complementary party consisting of the combined
opposition. For any given party, the distribution of the relative power of the
complementary party is the complement of the distribution of the relative power of
the given party. Applied to the present case, the percentage vote is taken as a measure
of relative power. Taking p and q as the proportional vote for the given party and for
the complementary party respectively, we have p+q=1 and q=1-p. The peaks of the
complementary party correspond to the troughs of the given party; and the troughs of
the complementary party correspond to the peaks of the given party. [Slide 17]
Thus in Slide 17, the peaks of the complementary party to the SNP are Lothian and
Glasgow, encompassing Scotland’s two major cities; and the troughs of the
complementary party to the SNP are North East and Central. [As noted earlier for
certain types of situation the number of troughs should be the same as the number of
peaks (possibly plus or minus one)]
Slide 17
The complementary party to the SNP
Figure 3 The percentage vote for the complementary party to the SNP across the eight
regions, Scottish Parliament 2011: two peaks (maxima) and two troughs (minima)
.
.
.
.
.
.
.
.
.
.
|
|
|
|
|
West 58 Glasgow 60
|
|
|
|
Highlands
& Islands 52
|
|
|
North East 47
Mid &……... Fife 55
|
|
|
|
|
Central 54
Lothian 61
|
|
South 59 |
13
2.8 Profiles ... party vectors in regional space ... regional vectors in party space
Of course in reality the complementary party may be an aggregate of a number of
quite distinct parties. Each percentage refers to a party and a region. There are six
parties and eight regions. The percentages can be displayed in the regional profiles of
the six parties. Alternatively, there are dual representations of the data: six party
vectors in eight-dimensional regional space; and eight regional vectors in sixdimensional party space.
The regional profiles of the six parties
[Slide 18] Figure 4 provides an example giving the percentage votes for the six parties
across the eight regions in the 2007 election. The regions are ordered left to right
according to geography: from south to north, and from west to east – except that
Glasgow is switched to be besides Lothian. Notice that Labour and SNP dominate the
other four parties, Labour ahead of SNP in the central belt and SNP ahead of Labour
in the north. Also each party is associated with a distinct peak: Conservatives in
South; Labour in Central; ‘Others’ in Glasgow; Greens in Lothian; SNP in North
East; and Liberals in Highlands & Islands. Note too the overall shape of each party
profile exhibiting just one or two peaks, just one or two troughs. The constraint that
the percentages should add to one fosters the formation of complementary profiles.
The Labour profile complements the Liberal Democrat profile. The Conservative
profile complements the Others profile.
The figure shows the greater complexity which attends the presence of more than two
parties and the presence of profiles with more than one maximum – either one
maximum in the middle and one at an end point or two maxima in the middle or in the
case of the Greens, a maximum in the middle and a maximum at each end point. Note
though that the ordering in the profile determines the number of peaks and troughs.
Slide 18
The regional profiles of the six parties
north-to-south and east-to-west model of % party votes in 2007
0.45
0.4
0.35
% party votes, 2007
0.3
SNP
Labour
Conservative
Liberal
Greens
Others
0.25
0.2
0.15
0.1
0.05
0
South
West
Central
Glasgow
Lothian
Mid & Fife
North East
Highlands &
Islands
the eight regions, north to south and east to west
14
Six party vectors in eight-dimensional regional space
We now wish to represent each party by a vector of proportions, one proportion for
each region. So we have six points in an eight-dimensional space. This is difficult to
present visually. Instead we present a reduced two-dimensional space. [Slide 19] Each
party is a point, a two-dimensional vector. The first dimension is the region-based
correlation of the party with Labour; and the second dimension is the region-based
correlation of the party with the Conservatives. Parties which are close together have
similar regional profiles (e.g. Labour and Others; and Lib Dem and SNP).; and parties
which are far apart have different regional profiles, possibly ‘opposite’ profiles (e.g.
Labour and Liberal; and Conservatives and Others).
Slide 19
Eight regional vectors in six-dimensional party space
We now wish to represent each region by a vector of proportions, one proportion for
each party. So we have eight points in a six-dimensional space. This is difficult to
present visually. Instead we present a reduced two-dimensional space. [Slide 20] Each
region is a point, a two-dimensional vector. The first dimension is the party-based
correlation of the region with Glasgow; and the second dimension is the party-based
correlation of the region with Highlands & Islands. Regions which are close together
have similar party profiles (e.g. Glasgow, Central and West; and Highlands & Islands
and North East); and regions which are far apart have different party profiles (e.g.
Glasgow and Highlands & Islands).
15
3 Temporal variation
We now turn to temporal variation. [Slide 21] As we noted in the introduction there
has been dramatic change, with the SNP vote doubling over the past decade. Whereas
the SNP experienced constant gains across regions in 2011, the LibDems experienced
proportional decline. Alongside this dramatic change there has been stability of the
spatial profile: peak regions and trough regions have remained more or less
unchanged. This has implications for the nature of the change function.
3.1 Stable configurations over the years
3.2 Constant and proportional change
3.3 The change function
3.4 The micro basis for macro change ... complexity dynamics
3.1 Stable configurations over the years
The spatial distribution of the SNP vote has remained much the same. And indeed this
is the case for the other parties also. For each party the correlation between the
regional profiles for successive elections is around 0.9. The high peak and low trough
regions for each party have stayed much the same over the past decade. [Slide 22]
Thus the primary location of the Liberal Democrats and the SNP is in the north; the
primary location of Labour, the rest and the Greens is in the central belt; and the
primary location of the Conservatives is in the south. Thus Labour and ‘the rest’ are
interior parties and the others are perimeter parties. The minima for Conservatives and
for Liberal Democrats are Central and Glasgow, reflecting a perimeter v. interior
party (Labour) competition. The minima for SNP are Lothian and the South reflecting
a perimeter v perimeter conflict.
16
Slide 22
Stable configurations over the years
Table High peak and low trough regions for each party over four elections, 1999,
2003, 2007, 2011
.
LD
SNP
Lab
Highlands & Islands HHHH L-LL -L-North East
-H-- HHHH -L-Mid & Fife
Central
Glasgow
West
Lothian
LLLL
South
rest
Green Cons
L-LL? LLLL--L
-HHH ---H LLLL --LL
H--H HHHH
LLLL
---L L-L-LLL
---H HHHH
L---
L---
L-L-
HHHH
3.2 Constant and proportional change
The high correlation between spatial variation in the vote at one point in time and at
another point in time indicates a relationship – but it does not tell us the mathematical
form of this relationship. The empirical data reveals two types of relationship:
constant gain or loss and proportional gain or loss. [Slide 23]
Figure below shows that the Scottish Nationalists’ remarkable gain between 2007 and
2011 of around 13% was fairly uniform across all eight Scottish regions.
The Times (2011) Briton of the Year. Thursday 27 December. Pp. 1, 2, 12, 13.
Slide 23
Figure Gain in the percentage vote for the SNP across all eight regions, Scottish
Parliament 2007 to 2011
dp=0.13: Change in SNP percentage vote across regions, 2007 to 2011
0.18
0.16
0.14
change in % vote
0.12
0.1
0.08
0.06
0.04
0.02
0
Central
Glasgow
West
South
Lothian
Mid & Fife
North East
Highlands &
Islands
region
17
Whereas the SNP gains were uniform the Lib Dem losses in Scotland showed great
variation. Following the old rugby maxim of ‘the bigger they are the harder they fall’,
the percentage loss was greatest where their proportion of the vote had been highest in
2007. See [Slide 24]
Slide 24
Figure 2 Loss in the percentage vote for the Lib Dems 2007 to 2011, as a function of
their percentage vote in 2007, across all eight regions, Scottish Parliament
dp = - 0.5p: proportional change in LibDem vote across the regions, 2007 to 2011
0
0
0.05
0.1
0.15
0.2
0.25
-0.01
dp: the change in % vote 2007 to 2011
-0.02
-0.03
-0.04
-0.05
-0.06
-0.07
-0.08
-0.09
p: the LibDem % vote in 2007
3.3 The change function
The above results have implications for the nature of the change function. Before
considering this however note that if we are just looking at the data then there are just
four points of time, the times of the four Scottish Parliament elections. Underlying the
data though is the notion that these are four points in what is a continuous stream of
time and that at each point in time there are aggregate voting intentions for the parties
– as is made evident in the regular opinion polls.
First let us suppose that the change function is density-dependent, in other words
change dp in the percentage vote p is a function of p, dp=f(p). If the change function f
is to be profile-preserving, then it needs to take the particular form of a monotonic
function, either increasing or decreasing. Two simple types of function are constant
change and proportional change.
Since ∑p=1, ∑dp=0. So the change functions are density-interdependent, dp=f(p).
The simplest case is the relationship between the change dp in the vote for a party and
the change dq in the vote for the complementary party. If p satisfies dp=f(p), then q
satisfies dq=-f(1-q). This prompts the thought that the function f may have some kind
of symmetry. For example: if p satisfies dp=k, then q satisfies dq=-k; and if p satisfies
dp=a+bp, then q satisfies dq=-(a+b)+bq.
However beyond the simplest case density-interdependence is more complicated.
18
Between 2007 and 2011, the SNP exhibited constant change across the regions,
dp(SNP)=+0.13; and the LibDems exhibited proportional change across the regions,
dp(LD)=-0.5p(LD). So the density-interdependent change equation for the group
‘neither SNP nor LibDem’ is dp(neither)=-0.13+0.5p(LD).
Slide 25
temporal variation
dramatic change - and stability of the spatial profile
density-dependent
dp=f(p)
profile-preserving implies monotonic
Since ∑p=1, ∑dp=0
density-interdependent, dp=f(p)
constant change
dp(SNP)=+0.13
proportional change
dp(LD)=-0.5p(LD)
dp(neither)=-0.13+0.5p(LD)
3.4 The micro basis for macro change ... complexity dynamics
All this needs further conceptualisation! And this is to be found in the micro basis for
macro change. [Slide 10] Each macro percentage is an aggregate of micro binaries for
a population of individuals, p=∑b/N. Each macro percentage vector is an aggregate of
micro binary vectors for a population of individuals, p=∑b/N. So change is a
population binary vector process. The change vector dp is composed of ‘births’,
‘deaths’ and ‘migrations’ – flows p(i,j) from voting for party i to voting for party j. In
models of migration such as the gravity model, flows are density-dependent. In
particular we might have dp(i)=ap(i)p(j).
Consider the concept of an individual’s propensity to change. Some years ago I
analysed data in a study by Yang, Goldstein and Heath (2000), looking at an
individual’s propensity to change their voting for or against the Conservative party.
Inhomogeneous, history-dependent, dynamic propensities were modelled by a
weighted-average linear one-step auto-dependent process, p(t+1)=0.5b(t)+0.5p(t) ...
very approximately. In a monograph on participation and performance I discussed
how stochastic explanatory variable dynamics drives criterion variable dynamics
drives binary choice dynamics drives propensity dynamics. Choice behaviour is
explained by value. Whereas this model assumes independence between individuals’
values, contagion opinion models such as Lux (1995) model of the stockmarket posits
interdependence. An individual’s propensity depends not just on their previous
propensity and behaviour but also on the population average behaviour and on the
individual’s susceptibility to others. This gives rise to a macro equation which
generates complex dynamics in aggregate population behaviour, exhibiting dramatic
fluctuations. Such a model could well explain the dramatic fluctuations which are to
be found in Scottish voting behaviour, the dramatic rise of the SNP and the collapse
of the Liberal Democrats. (Burt, 2010, pp. 206, 207)
19
Slide 26
The micro basis for macro change ... complexity dynamics
p=∑b/N; p=∑b/N
change is a population binary vector process
dp =‘births’+‘deaths’+‘migrations/flows’
flows p(i,j) =g(p)
gravity: dp(i)=ap(i)p(j)
inhomogeneous, history-dependent, dynamic propensities
weighted-average linear one-step auto-dependent process
p(t+1)=0.5b(t)+0.5p(t)
stochastic explanatory, criterion, binary choice, propensity dynamics
individuals’ values: independence or interdependence?
contagion of micro opinion: p(t+1)=h(b(t),p(t),v,x^(t))
macro opinion: dx^(t)=2v[Tanh(z)-x^]Cosh(z)
We now consider what underlies individuals’ opinions. The percentage votes come
from individuals’ votes for parties. And where do individuals’ votes for parties come
from? They come from individuals’ beliefs about and attitudes to the parties. These
beliefs about and attitudes to the parties in turn derive in part from individual’s beliefs
about and attitudes to policies and individuals’ beliefs about and attitudes to parties’
beliefs about and attitudes to policies. For this reason the dominant model in the
literature takes policy space as its central concept and locates individuals and parties
in policy space. This provides an answer to the question, ‘where do individuals’ votes
for parties come from?’. They come from individuals choosing to vote for the party
which is closest to them in policy space. This partitions policy space into party
subspaces. Returning to our original question where do percentage votes come from?
They come from the distribution of individuals in policy space and the proportions of
individuals in party subspaces. [Slide 27]
Slide 27
policy space, the micro-macro linkage
micro
macro
overall percentage votes for parties
area percentage votes for parties
individuals’ votes for parties
individuals’ beliefs about and attitudes to the parties
individuals and parties located in policy space
vote for party closest in policy space
partitions policy space into party subspaces
distribution of individuals in policy space
proportions of individuals in party subspaces
We can apply the same line of reasoning to spatial and temporal variation in
percentage votes for parties. [Slide 28] Micro variation underlies macro variation.
These arise as a result of:
spatial and temporal variation in individuals’ votes for parties
spatial and temporal variation in individuals’ beliefs about and attitudes to the parties
spatial and temporal variation in the location of individuals and parties in policy space
spatial and temporal variation in the distribution of individuals in policy space
spatial and temporal variation in the proportions of individuals in party subspaces
20
4 Spatial dynamics: the discourse-location-value-choice linkage
Voting choices depend on the value of the parties which depend on the location of the
valuer which depend on the propositions of the valuer which depend on social
discourse. [Slide 29] Media discourse, geographical location, stockmarket behaviour
and election behaviour provide powerful examples of social discourse, social location,
social value and social choice dynamics. A powerful model of social discourse is what
I call dynamic social propositional calculus (Burt, 2010, 2011). A powerful model of
social location is provided by geometric models. A powerful model of social value is
provided by Lux’s complexity model of stockmarket prices (Lux, 1995 et seq.,
discussed in Burt, 2010). A powerful model of social choice is provided by the
standard rational choice models of voting behaviour.
Slide 29
Social discourse, social location, social value and social choice dynamics
social discourse
media
dynamic social propositional calculus
social location
geography
geometry
social value
stockmarkets
opinion contagion
social choice
elections
rational choice
5 Economy, technology, military, cultural
Lux’s model of stockmarket prices envisages two types of trader: fundamentalists and
naive traders. Naive traders just follow opinion contagion whereas fundamentalists
follow fundamental value based on economic and other facts. It may be that there are
naive Scottish voters and fundamentalist Scottish voters. Certainly there is naive
Scottish discourse and fundamentalist Scottish discourse. Bannockburn in 1314,
Flodden in 1513 – and a Scottish referendum in 2013 or 2014. The dates of ancient
battles were repeated in the pages of the Times with the suggestion that the
centenaries would arouse the passions of voters and affect the result of the referendum
(The Times, 2012, Jan 9, pp. 15, 20; Jan 10, pp. 2, 7; Jan 11, pp. 22, 25). Contrasting
with this naive discourse other articles in the media have referred to fundamentals
such as how the transition process would work and what the consequences might be in
the economic, technological and military spheres.
6 Conclusion
Let me finish by reviewing the topics we have covered. [Slide 30]
The mathematical object we have been studying consists of: a set X with a contiguity
relation R and a distance function d; and a mapping f from X to an n-dimensional
Euclidean space Rn. This object corresponds to the Scottish election data set and it
also corresponds to the situation in loss-of-power gradient theory. The question we
have been asking, ‘what can a loss-of-power gradient look like?’ corresponds to the
question ‘what is the nature of the function f?’. In particular, is the function f a simple
formula or is it in some sense configurational? The answer is both. A simple formula
can be a useful approximation. However the underlying reality is that f is
configurational and this is most clearly demonstrated by the existence of more than
one peak. There is a systematic method for analysing the configuration, what might be
called the ‘islands-and-troughs’ method.
21
One of my aims has been to inform the literature on the loss-of strength gradient.
Much of the time this literature assumes a single-peaked spatial distribution of
strength. The spatial distribution of percentage votes in Scotland exhibits multiple
peaks. So the loss-of-strength gradient cannot be an exact representation – however it
can provide an approximate representation. There will be cases where this is
misleading. So single-peakedness is a bit like linearity: in most studies it is assumed
rather than checked; and sometimes the assumption is correct or at least provides a
reasonable approximation but at other times can mislead.
We have also studied a more complicated mathematical object: the set X is a partition
of some underlying space S, and we are interested in the set P of all possible partitions
of S, what might be called the partition power set. Scaling formula relate the findings
at different scales of analysis based on different partitions, an example being the
formula for the number of peaks. There is a structure of configurations at different
levels, possibly irregular both within and between levels and so reminiscent of the
spatial patterns discussed by Jensen (1998) in his book on complexity and fractals.
The presence of multiple parties requires consideration of percentage vectors and the
representation of temporal variation requires change functions which preserve the
spatial profile and which are density-interdependent. Also there is a micro basis for
macro change. There is a flow of individuals from one party to another. Individual
change is characterised by inhomogeneous, history-dependent, dynamic propensities.
Propensity dynamics is driven by stochastic explanatory, criterion and binary choice
dynamics. The explanatories include individuals’ values and they may be
interdependent giving rise to complex dynamics in aggregate population behaviour
which can exhibit dramatic fluctuations.
Individuals’ values can be represented as population distributions in policy space and
this space is partitioned into party subspaces. Underlying the policy space
representation is policy discourse and my personal view is that this needs to be
handled by what I refer to as dynamic social propositional calculus. Values and
discourse are not self-contained – they refer to society with its economic, political,
cultural, technological, military and environmental dimensions.
It is in terms of these abstract ideas that we should understand the battle for hearts and
minds in Scotland and in Iraq, Libya, Syria and Afghanistan and indeed to any society
whether or not violence is present, whether or not there is a democracy.
Slide 30
Conclusion
.(1) set X, contiguity relation R, distance function d, mapping f from X to Rn.
.(2) corresponds to the Scottish election data set
.(3) corresponds to situation in loss-of-power gradient theory
.(4) what is the nature of the function f?
.(5) is the function f a simple formula or is it in some sense configurational?
.(6) analysing the configuration: the ‘islands-and-troughs’ method
.(7) set X is a partition of S; partition power set P
.(8) scale of analysis, n ... peaks, c=f(n,L) ... spatial complexity
.(9) more than one party ... profiles ... correlation spaces
.(10) profile-preserving, density-interdependent change
.(11) micro basis for macro change ... complexity dynamics
.(12) Spatial dynamics: the discourse-location-value-choice linkage
.(13) Economy, technology, military, cultural
.(14) Non-violent, democratic Scotland ... anywhere violent/not, democratic/not
22
References
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Diehl, 1991; Starr, 2005; Buhaig, 2010
Micro-macro contagion model of stockmarket: Lux (1995);
Dynamic social propositional calculus: Burt (2011); Burt (2010), Chapter 3
Mathematical logic, artificial intelligence and ordinary language
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23