Η Επιστήμη των Δικτύων: Μια Πολύ Σύντομη Εισαγωγική Παρουσίαση - 8 Οκτωβρίου 2014

H Epist mh twn DiktÔwn
Mia PolÔ SÔntomh Eisagwgik ParousÐash
Mwus c A. MpountourÐdhc
Tm ma Majhmatik¸n PanepisthmÐou Patr¸n
mboudour@upatras.gr
Okt¸brioc 2014
Mwus c A. MpountourÐdhc H Epist mh twn DiktÔwn

Perieqìmena
Basikèc 'Ennoiec DiktÔwn
Mia Panoramik 'Ekjesh Diktuak¸n 'Ergwn
LÐga Endeiktikˆ ParadeÐgmata Diktuak¸n Upologism¸n
EndeiktikoÐ Pìroi

Basikèc 'Ennoiec DiktÔwn

Ti eÐnai èna dÐktuo;
Praktikˆ, ètsi ìpwc katalabaÐnoume ennoiologikˆ ti
eÐnai èna dÐktuo:
'Ena dÐktuo eÐnai èna sÔnolo paragìntwn forèwn
drˆshc, pou onomˆzontai dr¸ntec (actors), oi opoÐoi
sqetÐzontai metaxÔ touc me kˆpoia morf¸mata
diadrastik c sumperiforˆc, pou onomˆzontai
desmoÐ (ties) sqèseic diˆdrashc (interactions).
Tupikˆ, ètsi ìpwc analÔetai majhmatikˆ (sthn JewrÐa
Grˆfwn [Graph Theory]) kai anaparÐstatai mèsw
grafik¸n optikopoi sewn (visualizations):
'Ena dÐktuo eÐnai èna sÔnolo kìmbwn ( koruf¸n
shmeÐwn), oi opoÐoi sundèontai metaxÔ touc me
kˆpoia sugkekrimèna morf¸mata sundèsmwn (links).

Kathgoriopoi seic Diktuak¸n Dr¸ntwn
Oi dr¸ntec enìc diktÔou mporeÐ na eÐnai:
'Atoma (ˆnjrwpoi) me diaforetikˆ dhmografikˆ
qarakthristikˆ (ìpwc fÔlo, ful –èjnoc, uphkoìthta,
hlikÐa, ekpaÐdeush, ergasÐa, oikonomik katˆstash, katoikÐa
klp.) se diaforetikèc yuqo–swmatikèc katastˆ-
seic [pq., asjèneiec, ugeÐa, bˆroc, eutuqÐa klp.] me diafo-
retikèc idèec–pepoij seic–topojet seic–protim seic
gia kˆpoia politistikˆ politikˆ oikonomikˆ klp. zht mata.
Omˆdec atìmwn (ìpwc organ¸seic, etairÐec, jesmikˆ s¸mata,
krˆth klp.).
OrganismoÐ (zwikoÐ biologikoÐ).
Ulikˆ prˆgmata (ìpwc biblÐa, ergasÐec, episthmonikoÐ
klˆdoi, mèsa epikoinwnÐac–plhrofìrhshc, teqnourg mata
[artifacts], emporeÔmata, mhqanèc, upologistèc, diadiktuakˆ
sˆðt/selÐdec klp.).
Sunajroistikˆ gegonìta (ìpwc sumfwnÐec, yhfoforÐec,
ekjèseic, diadhl¸seic diamarturÐac, sumbˆnta, peristˆseic,
taktikèc sunant seic se q¸rouc epikoinwnÐac atìmwn
organ¸sewn gia sugkekrimènouc skopoÔc klp.).

Kathgoriopoi seic Diktuak¸n Diadrˆsewn
Oi diadrˆseic se èna dÐktuo mporeÐ na eÐnai:
Koinwnikèc sqèseic metaxÔ atìmwn (ìpwc filÐac,
suggèneiac, sunaisjhmatik c fÔshc, sexoualik c sqèshc,
arèskeiac–dusarèskeiac, empistosÔnhc–duspistÐac klp.).
Koinwnikèc sqèseic allhlexˆrthshc (ìpwc didˆsko-
nta–didaskìmenou, proðstˆmenou–ufistˆmenou, sunergasÐac,
upost rixhc, allhlobo jeiac, paroq c sumboul¸n,
emporik¸n–oikonomik¸n sunallag¸n klp.).
Koinwnikèc sqèseic antipalìthtac (ìpwc diafwnÐac,
antiparajèsewn, èqjrac, fìbou, antagwnismoÔ klp.).
'Emmesec sqèseic diamoirasmoÔ summetoq c se
koinˆ gegonìta (ìpwc se organ¸seic, sumbˆnta,
lèsqec–klamp, sullìgouc, sumboÔlia, sqoleÐa,
jesmoÔc–idrÔmata, katagwg c diamon c se gewgrafikèc
perioqèc, sun–dhmosieÔsewn, bibliografik¸n anafor¸n,
diˆdoshc gn¸shc, koinwnik c epirro c, koin¸n asqoli¸n,
epanalambanìmenwn sunhjei¸n, ìpwc qr shc narkwtik¸n, klp.,
metˆdoshc [contagion] asjenei¸n k.ˆ. diˆqushc i¸n k.ˆ. klp.).

Qarakthristikˆ Dr¸ntwn kai Diadrˆsewn
Diaforetikèc kathgorÐec ( tÔpoi) dr¸ntwn diadrˆsewn mporoÔn
na enopoihjoÔn se omˆdec, stic opoÐec orÐzetai èna qarakthristikì
(attribute), pou paÐrnei diaforetikèc timèc se kˆje omˆda.
Genik¸c, kˆje qarakthristikì mporeÐ na jewrhjeÐ wc mia metablht ,
eÐte posotik (suneq¸n diakrit¸n tim¸n) poiotik (diataktik¸n
[ordinal] onomastik¸n] [nominal] tim¸n). P.q.:
'Arrenec kai j leic dr¸ntec omadopoioÔntai kˆtw apì to
(poiotikì) onomastikì qarakthristikì tou fÔlou.
Dr¸ntec diaforetikoÔ bˆrouc omadopoioÔntai kˆtw apì to
(posotikì) suneqèc qarakthristikì tou bˆrouc.
Diadrˆseic hlektronik c epikoinwnÐac omadopoioÔntai kˆtw apì
to (posotikì) diakritì qarakthristikì tou pl jouc thc
suqnìthtac twn antallassìmenwn mhnumˆtwn (se kˆpoia
perÐodo).
Diadrˆseic diaforetik¸n bajm¸n thc sqèshc filÐac omadopoioÔ-
ntai kˆtw apì to (poiotikì) diataktikì qarakthristikì thc
diabˆjmishc thc èntashc thc sqèshc filÐac.
Diadrˆseic sumpˆjeiac–antipˆjeiac omadopoioÔntai kˆtw apì
to (poiotikì) diataktikì (duadikì) qarakthristikì tou
prìshmou (jetikoÔ arnhtikoÔ) thc sqèshc.
Oi antÐstoiqoi grˆfoi eÐnai oi grˆfoi me bˆrh (timèc) (weighted–
valued graphs) stouc kìmbouc stouc sundèsmouc. Eidikˆ sto teleu-
taÐo parˆdeigma, onomˆzontai proshmasmènoi grˆfoi (signed graphs).
'Ena dÐktuo, sto opoÐo oi Ðdioi dr¸ntec diathroÔn perissìterec thc
miac diaforetikèc diadrˆseic onomˆzetai polusqidèc (multiplex).

Diktuakèc AnalÔseic
1 Koinwnikˆ Diktuakˆ Dedomèna
Erwthmatolìgia kai sunenteÔxeic
Istorikˆ arqeÐa kai arqeÐa tÔpou
Bibliometrikˆ kai episthmometrikˆ dedomèna
Dedomèna apì to Internet (mhnÔmata, istoselÐdec,
mplogk, koinwnikˆ mèsa)
Sqesiakˆ Megˆla Dedomèna (Big Data) kai Anoiktˆ
Dedomèna (Open Data)
2 Diktuakˆ Mètra
BajmoÐ kìmbwn – Istogrˆmmata
Kentrikìthtec kìmbwn
Suntelest c suss¸reushc kai metabatikìthta
Amoibaiìthta sundèsmwn
Apostˆseic kìmbwn

3 DiktuakoÐ diamerismoÐ
Sunektikèc sunist¸sec kai klÐkec
k–pur nec
Pur nac–perifèreia
IsodunamÐec kìmbwn (domik kai kanonik )
OmadopoÐhsh se mplok Blockmodeling
Koinìthtec (Communities)
Taxinomhsimìthta (assortativity) kai anˆmeixh (mixing)
4 Qronik¸c Exart¸mena DÐktua
Troqièc metabˆsewn
5 Statistik JewrÐa DiktÔwn
Exponential Random Graph Models (ERGM)
6 Montelopoi seic DiktÔwn
Koinwnik epirro
Diˆqush (montèla SIR kai SIS)
TuqaÐoi grˆfoi Erd¨os–Re´ nyi
DÐktua mikr¸n kìsmwn (small–worlds)
DÐktua qwrÐc klÐmaka (scale–free)
Auxanìmena tuqaÐa dÐktua kai to montèlo
Bara´ basi–Albert

Mia Panoramik 'Ekjesh Diktuak¸n
'Ergwn

To DÐktuo twn Flwrentian¸n OÐkwn
Sq ma: To dÐktuo twn gˆmwn metaxÔ twn megˆlwn OÐkwn thc
FlwrentÐac tou mesaÐwna (Padgett & Ansell, Robust action and the rise of the
Medici, 1400–1434, American Journal of Sociology, 1993, 98(6): 12591319).

To DÐktuo twn Sten¸n Sunergat¸n tou Santˆm
Sq ma: To dÐktuo tou eswterikoÔ kÔklou twn sten¸n sunergat¸n tou
Santˆm Qouseòn (Baraba´ si et al., Network Science Book,
http://barabasilab.neu.edu/networksciencebook/).

DÐktuo FilÐac Mel¸n Lèsqhc Karˆte
Sq ma: DÐktuo filÐac twn 34 mel¸n miac Panepisthmiak c lèsqhc karˆte
(Zachary, An information flow model for conflict and fission in small groups,
Journal of Anthropological Research, 1977, 33: 452473).

DÐktuo Sunaisjhmatik¸n Sqèsewn
Sq ma: DÐktuo sunaisjhmatik¸n–erwtik¸n sqèsewn (Bearman et al.,
Chains of affection: The structure of adolescent romantic and sexual networks,
American Journal of Sociology, 2004, 110: 4491)
se optikopoÐhsh tou Mark
Newman (http://www-personal.umich.edu/~mejn/networks/).

To dÐktuo twn AjlÐwn tou BÐktwroc Ougk¸
Sq ma: To dÐktuo twn sqèsewn metaxÔ twn kÔriwn qarakt rwn twn
AjlÐwn tou BÐktwroc Ougk¸ (ta qr¸mata antistoiqoÔn se koinìthtec, pou
upologÐsjhkan ek twn ustèrwn) (Newman & Girvan, Finding and evaluating
community structure in networks, Physical Review E, 2004, 69: 026113).

KuriarqoÔntec kìmboi kai koinìthtec sto
dÐktuo twn AjlÐwn

DÐktuo PaqÔsarkwn Atìmwn
Sq ma: DÐktuo paqÔsarkwn atìmwn (Christakis & Fowler, The spread of
obesity in a large social network over 32 years, New Englnd Journal of Medicine,
2007, 357(4): 370379)
[Mègejoc kìmbwn BMI, kÐtrinoi paqÔsarkoi,
prˆsino mh paqÔsarkoi, mwb sundèseic filÐa, portokalÐ suggeneÐc.]

DÐktuo Eutuqismènwn Atìmwn
Sq ma: DÐktuo eutuqismènwn atìmwn (Fowler & Christakis, Dynamic spread
of happiness in a large social network: Longitudinal analysis over 20 years in the
Framingham Heart Study, British Medical Journal, 2008, 337(768): a2338).
Kìmboi tetragwnikoÐ gunaÐkec, kuklikoÐ ˆndrec, mple ligìtero eutuqeÐc,
kÐtrino perissìtero eutuqeÐc, kìkkinec sundèseic filÐa, maÔrec suggeneÐc.]

DÐktuo FilÐac Majht¸n tou Faux Magnolia High School
Sq ma: To dÐktuo filÐac 1461 majht¸n tou Faux Magnolia High School
qwrÐc apomonwmènouc kìmbouc (Goudreau et al., A statnet Tutorial, Journal of
Statistical Software, 2008, 24(9): 127).

DÐktuo FilÐac Majht¸n tou Faux Magnolia High School me to
Qarakthristikì tou 'Etouc FoÐthshc twn Majht¸n
Sq ma: To dÐktuo filÐac 1461 majht¸n tou Faux Magnolia High School me
to qarakthristikì tou ètouc foÐthshc twn majht¸n kai qwrÐc
apomonwmènouc kìmbouc (Goudreau et al., A statnet Tutorial, Journal of
Statistical Software, 2008, 24(9): 127).

DÐktuo FilÐac me to Qarakthristikì thc Ful c twn Majht¸n
Sq ma: 'Ena dÐktuo filÐac majht¸n me to qarakthristikì thc ful c twn majht¸n (kÐtrino =
leukoÐ, prˆsino = maÔroi, kìkkino = ˆllhc ful c) kai qwrÐc apomonwmènouc kìmbouc (Moody, Race,
school integration, and friendship segregation in America, American Journal of Sociology, 2001, 107:
679716)
se optikopoÐhsh tou Mark Newman (http://www-personal.umich.edu/~mejn/networks/).

DÐktuo SunergasÐac sthn Santa Fe
Sq ma: DÐktuo sunergasÐac episthmìnwn tou InstitoÔtou Santa Fe (Girvan
& Newman, Community structure in social and biological networks, Proceedings of
the National Academy of Sciences of the USA, 2002, 99: 82718276).

DÐktuo Parapomp¸n sthn KoinwniologÐa
Sq ma: DÐktuo bibliografik¸n parapomp¸n sthn KoinwniologÐa apì dedomèna tou Jim Moody
(http://orgtheory.wordpress.com/2009/08/14/sociologys-citation-core/).

To DÐktuo twn Qwr¸n me Megˆlo Qrèoc
Sq ma: To dÐktuo twn qwr¸n me megˆla qrèh (Baraba´ si et al., Network
Science Book, http://barabasilab.neu.edu/networksciencebook/).

Trofikìc Istìc tou Oikosust matoc miac LÐmnhc
Sq ma: To dÐktuo tou trofikoÔ istoÔ (food web) tou oikosust matoc thc
LÐmnhc Little Rock tou Wisconsin (Martinez, Artifacts or attributes? Effects of
resolution on the Little Rock Lake food web, Ecological Monographs, 1991, 61:
367392).

DÐktuo Fainotupik¸n Asjenei¸n
Sq ma: DÐktuo fainotupik¸n asjenei¸n (Hidalgo, Blumm, Baraba´ si &
Christakis, A Dynamic Network Approach for the Study of Human Phenotypes,
PLOS Computational Biology, http://www.ploscompbiol.org/article/info%
3Adoi%2F10.1371%2Fjournal.pcbi.1000353).

To Internet
Sq ma: To dÐktuo twn ISPs tou Internet (Cheswick & Burch, Internet Atlas Gallery,
http://www.caida.org/projects/internetatlas/gallery/ches/data.xml).

To Facebook
Sq ma: To dÐktuo epikoinwnÐac tou Facebook (Baraba´ si et al., Network
Science Book, http://barabasilab.neu.edu/networksciencebook/).

DÐktuo Twitter
Sq ma: H gigantiaÐa sunektik sunist¸sa enìc diktÔou Twitter gia
kˆpoia RTs (retweets) pou èginan anaforikˆ me ta gegonìta diamarturÐac
sthn TourkÐa ton IoÔnio tou 2013.

DÐktuo LinkedIn
Sq ma: Oi koinoÐ – moirasmènoi (shared) – fÐloi gia duo qr stec tou
LinkedIn.

DÐktuo (Dia–)Kleidwmènwn DS Etairi¸n
(Interlocking Directorates)
Sq ma: DÐktuo (Dia–)Kleidwmènwn (Interlocking Directorates) Dioikhtik¸n SumboulÐwn Etairi¸n
apì koinèc summetoqèc steleq¸n se autèc
(http://orgtheory.wordpress.com/2011/08/19/theyrule-net-interlocking-boards/,
http://theyrule.net/).

To DÐktuo BioteqnologÐac–BiomhqanÐac stic
HPA
Sq ma: To dÐktuo twn sqèsewn BioteqnologÐac–BiomhqanÐac stic HPA (Powell, White, Koput &
Owen–Smith, Network Dynamics and Field Evolution: The Growth of Interorganizational Collaboration in the
Life Sciences, American Journal of Sociology, 2005, 110(4): 11321205,
http://eclectic.ss.uci.edu/~drwhite/Movie/).

DÐktuo Summetoq¸n se Koinwnikèc Ekdhl¸seic
Gunaik¸n tou Nìtou
Sq ma: DÐktuo summetoq¸n–maz¸xewn se 14 sumbˆnta koinwnik¸n
ekdhl¸sewn 18 gunaik¸n tou Amerikˆnikou Nìtou (Davis, Gardner & Gardner,
Deep Douth, University of Chicago Press, 1941).

DÐktuo Yhfofori¸n sto An¸tero Dikast rio
twn HPA
Sq ma: DÐktuo yhfofori¸n sto An¸tero Dikast rio twn HPA me tic suneqìmenec grammèc na
sumbolÐzoun jetikèc y fouc kai tic diakoptìmenec grammèc arnhtikèc y fouc (Mrvar & Doreian,
Partitioning signed twomode
networks, Journal of Mathematical Sociology, 2009, 33: 196221).

DÐktuo Epitrop¸n sthn Boul twn
Antipros¸pwn twn ARTICLE HPA
IN PRESS
BUDGET
VETERANS’ AFFAIRS
ARMED SERVICES
AGRICULTURE
420 M.A. Porter et al. / Physica A 386 (2007) 414–438
APPROPRIATIONS
INTELLIGENCE
HOUSE ADMINISTRATION
ENERGY/COMMERCE
OFFICIAL CONDUCT
HOMELAND SECURITY
GOVERNMENT REFORM
WAYS AND MEANS
INTERNATIONAL RELATIONS
TRANSPORTATION
SMALL BUSINESS
EDUCATION
SCIENCE
FINANCIAL SERVICES
RULES
RESOURCES
JUDICIARY
Sq ma: Fig. 4. (Color) Network of committees (squares) and subcommittees (circles) in the 108th US House of Representatives, color-coded by
the DÐktuo parent standing epitrop¸n and select committees. (tetrˆgwna(The depicted ) labels kai indicate upo–the epitrop¸n parent committee (of kÔkloieach group ) sthn but do not 108identify h Boul the
twn
Antipros¸pwn location twn of that HPA committee (Porter, in the plot.) Mucha, As with Fig. Newman 2, this visualization & Friend, was produced Community using a variant of the Kamada–Kawai spring
embedder, with link strengths (again indicated by darkness) determined by normalized interlocks. Observe structure again that subcommittees in the United of the
States
House of Representatives, same parent committee Physica are closely A, connected 2007, to each 386: other.
414438).
Oi sundèseic anaparistoÔn koinèc
summetoqèc bouleut¸n se (upo)epitropèc.
Security Committee shares only one common member (normalized interlock 2.4) with the Intelligence Select
Committee (located near the 1 o’clock position in Fig. 5) and has no interlock at all with any of the four

LÐga Endeiktikˆ ParadeÐgmata
Diktuak¸n Upologism¸n

BasikoÐ SumbolismoÐ JewrÐac Grˆfwn
'Enac grˆfoc G eÐnai èna zeÔgoc (V, E), ìpou to V eÐnai
èna sÔnolo koruf¸n ( kìmbwn shmeÐwn) kai to E
eÐnai èna sÔnolo akm¸n ( gramm¸n sundèsmwn
sundèsewn).
'Etsi, o grˆfoc grˆfetai wc G = (V, E) ki, ìtan qreiˆzetai na epishmˆnoume
ìti to V eÐnai to sÔnolo koruf¸n tou grˆfou G, grˆfoume V = V(G) kai,
parìmoia, ìti to E eÐnai to sÔnolo akm¸n tou G, grˆfoume E = E(G).
Kˆje akm e 2 E enìc grˆfou G = (V, E) antistoiqeÐ se duo korufèc tou
sunìlou V, oi opoÐec apoteloÔn ta duo ˆkra thc akm c. 'Otan ta ˆkra thc
akm c e 2 E eÐnai oi korufèc u kai v 2 V, grˆfoume e = (u, v). Shmeiwtèon ìti
oi akmèc den èqoun kateÔjunsh, dhlad , e = (u, v) = (v, u).
O grˆfoc G = (V, E) me V = fv1, v2, v3, v4g kai E = f(v1, v3), (v2, v3), (v3, v4)g:
a pair (V,E), where V is a set of vertices (also called points), and E called lines).
number of vertices is called the order of a graph and the number of edges is called graph.
graph G = (V,E) the vertex set V is often denoted V (G) and the edge set E e ∈ E is associated with a pair of points from V . If u and v are associated they are called the endpoints of e, we often write uv or {u, v} to represent 2
v1
v2 ❅
❅
v4
v3
❅
❅
V = {v1, Mwus c v2, Av3}, . MpountourÐdhc E = {{v3, H Epist mh v4}, {v2, twn v3}, DiktÔwn
{v1, v3}}

observe that G1 ⊕ G2 = G1 ∪ G2. However, usually for ring sums we have the same vertex
TÔpoi Grˆfwn
= V2, but different edge sets E1= E2, whereas for unions we often want disjoint unions, = ∅.
be aware that notations for these operations vary. In particular, some authors use G1 ∨ G2,
join and take G1 + G2 to be a disjoint union.
'Enac brìqoc (loop) eÐnai mia akm pou en¸nei mia koruf v me ton
eautì thc, e = (v, v).
Duo ( perissìterec) akmèc onomˆzontai parˆllhlec an ta ˆkra touc
eÐnai oi Ðdiec korufèc.
'Enac grˆfoc qwrÐc brìqouc kai qwrÐc parˆllhlec akmèc onomˆzetai
aplìc, en¸ diaforetikˆ onomˆzetai pollaplìc grˆfoc (multi–graph).
'Enac grˆfoc onomˆzetai grˆfoc me bˆrh (weighted graph) kai
sumbolÐzetai wc G = (V, E,w), an se kˆje akm tou e antistoiqeÐ èna
bˆroc mia tim w(e) 2 R.
Directed Graphs
Definition 10
directed graph or digraph is a pair (V,E), where V is a set of points (also called vertices),
and E is a set of ordered pairs of points from V called arcs.
'Enac kateujunìmenoc grˆfoc digrˆfoc G eÐnai èna zeÔgoc (V, E), ìpou to
V eÐnai èna sÔnolo koruf¸n ( kìmbwn shmeÐwn) kai to E eÐnai èna
sÔnolo tìxwn me to kˆje tìxo e 2 E na antistoiqeÐ se èna diatetagmèno
zeÔgoc koruf¸n (u, v) ètsi ¸ste na kateujÔnetai apì thn koruf u proc
thn koruf v.
O kateujunìmenoc grˆfoc G = (V, E) me V = fv1, v2, v3, v4g kai E = f(v1, v3), (v3, v2), (v4, v3)g:
Each arc e ∈ E is associated with an ordered pair of points from V . If u and v are associated
with the edge e they are called the endpoints of e, we often write uv or (u, v) to represent the
arc e.
Example 11
v1
❅ ✻
❅
v4
v2
v3
❅
❅❘
✲
V Mwus c = {v1, A. v2, MpountourÐdhc v3}, E = {(v4, H v3), Epist mh (v3, twn v2), DiktÔwn
(v1, v3)}

DimereÐc Grˆfoi
'Enac grˆfoc onomˆzetai dimer c
(bipartite), ìtan upˆrqei ènac
diamerismìc tou sunìlou twn
koruf¸n tou V se duo mèrh
(tm mata), to U kai to W, dhlad ,
V = U [W (ìpou U W = ?), ètsi
¸ste ìlec oi akmèc na phgaÐnoun
apì to U sto W kai na mhn
upˆrqei kamiˆ akm oÔte metaxÔ
koruf¸n tou U oÔte metaxÔ
koruf¸n tou W.
Probolèc dimeroÔc grˆfou:
u1
u3
u2
u4
u1
u2
u3
u4
w1
w2
w3 w2
w1
w3
1
2
2
1
1 2
1
2

BajmoÐ Kìmbwn
GeÐtonec: 'Estw o mh kateujunìmenoc grˆfoc G = (V, E) kai i, j 2 V
duo korufèc tou. H j lègetai geÐtonac thc i ìtan (i, j) 2 E.
PÐnakac GeitnÐashc (Adjacency Matrix): EÐnai ènac
(summetrikìc) pÐnakac A = fAgi,j2V tˆxhc jVj jVj tètoioc ¸ste
A = 1, ìtan i, j geÐtonec, A = 0, diaforetikˆ.
BajmoÐ: Sto mh kateujunìmeno grˆfo G, o bajmìc miac koruf c i,
pou sumbolÐzetai wc ki, orÐzetai san to pl joc twn geitìnwn tou i,
dhlad , to pl joc twn sundèsewn pou prospÐptoun sto i.
Profan¸c, isqÔei:
ki =
X
j2V
A =
X
i2V
A
ki, epiplèon,
X
i2V
ki =
X
i,j2V
A = 2jEj

'Estw t¸ra o kateujunìmenoc grˆfoc G = (V, E), gia ton opoÐon o
antÐstoiqoc pÐnakac geitnÐashc A = fAg eÐnai mh summetrikìc.
O bajmìc eisìdou thc koruf c i tou G, pou sumbolÐzetai wc kin i ,
orÐzetai san to pl joc twn sundèsewn pou xekinoÔn apì geÐtonec
tou i kai kateujÔnontai proc ton i, dhlad ,
kin
i =
X
j2V
A
O bajmìc exìdou thc koruf c i tou G, pou sumbolÐzetai wc kout i ,
orÐzetai san to pl joc twn sundèsewn pou xekinoÔn apì ton i kai
kateujÔnontai proc geÐtonec tou i, dhlad ,
kout
i =
X
i2V
A
Profan¸c, isqÔei:
X
i2V
kin
i =
X
j2V
kout
i =
X
i,j2V
A = jEj

BajmoÐ kìmbwn sto dÐktuo karˆte

GenikoÐ TÔpoi Diktuak¸n Katanom¸n Bajm¸n
Sq ma: Diwnumik Katanom ( Katano-
m Poisson) gia tuqaÐouc grˆfouc Erd¨os–Re´ nyi
Sq ma: Katanom Nìmou DÔnamhc (Power
Law) gia dÐktua qwrÐc klÐmaka (scale–free
networks)

Kentrikìthtec Kìmbwn:
1. Kentrikìthta BajmoÔ (Degree Centrality)
Oi orismoÐ OLWN twn kentrikìthtwn pou ja d¸soume
ed¸ kai sth sunèqeia aforoÔn mh kateujunìmenouc
(aploÔc) grˆfouc.
H kentrikìthta bajmoÔ (degree centrality) xi tou kìmbou
i isoÔtai proc ton bajmì ki tou kìmbou autoÔ:
xi = ki
x8 = 5

2. Kentrikìthta Endiamesìthtac
(Betweenness Centrality)
H kentrikìthta endiamesìthtac (betweenness centrality) xi tou kìmbou
i isoÔtai proc:
xi =
X
s6=i6=t2V
nist
gst
ìpou nist eÐnai to pl joc twn gewdaitik¸n diadrom¸n metaxÔ twn
kìmbwn s kai t, pou pernoÔn apì ton kìmbo i, kai gst eÐnai to sunolikì
pl joc twn gewdaitik¸n diadrom¸n metaxÔ twn kìmbwn s kai t.
n23
,23 = 2
g3,23 = 4
x2 = 0.1436

3. Kentrikìthta EggÔthtac
(Closeness Centrality)
Se ènan grˆfo G, gia kˆje duo kìmbouc i, j, h (gewdaitik ) apìstas
touc d(i, j) orÐzetai wc to m koc thc suntomìterhc diadrom c apì to i
sto j, efìson oi kìmboi autoÐ eÐnai sundedemènoi, en¸ d(i, j) = 1,
diaforetikˆ (kai fusikˆ, d(i, i) = 0). (H ‘‘suntomìterh diadrom ’’
metaxÔ duo kìmbwn eÐnai h diadrom pou èqei to elˆqisto m koc
anˆmesa se ìlec tic diadromèc metaxÔ twn duo kìmbwn.)
Se ènan grˆfo me n kìmbouc, h kentrikìthta eggÔthtac (closeness
centrality) xi tou kìmbou i isoÔtai proc:
xi =
n P
j2V d(i, j)
x0 = 0.5689
x2 = 0.5593
x33 = 0.55
x31 = 0.5409

4. Kentrikìthta IdiodianÔsmatoc
(Eigenvector Centrality)
H kentrikìthta idiodianÔsmatoc (eigenvector centrality) xi tou kìmbou i
isoÔtai proc:
xi = 1
1
X
j2V
Axj
ìpou A eÐnai o pÐnakac geitnÐashc (adjacency matrix) tou grˆfou kai
xi eÐnai oi sunist¸sec tou idiadianÔsmatoc tou A, pou antistoiqoÔn
sth megalÔterh idiotim tou 1.
x33 = 0.3734
x0 = 0.3555
x2 = 0.3172
x32 = 0.3086

Oi 4 kentrikìthtec twn kìmbwn tou diktÔou karˆte
Nodes Degree C. Betweenness C. Closeness C. Eigenvector C.
0 0.484848485 0.437635281 0.568965517 0.355490721
1 0.272727273 0.053936688 0.485294118 0.265959605
2 0.303030303 0.143656806 0.559322034 0.317192417
3 0.181818182 0.011909271 0.464788732 0.211179694
4 0.090909091 0.000631313 0.379310345 0.075968879
5 0.121212121 0.029987374 0.38372093 0.079483113
6 0.121212121 0.029987374 0.38372093 0.079483113
7 0.121212121 0 0.44 0.170959892
8 0.151515152 0.055926828 0.515625 0.227404355
9 0.060606061 0.000847763 0.434210526 0.102674504
10 0.090909091 0.000631313 0.379310345 0.075968879
11 0.03030303 0 0.366666667 0.052855817
12 0.060606061 0 0.370786517 0.084254727
13 0.151515152 0.045863396 0.515625 0.226473112
14 0.060606061 0 0.370786517 0.10140365
15 0.060606061 0 0.370786517 0.10140365
16 0.060606061 0 0.284482759 0.023635566
17 0.060606061 0 0.375 0.092399699
18 0.060606061 0 0.370786517 0.10140365
19 0.090909091 0.032475048 0.5 0.147912918
20 0.060606061 0 0.370786517 0.10140365
21 0.060606061 0 0.375 0.092399699
22 0.060606061 0 0.370786517 0.10140365
23 0.151515152 0.017613636 0.392857143 0.150118912
24 0.090909091 0.002209596 0.375 0.057052326
25 0.090909091 0.003840488 0.375 0.059206342
26 0.060606061 0 0.362637363 0.075579616
27 0.121212121 0.022333454 0.458333333 0.133477386
28 0.090909091 0.001794733 0.452054795 0.131077964
29 0.121212121 0.002922078 0.38372093 0.134961122
30 0.121212121 0.014411977 0.458333333 0.174758637
31 0.181818182 0.138275613 0.540983607 0.191034394
32 0.363636364 0.145247114 0.515625 0.308643749
33 0.515151515 0.304074976 0.55 0.373362539

Kentrikìthta Mikr tim Megˆlh tim
Degree LÐgoi geÐtonec (sundèseic) PolloÐ geÐtonec (sundèseic)
Betweenness Mikrìc èlegqoc ro c Megˆloc èlegqoc ro c
Closeness Proc thn perifèreia Proc to kèntro
Eigenvector LÐgoi lÐgo shmantikoÐ geÐtonec PolloÐ polÔ shmantikoÐ geÐtonec
To dÐktuo twn stratiwtik¸n tou David Krackhardt:

Suntelest c Suss¸reushc
O suntelest c suss¸reushc (clustering coefficient) Ci tou kìmbou i
orÐzetai wc:
Ci =
2i
ki(ki 1)
ìpou i eÐnai to pl joc twn sundèsewn metaxÔ twn geitonik¸n
kìmbwn tou i kai d i proc opoiod pote ˆllo kìmbo ki eÐnai to
pl joc twn geitonik¸n kìmbwn tou i.
C23 =
2 4
5 4 = 0.4

Diktuak Metabatikìthta
O sunolikìc suntelest c suss¸reushc (global clustering coefficient)
(ìlou) tou grˆfou G orÐzetai wc h mèsh tim twn suntelest¸n
suss¸reushc twn kìmbwn tou:
C(G) =
1
jVj
X
i
Ci
H metabatikìthta (transitivity) tou grˆfou G orÐzetai wc to phlÐko:
T(G) =
pl joc trig¸nwn
pl joc sundedemènwn triˆdwn
C(G) = 0.16
T(G) = 0.19

Amoibaiìthta Sundèsewn se Kateujunìmeno Grˆfo
Se ènan kateujunìmeno grˆfo, o suntelest c amoibaiìthtac
sundèsewn/desm¸n (link/tie mutuality coefficient) orÐzetai wc ex c:
M(G) =
pl joc antapodidìmenwn sundèsewn Er
pl joc ìlwn twn sundèsewn/tìxwn E
Er(G) = 64
E(G) = 195
M(G) = 0.3282

Apostˆseic Kìmbwn se Grˆfo
SumbolÐzontac me d(i, j) th (gewdaitik ) apìstash ston grˆfo G
metaxÔ twn duo kìmbwn i, j (dhlad , wc m koc thc suntomìterhc
diadrom c apì to i sto j, efìson oi kìmboi autoÐ eÐnai sundedemènoi,
en¸ d(i, j) = 1, ìpou h ‘‘suntomìterh diadrom ’’ metaxÔ duo kìmbwn
eÐnai h diadrom pou èqei to elˆqisto m koc anˆmesa se ìlec tic
diadromèc metaxÔ twn duo kìmbwn), to mèso m koc twn suntomìterwn
diadrom¸n (average shortest path length) ston grˆfo autì, orÐzetai wc:
a =
1
jVj(jVj 1)
X
i,j2V
d(i, j)
a = 2.4082

Mètra Diˆforwn Empeirik¸n DiktÔwn TABLE I. The general characteristics of several real networks. For each network we indicated the number of nodes, the average
degree !k, the average path length ! and the clustering coefficient C. For a comparison we have included the average path
length !rand and clustering coefficient Crand of a random graph with the same size and average degree. The last column
identifies the symbols in Figs. 8 and 9.
Network Size !k ! !rand C Crand Reference Nr.
WWW, site level, undir. 153, 127 35.21 3.1 3.35 0.1078 0.00023 Adamic 1999 Internet, domain level 3015 - 6209 3.52 - 4.11 3.7 - 3.76 6.36 - 6.18 0.18 - 0.3 0.001 Yook et al. 2001a,
Pastor-Satorras et al. 2001 Movie actors 225, 226 61 3.65 2.99 0.79 0.00027 Watts, Strogatz 1998 LANL coauthorship 52, 909 9.7 5.9 4.79 0.43 1.8 × 10−4 Newman 2001a,b MEDLINE coauthorship 1, 520, 251 18.1 4.6 4.91 0.066 1.1 × 10−5 Newman 2001a,b SPIRES coauthorship 56, 627 173 4.0 2.12 0.726 0.003 Newman 2001a,b,c NCSTRL coauthorship 11, 994 3.59 9.7 7.34 0.496 3 × 10−4 Newman 2001a,b Math coauthorship 70, 975 3.9 9.5 8.2 0.59 5.4 × 10−5 Barabási et al. 2001 Neurosci. coauthorship 209, 293 11.5 6 5.01 0.76 5.5 × 10−5 Barabási et al. 2001 E. coli, substrate graph 282 7.35 2.9 3.04 0.32 0.026 Wagner, Fell 2000 10
E. coli, reaction graph 315 28.3 2.62 1.98 0.59 0.09 Wagner, Fell 2000 11
Ythan estuary food web 134 8.7 2.43 2.26 0.22 0.06 Montoya, Solé 2000 12
Silwood park food web 154 4.75 3.40 3.23 0.15 0.03 Montoya, Solé 2000 13
Words, cooccurence 460.902 70.13 2.67 3.03 0.437 0.0001 Cancho, Solé 2001 14
Words, synonyms 22, 311 13.48 4.5 3.84 0.7 0.0006 Yook et al. 2001 15
Power grid 4, 941 2.67 18.7 12.4 0.08 0.005 Watts, Strogatz 1998 16
C. Elegans 282 14 2.65 2.25 0.28 0.05 Watts, Strogatz 1998 17
TABLE II. The scaling exponents characterizing the degree distribution of several scale-free networks, for which P(k) follows
a power-law (2). We indicate the size of the network, its average degree !k and the cutoff for the power-law scaling. For
directed networks we list separately the indegree (#in) and outdegree (#out) exponents, while for the undirected networks,
marked with a star, these values are identical. The columns lreal , lrand and lpow compare the average path length of real
networks with power-law degree distribution and the prediction of random graph theory (17) and that of Newman, Strogatz
and Watts (2000) (62), as discussed in Sect. V. The last column identifies the symbols in Figs. 8 and 9.

EndeiktikoÐ Pìroi

Pìroi gia Eisagwg sta Koinwnikˆ DÐktua
1 BiblÐa
Mark Newman, Networks: An Introduction, Oxford
University Press, 2010.
Stanley Wasserman and Katherine Faust, Social Network
Analysis: Methods and Applications, Cambridge
University Press, 1994.
2 'Arjra Episkìphshc
Mark Newman, The structure and function of complex
networks: http://arxiv.org/pdf/cond-mat/0303516v1
Laszlo Bara´ basi et al., Network Science Book:
http://barabasilab.neu.edu/networksciencebook/
Robert A. Hanneman and Mark Riddle, Introduction to
social network methods:
http://faculty.ucr.edu/~hanneman/nettext/

3 Logismikì
NetworkX: http://networkx.github.io/
Gephi: http://gephi.github.io/
Pajek: http://pajek.imfm.si/doku.php
UCInet:
https://sites.google.com/site/ucinetsoftware/home
iGraph: http://igraph.sourceforge.net/

Η Επιστήμη των Δικτύων: Μια Πολύ Σύντομη Εισαγωγική Παρουσίαση - 8 Οκτωβρίου 2014

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