arXiv:1906.08946v1  [cs.SI]  21 Jun 2019
Appliance of network theory in economic
geography
Alexandra Barina1,3, Gabriel Barina1 and Mihai Udrescu1,2
1 Department of Computer and Information Technology, Politehnica University of
Timi¸soara, Timi¸soara, Romania
2 Timi¸soara Institute of Complex Systems, Timi¸soara, Romania
3 Corresponding author at Faculty of Automation and Computers, University
Politehnica of Timisoara, Bd. Vasile Parvan 2, 300223, Timisoara, Romania, email:
vadasanmaria@gmail.com
Abstract. A continuously evolving geography requires a good under-
standing in networks. As such, this paper accounts for theories and
applications of complex networks and their role both in geography in
general, as well as in determining various geographical network trajecto-
ries. It assesses how links between agents lead to an evolutionary process
of network retention, as well as network variation, and how geography
inﬂuences these mechanisms.
Keywords: social network analysis, complex networks, network trajec-
tory, economic geography, evolutionary geography, innovation
1
Introduction
The limited insight oﬀered by the neoclassical growth theory [1] pertaining both
regional growth [2,3] and geographical agglomeration [4,5,6,7] has brought forth
completely new approaches to economic geography [8,9]. Integrating both growth
theories and endogenous explanations pertaining to regional economy develop-
ment [10,11,12], valuable insight is thus gained by economic geographers. Shift-
ing their attention towards the understanding of geographical proximity of both
technological and economical innovations [13], geographers are interested in un-
derstanding how innovation is made and why innovative practice often concen-
trates in certain geographical proximities [7].
As a result, one way of analyzing certain regional economical developments
[14] is to apply network science and identify certain interactions in generated
economic network. As such, geographers – especially the ones from the economic
department – have become gradually receptive to network science, due to its
various applications in analyzing economic outcomes [15,16]. Just like evolution-
ary economics, network theory uses regional clusters to study various conditions
and outcomes in a dynamic fashion [17]. In the dynamic ﬁeld of geography, net-
works have been applied with success over the course of many years already
and have coined speciﬁc terminologies pertaining geographical clusters and even
globalization.

Following a critical review published by economic geographer Gernot Grabher
[18], it has come to the attention however, that the application of network science
in the ﬁeld of geography has been done either superﬁcially or selectively. As such,
in this paper we address the development of network science and its application
in geography. Using emerging research on network evolution [19,20], we aim at
integrating concepts into the ﬁeld of economic geography. As a result, we propose
the following objectives:
– Deﬁne a geographical network trajectory [21] which incorporates the appli-
cation of network science in order to study the evolution of networks across
regions.
– Analyze retention-, and variation-mechanisms in network structures which
are endogenous to network evolution.
– Determine models of variation in network trajectories by taking into consid-
eration various regional innovations.
The purpose of this paper is to discuss the contribution of network theory
to an evolutionary economic geography, as well as to prove that innovation and
regional growth results from the bridging of several (unconnected) networks. As
such, after addressing basic concepts in section 2, section 3 presents the exist-
ing mechanisms responsible for selection-, retention-, and variation-mechanisms.
Conclusively, section 4 ends this paper with a general discussion pertaining
sources of innovation for regional growth.
2
Theoretical Background
In this section we oﬀer theoretical background pertaining social network analysis
and its application in geography, with particular focus on the economic side of
it.
2.1
Social Network Analysis
Social network analysis is the process of investigating various social structures by
using networks and graph theory [22]. As such, a social network is a construction
formed by individuals (actors, agents) with bidirectional connections (relation-
ships, friendships) between them, eﬀectively resembling a real-world structure of
our own society. Social networks derive from complex networks, together with
which they form part of the nascent ﬁeld of network science [23,24], ever since
their ﬁrst introduction to literature in the 1970’s [25]. Based on graph theory
and network theory, empirical observations of real-world networks and sociology,
their main purpose is to model the various relationships from our own society.
Underpinned by empirical studies of real-world systems, complex networks
have gained increased interest due to their applicability in various social [26,27,28]
and scientiﬁc ﬁelds, such as biology, economy [29] and geography [30]. For in-
stance, in the economic area it is highly important to understand the mechanism
in which certain economic agents handle better than others [31].

2.2
Geography
The ﬁeld of geography can be linked to complex networks by factoring in two
main properties, namely proximity and place, both very important views of
geography when taking into consideration the concept of geographical network
trajectory.
Proximity, a latent eﬀect of various economic processes, inﬂuences network
formation in a direct manner; as such, the most widely used approach in eco-
nomic geography aims at addressing this issue. It is directly inﬂuenced by two
underlying social technologies present throughout all existing relations of eco-
nomic geography, namely communication technology [32] and transport technol-
ogy [33]. As such, only by analyzing communication preferences of social actors
and mobility opportunities can we also determine the existing relation between
physical space and economic interaction; in other words, constraints of proxim-
ity only apply if face-to-face is the only means of communication while travel is
denied.
By using the notion of resource bundle from the theory of the growth [34],
a place – the second property by which we can establish a link between net-
works and geography – can be considered both a resource and opportunity. As
such, a place-speciﬁc resource oﬀers a source of contextuality, diﬀerence and
contingency for economic development [35,36]. Conclusively, by taking into con-
sideration structural (e.g., social capital, structural holes, etc.), material, social
and economical resources, places do not coerce the formation of a given network,
but various social interactions do inﬂuence the geography (of the network) [10].
2.3
Geographical network trajectory
The main focus of any theory pertaining network evolution is how structural lay-
out of a given network at time 1 aﬀects the interactions among agents (nodes)
and/or the formation of new relationships (ties) at time 2 [37]. As such, the
concept that combines network, geography and evolution in order to analyze
the above scenario is the geographical network trajectory [38]. It describes from
a geographic point of view the formation and dissolution of relationships (ties)
within a given network. This perspective eﬀectively shifts the analysis of single
relations to the analysis of the overall relations from within a network. As such,
a theory behind network evolution takes into consideration not only the appear-
ance of a new tie, but also the impact that the system imposes on the creation
of additional relationships.
3
The evolutionary process in geographical network
trajectory
A given evolutionary system can be deﬁned by the following principles: selection,
retention as well as variation [39,40]. As such, in the following sub-sections we
present these principles in the context of network theory.

3.1
Selection
The selection mechanism is based on environment. In the ﬁeld of economics,
for instance, it is the market competition that dictates the existence and layout
of various producers [41]. In ﬁeld of social network analysis however, selection
refers to the presence of relationships (ties) between agents (nodes) of the same
network [42,43,44,45,46,47]. This selection is the result not only of an external
environment, but also of the various decisions made by the mutual agents linked
together. This implies that the selection mechanism is based on both the re-
quirement of constant adaptation, as well on a strategy focused on improving
the relations between both agents, to the point of mutual beneﬁt.
In all economical networks new relationships occur either between two pro-
ducers who already have a shared history, or new producers, without any previ-
ous relationships. As such, a theory of network evolution must take into consid-
eration both the emergence of dynamic links, as well as possible disappearance
of these; both cases imply the presence of another factor, namely cost.
Creating and/or maintaining relations is a costly resource, and is greatly
dependent on each producer as a limiting factor. As such, producers analyze
the presence of relations mainly from a utility perspective, and how they would
beneﬁt from it. One such beneﬁt would be the possibility of accessing external
resources [48]; this, however, would also automatically decrease the producers’
attractiveness for new relations. In this case, tie selection can be deﬁned as the
competitive allocation of limited relationships where the presence of such a rela-
tionship automatically implies possible lost opportunities. This suggests that tie
selection is a competitive process which hinges on both exogenous changes (e.g.,
regulatory markets) [49] as well as endogenous dynamics (e.g., one producer has
become more attractive because of its alliance with a third party) [50,51]. In
what follows, we further address these two mechanisms, with focus on endoge-
nous mechanisms of network evolution like retention and variation of network
structures.
3.2
Retention
The retention mechanism of a network refers to the inclination towards a given
(future) tie selection based on past decisions and their outcome. This mechanism
stems from either persistent (i.e., slow-decaying) or new, path-dependent ties.
According to sociologist Ronald S. Burt, the decay is a power function of time, in
which decay probability decreases the older a tie and/or node is [52]. However, as
mentioned before, retention also comes from the appearance of path-dependent
ties. As such, while studies on tie-decays analyze the life-span of existing rela-
tionships, another approach is to analyze where the next tie is most likely to
form, even without (signiﬁcant) change in layout, centrality or density distribu-
tions or even fragmentation [53]. As such, the existing literature on the retention
mechanism describes three main theories, described in what follows.

Preferential attachment This theory states that when creating new ties in
a scale-free network topology [54], agents with a high degree are capable of
increasing their degree even more and at a much faster pace, while agents with
a small degree will inevitably stagnate in the process [55,56]. From this, we can
deduce that well-positioned agents have a clear, accumulative advantage (over
time) compared to the un-connected and peripheral agents [44,57]. However, in
a real-life scenario, this does have a limitation, in that agents can only maintain
a limited number of relationships, and as such, the process of centralization is,
in fact, empirically ﬁnite [58].
Embedding This theory suggests that new relationships are formed based on
the trust of the old ones. The increased interconnection to which this leads
is the process of social embedding [44]. As a result, this leads to persistent
network structures, as well as the formation of new mentalities within groups of
interconnected agents [71].
Multi-connectivity This theory pertains to the fact that networks expend by
means of a process meant to induce diversity. Creating multiple independent
ties between them, agents gain a cumulative advantage of multi-connectivity,
and become more attractive for the formation of new relationships [59]. As these
new links form, they reinforce clusters of agents, who become more cohesive over
time. This led Walker [60] et. al. to argue in favor of path-dependence in network
growth [61,62].
As a whole, the previously mentioned theories (i.e., preferential attachment,
embedding and multi-connectivity) represent a cumulative path-dependent re-
tention mechanisms found in networks of various types. Additionally, geograph-
ical location also aﬀects the evolution of the network trajectory [43,63,64,65];
however, the impact is only limited to a certain regional level, where face-to-face
communication is allowed [59]. New relationships are more likely to form when
two agents are located in the same region (i.e., co-located), even if the involved
agents are not central within a given network [66,67]. As a result, there is numer-
ous evidence for retention mechanisms in networks from a geographical point of
view.
Complex networks successfully convey important aspects of cumulative, path-
dependent evolution over time between individual agents as well as between
groups of agents (clusters). However, these retention mechanisms may also lead
to a situation known as local lock-in [11,68], in which certain already-existing
patterns prohibit the appearance of new ones. Lock-ins, however, can be over-
come by the emergence of new variation.
3.3
Variation
As we can see, variation is the result of endogenous network mechanisms for
both tie formation and dissolution. In a network, variation opposes an exist-
ing network trajectory in favor of selection of new ties. As such, even though

variation refers to changes in network structure, it is deﬁned at the level of tie
selection, which in turn, aﬀects the re-organization of the network [69]. Empiri-
cal data has shown that creating and maintaining an active relationship with an
agent outside of ones own network would shorten the given agents membership
duration [70]; it is due to this phenomenon that variations happen in social and
economic networks, which countervail the previously-mentioned retention mech-
anisms. Then again, once such a link is established, the cumulative retention
mechanisms (i.e., preferential attachment, embedding and multi-connectivity)
create new opportunities for new links, thus eﬀectively countering existing pat-
terns of path-dependence [71,72].
4
Conclusions
Evolutionary economic networks are greatly inﬂuenced by retention-, and variation-
mechanisms which, in turn, create path-dependent network trajectories [11] by
implicating certain path-dependence attributes (e.g., preferential attachment,
embeddedness, multi-connectivity, etc.) [63]; moreover, evolutionary theory of
economic growth must also be able to explain innovation in regional economic
development. However, since evolutionary theory in ﬁelds like economy [8], soci-
ology [73], geography [8,74] is still in its infancy, the observations drawn in this
paper are exploratory only.
As a result of limited application of evolutionary theory, geographers have
turned to network theory from the new domain of social network analysis. As
such, rather than oﬀering a coherent theory pertaining regional evolution, net-
work theory oﬀers a new, evolutionary perspective to economic geography. Of-
fering a way of promoting economic growth and innovation, it also represents
a means to analyze the social values and economic interests between regional
structures and individuals.
The analysis of network evolution, including from an economic point of view,
is important due to the emergence or disappearance of both agents (nodes) and
relationships (ties) simultaneously. As such, when new relationships form, the
causes and consequences of network growth must also be addressed: what is the
relationship between a path-dependent network trajectory and the growth rate
of a network? Apart from this simple question, there are many other questions
(economic) geographers need answers to, however an evolutionary analysis of
changes within a network requires not only relational-, but also longitudinal data
spanning over extensive time-periods [71]. Seeing that this kind of data can only
be analyzed using network theory, it oﬀers a perfect incentive for geographers to
apply methodologies oﬀered only by social network analysis.
References
1. Solow, R.M., 1999. Neoclassical growth theory. Handbook of macroeconomics, 1,
pp.637-667.

2. Sala-i-Martin, X.X., 1996. Regional cohesion: evidence and theories of regional
growth and convergence. European Economic Review, 40(6), pp.1325-1352.
3. Le Gallo, J. and Fingleton, B., 2019. Regional growth and convergence empirics.
Handbook of regional science, pp.1-28.
4. Fujita, M. and Thisse, J.F., 2003. Does geographical agglomeration foster economic
growth? And who gains and loses from it?. The Japanese Economic Review, 54(2),
pp.121-145.
5. Bertinelli, L. and Decrop, J., 2005. Geographical agglomeration: Ellison and
Glaeser’s index applied to the case of Belgian manufacturing industry. Regional
Studies, 39(5), pp.567-583.
6. Dupont, V., 2007. Do geographical agglomeration, growth and equity conﬂict?.
Papers in Regional Science, 86(2), pp.193-213.
7. Stelder, D., 2005. Where do cities form? A geographical agglomeration model for
Europe. Journal of Regional Science, 45(4), pp.657-679.
8. Boschma, R.A. and Lambooy, J.G., 1999. Evolutionary economics and economic
geography. Journal of evolutionary economics, 9(4), pp.411-429.
9. Boschma, R.A. and Frenken, K., 2017. Why is economic geography not an evolu-
tionary science? Towards an evolutionary economic geography. In Economy (pp.
127-156). Routledge.
10. Frenken, K. and Boschma, R.A., 2007. A theoretical framework for evolutionary
economic geography: industrial dynamics and urban growth as a branching process.
Journal of economic geography, 7(5), pp.635-649.
11. Martin, R. and Sunley, P., 2006. Path dependence and regional economic evolution.
Journal of economic geography, 6(4), pp.395-437.
12. Kogler, D.F., 2015. Evolutionary economic geographyTheoretical and empirical
progress.
13. Wiebe, K.S. and Lutz, C., 2016. Endogenous technological change and the policy
mix in renewable power generation. Renewable and Sustainable Energy Reviews,
60, pp.739-751.
14. Storper, M., 1997. The regional world: territorial development in a global economy.
Guilford press.
15. Granovetter, M., 2005. The impact of social structure on economic outcomes. Jour-
nal of economic perspectives, 19(1), pp.33-50.
16. Gell, M., Pellizzari, M., Pica, G. and Rodrguez Mora, J.V., 2018. Correlating social
mobility and economic outcomes. The Economic Journal, 128(612), pp.F353-F403.
17. Dez-Vial, I. and Montoro-Snchez, ., 2018. How Local Knowledge Networks and
Firm Internal Characteristics Evolve Across Time Inside Science Parks. In Ag-
glomeration and Firm Performance (pp. 139-153). Springer, Cham.
18. Grabher, G., 2006. Trading routes, bypasses, and risky intersections: mapping the
travels ofnetworks’ between economic sociology and economic geography. Progress
in human geography, 30(2), pp.163-189.
19. Ferguson, N., 2017. The False Prophecy of Hyperconnection: How to Survive the
Networked Age. Foreign Aﬀ., 96, p.68.
20. Knudsen, T.R., 2004. General selection theory and economic evolution: The price
equation and the replicator/interactor distinction. Journal of Economic Method-
ology, 11(2), pp.147-173.
21. Boschma, R. and Frenken, K., 2018. Evolutionary economic geography (pp. 213-
229). Oxford: Oxford University Press.
22. Kim, J. and Hastak, M., 2018. Social network analysis: Characteristics of online
social networks after a disaster. International Journal of Information Management,
38(1), pp.86-96.

23. Borgatti, S.P., Mehra, A., Brass, D.J. and Labianca, G., 2009. Network analysis in
the social sciences. science, 323(5916), pp.892-895.
24. Easley, D. and Kleinberg, J., 2010. Networks, crowds, and markets (Vol. 8). Cam-
bridge: Cambridge university press.
25. Kenett, D.Y., Perc, M. and Boccaletti, S., 2015. Networks of networksAn intro-
duction. Chaos, Solitons & Fractals, 80, pp.1-6.
26. Topirceanu, A., et al. ”Tolerance-based interaction: A new model targeting opinion
formation and diﬀusion in social networks.” PeerJ Computer Science 2, 2016
27. Duma, A., and Topirceanu, A. ”A network motif based approach for classifying on-
line social networks.” Applied computational intelligence and informatics (SACI),
2014 IEEE 9th international symposium on, pp. 311-315. IEEE, 2014
28. Topirceanu, A., Udrescu, M. and Vladutiu, M., 2014. Genetically optimized realis-
tic social network topology inspired by facebook. In Online Social Media Analysis
and Visualization (pp. 163-179). Springer, Cham.
29. Topirceanu, A., Barina, G. and Udrescu, M., 2014, September. Musenet: Collabo-
ration in the music artists industry. In 2014 European Network Intelligence Con-
ference (pp. 89-94). IEEE.
30. Balland, P.A. and Rigby, D., 2017. The geography of complex knowledge. Economic
Geography, 93(1), pp.1-23.
31. Jackson, M.O., 2008, December. Average distance, diameter, and clustering in so-
cial networks with homophily. In International Workshop on Internet and Network
Economics (pp. 4-11). Springer, Berlin, Heidelberg.
32. Storper, M. and Venables, A.J., 2004. Buzz: face-to-face contact and the urban
economy. Journal of economic geography, 4(4), pp.351-370.
33. Marquis, C. (2003) The pressure of the past: Network imprinting in intercorporate
communities. Administrative Science Quarterly, 48: 655-89. Marsden, P. V. (1990)
Network data and measurement. Annual Review of Sociology, 16: 435-63.
34. Penrose, E., 1959. The theory of the growth of the ﬁrm. JohnW iley & Sons, New
York.
35. Sayer, A. (1991) Behind the locality debate: Deconstructing geography’s dualisms.
Environment and Planning A, 23: 283-308.
36. Bathelt, H. and Glckler, J., 2005. Resources in economic geography: from sub-
stantive concepts towards a relational perspective. Environment and Planning A,
37(9), pp.1545-1563.
37. Kenis, P. and Knoke, D., 2002. How organizational ﬁeld networks shape interorga-
nizational tie-formation rates. Academy of Management Review, 27(2), pp.275-293.
38. Kilduﬀ, M., Tsai, W. (2003) Social Networks and Organizations. London: Sage.
39. Nelson, R. R., Winter, S. G. (2002) Evolutionary theorizing in economics. Journal
of Economic Perspectives, 16: 23-46.
40. Hodgson, G.M. and Lamberg, J.A., 2018. The past and future of evolutionary
economics: some reﬂections based on new bibliometric evidence. Evolutionary and
Institutional Economics Review, 15(1), pp.167-187.
41. Knudsen, T. (2002) Economic selection theory. Journal of Evolutionary Economics,
12: 443-70.
42. Gulati, R. (1995) Does familiarity breed trust? The implications of repeated ties
for contractual choice in alliances. Academy of Management Journal, 38: 85-112.
43. Stuart, T. E. (1998) Network positions and propensities to collaborate: an investi-
gation of strategic alliance formation in a high-technology industry. Administrative
Science Quarterly, 43: 668-98.
44. Gulati, R., Gargiulo, M. (1999) Where do interorganizational networks come from?
American Journal of Sociology, 104: 1439-93.

45. Ahuja, G. (2000) Collaboration networks, structural holes, and innovation: A lon-
gitudinal study. Administrative Science Quarterly, 45: 425-55.
46. Venkatraman, N., Lee, C.-H. (2004) Preferential linkage and network evolution:
a conceptual model and empirical test in the US video game sector. Academy of
Management Journal, 47: 876-92.
47. Sol, R.V., FerrerCancho, R., Montoya, J.M. and Valverde, S., 2002. Selection, tin-
kering, and emergence in complex networks. Complexity, 8(1), pp.20-33.
48. Pfeﬀer, J., Salancik, G. R. (1978) The External Control of Organizations. New
York: Harper and Row.
49. Derrien, F. and Kecsks, A., 2013. The real eﬀects of ﬁnancial shocks: Evidence from
exogenous changes in analyst coverage. The Journal of Finance, 68(4), pp.1407-
1440
50. Hallegatte, S. and Ghil, M., 2008. Natural disasters impacting a macroeconomic
model with endogenous dynamics. Ecological Economics, 68(1-2), pp.582-592.
51. Galster, G., Cutsinger, J. and Lim, U., 2007. Are neighbourhoods self-stabilising?
Exploring endogenous dynamics. Urban Studies, 44(1), pp.167-185.
52. Burt, R. (2000) Decay functions. Social Networks, 22: 1-28.
53. Vanacker, T., Manigart, S. and Meuleman, M., 2014. Pathdependent evolution
versus intentional management of investment ties in sciencebased entrepreneurial
ﬁrms. Entrepreneurship Theory and Practice, 38(3), pp.671-690.
54. Barabsi, A. L., Jeong, H., Nda, Z., Ravasz, E., Schubert, A., Vicsek, T. (2002)
Evolution of the social network of scientiﬁc collaborations. Physica A, 311: 590-
614.
55. Barabsi, A.-L., Reka, A. (1999) Emergence of scaling in random networks. Science,
286: 509-12.
56. Wang, X.F. and Chen, G., 2003. Complex networks: small-world, scale-free and
beyond. IEEE circuits and systems magazine, 3(1), pp.6-20.
57. Powell, W. W., Koput, K. W., Smith-Doerr, L. (1996) Interorganizational collab-
oration and the locus of innovation: networks of learning in biotechnology. Admin-
istrative Science Quarterly, 41: 116-45.
58. Holme, P., Edling, C. R., Liljeros, F. (2004) Structure and time evolution of an
internet dating community. Social Networks, 26: 155-74.
59. Powell, W. W., White, D., Koput, K. W., Owen-Smith, J. (2005) Network dynamics
and ﬁeld evolution: The growth of interorganizational collaboration in the life
sciences. American Journal of Sociology, 110: 1132-205.
60. Walker, G., Kogut, B., Shan, W. (1997) Social capital, structural holes and the
formation of an industry network. Organization Science, 8: 109-25.
61. Hite, J.M. and Hesterly, W.S., 2001. The evolution of ﬁrm networks: From emer-
gence to early growth of the ﬁrm. Strategic management journal, 22(3), pp.275-286.
62. Cooke, P., Asheim, B., Boschma, R., Martin, R., Schwartz, D. and Todtling, F.
eds., 2011. Handbook of regional innovation and growth. Edward Elgar Publishing.
63. Glckler, J., 2007. Economic geography and the evolution of networks. Journal of
Economic Geography, 7(5), pp.619-634.
64. Sorenson, O., Stuart, T. E. (2001) Syndication networks and the spatial distribu-
tion of venture capital investments. American Journal of Sociology, 106: 154688.
65. Powell, W. W., Koput, K. W., Bowie, J. I., Smith-Doerr, L. (2002) The spatial
clusering of science and capital: accounting for biotech ﬁrm-venture capital rela-
tionships. Regional Studies, 36: 291-306.
66. Owen-Smith, J., Powell, W. W. (2004) Knowledge networks as channels and con-
duits: The eﬀects of spillovers in the Boston biotechnology community. Organiza-
tion Science, 15: 5-21.

67. Schutjens, V., Stam, E. (2003) The evolution and nature of young ﬁrm networks:
a longitudinal perspective. Small Business Economics, 21: 114-34.
68. Hassink, R. (2005) How to unlock regional economies from path dependency? From
learning region to learning cluster. European Planning Studies, 13: 521-35.
69. Burt, R. S. (2004) Structural holes and good ideas. American Journal of Sociology,
110: 349-99.
70. McPherson, J. M., Popielarz, P. A., Drobnic, S. (1992) Social networks and orga-
nizational dynamics. American Sociological Review, 57: 153-70.
71. Baum, J. A., Shipilov, A. V., Rowley, T. J. (2003) Where do small worlds come
from? Industrial and Corporate Change, 12: 697-725.
72. Rowley, T. J., Greve, H. R., Rao, H., Baum, J. A. C., Shipilov, A. V. (2005) Time to
break up: Social and instrumental antecedents of ﬁrm exits from exchange cliques.
Academy of Management Journal, 48: 499-520.
73. Dietz, T., Burns, T.R. and Buttel, F.H., 1990, June. Evolutionary theory in soci-
ology: An examination of current thinking. In Sociological Forum (Vol. 5, No. 2,
pp. 155-171). Kluwer Academic Publishers-Plenum Publishers
74. Boschma, R. and Martin, R., 2007. Constructing an evolutionary economic geog-
raphy.

arXiv:1906.08946v1  [cs.SI]  21 Jun 2019
Springer LNCS Example Paper
Ivar Ekeland1, Roger Temam2 Jeﬀrey Dean, David Grove, Craig Chambers,
Kim B. Bruce, and Elsa Bertino
1 Princeton University, Princeton NJ 08544, USA,
I.Ekeland@princeton.edu,
WWW home page: http://users/~iekeland/web/welcome.html
2 Universit´e de Paris-Sud, Laboratoire d’Analyse Num´erique, Bˆatiment 425,
F-91405 Orsay Cedex, France
Abstract. The abstract should summarize the contents of the paper
using at least 70 and at most 150 words. It will be set in 9-point font
size and be inset 1.0 cm from the right and left margins. There will be
two blank lines before and after the Abstract. . . .
Keywords: computational geometry, graph theory, Hamilton cycles
1
Fixed-Period Problems: The Sublinear Case
With this chapter, the preliminaries are over, and we begin the search for periodic
solutions to Hamiltonian systems. All this will be done in the convex case; that
is, we shall study the boundary-value problem
˙x = JH′(t, x)
x(0) = x(T )
with H(t, ·) a convex function of x, going to +∞when ∥x∥→∞.
1.1
Autonomous Systems
In this section, we will consider the case when the Hamiltonian H(x) is au-
tonomous. For the sake of simplicity, we shall also assume that it is C1.
We shall ﬁrst consider the question of nontriviality, within the general frame-
work of (A∞, B∞)-subquadratic Hamiltonians. In the second subsection, we shall
look into the special case when H is (0, b∞)-subquadratic, and we shall try to
derive additional information.
The General Case: Nontriviality. We assume that H is (A∞, B∞)-sub-
quadratic at inﬁnity, for some constant symmetric matrices A∞and B∞, with
B∞−A∞positive deﬁnite. Set:
γ : = smallest eigenvalue of B∞−A∞
(1)
λ : = largest negative eigenvalue of J d
dt + A∞.
(2)

Theorem 1 tells us that if λ + γ < 0, the boundary-value problem:
˙x = JH′(x)
x(0) = x(T )
(3)
has at least one solution x, which is found by minimizing the dual action func-
tional:
ψ(u) =
Z T
o
1
2
 Λ−1
o u, u

+ N ∗(−u)

dt
(4)
on the range of Λ, which is a subspace R(Λ)2
L with ﬁnite codimension. Here
N(x) := H(x) −1
2 (A∞x, x)
(5)
is a convex function, and
N(x) ≤1
2 ((B∞−A∞) x, x) + c
∀x .
(6)
Proposition 1. Assume H′(0) = 0 and H(0) = 0. Set:
δ := lim inf
x→0 2N(x) ∥x∥−2 .
(7)
If γ < −λ < δ, the solution u is non-zero:
x(t) ̸= 0
∀t .
(8)
Proof. Condition (7) means that, for every δ′ > δ, there is some ε > 0 such that
∥x∥≤ε ⇒N(x) ≤δ′
2 ∥x∥2 .
(9)
It is an exercise in convex analysis, into which we shall not go, to show that
this implies that there is an η > 0 such that
f ∥x∥≤η ⇒N ∗(y) ≤1
2δ′ ∥y∥2 .
(10)
Fig. 1. This is the caption of the ﬁgure displaying a white eagle and a white horse on
a snow ﬁeld

Since u1 is a smooth function, we will have ∥hu1∥∞≤η for h small enough,
and inequality (10) will hold, yielding thereby:
ψ(hu1) ≤h2
2
1
λ ∥u1∥2
2 + h2
2
1
δ′ ∥u1∥2 .
(11)
If we choose δ′ close enough to δ, the quantity
  1
λ + 1
δ′

will be negative, and
we end up with
ψ(hu1) < 0
for h ̸= 0 small .
(12)
On the other hand, we check directly that ψ(0) = 0. This shows that 0 cannot
be a minimizer of ψ, not even a local one. So u ̸= 0 and u ̸= Λ−1
o (0) = 0.
⊓⊔
Corollary 1. Assume H is C2 and (a∞, b∞)-subquadratic at inﬁnity. Let ξ1,
. . . , ξN be the equilibria, that is, the solutions of H′(ξ) = 0. Denote by ωk the
smallest eigenvalue of H′′ (ξk), and set:
ω := Min {ω1, . . . , ωk} .
(13)
If:
T
2π b∞< −E

−T
2πa∞

< T
2πω
(14)
then minimization of ψ yields a non-constant T -periodic solution x.
We recall once more that by the integer part E[α] of α ∈IR, we mean the
a ∈ZZ such that a < α ≤a + 1. For instance, if we take a∞= 0, Corollary 2
tells us that x exists and is non-constant provided that:
T
2π b∞< 1 < T
2π
(15)
or
T ∈
2π
ω , 2π
b∞

.
(16)
Proof. The spectrum of Λ is 2π
T ZZ + a∞. The largest negative eigenvalue λ is
given by 2π
T ko + a∞, where
2π
T ko + a∞< 0 ≤2π
T (ko + 1) + a∞.
(17)
Hence:
ko = E

−T
2πa∞

.
(18)
The condition γ < −λ < δ now becomes:
b∞−a∞< −2π
T ko −a∞< ω −a∞
(19)
which is precisely condition (14).
⊓⊔

Lemma 1. Assume that H is C2 on IR2n\{0} and that H′′(x) is non-degenerate
for any x ̸= 0. Then any local minimizer ex of ψ has minimal period T .
Proof. We know that ex, or ex + ξ for some constant ξ ∈IR2n, is a T -periodic
solution of the Hamiltonian system:
˙x = JH′(x) .
(20)
There is no loss of generality in taking ξ = 0. So ψ(x) ≥ψ(ex) for all ex in
some neighbourhood of x in W 1,2  IR/T ZZ; IR2n
.
But this index is precisely the index iT (ex) of the T -periodic solution ex over
the interval (0, T ), as deﬁned in Sect. 2.6. So
iT (ex) = 0 .
(21)
Now if ex has a lower period, T/k say, we would have, by Corollary 31:
iT (ex) = ikT/k(ex) ≥kiT/k(ex) + k −1 ≥k −1 ≥1 .
(22)
This would contradict (21), and thus cannot happen.
⊓⊔
Notes and Comments. The results in this section are a reﬁned version of [1]; the
minimality result of Proposition 14 was the ﬁrst of its kind.
To understand the nontriviality conditions, such as the one in formula (16),
one may think of a one-parameter family xT , T ∈
 2πω−1, 2πb−1
∞

of periodic
solutions, xT (0) = xT (T ), with xT going away to inﬁnity when T →2πω−1,
which is the period of the linearized system at 0.
Table 1. This is the example table taken out of The TEXbook, p. 246
Year
World population
8000 B.C.
5,000,000
50 A.D.
200,000,000
1650 A.D.
500,000,000
1945 A.D.
2,300,000,000
1980 A.D.
4,400,000,000
Theorem 1 (Ghoussoub-Preiss). Assume H(t, x) is (0, ε)-subquadratic at
inﬁnity for all ε > 0, and T -periodic in t
H(t, ·)
is convex ∀t
(23)
H(·, x)
is T −periodic ∀x
(24)
H(t, x) ≥n (∥x∥)
with n(s)s−1 →∞as s →∞
(25)

∀ε > 0 ,
∃c : H(t, x) ≤ε
2 ∥x∥2 + c .
(26)
Assume also that H is C2, and H′′(t, x) is positive deﬁnite everywhere. Then
there is a sequence xk, k ∈IN, of kT -periodic solutions of the system
˙x = JH′(t, x)
(27)
such that, for every k ∈IN, there is some po ∈IN with:
p ≥po ⇒xpk ̸= xk .
(28)
⊓⊔
Example 1 (External forcing). Consider the system:
˙x = JH′(x) + f(t)
(29)
where the Hamiltonian H is (0, b∞)-subquadratic, and the forcing term is a
distribution on the circle:
f = d
dtF + fo
with F ∈L2  IR/T ZZ; IR2n
,
(30)
where fo := T −1 R T
o f(t)dt. For instance,
f(t) =
X
k∈IN
δkξ ,
(31)
where δk is the Dirac mass at t = k and ξ ∈IR2n is a constant, ﬁts the pre-
scription. This means that the system ˙x = JH′(x) is being excited by a series
of identical shocks at interval T .
Deﬁnition 1. Let A∞(t) and B∞(t) be symmetric operators in IR2n, depending
continuously on t ∈[0, T ], such that A∞(t) ≤B∞(t) for all t.
A Borelian function H : [0, T ] × IR2n →IR is called (A∞, B∞)-subquadratic
at inﬁnity if there exists a function N(t, x) such that:
H(t, x) = 1
2 (A∞(t)x, x) + N(t, x)
(32)
∀t ,
N(t, x)
is convex with respect to x
(33)
N(t, x) ≥n (∥x∥)
with n(s)s−1 →+∞as s →+∞
(34)
∃c ∈IR :
H(t, x) ≤1
2 (B∞(t)x, x) + c
∀x .
(35)
If A∞(t) = a∞I and B∞(t) = b∞I, with a∞≤b∞∈IR, we shall say that
H is (a∞, b∞)-subquadratic at inﬁnity. As an example, the function ∥x∥α, with
1 ≤α < 2, is (0, ε)-subquadratic at inﬁnity for every ε > 0. Similarly, the
Hamiltonian
H(t, x) = 1
2k ∥k∥2 + ∥x∥α
(36)
is (k, k + ε)-subquadratic for every ε > 0. Note that, if k < 0, it is not convex.

Notes and Comments. The ﬁrst results on subharmonics were obtained by Ra-
binowitz in [5], who showed the existence of inﬁnitely many subharmonics both
in the subquadratic and superquadratic case, with suitable growth conditions
on H′. Again the duality approach enabled Clarke and Ekeland in [2] to treat
the same problem in the convex-subquadratic case, with growth conditions on
H only.
Recently, Michalek and Tarantello (see [3] and [4]) have obtained lower bound
on the number of subharmonics of period kT , based on symmetry considerations
and on pinching estimates, as in Sect. 5.2 of this article.
References
1. Clarke, F., Ekeland, I.: Nonlinear oscillations and boundary-value problems for
Hamiltonian systems. Arch. Rat. Mech. Anal. 78, 315–333 (1982)
2. Clarke, F., Ekeland, I.: Solutions p´eriodiques, du p´eriode donn´ee, des ´equations
hamiltoniennes. Note CRAS Paris 287, 1013–1015 (1978)
3. Michalek, R., Tarantello, G.: Subharmonic solutions with prescribed minimal period
for nonautonomous Hamiltonian systems. J. Diﬀ. Eq. 72, 28–55 (1988)
4. Tarantello, G.: Subharmonic solutions for Hamiltonian systems via a ZZp pseudoin-
dex theory. Annali di Matematica Pura (to appear)
5. Rabinowitz, P.: On subharmonic solutions of a Hamiltonian system. Comm. Pure
Appl. Math. 33, 609–633 (1980)
