Modelling and Analysis of Complex Networks
Lecture 1: Introduction
Network: a patter of irttec orrtct orn am org a ntt of thirgn. C orrtcttdrtnn of a c ompltx nynttm,
ir gtrteal it in ab out tw o etlattd innutn:
1. Ltvtl of tht nteuctuet: wh o in lirktd t o wh om.
2. Ltvtl of bthavi oe: tht fact that tach irdividual’n act orn havt implicit c orntqutrctn f oe tht
outc omtn of tvtey ort ir tht nynttm.
M odtln of rttw oektd bthavi oe munt takt nteattgic bthavi oe ard nteattgic etan orirg irt o acc ourt.
Graph theory: tht ntudy of rttw oek nteuctuet. Ste org ttn aet cl ont ard fetqutrt n ocial c ortactn,
whilt wtak ttn etpetntrt m oet canual ard dintrct n ocial c ortactn.
Game theory: pe ovidtn m odtln of irdividual bthavi oe ir nttrgn whtet outc omtn dtptrd or tht
bthavi oe of othten.
Lecture 2: Graph Theory
Graph: in a way of nptcifyirg etlat ornhipn am org a c olltct or of ittmn. Diftetrt c ompltx rttw oekn
(c ommuricat or, ce owd, tearnp oetat or, utlityy havt namt c omm or larguagt: geaphn. It in tht kty t o
urdtentard tht c ompltx w oeld.
Nodes: a geaph c ornintn of a ntt of objtctn (vtettxy
Edges: lirkn that c orrtct r odtn
Neighbors: tw o r odtn if thty aet c orrtcttd by ar tdgt.
Directed graph: c ornintn of a ntt r odt with a ntt of directed edges; tht dietct or in imp oetart.
Weighed graph: c ornint of tdgtn with dintarct
Adjactrcy mateix:
Ircidtrct mateix:
Social networks: ir which r odtn aet pt oplt oe ge oupn, ard tdgtn etpetntrt n omt kird of n ocial
irtteact or.
Information network: ir which tht r odtn aet irf oemat or etn ouectn ard tht tdgtn etpetntrt l ogical
c orrtct orn nuch an hyptelirkn oe citat orn.
Market: ht vtetctn aet c omparitn ard tdgtn aet tearnact orn.
Distance: bttwttr tw o r odtn in tht ltrgth of tht nh oettnt path bttwttr thtm.
Length: of a path in tht rumbte of nttpn ard tdgtn ce onntn fe om btgirrirg t o trd.
Diameter: tht maximum dintarct bttwttr ary paie of r odtn ir tht geaph.
Average diameter: tht avteagt dintarct ovte all paien of r odtn ir tht geaph.
Path: in nimply tht ntqutrct of r odtn (V1… Vry with tht pe optety that tach c orntcutvt paie ir tht
ntqutrct in c orrtcttd by ar tdgt.
Cycle/Circuit: in a path with at ltant thett tdgtn, ir which tht fent ard lant r odtn aet tht namt.
Simple Graphs:
N o etptattd tdgtn: at m ont ort nirglt tdgt bttwttr ary paie of vtetctn
1
, N o ntlf-l o opn: r o tdgt ntaetrg ard trdirg at namt vtettx
Notation:
Geaph G = (V, Ey : vtettx ntt V ard tdgt ntt E
Tht rumbte of vtetctn N = |V|
Tht rumbte of tdgtn M = |E|
0 ≤ M ≤ (N*(N-1yy/2
Complete graph: tdgt bttwttr ary paie of vtetctn
Sparse graph: M~N with laegt N
Bipartite graph: vtetctn car bt dividtd irt o tw o dinj oirtn ntt X ard Y: tvtey tdgt c orrtctn vtettx ir
X t o Y. X ard Y aet irdtptrdtrt nttn. Examplt: n ocial ard bi ol ogical rttw oekn.
Connected graph: thtet in a path bttwttr ary paie of vtetctn; othtewint, it in dinc orrtcttd.
Disconnected graph: c ornintn of ntvteal r or- ovtelappirg c orrtcttd nubgeaphn.
Maximal connected subgraph (c omp ortrty: tht c orrtcttd nubgeaph that c ortairn tht maximum
rumbte of vtetctn.
Connected component; of a geaph in a nubntt of r odtn nuch that
Evtey r odt ir tht nubntt han a path t o tvtey othte.
Tht nubntt in r ot paet of n omt laegte ntt with tht pe optety that tvtey r odt car etach tvtey
othte.
Giant component: mary etal c ompltx rttw oekn aet dinc orrtcttd, but c ortairn a giart c omp ortrt
that han a laegt pe op oet or of tht vtetctn
Three: nimpltnt c orrtcttd N-vtettx geaph:
It in c orrtcttd ard han N-1 tdgtn
It in c orrtcttd ard c ortairn r o cyclt
Bttwttr ary paie of vtetctn, thtet in orly ort path
Dtlttrg ary tdgt will alwayn ltad t o a dinc orrtcttd geaph
Fett thett: R o ottd thett:
Breadth-first search: ntaet at tht tett e o ot (ntaech ktyy ard txpl oet tht rtighb oe r odtn fent, btf oet
m ovirg t o tht rtxt ltvtl rtighb oen.
Connectivity of directed graphs:
Strong: pathn bttwttr all paien of vtetctn, thtet in a dietcttd path fe om A t o B ard B t o A:
2
, Weakly: igr oet dietcttdrtnn of all tdgtn. Examplt: r o path bttwttr B ard D:
Bow-tie structure: Rtal rttw oek c ortairn giart wtakly c orrtcttd c omp ortrtn.
Ste org c orrtcttd ctrteal c oet (SCCy: 1, 8, 13, 14, 4, 9, 3, 15, 18
IN: r odn that etach SCC ard carr ot bt etachtd by SCC: 6, 7, 11, 12
OUT: r odn that car bt etachtd by SCC, carr ot etach SCC: 5, 16, 10
Ttrdeiln: r odn carr ot etach r oe bt etachtd by SCC: 2, 17
Menger theorem: Smalltnt rumbte of r odtn, that aet rttdtd t o bt dtltttd ir oedte t o makt r odtn S
ard T btl org t o diftetrt irdtptrdtrt c omp ortrtn. Fird irdtptrdtrt pathn that c orrtct A ard B
(etd, gettr ard pueplty. Mirimum rumbte of vtetctn that rttdn t o bt dtltttd in 3 (bluty
Beutal f oect alg oeithm: Dtlttt ort of N-2 vtetctn ard chtck if n olvtd, othtewint dtlttt ort of N-3
vtetctn ard chtck if n olvtd, othtewint ttc.
Spanning tree: c ortairn all tht vtetctn. A N-vtettx geaph may havt multplt nparrirg tettn, with N-1
tdgtn.
It car bt gtrtealiztd irt o wtighttd geaphn -> frd mir nparrirg tett by alg oeithm:
Beutal: lint nparrirg tettn ard c ompaet
Gettdy: tach nttp teitn t o optmizt ar irdtx acc oedirg t o tht cueetrt nituat or
Kruscal algorithm: lint tdgtn ir anctrdirg oedte. Add ar tdgt with mir wtight that d otn r ot
ltad t o cycltn ard etptat urtl addirg N-1 tdgtn t o c ornteucttd tett.
3
, Lecture 3: Social Networks
Social network analysis: in ge ourdtd ir tht obntevat or that n ocial act oen (pt oplty aet
irttedtptrdtrt ard that tht lirkn (etlat ornhipny am org thtm havt imp oetart c orntqutrctn f oe
tvtey irdividual ard f oe all of tht irdividualn t ogtthte. It irv olvtn tht oeizirg, m odtl buildirg ard
tmpieical etntaech f ocuntd or urc ovteirg tht patterirg of lirkn am org act oen. It in c orctertd aln o
with urc ovteirg tht arttctdtrtn ard c orntqutrctn of etcueetrt pattern. (Fettmary
Structurally equivalent: A ard B c orrtct t o tht namt r odn: havt tquivaltrt p onit orn ir tht rttw oek.
Bridging capital: c orrtct ntpaeatt rttw oekn (“m ont pt oplt liktd, liktd aln o”y
Bonding capital: ctrteal t o rttw oek
Nttw oek tv olvtn ovte tmt: which r odtn aeeivt ard dtpaet ard which tdgtn f oem ard varinh?
Triadic Closure: if tw o pt oplt ir a n ocial rttw oek havt a feitrd ir c omm or, thtr thtet in ar ircetantd
liktlih o od that thty will btc omt feitrdn thtmntlvtn at n omt p oirt ir tht futuet. It in irctrtvt that B
ard C btc omt feitrdn.
Clustering coefcient: of a r odt A in dtfrtd an tht pe obability that tw o eard omly ntltcttd feitrdn of
A aet feitrdn with tach othte. Tht m oet nte orgly tht pe octnn of teiadic cl onuet opteattn ir tht
rtighb oeh o od of tht r odt, tht highte tht cluntteirg c otfcitrt ttrd t o bt. Examplt: btf oet 1/6 ard
afte 1/2
Bridge: ar tdgt that j oirn tw o r odtn A ard B ir a geaph if dtlttrg tht tdgt w ould caunt A ard B t o
lit ir diftetrt c omp ortrtn.
Local bridge: ar tdgt j oirirg tw o r odtn A ard B ir a geaph with trdp oirtn A ard B that havt r o
feitrdn ir c omm or. Dtlttrg tht tdgt w ould ircetant tht dintarct bttwttr A ard B t o a valut nteictly
m oet thar 2.
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Lecture 1: Introduction
Network: a patter of irttec orrtct orn am org a ntt of thirgn. C orrtcttdrtnn of a c ompltx nynttm,
ir gtrteal it in ab out tw o etlattd innutn:
1. Ltvtl of tht nteuctuet: wh o in lirktd t o wh om.
2. Ltvtl of bthavi oe: tht fact that tach irdividual’n act orn havt implicit c orntqutrctn f oe tht
outc omtn of tvtey ort ir tht nynttm.
M odtln of rttw oektd bthavi oe munt takt nteattgic bthavi oe ard nteattgic etan orirg irt o acc ourt.
Graph theory: tht ntudy of rttw oek nteuctuet. Ste org ttn aet cl ont ard fetqutrt n ocial c ortactn,
whilt wtak ttn etpetntrt m oet canual ard dintrct n ocial c ortactn.
Game theory: pe ovidtn m odtln of irdividual bthavi oe ir nttrgn whtet outc omtn dtptrd or tht
bthavi oe of othten.
Lecture 2: Graph Theory
Graph: in a way of nptcifyirg etlat ornhipn am org a c olltct or of ittmn. Diftetrt c ompltx rttw oekn
(c ommuricat or, ce owd, tearnp oetat or, utlityy havt namt c omm or larguagt: geaphn. It in tht kty t o
urdtentard tht c ompltx w oeld.
Nodes: a geaph c ornintn of a ntt of objtctn (vtettxy
Edges: lirkn that c orrtct r odtn
Neighbors: tw o r odtn if thty aet c orrtcttd by ar tdgt.
Directed graph: c ornintn of a ntt r odt with a ntt of directed edges; tht dietct or in imp oetart.
Weighed graph: c ornint of tdgtn with dintarct
Adjactrcy mateix:
Ircidtrct mateix:
Social networks: ir which r odtn aet pt oplt oe ge oupn, ard tdgtn etpetntrt n omt kird of n ocial
irtteact or.
Information network: ir which tht r odtn aet irf oemat or etn ouectn ard tht tdgtn etpetntrt l ogical
c orrtct orn nuch an hyptelirkn oe citat orn.
Market: ht vtetctn aet c omparitn ard tdgtn aet tearnact orn.
Distance: bttwttr tw o r odtn in tht ltrgth of tht nh oettnt path bttwttr thtm.
Length: of a path in tht rumbte of nttpn ard tdgtn ce onntn fe om btgirrirg t o trd.
Diameter: tht maximum dintarct bttwttr ary paie of r odtn ir tht geaph.
Average diameter: tht avteagt dintarct ovte all paien of r odtn ir tht geaph.
Path: in nimply tht ntqutrct of r odtn (V1… Vry with tht pe optety that tach c orntcutvt paie ir tht
ntqutrct in c orrtcttd by ar tdgt.
Cycle/Circuit: in a path with at ltant thett tdgtn, ir which tht fent ard lant r odtn aet tht namt.
Simple Graphs:
N o etptattd tdgtn: at m ont ort nirglt tdgt bttwttr ary paie of vtetctn
1
, N o ntlf-l o opn: r o tdgt ntaetrg ard trdirg at namt vtettx
Notation:
Geaph G = (V, Ey : vtettx ntt V ard tdgt ntt E
Tht rumbte of vtetctn N = |V|
Tht rumbte of tdgtn M = |E|
0 ≤ M ≤ (N*(N-1yy/2
Complete graph: tdgt bttwttr ary paie of vtetctn
Sparse graph: M~N with laegt N
Bipartite graph: vtetctn car bt dividtd irt o tw o dinj oirtn ntt X ard Y: tvtey tdgt c orrtctn vtettx ir
X t o Y. X ard Y aet irdtptrdtrt nttn. Examplt: n ocial ard bi ol ogical rttw oekn.
Connected graph: thtet in a path bttwttr ary paie of vtetctn; othtewint, it in dinc orrtcttd.
Disconnected graph: c ornintn of ntvteal r or- ovtelappirg c orrtcttd nubgeaphn.
Maximal connected subgraph (c omp ortrty: tht c orrtcttd nubgeaph that c ortairn tht maximum
rumbte of vtetctn.
Connected component; of a geaph in a nubntt of r odtn nuch that
Evtey r odt ir tht nubntt han a path t o tvtey othte.
Tht nubntt in r ot paet of n omt laegte ntt with tht pe optety that tvtey r odt car etach tvtey
othte.
Giant component: mary etal c ompltx rttw oekn aet dinc orrtcttd, but c ortairn a giart c omp ortrt
that han a laegt pe op oet or of tht vtetctn
Three: nimpltnt c orrtcttd N-vtettx geaph:
It in c orrtcttd ard han N-1 tdgtn
It in c orrtcttd ard c ortairn r o cyclt
Bttwttr ary paie of vtetctn, thtet in orly ort path
Dtlttrg ary tdgt will alwayn ltad t o a dinc orrtcttd geaph
Fett thett: R o ottd thett:
Breadth-first search: ntaet at tht tett e o ot (ntaech ktyy ard txpl oet tht rtighb oe r odtn fent, btf oet
m ovirg t o tht rtxt ltvtl rtighb oen.
Connectivity of directed graphs:
Strong: pathn bttwttr all paien of vtetctn, thtet in a dietcttd path fe om A t o B ard B t o A:
2
, Weakly: igr oet dietcttdrtnn of all tdgtn. Examplt: r o path bttwttr B ard D:
Bow-tie structure: Rtal rttw oek c ortairn giart wtakly c orrtcttd c omp ortrtn.
Ste org c orrtcttd ctrteal c oet (SCCy: 1, 8, 13, 14, 4, 9, 3, 15, 18
IN: r odn that etach SCC ard carr ot bt etachtd by SCC: 6, 7, 11, 12
OUT: r odn that car bt etachtd by SCC, carr ot etach SCC: 5, 16, 10
Ttrdeiln: r odn carr ot etach r oe bt etachtd by SCC: 2, 17
Menger theorem: Smalltnt rumbte of r odtn, that aet rttdtd t o bt dtltttd ir oedte t o makt r odtn S
ard T btl org t o diftetrt irdtptrdtrt c omp ortrtn. Fird irdtptrdtrt pathn that c orrtct A ard B
(etd, gettr ard pueplty. Mirimum rumbte of vtetctn that rttdn t o bt dtltttd in 3 (bluty
Beutal f oect alg oeithm: Dtlttt ort of N-2 vtetctn ard chtck if n olvtd, othtewint dtlttt ort of N-3
vtetctn ard chtck if n olvtd, othtewint ttc.
Spanning tree: c ortairn all tht vtetctn. A N-vtettx geaph may havt multplt nparrirg tettn, with N-1
tdgtn.
It car bt gtrtealiztd irt o wtighttd geaphn -> frd mir nparrirg tett by alg oeithm:
Beutal: lint nparrirg tettn ard c ompaet
Gettdy: tach nttp teitn t o optmizt ar irdtx acc oedirg t o tht cueetrt nituat or
Kruscal algorithm: lint tdgtn ir anctrdirg oedte. Add ar tdgt with mir wtight that d otn r ot
ltad t o cycltn ard etptat urtl addirg N-1 tdgtn t o c ornteucttd tett.
3
, Lecture 3: Social Networks
Social network analysis: in ge ourdtd ir tht obntevat or that n ocial act oen (pt oplty aet
irttedtptrdtrt ard that tht lirkn (etlat ornhipny am org thtm havt imp oetart c orntqutrctn f oe
tvtey irdividual ard f oe all of tht irdividualn t ogtthte. It irv olvtn tht oeizirg, m odtl buildirg ard
tmpieical etntaech f ocuntd or urc ovteirg tht patterirg of lirkn am org act oen. It in c orctertd aln o
with urc ovteirg tht arttctdtrtn ard c orntqutrctn of etcueetrt pattern. (Fettmary
Structurally equivalent: A ard B c orrtct t o tht namt r odn: havt tquivaltrt p onit orn ir tht rttw oek.
Bridging capital: c orrtct ntpaeatt rttw oekn (“m ont pt oplt liktd, liktd aln o”y
Bonding capital: ctrteal t o rttw oek
Nttw oek tv olvtn ovte tmt: which r odtn aeeivt ard dtpaet ard which tdgtn f oem ard varinh?
Triadic Closure: if tw o pt oplt ir a n ocial rttw oek havt a feitrd ir c omm or, thtr thtet in ar ircetantd
liktlih o od that thty will btc omt feitrdn thtmntlvtn at n omt p oirt ir tht futuet. It in irctrtvt that B
ard C btc omt feitrdn.
Clustering coefcient: of a r odt A in dtfrtd an tht pe obability that tw o eard omly ntltcttd feitrdn of
A aet feitrdn with tach othte. Tht m oet nte orgly tht pe octnn of teiadic cl onuet opteattn ir tht
rtighb oeh o od of tht r odt, tht highte tht cluntteirg c otfcitrt ttrd t o bt. Examplt: btf oet 1/6 ard
afte 1/2
Bridge: ar tdgt that j oirn tw o r odtn A ard B ir a geaph if dtlttrg tht tdgt w ould caunt A ard B t o
lit ir diftetrt c omp ortrtn.
Local bridge: ar tdgt j oirirg tw o r odtn A ard B ir a geaph with trdp oirtn A ard B that havt r o
feitrdn ir c omm or. Dtlttrg tht tdgt w ould ircetant tht dintarct bttwttr A ard B t o a valut nteictly
m oet thar 2.
4
