Characterisation of individual nodes in the mesoscale of

complex networks14th Mathematics of Networks meeting (MoN14)

Florian Klimm 21.9.2015

Structure

1. Biological neuronal networks

2. Climate networks

3. Multilayer transportation network

Neuronal networks are extremely complex systems

consisting of from 279 to ~86 billion neurons linked with to 8,000 to 10^15 synapses

degree0 60

A B

Segregation & Integrationenable parallel information processing

Tononi, G., Sporns, O., & Edelman, G. M. (1994). A measure for brain complexity: relating functional segregation and integration in the nervous system. Proceedings of the National Academy of Sciences, 91(11), 5033-5037.

community detection as mesoscale analysis

Local Mesoscale Globaldegree k

clustering c communitystructure

diameterbetweenness

Local Mesoscale Global

Local contribution to mesoscale

degree kclustering c

communitystructure

diameterbetweenness

participation pdispersion d

we measure the position of individual nodes in the mesoscale

Measuring segregation and integration in a network

-> role of nodes in the mesoscale

Hubness

Participation

Peripheral Connector Kinless

hubs

non-hubs

Guimera, R., & Amaral, L. A. N. (2005). Functional cartography of complex metabolic networks. Nature, 433(7028), 895-900.

participation vector for each node M elements (= number of modules)

A

B

C H

G

D F

E

K

IJ

(number of neighbours in this module)(number of potential neighbours in this module)

(0,0,2/3)(2/3,0,0)

(1/3,1/5,1/4)

L N(0,1/3,1/4)

Klimm, F., Borge-Holthoefer, J., Wessel, N., Kurths, J., & Zamora-López, G. (2014). Individual nodeʼs contribution to the mesoscale of complex networks. New Journal of Physics, 16(12), 125006.

participation vector for each node M elements (= number of modules)

normalisation of each vector

A

B

C H

G

D F

E

K

IJ

(number of neighbours in this module)(number of potential neighbours in this module)

(0,0,2/3)(2/3,0,0)

(1/3,1/5,1/4)

L N(0,1/3,1/4)

(1,0,0)

(0.43,0.25,0.32)(0,0,1)

(0,0.6,0.4)

participation index reduction to scalar value necessary

A

B

C H

G

D F

E

K

IJ

L N

0

0.90

0.4

degree is not sufficient to define hubness

hubness compares node’s degree with a random null model

We analyse different networks with this 2-dimensional mapping

-> role of nodes in the mesoscale

Hubness

Participation

Peripheral Connector Kinless

hubs

non-hubs

Segregation

Integration

participation p

glo

balhubnes

sh

(g)

0 0.5 1−4

0

4

random networkno hubs and no modularity

occu

rren

ce#

0

20

40

60

80

100

participation p

glo

balhubnes

sh

(g)

0 0.5 1−5

0

20

scale-free networkshubs but still no modularity

1234

5

67

89101112

1314

15

16

171819

20

high degreelow degree

occu

rren

ce#

0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1−4

−2

0

2

4

pex = 0 0.025 0.1 0.2 0.3

participation p

glo

balhubnes

sh

(g)

modular networksmodularity is tunebale but no hubs

participation p

glo

balhubnes

sh

(g)

0 0.5 1−3

0

7

modular network with SF attachmentshows segregated modules and connecting hubs

occu

rren

ce#

0

20

40

60

80

100

participation p

glo

balhubnes

sh

(g)

0 0.5 1−4

0

20

both features coexist in real world neuronal data!

for example in the roundworm C. elegans

0.1 mm

(a)

(b)

0

50

100 liatdaeh

degree

k

(c)

occu

rren

ce#

0

20

40

60

80

100

Climate networks

Stolbova, V., Martin, P., Bookhagen, B., Marwan, N., & Kurths, J. (2014). Topology and seasonal evolution of the network of extreme precipitation over the Indian subcontinent and Sri Lanka. Nonlinear Processes in Geophysics, 21(4), 901-917.

Time series data for each node (e.g.

Precipitation) weighted network unweighted network

thresholding

your favourite correlation measure

Climate networks

Stolbova, V., Martin, P., Bookhagen, B., Marwan, N., & Kurths, J. (2014). Topology and seasonal evolution of the network of extreme precipitation over the Indian subcontinent and Sri Lanka. Nonlinear Processes in Geophysics, 21(4), 901-917.

Time series data for each node (e.g.

Precipitation) weighted network unweighted network

5% strongest links

event synchronisation

A

B

C

1 2 3 4 5 6 7 8t

Indian Summer Monsoon902 V. Stolbova et al.: Network of extreme precipitation over the Indian subcontinent

Summer Monsoon (ISM) season; June–September), and thepost-monsoon season (October–December). During the In-dian Summer Monsoon season, the Indian subcontinent re-ceives more than 80% of its annual rainfall across the region(Bookhagen and Burbank, 2010). During the post-monsoonseason, the strongest rainfall covers the southern tip of In-dia and most of Sri Lanka. Pre-monsoon months (March–May) are characterized by light rainfall over the Himalaya,the southwestern part of India, and Myanmar (Burma) (seeFig. 1).The ISM is driven by heating of the high-altitude Tibetan

Plateau that leads to a near-surface low-pressure system thatattracts moist air from the surrounding oceans, especially theBay of Bengal (Flohn, 1957; Gadgil, 2003; Webster, 1997).On large scales, the ISM is linked to the seasonal heatingof land masse and interacts with the westerly jet stream.An important interaction occurs when the Somali jet streamcrosses the Arabian Sea, changing the direction of ocean cur-rents, and when colder water wells up from lower ocean lay-ers, causing a decrease in temperature. Additionally, a low-pressure trough where the southern and northern trade windsmeet, a border known as the Intertropical Convergence Zone(ITCZ), moves north. As a result, tropical storms, depres-sions, cyclones, squall lines and daily cycles become moreerratic and thus less predictable. On the eastern peninsula,the differential heating between the Bay of Bengal and thesurrounding land masses (including the Tibetan Plateau) con-stitutes the moisture source of monsoonal rain. These com-bined factors create a low-pressure “monsoon trough” to thesouth and parallel to the Himalayan mountains. It is thistrough’s fluctuations that largely influence when monsoonalrains start and stop, generally considered active and breakphases (Gadgil, 2003).October–December is a period during which the southern

part of the Indian peninsula and Sri Lanka receive a majoramount of their yearly rainfall. This season is also widelyknown as the Northeast Monsoon (NEM), winter monsoon(Rao, 1999), or post-monsoon season, which we use in thispaper (Singh and Sontakke, 1999), as it refers to a reversalof wind direction. Although differential heating and the re-sultant thermal circulation are responsible for both the ISMand post-monsoon seasons, the two monsoon seasons differsubstantially in several respects (Wang, 2006). First of all,the heat source region of the post-monsoon season is muchcloser to the equator (Krishnamurti, 1971), where the effectof the Earth’s rotation is diminished. Also, the circulation ofthe post-monsoon season encompasses a larger meridionaldomain, such that the tropical region has a strong interactionwith the extratropical region. During the post-monsoon sea-son, the ITCZ moves south, and jet streams return to theirwinter locations. The westerly jet stream splits into two: thenorthern jet stream runs north of the high-elevation Tibetanplateau and continues eastward above China; the southern jetstream runs to the south of the Himalaya. The area of max-imum heating moves south to Indonesia, and low-pressure

Figure 1. Regional overview of the Indian subcontinent. Major to-pographic and political features referred to in the text are labeledin white, and the background image is based on the Shuttle RadarTopography Mission (SRTM30) gridded digital elevation model,which is available at GEBCO (2014).

cyclones migrate into the Pacific Ocean. During the post-monsoon season, rainfall across northwestern India, Pak-istan, Bangladesh and Nepal is caused by winter westerlies.Precipitation during the winter months is crucial for agricul-ture, particularly for the rabi crops.The pre-monsoon season is usually determined to be the

period before the ISM (March–May), and is characterized bychanges in the temperature and pressure gradients, as well aswind direction – from predominantly northwesterly to south-westerly winds. During this season, climatologists try to pre-dict the onset, strength, and internal variability of the devel-oping ISM. In this study, we focus on special features of theISM and pre-monsoon seasons, including spatial distributionof the extreme precipitation, temperature and pressure, whichdetermine the coming ISM season.

1.2 Climate networks as a tool for ISM analysis

Various approaches exist for studying Indian monsoon dy-namics, such as numerical modeling using, for instance, At-mospheric General Circulation Models (AGCMs) (KrishnaKumar, 2005; Waliser et al., 2003), statistical analysis of ob-servational or reanalysis data (Revadekar and Preethi, 2012;Rajkumari and Narasimha, 1996), or recurrence analysis todetect regime transitions (Marwan et al., 2009, 2013) orcoupling directions with intersystem recurrence networks(Feldhoff et al., 2013).The application of complex network theory to analyze dif-

ferent climate phenomena is a new but rapidly growing areaof research, where a number of studies have been carriedout recently (Donges et al., 2009a, b; Malik et al., 2010,2011; Tsonis and Roebber, 2004; Tsonis et al., 2006, 2008,

Nonlin. Processes Geophys., 21, 901–917, 2014 www.nonlin-processes-geophys.net/21/901/2014/

Indian Summer Monsoon902 V. Stolbova et al.: Network of extreme precipitation over the Indian subcontinent

Summer Monsoon (ISM) season; June–September), and thepost-monsoon season (October–December). During the In-dian Summer Monsoon season, the Indian subcontinent re-ceives more than 80% of its annual rainfall across the region(Bookhagen and Burbank, 2010). During the post-monsoonseason, the strongest rainfall covers the southern tip of In-dia and most of Sri Lanka. Pre-monsoon months (March–May) are characterized by light rainfall over the Himalaya,the southwestern part of India, and Myanmar (Burma) (seeFig. 1).The ISM is driven by heating of the high-altitude Tibetan

Plateau that leads to a near-surface low-pressure system thatattracts moist air from the surrounding oceans, especially theBay of Bengal (Flohn, 1957; Gadgil, 2003; Webster, 1997).On large scales, the ISM is linked to the seasonal heatingof land masse and interacts with the westerly jet stream.An important interaction occurs when the Somali jet streamcrosses the Arabian Sea, changing the direction of ocean cur-rents, and when colder water wells up from lower ocean lay-ers, causing a decrease in temperature. Additionally, a low-pressure trough where the southern and northern trade windsmeet, a border known as the Intertropical Convergence Zone(ITCZ), moves north. As a result, tropical storms, depres-sions, cyclones, squall lines and daily cycles become moreerratic and thus less predictable. On the eastern peninsula,the differential heating between the Bay of Bengal and thesurrounding land masses (including the Tibetan Plateau) con-stitutes the moisture source of monsoonal rain. These com-bined factors create a low-pressure “monsoon trough” to thesouth and parallel to the Himalayan mountains. It is thistrough’s fluctuations that largely influence when monsoonalrains start and stop, generally considered active and breakphases (Gadgil, 2003).October–December is a period during which the southern

part of the Indian peninsula and Sri Lanka receive a majoramount of their yearly rainfall. This season is also widelyknown as the Northeast Monsoon (NEM), winter monsoon(Rao, 1999), or post-monsoon season, which we use in thispaper (Singh and Sontakke, 1999), as it refers to a reversalof wind direction. Although differential heating and the re-sultant thermal circulation are responsible for both the ISMand post-monsoon seasons, the two monsoon seasons differsubstantially in several respects (Wang, 2006). First of all,the heat source region of the post-monsoon season is muchcloser to the equator (Krishnamurti, 1971), where the effectof the Earth’s rotation is diminished. Also, the circulation ofthe post-monsoon season encompasses a larger meridionaldomain, such that the tropical region has a strong interactionwith the extratropical region. During the post-monsoon sea-son, the ITCZ moves south, and jet streams return to theirwinter locations. The westerly jet stream splits into two: thenorthern jet stream runs north of the high-elevation Tibetanplateau and continues eastward above China; the southern jetstream runs to the south of the Himalaya. The area of max-imum heating moves south to Indonesia, and low-pressure

Figure 1. Regional overview of the Indian subcontinent. Major to-pographic and political features referred to in the text are labeledin white, and the background image is based on the Shuttle RadarTopography Mission (SRTM30) gridded digital elevation model,which is available at GEBCO (2014).

cyclones migrate into the Pacific Ocean. During the post-monsoon season, rainfall across northwestern India, Pak-istan, Bangladesh and Nepal is caused by winter westerlies.Precipitation during the winter months is crucial for agricul-ture, particularly for the rabi crops.The pre-monsoon season is usually determined to be the

period before the ISM (March–May), and is characterized bychanges in the temperature and pressure gradients, as well aswind direction – from predominantly northwesterly to south-westerly winds. During this season, climatologists try to pre-dict the onset, strength, and internal variability of the devel-oping ISM. In this study, we focus on special features of theISM and pre-monsoon seasons, including spatial distributionof the extreme precipitation, temperature and pressure, whichdetermine the coming ISM season.

1.2 Climate networks as a tool for ISM analysis

Various approaches exist for studying Indian monsoon dy-namics, such as numerical modeling using, for instance, At-mospheric General Circulation Models (AGCMs) (KrishnaKumar, 2005; Waliser et al., 2003), statistical analysis of ob-servational or reanalysis data (Revadekar and Preethi, 2012;Rajkumari and Narasimha, 1996), or recurrence analysis todetect regime transitions (Marwan et al., 2009, 2013) orcoupling directions with intersystem recurrence networks(Feldhoff et al., 2013).The application of complex network theory to analyze dif-

ferent climate phenomena is a new but rapidly growing areaof research, where a number of studies have been carriedout recently (Donges et al., 2009a, b; Malik et al., 2010,2011; Tsonis and Roebber, 2004; Tsonis et al., 2006, 2008,

Nonlin. Processes Geophys., 21, 901–917, 2014 www.nonlin-processes-geophys.net/21/901/2014/

TRMM

A

B

C

D

G

E

F

H

4500 m

community detection and participation are useful in climate networks

extreme precipitation synchronisation during Indian Summer Monsoon

TRMM

part

icip

ati

on

p

0

0.2

0.4

0.6

0.8

1

layer 1

layer 2

layer 3

layer 4

multilayer network

time-dependentnetwork

t=1

t=2

t=3

t=4

Also applicable to multilayer or time-dependent networks

• multilayer networks consist of different kind of edges (e.g. social interactions or means of transportation)

• time-dependent networks change the adjacency matrix at each time step and can be represented as multilayer networks

or

airlines are spatially organisedhub and spoke

20 40 60 800

20

40

60

degree k

#of

airports

Istanb

ul

Ank

ara

hubspoke

20 40 60 800

20

40

60

degree k

#of

airports

Cope

nhag

enStoc

kholm

Oslo

threefoldedhub structure

airports have diverse roles in this multilayer organisation

participation p

degree k

1 40 80 160

0 1

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1

2

5

10

20

50

100

200

A Coruna

Alg he ro

Alica nte

Allg a u

Anta lya

Arla nda

Aro

As turia s

Ata turk

Aure l Vla icu

Ba lice

Ba ra ja sBa rce lona

Be lfa s t Intl

Be n Gurion

Be og ra d

Be rg a mo Orio Al S e rio

Birming ha m

Bla g na c

Bodo

Bolog na

Bris tol

Brus s e ls Na tl

Ca mpo De ll Oro

Cluj Na poca

Cork

Dublin

Edinburg h

Ela z ig

Es e nbog a

Exe te r

Fa ro

Fire nz e

Fle s la nd

Fra nkfurt Ha hn

Frie drichs ha fe n

Ga twick

Ge nova S e s tri

Girona

Gla s g ow

Gue rns e y

Guipa va s

Ha mburg

He nri Coa nda

Inns bruck

J a s ionka

J e re z

J e rs ey

Ka s trup

Ka una s Intl

Kos

Le Bourg e t

Le ipz ig Ha lle

Lie g e

Lina te

Luton

Luxe mbourg

M R S te fa nik

Ma la g a

Ma lpe ns a

Mos sNie de rrhe in

P is a

P re s twick

Re us

Rig a Intl

Rotte rda m

Ruz yne

S a biha Gokce n

S a lz burg

S a ra je vo

S a ve

S chiphol

S chwe cha t

S e villa

S ka vs ta S ofia

S ola

S outha mpton

S plit

S ta ns te d

Tille

Tira na Rina s

Torp

Ume a

Ve ne z ia Te s s e ra

Vig ra

VitoriaZa g re b

participation

degr

ee

Conclusion• nodes have diverse roles in the mesoscale

structure of networks

• participation and hubness are one way of deciphering those

• structures for segregation and integration are present in neuronal networks

• borders between modules can be investigated

• multilayer variant is applicable

Local

Mesoscale

Global

Local contribution to mesoscale

Thank you! florian.klimm@dtc.ox.ac.uk

Klimm, F., Borge-Holthoefer, J., Wessel, N., Kurths, J., & Zamora-López, G. (2014). Individual nodeʼs contribution to the mesoscale of complex networks. New Journal of Physics, 16(12), 125006. Klimm, F., Stolbova, V., Kurths, J., & Zamora-López, G. Mesoscale analysis of the network of extreme precipitation during the Indian Summer Monsoon (to be submitted to Nonlinear Processes in Geophysics) Klimm, F., Kurths, J., & Zamora-López, G. Roles of nodes inside the multilayer structure of networks (in preparation)