Ανάλυση Δικτύων με το NetworkX της Python: Μια προκαταρκτική (αλλά ημιτελής ως το καλοκαίρι 2014) εισαγωγή

Anˆlush DiktÔwn
me to NetworkX thc Python
Mwus c A. MpountourÐdhc
Tm ma Majhmatik¸n PanepisthmÐou Pˆtrac
mboudour@upatras.gr
22 NoembrÐou 2012
Mwus c A. MpountourÐdhc Anˆlush DiktÔwn me to NetworkX thc Python

PÐnakac Perieqomènwn
1 Python kai NetworkX
2 Eisagwg Grˆfwn
Mh Kateujunìmenoi Grˆfoi
Kateujunìmenoi Grˆfoi
Grˆfoi me Bˆrh Akm¸n
DimereÐc Grˆfoi
Qarakthristikˆ (Attributes) Kìmbwn kai Akm¸n
TÔpoi Grˆfwn kai Sqesiak¸n Dedomènwn
Genn torec Prokataskeuasmènwn Grˆfwn
3 Diktuakˆ Mètra
BajmoÐ Kìmbwn
Kentrikìthtec Kìmbwn
Suntelest c Suss¸reushc kai Metabatikìthta
Amoibaiìthta
Apostˆseic

4 DiktuakoÐ DiamerismoÐ
Sunist¸sec kai KlÐkec
k–Pur nec (k–Cores)
Blockmodeling
Koinìthtec (Communities)
5 Diktuakˆ Montèla
Anameiximìthta (Assortativity)
Epirro kai Diˆqush
DÐktua Mikr¸n Kìsmwn (Small–Worlds)
DÐktua QwrÐc KlÐmaka (Scale–Free)

Python kai NetworkX
H Python eÐnai mia ermhneutik gl¸ssa, dhlad ,
ekteleÐtai qwrÐc na qreiˆzetai na perˆsei apì thn
diadikasÐa thc sÔntaxhc.
H Python diatÐjetai gia katèbasma (mazÐ me odhgÐec
egkatˆstashc) gia ìla ta leitourgikˆ sust mata
apì to http://www.python.org/download/. IdiaÐtera
gia ta Windows sunistˆtai h ActivePython.
To NetworkX eÐnai èna pakèto thc Python gia thn
dhmiourgÐa kai ton qeirismì grˆfwn kai diktÔwn.
To pakèto, h tekmhrÐwsh ki ˆlloi pìroi tou
NetworkX eÐnai diajèsimoi sto
http://networkx.lanl.gov/, ìpwc kai sthn apoj kh
pakètwn thc Python PyPI.

Mia apeujeÐac (one–click) egkatˆstash thc Python,
tou NetworkX ki ˆllwn monˆdwn thc Python mporeÐ
na gÐnei apì thn dianom thc Python Enthought,
pou dÐnetai dwreˆn gia akadhmaðk qr sh.
Pollèc qr simec plhroforÐec gia thn egkatˆstash
twn Python, NetworkX ki ˆllwn aparaÐthtwn
upologistik¸n pìrwn dÐnontai sth selÐda twn
Resources tou maj matoc Comp 200 tou
PanepisthmÐou tou Rice.
'Oloi oi k¸dikec twn diafanei¸n ed¸ èqoun
dokimasjeÐ pˆnw sthn èkdosh 2.7.3 thc Python kai
thn èkdosh 1.7 tou NetworkX.

Sto prooÐmio thc graf c (script) ìlwn twn
ekteloÔmenwn ed¸ kwdÐkwn thc Python ja prèpei
na eisˆgontai oi ex c grammèc:
import networkx as nx
import matplotlib . pyplot as plt
import pylab
Sthn antigraf ki epikìllhsh twn entol¸n pou
dÐnontai sth sunèqeia prèpei na proseqjeÐ h
stoÐqish stic allagèc gramm¸n, giatÐ h Python
eÐnai euaÐsjhth wc proc thn autìmath topojèthsh
esoq¸n (indentation).

Eisagwg Grˆfwn
Pr¸ta, dÐnontai oi entolèc gia mh kateujunìmenouc
grˆfouc.
Arqikˆ, dhmiourgeÐtai ènac kenìc grˆfoc:
G = nx. Graph ()
Metˆ, eisˆgontai arqikˆ oi kìmboi, p.q., oi 5
kìmboi '1', '2', '3', '4' kai '5' mpaÐnoun wc ex c:
G. add_nodes_from ([1,2,3,4,5])
Bèbaia, antÐ gia arijmoÔc, oi kìmboi mporoÔn na
eisˆgontai wc onìmata lèxeic, p.q.:
G. add_nodes_from ([ 'John ', 'Mary '])
G. add_node (" London ")
G. add_nodes_from ([ 'a','b','c','d','e'])

Tèloc, eisˆgontai oi akmèc, p.q., gia touc 5
kìmbouc 1–5 na 6 sundèseic:
G. add_edges_from ([(1,2),(1,4),(2,3),(3,4),(3,5),(4,5 )])
Oi lÐstec kai to pl joc twn eisaqjèntwn kìmbwn
kai akm¸n dÐnontai apì tic entolèc:
G. nodes ()
G. number_of_nodes ()
G. edges ()
G. number_of_edges ()

Gia to sqediasmì tou eisaqjèntoc grˆfou, p.q,
tou paradeÐgmatoc autoÔ tou grˆfou me 5
kìmbouc kai 6 akmèc, ekteloÔntai oi entolèc:
plt . figure ()
nx. draw (G)
plt . show ( block = False )
H teleutaÐa entol dèqetai diˆforec paramètrouc,
ìpwc, p.q., gia diaforetikì tÔpo sqedÐou (layout),
mègejoc kai qr¸ma kìmbwn kai afaÐresh twn
onomˆtwn (ids) twn kìmbwn:
nx. draw_spring (G, node_size =100 , node_color ='# A0CBE2 ',
with_labels = False )

ParaleÐpontac parametropoi seic, o sqediasmìc
tou grˆfou tou paradeÐgmatoc dÐnei to sq ma:

Oi kìmboi mporoÔn na topotethjoÔn se stajerèc jèseic,
efìson o grˆfoc eÐqe eisaqjeÐ me tic suntetagmènec
touc, pou mporeÐ na gÐnei, p.q., sto parˆdeigma autì
wc ex c:
pos ={1:(0,0),2:(1,0),4:(0,1),3:(1,1),5:(0.5,2.0)}
nx. draw (G, pos )

Eisagwg Kateujunìmenwn Grˆfwn
GÐnetai wc ex c, p.q., sto parˆdeigma autì:
G=nx. DiGraph ()
G. add_nodes_from (["A","B","C","D"])
G. add_edges_from ([( "A","B"), ("C","A"), ("C","B"),
("B","D")])

Eisagwg Grˆfwn me Bˆrh Akm¸n
GÐnetai, p.q., sto parˆdeigma autì me tic entolèc:
G=nx. Graph ()
G. add_weighted_edges_from ([( 'a','b',4),( 'a','c',8),
('a','d',5),( 'c','d',3)])
kai o sqediasmìc wc ex c:
pos =nx. spring_layout (G)
edge_labels = dict ([((u,v ,),d['weight '])
for u,v,d in G. edges ( data = True )])
nx. draw_networkx_nodes (G,pos , node_size = 700)
nx. draw_networkx_edges (G, pos)
nx. draw_networkx_labels (G,pos , font_size =20)
nx. draw_networkx_edge_labels (G,pos ,
edge_labels = edge_labels , font_size =20)

AntÐ thc parˆjeshc twn tim¸n twn bar¸n, oi akmèc mporoÔn na
sqediasjoÔn me eÔrh anˆloga twn tim¸n twn bar¸n wc ex c:
nx. draw_networkx_nodes (G,pos , node_size = 700)
edgewidth =[]
for (u,v,d) in G. edges ( data = True ):
edgewidth . append (d['weight '])
nx. draw_networkx_edges (G,pos , edge_color ='b',
width = edgewidth )
nx. draw_networkx_labels (G,pos , font_size =20)

Epiplèon, mporoÔn na krathjoÔn sto sqèdio mìno oi akmèc me bˆrh
megalÔtera kˆpoiac endiˆmeshc tim c, p.q., 4:
elarge = [(u,v) for (u,v,d) in G. edges ( data = True )
if d['weight '] >4]
esmall = [(u,v) for (u,v,d) in G. edges ( data = True )
if d['weight '] <=4]
nx. draw_networkx_edges (G,pos , edgelist = elarge ,
edge_color ='b',width = edgewidth )
nx. draw_networkx_edges (G,pos , edgelist = esmall , width =6,
alpha =0.5, edge_color ='g',style ='dashed ')

Eisagwg Dimer¸n Grˆfwn
H eisagwg dimer¸n grˆfwn gÐnetai ìpwc, p.q.,
sto parakˆtw parˆdeigma:
from networkx . algorithms import bipartite
G = nx. Graph ()
G. add_nodes_from ([1,2,3,4], bipartite =0)
G. add_nodes_from ([ 'a','b','c'], bipartite =1)
G. add_edges_from ([(1,'a'),(1,'b'),(2,'a'),(2,'b'),
(2,'c'),(3,'c'),(4,'b'),(4,'c')])
pos ={1:(0,0),
2:(0,1),
3:(0,2),
4:(0,3),
'a':(1,0.5),
'b':(1,1.5),
'c':(1,2.5)}
O èlegqoc an prìkeitai perÐ dimeroÔc grˆfou
(True) ìqi (False) gÐnetai me thn entol :
print bipartite . is_bipartite (G)
True

O sqediasmìc tou dimeroÔc grˆfou gÐnetai wc
ex c:
mode1 , mode2 = bipartite . sets (G)
nx. draw_networkx_nodes (G,pos , nodelist = list ( mode1 ),
node_color ='b',node_size = 700)
nx. draw_networkx_nodes (G,pos , nodelist = list ( mode2 ),
node_color ='g',node_size = 700)
nx. draw_networkx_edges (G, pos)
nx. draw_networkx_labels (G,pos , font_size =20 ,
font_color ='# FFFFFF ')
plt . axis ('off ')

Qarakthristikˆ (Attributes) Kìmbwn kai Akm¸n

TÔpoi Grˆfwn kai Sqesiak¸n Dedomènwn

Genn torec Prokataskeuasmènwn Grˆfwn

BajmoÐ Kìmbwn
Pr¸ta, gia mh kateujunìmenouc grˆfouc: O grˆfoc
G = (V, E) sto sÔnolo koruf¸n V eÐnai mh kateujunìmenoc, ìtan to
sÔnolo twn akm¸n tou E eÐnai èna summetrikì uposÔnolo tou V V.
GeÐtonec kìmbwn: Gia duo korufèc i, j 2 V tou G, h j lègetai
geÐtonac thc i ìtan (i, j) 2 E.
G. nodes () # List of nodes
G. number_of_nodes () # Number of nodes
len (G) # Number of nodes
G. order () # Number of nodes
G. number_of_edges () # Number of of edges
G. neighbors (1) # Neighbors of node 1
## To get the list of neighbors of all nodes :
for node in G. nodes ():
print node , G. neighbors ( node )
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.
A = nx. to_numpy_matrix (G)

BajmoÐ kìmbwn: 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
G. degree (1) # Degree of node 1
len (G. neighbors (1) # Degree of node 1
G. degree () # List of degrees of all nodes
## To get the table of degrees of all nodes :
print node , G. degree ( node )

BajmoÐ Kìmbwn Kateujunìmenwn Grˆfwn
'Estw o kateujunìmenoc grˆfoc G = (V, E) sto sÔnolo koruf¸n V,
ìpou t¸ra to sÔnolo twn akm¸n tou E eÐnai èna mh summetrikì
uposÔnolo tou V V ki 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
G. in_degree (1) #In - Degree of node 1
G. out_degree (1) #Out - Degree of node 1
G. in_degree () # List of in - degrees of all nodes
G. out_degree () # List of out - degrees of all nodes
## To get the table of in - degrees and out - degrees of all nodes :
print node , G. in_degree ( node ), G. out_degree ( node )

Katanomèc Bajm¸n
Istìgramma katanom c bajm¸n (gia mh kateujunìmeno grˆfo):
degrees = G. degree ()
values = sorted ( set ( degrees . values ()))
hist = [ degrees . values (). count (x) for x in values ]
fig = plt . figure ()
ax = fig . add_subplot ( 111 )
plt . plot ( values ,hist ,'ro -')
#ax . set_xscale (' log ') # Logarithmic x scale
#ax . set_yscale (' log ') # Logarithmic y scale
plt . title ( Degree Distribution )
plt . xlabel ('Degree ')
plt . ylabel ('Number of nodes ')
Istìgramma katanom c bajm¸n (gia kateujunìmeno grˆfo):
## in_degrees = G. in_degree ()
in_values = sorted ( set ( in_degrees . values ()))
in_hist = [ in_degrees . values (). count (x) for x in in_values ]
out_degrees = G. out_degree ()
out_values = sorted ( set ( out_degrees . values ()))
out_hist = [ out_degrees . values (). count (x) for x in out_values ]
fig = plt . figure ()
ax = fig . add_subplot ( 111 )
plt . plot ( in_values , in_hist ,'ro -')
plt . plot ( out_values , out_hist ,'bv -')
#ax . set_xscale (' log ') # Logarithmic x scale
#ax . set_yscale (' log ') # Logarithmic y scale
plt . titke (In - Degree Out - Degree Distributions )
plt . xlabel ('Degree ')
plt . ylabel ('Number of nodes ')

Parˆdeigma: To dÐktuo karˆte
G = nx. karate_club_graph ()

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)
H kentrikìthta eggÔthtac (closeness centrality) xi tou kìmbou i
isoÔtai proc:
xi =
n P
j2V d
ìpou n eÐnai to pl joc twn kìmbwn tou grˆfou kai d h gewdaitik
apìstash apì ton kìmbo i proc opoiod pote ˆllo kìmbo 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

Upologismìc Kentrikìthtwn
Entolèc NetworkX:
nx. degree_centrality (G)
nx. betweenness_centrality (G)
nx. closeness_centrality (G)
nx. eigenvector_centrality (G)
EktÔpwsh se pÐnaka:
centralities = []
centralities . append (nx. degree_centrality (G ));
centralities . append (nx. betweenness_centrality (G ));
centralities . append (nx. closeness_centrality (G));
centralities . append (nx. eigenvector_centrality (G ));
for node in G. nodes_iter ():
measures = (t). join ( map ( lambda f: str(f[ node ]),
centralities ))
print %s %s % (node , measures )

Parˆdeigma: To dÐktuo Karˆte
G = nx. karate_club_graph ()
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.
nx. clustering (G) # Clustering coefficients of all nodes
nx. clustering (G,n) # Clustering coefficient of node n
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
nx. average_clustering (G)
# Graph clustering coefficient
nx. transitivity (G)
# Graph transitivity
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
pl joc ìlwn twn sundèsewn/tìxwn
G. to_undirected ( True ). size ()
# Reciprocated edges
G. number_of_edges ()
# Total edges
Er(G) = 64
E(G) = 195
M(G) = 0.3282

Apostˆseic
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.)
To mèso m koc suntomìterhc diadrom c (average shortest path length)
orÐzetai wc ex c:
a =
1
jVj(jVj 1)
X
i,j2V
d(i, j)
a = nx. average_shortest_path_length (G)
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.

Sunist¸sec kai KlÐkec

Blockmodeling

Koinìthtec (Communities)

Anameiximìthta (Assortativity)

Epirro kai Diˆqush

DÐktua Mikr¸n Kìsmwn (Small–Worlds)

DÐktua QwrÐc KlÐmaka (Scale–Free)

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