since it only contain the initial $\tau$-iterations results and coefficient is
diminished by $T$. When the $T$ is large enough, the extra term is almost
negligible. Moreover, we can use some warm-up strategy, such as allreduce,
that forces all agents’ iterates in the first period are the same, i.e.
$\sum_{k=0}^{\tau-1}\mathbb{E}\,\left\|{\mathbf{x}}^{(k)}-{\bar{\mathbf{x}}}^{(k)}\right\|^{2}=0$.
Under this situation, we immediately obtain the following corollary.

###### Corollary 3

Under the same assumptions as lemma 6 and using the all-reduce warm-up
strategy at the first $\tau$ iterations, it holds

$\displaystyle\frac{1}{T+1}\sum_{k=0}^{T}\mathbb{E}\,\left\|{\mathbf{x}}^{(k)}-{\bar{\mathbf{x}}}^{(k)}\right\|^{2}\leq\frac{8\tau\gamma^{2}\rho_{\rm
max}^{2}}{(1-\beta)^{2}}(\sigma^{2}+4\tau b^{2})$ (103)

Proof. Replacing
$\sum_{k=0}^{\tau-1}\mathbb{E}\,\left\|{\mathbf{x}}^{(k)}-{\bar{\mathbf{x}}}^{(k)}\right\|^{2}$
by 0 gives the conclusion immediately. $\hfill\blacksquare$

Figure 12: The value of $\|\prod_{i=0}^{k-1}{\widehat{W}}^{(i)}\|^{2}$ evolves
with $k$ over different number of nodes.

Lastly, we revisit the quantity $\rho_{\rm max}^{2}$ by numerical experiment
here. Looking at the (90) again, we relax our bounds by simply taking the
maximum value of all $\|\prod_{i=t}^{k-1}{\widehat{W}}^{(i)}\|^{2}_{2}$. But
this value can be much smaller than $\rho_{\rm max}^{2}$. We just validate
them by the numerical experiment in Fig. 12.

###  D.4 Proof of the convergence Theorem 1

Finally, we are ready to present the convergence theorem about the
decentralized momentum SGD over one-peer exponential graph. Substituting the
conclusion of the descent lemma 4 into the consensus lemma6, we immediately
establish

$\displaystyle\frac{1}{T+1}\sum_{k=1}^{T}\mathbb{E}\,\|\nabla
f({\bar{\mathbf{x}}}^{(k)})\|^{2}\leq$
$\displaystyle\frac{2(1-\beta)}{\gamma(T+1)}(\mathbb{E}\,f({\bar{\mathbf{z}}}^{(0)})-f^{\star})+\frac{\gamma
L}{n(1-\beta)}\sigma^{2}+\frac{\beta L\gamma}{n(1-\beta)^{2}}\sigma^{2}$
$\displaystyle\;\;\;+\frac{8\tau\gamma^{2}L^{2}\rho_{\rm
max}^{2}}{(1-\beta)^{2}}(\sigma^{2}+4\tau
b^{2})+\frac{2L^{2}}{(T+1)}\sum_{k=0}^{\tau-1}\mathbb{E}\,\left\|{\mathbf{x}}^{(k)}-{\bar{\mathbf{x}}}^{(k)}\right\|^{2}$
(104)

where the learning rate $\gamma$ requires:

$\displaystyle\gamma\leq\min\left\\{\frac{1-\beta}{6L\tau},\frac{(1-\beta)^{2}}{2(1+\beta)L}\right\\}$
(105)

Simplifying and grouping the terms, we obtain

$\displaystyle\frac{1}{T+1}\sum_{k=1}^{T}\mathbb{E}\,\|\nabla
f({\bar{\mathbf{x}}}^{(k)})\|^{2}$ $\displaystyle\leq$
$\displaystyle\frac{2(1-\beta)}{\gamma(T+1)}\left(\mathbb{E}\,f({\bar{\mathbf{z}}}^{(0)})+\frac{\gamma
L^{2}}{1-\beta}\sum_{k=0}^{\tau-1}\mathbb{E}\,\left\|{\mathbf{x}}^{(k)}-{\bar{\mathbf{x}}}^{(k)}\right\|^{2}-f^{\star}\right)$
$\displaystyle\;\;\;+\frac{\gamma
L}{n(1-\beta)^{2}}\sigma^{2}+\frac{8\tau\gamma^{2}L^{2}\rho_{\rm
max}^{2}}{(1-\beta)^{2}}(\sigma^{2}+4\tau b^{2})$ $\displaystyle=$
$\displaystyle O\left(\frac{(1-\beta)}{\gamma
T}\right)+O\left(\gamma\frac{\sigma^{2}}{n(1-\beta)^{2}}\right)+O\left(\frac{\sigma^{2}\tau\gamma^{2}}{(1-\beta)^{2}}\right)+O\left(\frac{b^{2}\tau^{2}\gamma^{2}}{(1-\beta)^{2}}\right)$
(106)

If we set the learning rate as
$\gamma=O\left(\frac{\sqrt{n(1-\beta)^{3}}}{\sqrt{T}}\right)$, we have

$\displaystyle\frac{1}{T}\sum_{k=1}^{T}\mathbb{E}\,\|\nabla
f({\bar{\mathbf{x}}}^{(k)})\|^{2}=O\left(\frac{\sigma^{2}}{\sqrt{(1-\beta)nT}}\right)+O\left(\frac{n(1-\beta)\sigma^{2}\tau}{T}\right)+O\left(\frac{n(1-\beta)b^{2}\tau^{2}}{T}\right)$
(107)

Last, we derive the transient iteration complexity for the data-homogeneous
and data-heterogeneous scenarios, respectively.

$\displaystyle\frac{\sigma^{2}}{\sqrt{(1-\beta)nT}}=\frac{n(1-\beta)\sigma^{2}\tau}{T}\;\;$
$\displaystyle\Longrightarrow\;\;T=(1-\beta)^{3}n^{3}\tau^{2}\;\;\mbox{(data-
homogeneous)}$ (108)
$\displaystyle\frac{\sigma^{2}}{\sqrt{(1-\beta)nT}}=\frac{n(1-\beta)b^{2}\tau^{2}}{T}\;\;$
$\displaystyle\Longrightarrow\;\;T=(1-\beta)^{3}n^{3}\tau^{4}(b^{4}/\sigma^{4})\;\;\mbox{(data-
heterogeneous)}$ (109)

Absorbing the constants into $\Omega(\cdot)$ notation and replacing $\tau$ by
$\log_{2}(n)$, we establish the transient iteration complexity as stated in
Theorem 1. $\hfill\blacksquare$

###  D.5 Comparison with other commonly-used graphs

####  D.5.1 Comparison in per-iteration communication and transient iteration
(Homogeneous)

Table 7 summarizes the per-iteration communication and transient iteration
complexity of DmSGD with commonly-used topologies. The details of each
topology and its associated weight matrix $W$ can be referred to Sec. A.3.
Table 7 assumes homogeneous data distributions across all nodes. If the
logarithm term can be ignored when $n$ is large, it is observed that both
static and one-peer exponential graphs can achieve $\tilde{\Omega}(1)$ per-
iteration communication and $\tilde{\Omega}(n^{3})$ transient iteration
complexity, both of which are nearly best among all compared graphs. Table 7
is an extension of Table 1 by comparing with more topologies.

Table 7: Comparison in per-iteration communication time and transient iteration complexity between decentralized momentum SGD over various commonly-used topologes. The table assumes homogeneous data distributions across all nodes. | Per-iter. Comm. | Transient iter. complexity
---|---|---
ring | $\Omega(2)$ | $\Omega(n^{7})$
star graph | $\Omega(n)$ | $\Omega(n^{7})$
2D-Grid | $\Omega(4)$ | $\Omega(n^{5}\log^{2}_{2}(n))$
2D-Torus | $\Omega(4)$ | $\Omega(n^{5})$
$\frac{1}{2}$-random graph | $\Omega(\frac{n}{2})$ | $\Omega(n^{3})$
bipartite random match | $\Omega(1)$ | N.A.
static exponential | $\Omega(\log_{2}(n))$ | $\Omega(n^{3}\log_{2}^{2}(n))$
one-peer exponential | $\Omega(1)$ | $\Omega(n^{3}\log_{2}^{2}(n))$

####  D.5.2 Comparison in per-iteration communication and transient iteration
(Heterogeneous)

Table 8 summarizes the per-iteration communication and transient iteration
complexity of DmSGD with commonly-used topologies when data distributions are
heterogeneous. Compared to Table 7, it is observed that the transient
iteration complexity achieved by each topology in the heterogeneous scenario
is typically worse than that in the heterogeneous scenario. Again, if the
logarithm term can be ignored when $n$ is large, it is observed that both
static and one-peer exponential graphs can achieve $\tilde{\Omega}(1)$ per-
iteration communication and $\tilde{\Omega}(n^{3})$ transient iteration
complexity, both of which are nearly best among all compared graphs.

Table 8: Comparison in per-iteration communication time and transient iteration complexity between decentralized momentum SGD over various commonly-used topologes. The table assumes heterogeneous data distributions across all nodes. | Per-iter. Comm. | Transient iter. complexity
---|---|---
ring | $\Omega(2)$ | $\Omega(n^{11})$
star graph | $\Omega(n)$ | $\Omega(n^{11})$
2D-Grid | $\Omega(4)$ | $\Omega(n^{7}\log^{4}_{2}(n))$
2D-Torus | $\Omega(4)$ | $\Omega(n^{7})$
$\frac{1}{2}$-random graph | $\Omega(\frac{n}{2})$ | $\Omega(n^{3})$
bipartite random match | $\Omega(1)$ | N.A.
static exponential | $\Omega(\log_{2}(n))$ | $\Omega(n^{3}\log_{2}^{4}(n))$
one-peer exponential | $\Omega(1)$ | $\Omega(n^{3}\log_{2}^{4}(n))$

####  D.5.3 Exponential graphs endow DmSGD with smaller transient iterations:
numerical validation

In Tables 7 and 8, it is observed that exponential graphs endow DmSGD with
smaller transient iterations. In this subsection, we validate it with
numerical experiments.

We consider a distributed logistic regression problem with each local cost
function as

$\displaystyle
f_{i}(x)=\frac{1}{M}\sum_{m=1}^{M}\ln[1+\exp(-y_{i,m}h_{i,m})^{T}x],$ (110)

where $\\{h_{i,m},y_{i,m}\\}_{m=1}^{M}$ are local data samples at agent $i$
with $h_{i,m}\in\mathbb{R}^{d}$ being the feature vector and
$y_{i,m}\in\\{+1,-1\\}$ being the corresponding label. Each $h_{i,m}$ is
generated from the normal distribution ${\mathcal{N}}(0;10I_{d})$. To generate
$y_{i,m}$, we first generate an auxiliary random vector
$x^{\star}_{i}\in\mathbb{R}^{d}$ with each entry following
${\mathcal{N}}(0,1)$. Next, we generate $y_{i,m}$ from a uniform distribution
${\mathcal{U}}(0,1)$. If $y_{i,m}\leq 1/[1+\exp(-h_{i,m}^{T}x_{i}^{\star})]$
then $y_{i,m}$ is set as $+1$; otherwise $y_{i,m}$ is set as $-1$. We consider
a non-iid scenario in which $x_{i}^{\star}\neq x_{j}^{\star}\ \forall i,j$.
Each $x_{i}^{\star}$ is normalized. We set the number of nodes as 64.

Figure 13: Convergence curves of DmSGD with various topologies. It is observed
that DmSGD with exponential graphs has less transient iterations than with
other graphs.

Fig. 13 illustrates the convergence curves of DmSGD with different topologies
as well as parallel momentum SGD (PmSGD). The momentum parameter $\beta=0.8$.
The mean-square-error in the $y$-axis indicates
$\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}\|x_{i}^{(k)}-x^{\star}\|^{2}$. We set
$d=10$ and $M=14000$. The step-size $\gamma$ is initialized as $0.2$ and gets
decreased by half for every $1000$ iterations. We repeat all simulations $20$
times and illustrate the mean of all trials. It is observed that DmSGD with
static exponential graph converges closely to PmSGD, and it is with the
shortest transient iterations. Also, DmSGD with one-peer exponential graph is
observed to have slightly longer transient iterations than with the static
exponential graph. However, both exponential graphs endow DmSGD with shorter
transient iterations than grid and ring.

##  Appendix E More Experiments

###  E.1 Details of each topology and the weight matrix

The description of the tested topology and its associated weight matrix can be
referred to Sec. A.3.1.

###  E.2 DmSGD with exponential graphs when $n$ is not a power of $2$

Under the same experimental setting as in Sec. 6.2, this subsection examines
the performance of exponential graphs when $n$ is not a power of $2$. As shown
in Table 9, one-peer exponential graph can still endow DmSGD with similar, or
even better, training performance compared to its static counterpart.

Table 9: Comparison of top-1 validation accuracy(%) when using DmSGD with arbitrary numbers of nodes. nodes | 6(6x8 GPUs) | 9(9x8 GPUs) | 12(12x8 GPUs) | 15(15x8 GPUs)
---|---|---|---|---
static exp. | 76.21 | 75.93 | 75.73 | 76.03
one-peer exp. | 76.16 | 76.17 | 75.85 | 76.19

###  E.3 Performance with DSGD

In empirical studies, we conducted a few more experiments to validate how DSGD
performs over exponential graphs in deep learning. In Table 10, we repeated
the same experiment in Table 2 except for the parameter setting $\beta=0$ i.e.
eliminating the influence of momentum. It is observed that:

  * •

The accuracy performance of all DSGD scenarios has dropped over 7% compared to
the DmSGD scenarios. This highlights the critical role of the momentum in DSGD
for real deep learning experiments.

  * •

DSGD over the one-peer exponential graph achieves similar accuracy as the
static exponential graph, and both topologies enable DSGD with higher accuracy
than the ring topology. This is consistent with the two conclusions listed
above.

  * •

The training time of DSGD over different topologies is similar to DmSGD listed
in Table 2, and we, therefore, omitted it in the following table.

Table 10: Comparison of top-1 validation accuracy(%) when using DSGD with different topologies. nodes | 4(4x8 GPUs) | 8(8x8 GPUs) | 16(16x8 GPUs)
---|---|---|---
ring | 68.85 | 68.62 | 68.78
static exp. | 69.08 | 68.81 | 68.79
one-peer exp. | 69.01 | 68.94 | 68.85

###  E.4 Example code for implementation

For the implementation of decentralized methods, we utilize BlueFog, which is
an open-source high-performance decentralized deep training framework, to
facilitate the topology organization, weight matrix generation, and efficient
partial averaging.

⬇

1def neighbor_allreduce(tensor: torch.Tensor,

2 self_weight: float,

3 src_weights: Dict[int, float],

4 dst_weights: Dict[int, float]) -> torch.Tensor:

Listing 1: Neighbor allreduce functionality for communication.

One major functionality for decentralized communication is
_neighbor_allreduce_ , as listed in Listing 1, implementing the following
equation.

$\displaystyle
x^{(k+1)}_{i}=w_{ii}x^{(k)}_{i}+\sum_{j\in\mathcal{N}_{i}\backslash
i}w_{ij}x^{(k)}_{j},$ (111)

The argument _self_weight_ stands for $w_{ii}$, and $w_{ij}$ for communication
with the other node $j$ is achieved by either using _src_weights_ in pull-mode
or using _dst_weights_ in push-mode.

Listing 2 gives two utility functions for one-peer exponential graphs
generation. For each node, each call of function _GetOnePeerExpGraphGenerator_
provides the one-peer nodes connection information for communication. Passing
this connection information to _neighbor_allreduce_ is achieved through
updating the member variables of decentralized optimizer in
_UpdateOnePeerExpGraph_.

⬇

1# One-peer exponential graph generation

2def GetOnePeerExpGraphGenerator(size, self_rank):

3 tau = math.ceil(math.log2(size)) # Periodic cycle

4 index = 0

5 while True:

6 send_rank = (self_rank + 2**index) % size

7 recv_rank = (self_rank - 2**index) % size

8 yield send_rank, recv_rank

9 index += 1

10 index = index % tau

11

12# Graph update in each iteration

13one_peer_exp_graph_gen = GetOnePeerExpGraphGenerator(bf.size(), bf.rank())

14def UpdateOnePeerExpGraph(optimizer):

15 dst_rank, src_rank = next(one_peer_exp_graph_gen)

16 optimizer.dst_weights = {dst_rank: 1.0}

17 optimizer.src_weights = {src_rank: 0.5} # Corresponds to W matrix

18 optimizer.self_weight = 0.5 # Corresponds to the diagonal of W matrix

Listing 2: Utility functions for the generation of one-peer exponential
graphs.

With that, Listing 3 shows a simplified code for model training. The overall
code structure is similar as the traditional model training script, with few
modifications for decentralized environment. On line 5, a decentralized
optimizer wraps the original SGD optimizer. Under the hood, it registers the
neighbor_allreduce communication function through the hook mechanism. On line
11, the one-peer exponential graphs get updated in each iteration. After the
model forward propagation is computed locally, the backward propagation is
performed on line 19. Meanwhile, it also triggers communication using
_neighbor_allreduce_. In order to boost the training efficiency, the time of
computation and communication are overlapped as much as possible through the
multi-threading. Finally, line 21 updates the model until the communication
finishes.

⬇

1import bluefog.torch as bf

2... # Model and data preparation

3# Generate decentralized optimizer

4optimizer = optim.SGD(model.parameters(), lr=lr, momentum=momentum)

5optimizer = DmSGDOptimizer(optimizer, model=model)

6...

7for epoch in range(num_epochs):

8 # Training the model

9 for data, target in train_loader:

10 # Graph update in each iteration

11 UpdateOnePeerExpGraph(optimizer)

12 data, target = data.cuda(), target.cuda()

13 optimizer.zero_grad()

14 # Local forward propagation

15 output = model(data)

16 loss = F.cross_entropy(output, target_batch)

17 # Local backward propagation

18 # Meanwhile triggering neighbor_allreduce communication

19 loss.backward()

20 # Model update and wait until the communication finishes

21 optimizer.step()

22 # Validation

23 ...

Listing 3: Example of how to train a model using a DmSGD optimizer under a
one-peer exponential graph. The communication graph is updated in each
iteration.


# Decentralising the United Kingdom: the Northern Powerhouse strategy and
urban ownership links between firms since 2010

Natalia Zdanowska Luxembourg Institute of Socio-Economic Research, Luxembourg
Centre for Advanced Spatial Analysis, University College London, London,
United Kingdom  Robin Morphet Centre for Advanced Spatial Analysis, University
College London, London, United Kingdom

This paper explores a decentralisation initiative in the United Kingdom – the
Northern Powerhouse strategy (NPS) – in terms of its main goal: strengthening
connectivity between Northern cities of England. It focuses on economic
interactions of these cities, defined by ownership linkages between firms,
since the NPS's launch in 2010. The analysis reveals a relatively weak
increase in the intensity of economic regional patterns in the North, in spite
of a shift away from NPS cities'traditional manufacturing base. These results
suggest potential directions for policy-makers in terms of the future
implementation of the NPS.

Key words: Decentralisation, Northern Powerhouse strategy, cities,
interactions, ownership networks of firms, United Kingdom.

## Introduction

According to Sharon White, then (20th January 2015) second permanent secretary
at HM Treasury, the United Kingdom is almost the most centralised developed
country in the whole world (Agbonlahor, , 2015) thus identifying one of the
most important issues facing the UK. This reflected the 2014 view of OECD
chief executive Alexandra Jones who said that within the OECD, the UK is one
of the most centralised countries (Jones, , 2014; Charbit, , 2011). Strong
centralisation has not always characterised the British urban system. In the
19th century Manchester together with Liverpool was one of the most
industrialised conurbations in Europe, a major textile producer and, in 1830,
the first to have passenger railway connections (Elbaum and Lazonick, , 1984).
Manchester, the UK's second economic centre, has a significant history in
terms of innovation and social progress and, together with Liverpool, was the
first identified conurbation (Geddes, , 1915).

Despite many decentralisation initiatives proposed by the UK government to
devolve power to city regions (Sandford, , 2020), sub-regional governance
remains subordinate to a centralised model of decision making. The partial
move of the BBC from London to Manchester is one example of a decentralising
governmental initiative. Other examples such as City Deals (O' Brien and Pike,
, 2015) the establishment of Combined Authorities and the introduction of City
Mayors (Sandford, , 2020) must be seen in the context of the loss of spending
power of English Metropolitan Districts of over 20% between 2009/2010 and
2018/19 (Atkins, , 2020) as a result of reductions in the Revenue Support
Grant. Decentralisation is a central challenge particularly in the context of
the revenge of the places that don’t matter (Rodríguez-Pose, , 2017), of
rising populism and inequalities in the country (Carrascal-Incera et al., ,
2020). The Northern Powerhouse (NP) together with the Northern Powerhouse
strategy (NPS) is a central government proposal, albeit one with strong local
support. It was launched in 2010, to boost economic growth in the North of
England particularly in the core cities of Manchester, Liverpool, Leeds,
Sheffield, Hull and Newcastle. It corresponds to a vision of a globally
connected and competitive Northern Economy, exploiting strengthened
connections between and within the Northern cities (The Northern Powerhouse, ,
2016). The Prime Minister (Johnson, , 2019) pledged support for the NP saying
We are going to maximise the power of the North with more mayors across the
whole of the North. The NP Partnership Plan for 2050 claims that it will
improve the productivity of the North in its vision for the future (The
Northern Powerhouse, , 2016). Decentralising Britain is considered as the big
push towards inclusive prosperity Esteve et al.,  (2019) and reflects the view
that, to counterbalance the forces of growing globalisation, local economic
strengths should emerge as a countervailing force (Roudometof, , 2018).

This paper aims at exploring these local strengths (Roudometof, , 2018) within
the NP economy111Cheshire and Warrington, Cumbria, Greater Manchester, Humber,
Lancashire, Leeds City Region, Liverpool City Region, North East, Sheffield
City Region, Tees Valley, York, North Yorkshire and East Riding. North Wales
is defined as covering: Wrexham, Rhyl, Colwyn Bay, Llandudno and Bangor.. The
original contribution of this paper is to approach these issues within the
system of cities framework (Pumain, , 1997). In the latter we consider only
the theoretical aspects where the position and dynamics of cities in their
spatial and socio-economic dimensions are considered in the context of their
interactions with other cities. Preferential interactions between certain
cities of a national system can drive the emergence of sub-systems, which can
be relevant to test in the case of the NP economy. In addition, strengthening
the national system of cities and building relationships and partnerships
between cities are all outcomes of foresighting defined by the UK government
for the next fifty years (GoS, , 2016). In fact recent studies have stressed
the importance of urban foresight (Dixon et al., 2018a, ; Dixon and Tewdwr-
Jones, , 2021) and urban foresight techniques for city visioning (Dixon et
al., 2018b, ) in order to create opportunities for new investment in the local
urban economy and to elaborate strategic urban intelligence (Ravetz and Miles,
, 2016). From this perspective, a particular focus will be given to the city
of Manchester, to evaluate its regional role as the second ranked economic
centre of the UK. To conduct this research, we investigate transnational firm
networks defined by ownership links between firms at city-level, using the
Bureau Van Dijk's ORBIS and AMADEUS databases. We question how Manchester's
position has evolved within its inter-urban economic networks in a highly
centralized city-system, following the launch of the NP initiative in 2010
(Cox and Longlands, , 2016).

The paper is divided into four sections: literature review, hypotheses,
materials and methods, results and discussion.

## Literature review

Local embeddedness in economic geography has tended to conflate network and
territorial dimensions in focusing on the nature of a firm's linkages to, and
interactions with, other local actors, whether firms or cities. These
interactions are crucial for regional development and can contribute to
revitalizing their surrounding regions (Batty, , 2013; Clark et al., , 2018).
Strengthening these inter-urban ties is one of the NP's objectives, although
no clear critical assessment of this policy in terms of economic interactions
has been made since 2010. This obscures a clear city-vision for the NPS as a
whole, as has already been constructed for separate UK cities such as Reading
(Dixon et al., 2018b, ) or Newcastle (Tewdwr-Jones et al., , 2015) with the
aim of identifying those factors which could underpin a strong local economy
in the future. In this context urban (or city) foresight is understood as the
science of thinking about the future of cities. ”It draws on diverse methods
to give decision-makers comprehensive evidence about anticipated and possible
future change” (Dixon et al., 2018b, ). In this sense urban foresight in the
present paper is conducted by analysing the NP's economic interactions between
firms.

There are many reasons why a firm might want to possess a subsidiary in a
location remote from the parent firm. The parent firm may be seeking to ring
fence a particular operation (Bates Wells, , 2019) isolating both risks and
assets. For example, a subsidiary may be a locally well known brand or the
subsidiary may be developing innovative products with their concomitant risks.
The extent of ring fencing may vary according to whether or not the parent
company has guaranteed or indemnified its subsidiary. It may also vary in
terms of subsidiary autonomy from the parent. Autonomy may range from hands on
local management control to shared corporate policies or more remote
performance management. The latter may compare performance with other
subsidiaries of the parent rather than with more local competition. The
subsidiary sits somewhere in the competitive spectrum between the parent
institutional economy (Coase, , 1937) and the more or less, free market of its
local economy, and may seek to benefit from both. Investment in subsidiaries
by foreign firms may also take into account taxation benefits and the relative
stability of trading conditions under a different jurisdiction.

The rationale (Marshall, , 1930; Ellison et al., , 2010) for agglomeration
economies concentrates on the role of supply factors such as skilled labour,
knowledge spillovers and transport costs and the ability to access scale
economies. These are for the most part, treated statically. The advantages,
however, of a subsidiary organisation which accrue to a parent firm may
include its access to more dynamic factors such as a continually improving
pool of skilled labour, access to knowledgeably up to date capital investment,
access to an increasingly specialised supply chain of non-traded intermediates
and to an informed market (Porter, , 1994). The positive feedback of the
forward and backward linkages (Krugman, , 1991) which may be exploited by a
relatively autonomous subsidiary may well be sufficient to outweigh the
advantages of vertical linkages stemming from locating the subsidiary activity
within the parent firm. Another strong reason for locating in such an
environment is the need to develop trusted bilateral relationships (Mignot and
Vignes, , 2020) when progressing the development of innovative products.
Bilateral relationships in such markets require a degree of trust that is
unnecessary in auction markets where price is the dominant component of
information exchange. Granovetter,  (2005) describes the characteristics of
such networks in terms of norms and network density. Dense networks of
contacts tend to reinforce norms of conduct and the application of sanctions
for nonconformity as against the use of impersonal auction markets for supply.
Gammelgaard et al.,  (2012) found, when examining subsidiaries of foreign
firms, that increased autonomy improved inter organisational networks and that
increased inter organisational networks improved subsidiary performance. A
less sanguine view of the benefits of autonomy is given by Puck et al.,
(2016) who compared foreign owned subsidiaries with international joint
ventures in China and found that the wholly owned foreign subsidiaries
performed better. They argue the need to differentiate control from ownership
since control can vary considerably within similar levels of ownership. They
go on to argue that cultural distance will also affect the level of control
reducing it for wholly owned subsidiaries and increasing it for joint
ventures. Of course the cultural distance between China and a foreign firm
will be rather larger than the cultural distance between London and the NP but
the latter should not therefore be ignored.

Subsidiary performance is an active field of research which has relied largely
on cross sectional surveys. Autonomy and agglomeration benefits are recognised
as important components of cluster formation. Less studied is the temporal
dimension although its importance is acknowledged (Avnimelech and Feldman, ,
2010). Considering these structures and relationships begs the question of
what sort of evolution of arrangements we might expect to see in a NP which
was successfully transitioning into a counterweight to London. We might follow
the analysis of Avnimelech and Feldman,  (2010) for subsidiaries set up by
foreign firms. In this case we would argue for the support of inward
investment through subsidiary firm location on the grounds that this would
encourage the development of further more locally owned subsidiaries or start-
ups as off shoots of the foreign owned subsidiaries. The new firms would
benefit from the entrepreneurial and management experience gained in the
externally owned subsidiaries. Conversely we might follow the analysis of
manufacturing (Ellison et al., , 2010) which emphasised the Marshallian
agglomeration factors of the movement of ideas, goods and people and the need
to reduce the cost of such movements. In practice both will be necessary.
Transport improvements are not enough by themselves but are an essential part
of creating a more unified labour market across the NP. Facilitating knowledge
spillovers requires arrangements that encourage and are conducive to the like
minded entrepreneurs, researchers, politicians and business people meeting and
sharing ideas. So far as subsidiaries are concerned the time profile of
development could be an early dependence on externally owned firms setting up
subsidiaries to exploit local advantages followed by a generation of more
locally owned, more autonomous subsidiaries as part of the development of more
specialised supply trains in support of more innovative firms. Manchester in
particular, can look back on a long tradition of working in this way with its
innovations in railway transport in the early 19th century but also in its
organisation of support for the interaction of science and business in the
1840s (Kargon, , 1977). The organisations that led that process remain in
existence today in the form of the Literary and Philosophical Societies of
Liverpool, Manchester, Leeds, Sheffield, Newcastle and Hull together with many
of the associated specialist groups.

According to NESTA,  (2016) there is still a need for boosting links between
the city and entrepreneurs within the NP economy. NESTA recommends expansion
of international transport linkages to strengthen global business connections.
However, an analysis of the evolution of these business connections since the
launch of the NPS in 2010 was not conducted in NESTA's report. This paper
seeks to fill this gap in the literature through a temporal assessment of the
NPS within the system of cities framework, where preferential interactions
between certain cities can give rise to the emergence of national sub-systems
Pumain,  (1997) – in this case, the economy of the North of the England.

The spring 2016 report on the NPS by HM Government (HMGovernment, , 2016)
focuses mainly on transport issues and future road investments connecting NPS
cities, but does not highlight the state of business connections between these
cities. In the same perspective The Northern Powerhouse Independent Economic
Review (Pates et al., , 2016) identifies poor transport connectivity in the
North as a major problem. Existing economic sector strengths are presented :
advanced engineering, life sciences and pharmaceuticals, chemicals, energy and
environment, and financial and professional services. Agri-tech/food is a
sector with notable growth potential in the North (Pates et al., , 2016). Yet
no economic analysis of inter-urban interactions is conducted. The Greater
Manchester Combined Authority (GMCA) industrial strategy released in June 2019
HMGovernment,  (2019) also promotes global connectivity, supports innovative
and international enterprise and foreign direct investments and sustains
further internationalisation of the Greater Manchester business base. Four
specific sectors are to be targeted in the future: advanced manufacturing (by
creating the Graphene City at the University of Manchester and the Advanced
Materials City in the North East of Manchester), health innovation, the
already existing digital/media cluster and clean growth.

In this paper we wish to assess economic interactions within the NPS cities
and the evolution of their economic sector capabilities within inter-urban
ownership networks in the North of England. Literature on ownership networks
is very scarce. Because of the lack of precise geo-located data, research is
generally conducted either on aggregated data at regional or national level,
on panel data for several years (Demzetz and Lehn, , 1985) or for one specific
sector only (Franks and Mayer, , 1997; Bond et al., , 2006; Watts et al., ,
2007). Studies on ownership links of firms at city level have been conducted
for all types of sectors for different geographical areas (Gillespie, , 2017;
Martinus et al., , 2019; Rozenblat, , 2010; Rozenblat et al., , 2017;
Rozenblat and Pumain, , 2007; Śleszyński, , 2015; Zdanowska et al., , 2020;
Zdanowska, , 2020) with the objective of modeling growth of urban firm
networks (Raimbault et al., , 2020) . Ownership and industrial linkages have
already been studied in Northern England, but for regions only (Marshall, ,
1979).

We wish to identify polarisation and peripheralisation (Kühn, , 2015; Benedek
and Moldovan, , 2015) effects within ownership links at firm level in Northern
England. Polarisation is 'the attraction exerted by a place on a more or less
extended and heterogeneous one that is in a situation of dependency with
respect to this centre '(Elissalde, , 2004). This polarisation of the economy
is the main driver of the peripheralisation effect 'describing the production
of peripheries through social relations and their spatial implications '(Kühn,
, 2015). This effect is particularly relevant in the case of the United
Kingdom, given its centralised structure (Arcaute et al., , 2016).

## Hypotheses

The hypotheses that we seek to test within this paper are:

  * •

Hypothesis 1: The stated central and local policies for the NP have given rise
to an economic sub-system in the North of the UK observable at the level of
inter-urban interactions between firms;

  * •

Hypothesis 2: The economic structure of the Northern cities taken together,
offers a counterweight to the dominance of London and contributes to a
rebalancing of the centralised economy of the UK;

  * •

Hypothesis 3: The post 2010 industrial structure of NPS cities shows a shift
away from a traditional manufacturing base towards more advanced technologies
and diversification.

## Materials and Methods

In order to test these hypotheses, we use:

  * •

The population and boundaries of UK cities in 2018 defined as 138 Functional
Urban Areas with a common and harmonized definition of cities from the Global
Human Settlement (GHS) database produced by the European Commission (Schiavina
et al., , 2019).

  * •

The ownership links of capital between firms in 2010, 2013 and 2016 at city
level from the ORBIS dataset 222 The authors thank professor Céline Rozenblat
– Institut de géographie et durabilité, Université de Lausanne (IGD - UNIL) –
for providing access to the data. IGD-UNIL-BvD, ORBIS,  (2010, 2013, 2016) and
from AMADEUS for 2018333Accessed via University College London (UCL) Library
Services. AMADEUS is equivalent to ORBIS as it comes from the same BVD source,
AMADEUS being the extraction for European firms and ORBIS covering the
worldwide scale UCL-BvD, AMADEUS,  (2018). Produced by the Bureau Van Dijk
(BVD), both products indicate the exact geolocation, turnover and activity
codes for owners and owned companies in all sectors. Data is available for UK
companies and for the ones they own in the UK or overseas, as well as for UK
companies owned by companies outside the UK. For 2010, the ORBIS dataset is
composed of 203,403 links and 136,664 establishments, for 2013 of 186,321
links and 126 140 establishments and for 2016 of 232,620 links and 153,617
establishments. In AMADEUS 266,243 links and 289,808 establishments are
available for 2018.

The value of exploring Bureau Van Dijk's products, is that its variables
permit the reconstruction of ownership links between cities where companies
are localised. For the purpose of analysing specifically the city of
Manchester within the NPS, a decomposition of the ownership links was carried
out and led to the identification of capital control chains at several levels,
according to the following scheme: a firm (level L1) either in Manchester,
London, in another UK city, in a city of the European Union or in a city
outside Europe controls the capital of a firm in Manchester (level L2). The
latter itself owns the capital of another firm (level L3) in Manchester,
London, other cities of UK, in the European Union or outside Europe [see 3. in
Figure 1].

Figure 1: Defining the economic interurban ownership network of firms

The three-level subnetwork of ownership networks in UK contains 2,312 firms
and 1,562 ownership linkages. We have then calculated an index of intensity of
the revenues generated by these ownership links at city level which we call
'ownership links revenues'. The latter is proportional to the share of capital
owned by the foreign firm and the turnover generated by the owned firm,
expressed at the level of the city where the owned firm is located
(Śleszyński, , 2015). An aggregation of the chains, passing through the same
city at level L2, was carried out according to all ownership links revenues
generated in the given city [see 2 in Figure 1].

An aggregation into 21 first level NACE 444Statistical Classification of
Economic Activities in the European Community (in French ”Nomenclature
Statistique des activités Économiques dans la Communauté Européenne”
classification of activities has been applied for all analysis555(A)
Agriculture, Forestry and Fishing, (B) Mining and Quarrying, (C)
Manufacturing, (D) Electricity, Gas, Steam and Air Conditioning Supply, Water
Supply; (E) Sewerage, Waste Management and Remediation Activities, (F)
Construction, (G) Wholesale and Retail Trade; (H) Repair of Motor Vehicles and
Motorcycles, Transportation and Storage, (I) Accommodation and Food Service
Activities, (J) Information and Communication, (K) Financial and Insurance
Activities, (L) Real Estate Activities, (M) Professional, Scientific and
Technical Activities, (N) Administrative and Support Service Activities, (O)
Public Administration and Defence; Compulsory Social Security, (P) Education
(Q) Human Health and Social Work Activites (R) Arts, Entertainment and
Recreation, (S) Other Service Activities, Activities of Households as
Employers; (T) Undifferentiated Goods and Services Producing Activities of
Households for Own Use (U) Activities of extraterritorial organisations and
bodies.EUROSTAT,  (2019). Correspondence analysis is a multivariate
statistical technique which summarises a set of categorical data in two-
dimensional graphical form Hirschfeld,  (1935); Benzécri,  (1973); Greenacre,
(1983). It was applied in social science for the first time by Bourdieu,
(1973) and more recently in urban science for the analysis of trajectories of
cities (Paulus and Pumain, , 2000). In this paper correspondence analysis was
run to characterise the economic ownership specialisation of cities based on
the NACE 21 first level classification. The matrices used are described in the
relevant results subsections.

## Data / Empirics

### Geographical scope of ownership

Understanding the geographical scope of ownership of firms is of economy-wide
importance, as firms'market value is equivalent to the global GDP (De La Cruz
et al., , 2019). The geographical scope of ownership in the United Kingdom is
mainly nationally driven and dominated by the capital city. This is a
classical structure seen in many other countries (La Porta et al., , 1999). In
2016, 75 percent of firm owners in Manchester came from the UK and in 45
percent of cases from Manchester itself [Figure 2].

Figure 2: Multi-level ownership structure in 2016 and evolution since 2010;
Man-owned – Firms in Manchester owned by other firms; Man-owning – Firms in
Manchester owning other firms

The Manchester firms'ownership structure for 2016 is oriented in 85 percent of
cases towards the United Kingdom, although this has slightly diminished since
2010 (94 percent) and has become more internationally driven. In addition, in
2016, 18 percent of links coming from Manchester are oriented towards other UK
cities. This orientation has more than doubled since 2010 in terms of number
of links and at the same time the weight of London has diminished too. This
relevant result will be investigated in more detail below concentrating on the
interactions between cities covered by the NPS. We aim at testing if the NP
region defined by the government presents the potential characteristics of a
regional subsystem in the United Kingdom (hypothesis 1) and whether it offers
a counterweight to the dominance of London (hypothesis 2).

### Evolution of NPS inter-urban interactions

This section analyses the evolution of NP interactions within the UK. Firstly
we consider inter-urban interactions within the NP. The total number of links
between cities covered by the NPS –- here Manchester, Liverpool, Newcastle and
Leeds 666Other cities such as Hull and Sheffield covered by the NPS are not
included due to the lack of ownership data -– has been relatively stable
through time: 376 in 2010, 303 in 2013 and 400 in 2016. However, when
considering ownership links revenues generated from inter-urban ownership they
have diminished through time. They initially rose from 18 billion euros
ownership links revenues in 2010 to 24 billion in 2013 and then dropped to 9.5
billion in 2016 [Figure 3]
777 • Top 5 interactions of firms in Liverpool owning firms in Leeds in 2016:
Energy Acquisitions UK Limited owns 100 percent of Dyneley Holdings Limited
(force 1291788), Kingspan Investments Limited, owns 100 percent of Kingspan
Access Floors Holdings Limited (force 100655), Heweston Holdings Limited owns
100 percent of Kingspan Access Floors Limited (force 58164), Thermal Ceramics
UK Limited owns 34.96 of Jemmtec Limited (force 5978.859), Phoenix Medical
Supplies Limited owns 21.1 of RX Systems Limited (force 4987.196) • Top 5
interactions of firms in Newcastle owning firms in Leeds in 2016: DB (UK)
Investments Limited owns DB Cargo (UK) Holdings Limited (force 768508), Arriva
UK Trains Limited owns 100 percent of Arriva Trains Northern Limited (force
399709), Brompley Property Investments Limited owns 100 percent of BPT Limited
(force 157283.7), Crown Packaging UK PLC owns 100 percent of CarnauxMetalBox
Engineering Limited (force 120023), Nobia Holdings UK Limited owns 100 of
Gower Group Limited (force 75130). • Top 5 interactions of firms in Manchester
owning firms in Newcastle in 2016: Rolls Royce Industries Limited owns 100
percent of Rolls Royce Power Engineering PLC (force 1044233), GE UK GROUP owns
41.22 percent of PII Group Limited (force 42771.52, nace 9411), LF Europe
Limited owns 100 percent of Just Jamie and Paul Rich Limited (force 35429),
Lornamead Group Limited owns 100 percent of Natural White (UK) Limited (force
18903), IGE USA Group owns PII Group Limited (force 9110.479), Sterling Fluid
Systems (UK GROUP) limited owns Darwins Holdings Limited, force 100. .

Figure 3: Ownership links between Manchester, Liverpool, Newcastle and Leeds
in 2010 and 2016 Figure 4: Ownership revenues generated by London-NP and NP-
London ownership links in 2016 Figure 5: Ownership of firms in the United
Kingdom by firms based in London in 2018

The ownership strength between the NP cities has diminished probably due to an
intensification of links oriented towards London. In terms of the initial
hypotheses (1 and 2), it would seem that there has been little impact of the
policy since 2010 in terms of creating an economy of the North in order to re-
balance the UK economy and facilitate decentralisation.

When observing the ownership links coming from cities covered by the NPS and
oriented towards other UK cities in 2018 [Figure 4], the NP economy clearly
does not offer a counterbalance to the weight of firms in London. In fact the
later generate more than 50 billions pounds in revenues coming from ownership
in the NP, whereas firms in NP cities generate only 1 million pounds of
revenues from ownership of firms in London. Firms in London dominate the whole
UK territory in terms of ownership in 2018 [Figure 5]. To conclude this
section, we reject hypotheses 1 and 2 and confirm our statement regarding the
NP economy as being still far away from its initial objectives. In fact, the
2019 Centre for Cities report claims that NPS implementation is a job not even
half done yet Jeffrey,  (2019). These results confirm that polarisation and
peripheralisation effects are still ongoing in the United Kingdom. In the last
section we will test our third hypothesis regarding the evolution of the post
2010 industrial base of NP cities.

### Economic specialisation of cities within ownership networks

In this section we wish to examine the evolution of the economic
specialisation of NP's cities involved in ownership links between firms since
2010. At the same time this section will also verify the impact of the
industrial strategy in terms of specialisation in advanced manufacturing,
health, digital/media, clean growth, which are highlighted as sectors to be
focused on in the future by the Combined Authorities of the North and by the
NP.

In order to determine the main economic specialisation of owned firms in
Manchester, Liverpool, Newcastle and Leeds, a correspondence analysis has been
conducted using the ade4 and explor packages in R (Dray and Dufour, , 2007).
The analysis has been applied to a matrix of cities, in rows, and the economic
sectors of the companies owned in these cities in columns, all expressed in
total ownership links revenues by city and given sector [Figure 6].

Figure 6: Trajectories of the NP cities in terms of their ownership
specialisation in 2010, 2013, 2016

The evolution of the trajectories of the four cities of the NP indicate a
general upgrade in stages, of the domestic value chain (Cadestin et al., ,
2019), as the links coming from other firms have become more specialised in
advanced manufacturing, research and development and health. This justifies
the priorities of the NPS for the future and corresponds to the economic
sector capabilities of Northern cities as defined by the NPS and the GMCA's
industrial strategy.

The correspondence analysis shows that firms owned in Manchester have switched
from specialisation in manufacturing (C), to wholesale and trade (G) and
finally to professional scientific and technical activities (M) in 2016. This
confirms the GMCA's industrial strategy as investments in scientific
activities are related to the Manchester University strategy for the
production of graphene as well as the creation of a Graphene City in the North
(HMGovernment, , 2019). In fact Manchester now presents a strong advanced
manufacturing base in materials and textiles, chemicals, food and drink and is
the site of Trafford Park, one of Europe’s largest industrial parks
(HMGovernment, , 2019).

This switch in the global value chain is particularly relevant in the case of
Liverpool as firms owning capital have converted from Agriculture, Forestry
and Mining (A) in 2010 to Professional scientific and technical activities (M)
in 2013. Ownership in Newcastle switched from repair of motor vehicles and
motorcycles (H) to agriculture, forestry and mining (A) to education (P).
Leeds went from a specialisation in wholesale and trade (G) to electricity,
gas and steaming (D) to manufacturing (M) and education (P). This is a
relevant observation as it shows that owners of firms invest in scientific
activities and health as a priority and these sectors are the ones generating
the most revenues. These results weaken the findings in the previous section
relating to hypotheses 1 and 3 and confirm that there has been a shift away
from a traditional manufacturing base towards a more diversified
specialisation of cities and this might in part, be due to the impact of the
NPS since 2010.

However when coming back to hypothesis 2 the opposition between the Northern
cities and London is still very important in 2018 [Figure 7]. On one side
London concentrates firms owned in all kinds of sectors abut mainly Finance
and Insurance activities (K), whereas on the opposite side of the factorial
axis Leeds, Newcastle and Liverpool show specialisation in retail and
electricity, gas, steam and air conditioning supply. There is still an
enormous gap to fill and initiatives needed in order to create a stronger
economic subsystem in the North capable of counterbalancing the weight of
London and to meet the foresight for cities objectives in terms of
diversification of the economic base in order to reduce polarisation and
peripheralisation of the North of England.

Figure 7: NP cities and London in terms of their ownership specialisation in
2018

## Discussion

The analysis firstly shows that the geographical scope of ownership of firms
located in Manchester is mainly oriented towards firms in the city itself or
in London. Local links towards other UK cities outside London represent only
18 percent of total links in 2016. Secondly the analysis of the evolution of
these local links between Manchester, Liverpool, Leeds and Newcastle has
revealed a decrease of revenues generated by inter-urban ownership links and a
relatively stable number of interactions since 2010. The latter were polarised
on Liverpool, Manchester and Leeds leading to a peripheralisation of Newcastle
in 2016. Thirdly, findings show that Liverpool and Manchester have switched
from a traditional manufacturing base towards a specialisation in professional
and scientific activities within the ownership networks in this same period.
Despite the modernisation of the Manchester and Liverpool industrial
structure, it would seem that the NPS has had only a minor impact on economic
interaction between Northern cities since 2010.

This paper shows that the NPS has not yet created an integrated sub-system
between Northern cities of the United Kingdom – at least from the economic
perspective of ownership links between firms – and is still characterised by
peripheralisation and polarisation patterns when compared to London. The NP is
still far from contributing to decentralisation and foresight for cities
defined by the government. The evolution of the economic specialisation of
Liverpool and Manchester in the global value chain of production reveals that
this process of creating an economy of the North capable of withstanding the
effect of London is still ongoing. These findings are relevant for policy-
makers regarding the cities'foresight government objectives for the next fifty
years. An initiative such as the NP needs to be be continued and reinforced
for the future of cities to create a more resilient and balanced national
system of cities. A future extension of this research could include
transportation and commuting interactions between cities of the NP, as these
may well exhibit patterns of interaction that differ from those of the
economic networks. Furthermore adding the city of Hull in the NPS analysis
would bring a different perspective, as Hull presents a developing economic
structure in the North of England. This research suggests that one may think
differently about the NP from a network perspective with a focus on inter-
urban interactions rather than focusing on within city interactions. A further
extension of this research would be its application to case studies in other
countries'city-systems, where decentralisation is a major issue.

## Acknowledgments

The authors acknowledge support from the Urban Dynamics Lab under the EPSRC
Grant No. EP/M023583/1

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