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Temporal networks (with 0 transition time)

Temporal networks (with 0 transition time)

About semirings

John S. Baras, George Theodorakopoulos: Path Problems in Networks. (2010)

Slides from Sredin seminar 1220

Notes from Sredin seminar 1226 (in Slovene)

timer.zip ZIP with data and programs

Format of temporal networks files

project

Infinity in Python

>>> inf = float("inf")
>>> inf
inf
>>> inf+3
inf
>>> inf*inf
inf

⌘

In the following I present an implementation (March 10, 2014) of operations on temporal quantities restricted to the case of intervals of type 2 = [). In the examples I will only use the integer time values.

Addition and multiplication of temporal quantities

inf = float("inf")
a = [(1,5,2), (6,8,1), (11,12,3), (14,16,2), (17,18,5), (19,20,1)]
b = [(2,3,4), (4,7,3), (9,10,2), (13,15,5), (16,21,1)]
N = []
E = [(1,inf,1)]

def get(S):
   try: return(next(S))
   except StopIteration: return((inf,inf,0))

def plus(a,b): return(a+b) 
def times(a,b): return(a*b) 

def standard(a):
   if len(a) == 0: return(a)
   fc = inf; c = []
   for (sa,fa,va) in a:
      if fc == sa:
         if vc == va: fc = fa
         else:
            c.append((sc,fc,vc))
            vc = va; sc = sa; fc = fa
      else:
         if fc != inf: c.append((sc,fc,vc))
         sc = sa; fc = fa; vc = va
   c.append((sc,fc,vc))
   return(c)

def sum(a,b):
   if len(a) == 0: return(b)
   if len(b) == 0: return(a)
   c = []; A = a.__iter__(); B = b.__iter__()
   (sa,fa,va) = get(A); (sb,fb,vb) = get(B)
   while (sa<inf) or (sb<inf):
      if sa < sb:
         sc = sa; vc = va
         if sb < fa: fc = sb; sa = sb
         else: fc = fa; (sa,fa,va) = get(A)        
         c.append((sc,fc,vc))
      elif sa == sb:
         sc = sa; fc = min(fa,fb); vc = plus(va,vb)
         c.append((sc,fc,vc))
         sa = sb = fc; fA = fa
         if fA <= fb: (sa,fa,va) = get(A)
         if fb <= fA: (sb,fb,vb) = get(B)
      else:
         sc = sb; vc = vb
         if sa < fb: fc = sa; sb = sa
         else: fc = fb; (sb,fb,vb) = get(B)        
         c.append((sc,fc,vc))
   return(standard(c))

def prod(a,b):
   if len(a)*len(b) == 0: return([])
   c = []; A = a.__iter__(); B = b.__iter__()
   (sa,fa,va) = get(A); (sb,fb,vb) = get(B)
   while (sa<inf) or (sb<inf):
      if fa <= sb: (sa,fa,va) = get(A)
      elif fb <= sa: (sb,fb,vb) = get(B)
      else:
         sc = max(sa,sb); fc = min(fa,fb); vc = times(va,vb)
         c.append((sc,fc,vc))
         if fc == fa: (sa,fa,va) = get(A)
         if fc == fb: (sb,fb,vb) = get(B)          
   return(standard(c))         
         
print("a =",a); print("b =",b)
d = sum(a,b); print("d =",d)
e = prod(a,b); print("e =",e)

Results

plus = + ; times = * 
>>> 
a = [(1, 5, 2), (6, 8, 1), (11, 12, 3), (14, 16, 2), (17, 18, 5), (19, 20, 1)]
b = [(2, 3, 4), (4, 7, 3), (9, 10, 2), (13, 15, 5), (16, 21, 1)]
s = [(1, 2, 2), (2, 3, 6), (3, 4, 2), (4, 5, 5), (5, 6, 3), (6, 7, 4), (7, 8, 1), (9, 10, 2),
     (11, 12, 3), (13, 14, 5), (14, 15, 7), (15, 16, 2), (16, 17, 1), (17, 18, 6), (18, 19, 1),
     (19, 20, 2), (20, 21, 1)]
p = [(2, 3, 8), (4, 5, 6), (6, 7, 3), (14, 15, 10), (17, 18, 5), (19, 20, 1)]
>>> sum(a,N)
[(1, 5, 2), (6, 8, 1), (11, 12, 3), (14, 16, 2), (17, 18, 5), (19, 20, 1)]
>>> sum(N,N)
[]
>>> prod(a,N)
[]
>>> prod(N,N)
[]
>>> prod(a,E)
[(1, 5, 2), (6, 8, 1), (11, 12, 3), (14, 16, 2), (17, 18, 5), (19, 20, 1)]
>>> prod(E,E)
[(1, inf, 1)]
>>> print("a =",a); print("b =",b)
a = [(1, 5, 2), (6, 8, 1), (11, 12, 3), (14, 16, 2), (17, 18, 5), (19, 20, 1)]
b = [(2, 3, 4), (4, 7, 3), (9, 10, 2), (13, 15, 5), (16, 21, 1)]
>>> d = sum(a,b); print("d =",d)
d = [(1, 2, 2), (2, 3, 6), (3, 4, 2), (4, 5, 5), (5, 6, 3), (6, 7, 4), (7, 8, 1),
     (9, 10, 2), (11, 12, 3), (13, 14, 5), (14, 15, 7), (15, 16, 2), (16, 17, 1),
     (17, 18, 6), (18, 19, 1), (19, 20, 2), (20, 21, 1)]
>>> e = prod(a,b); print("e =",e)
e = [(2, 3, 8), (4, 5, 6), (6, 7, 3), (14, 15, 10), (17, 18, 5), (19, 20, 1)]

a:

b:

a+b:

a*b:

Connectivity semiring ({0,1},∨,∧,0,1).

def binary(a):
   c = []
   for (sa,fa,va) in a:
      if va > 0: c.append((sa,fa,1))
   return(c)

ba = binary(a); bb = binary(b)
def plus(a,b): return(max(a,b))    # OR
def times(a,b): return(min(a,b))   # AND

print("ba =",ba); print("bb =",bb)
d = sum(ba,bb); print("d =",d)
e = prod(ba,bb); print("e =",e)
print("E+ba =",sum(E,ba))
print("E+E =",sum(E,E))

we get

ba = [(1, 5, 1), (6, 8, 1), (11, 12, 1), (14, 16, 1), (17, 18, 1), (19, 20, 1)]
bb = [(2, 3, 1), (4, 7, 1), (9, 10, 1), (13, 15, 1), (16, 21, 1)]
d = [(1, 8, 1), (9, 10, 1), (11, 12, 1), (13, 21, 1)]
e = [(2, 3, 1), (4, 5, 1), (6, 7, 1), (14, 15, 1), (17, 18, 1), (19, 20, 1)]
E+ba = [(1, inf, 1)]
E+E = [(1, inf, 1)]

Shortest path semiring (ℝ⁺₀,min,+,∞,0).

plus = min ; times = +
>>> 
def plus(a,b): return(min(a,b)) 
def times(a,b): return(a+b)
E = [(1,inf,0)]

print("a =",a); print("b =",b)
d = sum(a,b); print("d =",d)
e = prod(a,b); print("e =",e)
print("E+a =",sum(E,a))
print("E+E =",sum(E,E))
>>> 
a = [(1, 5, 2), (6, 8, 1), (11, 12, 3), (14, 16, 2), (17, 18, 5), (19, 20, 1)]
b = [(2, 3, 4), (4, 7, 3), (9, 10, 2), (13, 15, 5), (16, 21, 1)]
d = [(1, 5, 2), (5, 6, 3), (6, 8, 1), (9, 10, 2), (11, 12, 3), (13, 14, 5), (14, 16, 2), (16, 21, 1)]
e = [(2, 3, 6), (4, 5, 5), (6, 7, 4), (14, 15, 7), (17, 18, 6), (19, 20, 2)]
E+a = [(1, inf, 0)]
E+E = [(1, inf, 0)]

In this semiring the absorption law holds: E + a = E .

Closure

Let (A,+,.,0,1) be a semiring. Then also the structure (A(T),⊕,⊙,N,E) of temporal quantities over (A,+,.,0,1) is a semiring; and so is the structure of square matrices (M_A(T),⊞,⊡,0,1) over (A(T),⊕,⊙,N,E).

N = [], E =[(1,inf,1)]

If the (A,+,.,0,1) is complete, so is the matrix semiring.

In a complete semiring we can compute the closure C = R^* of matrix R using the Fletcher's algorithm:

def closure(R):
   n = len(R); C = R
   for k in range(n):
      for u in range(n):
         for v in range(n): 
            C[u][v] = sum(C[u][v], prod(prod(C[u][k],star(C[k][k])),C[k][v]))
      C[k][k] = sum(E,C[k][k])
   return(C)

where star(a) is the procedure implementing the a^* operation. In the case of absorptive semiring a^* = 1, and the procedure can be simplified to:

def closure(R):
   n = len(R); C = R
   for k in range(n):
      for u in range(n):
         for v in range(n): 
            C[u][v] = sum(C[u][v], prod(C[u][k],C[k][v]))
      C[k][k] = sum(E,C[k][k])
   return(C)

def plus(a,b): return(min(a,b)) 
def times(a,b): return(a+b)
E = [(1,inf,0)]

W = [[ [] for i in range(6)] for j in range(6)]
W[0][1] = [(3,8,1), (8,9,5)]
W[4][0] = [(3,8,1), (8,9,5)]
W[0][1] = [(1,8,2), (9,10,4)]
W[1][2] = [(2,7,7), (8,10,3)]
W[1][5] = [(4,6,4), (8,10,5)]
W[4][5] = [(1,5,6), (5,7,3), (10,10,5)]
W[5][2] = [(4,8,1), (8,9,5)]
W[2][3] = [(1,5,5), (5,9,7)]
W[3][4] = [(1,3,4), (3,10,2)]

def listMatrix(M):
   n = len(M)
   for i in range(n):
      for j in range(n):
         print('(',i+1,',',j+1,') =',M[i][j])

print("matrix"); listMatrix(W)

C = closure(W)

print("closure"); listMatrix(C)

we get

matrix
( 1 , 2 ) = [(1, 8, 2), (9, 10, 4)]
( 2 , 3 ) = [(2, 7, 7), (8, 10, 3)]
( 2 , 6 ) = [(4, 6, 4), (8, 10, 5)]
( 3 , 4 ) = [(1, 5, 5), (5, 9, 7)]
( 4 , 5 ) = [(1, 3, 4), (3, 10, 2)]
( 5 , 1 ) = [(3, 8, 1), (8, 9, 5)]
( 5 , 6 ) = [(1, 5, 6), (5, 7, 3), (10, 10, 5)]
( 6 , 3 ) = [(4, 8, 1), (8, 9, 5)]

closure
( 1 , 1 ) = [(1, inf, 0)]
( 1 , 2 ) = [(1, 8, 2), (9, 10, 4)]
( 1 , 3 ) = [(2, 4, 9), (4, 6, 7), (6, 7, 9), (9, 10, 7)]
( 1 , 4 ) = [(2, 4, 14), (4, 5, 12), (5, 6, 14), (6, 7, 16)]
( 1 , 5 ) = [(2, 3, 18), (3, 4, 16), (4, 5, 14), (5, 6, 16), (6, 7, 18)]
( 1 , 6 ) = [(2, 3, 24), (3, 4, 22), (4, 6, 6), (6, 7, 21), (9, 10, 9)]
( 2 , 1 ) = [(3, 4, 15), (4, 5, 13), (5, 6, 15), (6, 7, 17), (8, 9, 17)]
( 2 , 2 ) = [(1, inf, 0)]
( 2 , 3 ) = [(2, 4, 7), (4, 6, 5), (6, 7, 7), (8, 10, 3)]
( 2 , 4 ) = [(2, 4, 12), (4, 5, 10), (5, 6, 12), (6, 7, 14), (8, 9, 10)]
( 2 , 5 ) = [(2, 3, 16), (3, 4, 14), (4, 5, 12), (5, 6, 14), (6, 7, 16), (8, 9, 12)]
( 2 , 6 ) = [(2, 3, 22), (3, 4, 20), (4, 6, 4), (6, 7, 19), (8, 10, 5)]
( 3 , 1 ) = [(3, 5, 8), (5, 8, 10), (8, 9, 14)]
( 3 , 2 ) = [(3, 5, 10), (5, 8, 12)]
( 3 , 3 ) = [(1, inf, 0)]
( 3 , 4 ) = [(1, 5, 5), (5, 9, 7)]
( 3 , 5 ) = [(1, 3, 9), (3, 5, 7), (5, 9, 9)]
( 3 , 6 ) = [(1, 3, 15), (3, 5, 13), (5, 7, 12)]
( 4 , 1 ) = [(3, 8, 3), (8, 9, 7)]
( 4 , 2 ) = [(3, 8, 5)]
( 4 , 3 ) = [(3, 4, 12), (4, 5, 9), (5, 7, 6)]
( 4 , 4 ) = [(1, inf, 0)]
( 4 , 5 ) = [(1, 3, 4), (3, 10, 2)]
( 4 , 6 ) = [(1, 3, 10), (3, 5, 8), (5, 7, 5)]
( 5 , 1 ) = [(3, 8, 1), (8, 9, 5)]
( 5 , 2 ) = [(3, 8, 3)]
( 5 , 3 ) = [(3, 4, 10), (4, 5, 7), (5, 7, 4)]
( 5 , 4 ) = [(3, 4, 15), (4, 5, 12), (5, 7, 11)]
( 5 , 5 ) = [(1, inf, 0)]
( 5 , 6 ) = [(1, 5, 6), (5, 7, 3), (10, 10, 5)]
( 6 , 1 ) = [(4, 5, 9), (5, 8, 11), (8, 9, 19)]
( 6 , 2 ) = [(4, 5, 11), (5, 8, 13)]
( 6 , 3 ) = [(4, 8, 1), (8, 9, 5)]
( 6 , 4 ) = [(4, 5, 6), (5, 8, 8), (8, 9, 12)]
( 6 , 5 ) = [(4, 5, 8), (5, 8, 10), (8, 9, 14)]
( 6 , 6 ) = [(1, inf, 0)]

Temporal centrality measures

The closure matrix can be used to compute selected temporal centrality measures. For an example let us consider the measure

C(v) = ∑ {u ∈ V∖{v} : 1 / d(v,u)} / (n-1)

Assuming that for u≠v, d(v,u)≥1, the largest possible value of the sum in C(v) is n-1, attained on the central vertex of star S_n or vertices of complete graph K_n. This gives the normalization factor 1/(n-1). Therefore C(v) ∈ [0,1].

def invert(a):
   c = []
   for (sa,fa,va) in a:
      c.append((sa,fa,1/va))
   return(c)

def central(v,C):
   cc = c = []; n = len(C)
   for t in range(n):
      if t!=v:
         c = sum(c,invert(C[v][t]))         
   for (s,f,v) in c: cc.append((s,f,v/(n-1)))
   return(cc)

ce = list(range(6))
for v in range(6): ce[v]=central(v,C)
for v in range(6): print(v+1,':',ce[v])

we get (reals were manually rounded at 4th decimal):

1 : [(1, 2, 0.1), (2, 3, 0.0083), (3, 4, 0.0091), (4, 5, 0.0143), (5, 6, 0.0125), (6, 7, 0.0095), (7, 8, 0.1),
     (9, 10, 0.0222)]
2 : [(2, 3, 0.0091), (3, 4, 0.01), (4, 5, 0.0154), (5, 6, 0.0133), (6, 7, 0.0105), (8, 9, 0.0118), (9, 10, 0.04)]
3 : [(1, 3, 0.0133), (3, 5, 0.0154), (5, 8, 0.0167), (8, 9, 0.0143)]
4 : [(1, 3, 0.02), (3, 4, 0.0167), (4, 5, 0.0222), (5, 7, 0.0333), (7, 8, 0.04), (8, 9, 0.0286), (9, 10, 0.1)]
5 : [(1, 3, 0.0333), (3, 4, 0.0133), (4, 5, 0.0167), (5, 7, 0.0182), (7, 8, 0.0667), (8, 9, 0.04), (10, 10, 0.04)]
6 : [(4, 5, 0.0182), (5, 8, 0.0154), (8, 9, 0.0105)]

Examples

def binMat(A):
   nr = len(A); nc = len(A[0])
   B = [[ [] for i in range(nc)] for j in range(nr)]
   for u in range(nr):
      for v in range(nc):
         B[u][v] = binary(A[u][v])
   return(B) 

def sumSyM(C,Rows,Cols,diag=False):
# extend for Rows and Cols are temporal partitions
   n = len(C); s = []
   for u in Rows:
      for v in Cols:
         if u>v: s = sum(s,C[v][u]) 
         elif u<v: s = sum(s,C[u][v]) 
         elif diag: s = sum(s,C[v][u])
   return(s)

def summary(a):
   vMin = tMin = inf; vMax = tMax = -inf
   for (sa,fa,va) in a:
      if sa < tMin: tMin = sa
      if fa > tMax: tMax = fa
      if va < vMin: vMin = va
      elif va > vMax: vMax = va
   return((tMin,tMax,vMin,vMax))

Drawing the temporal quantities

Test data

Small test data

Franzosi's Italy 1919-1922

import os, re, pickle, json
from timeR import *

os.chdir('C:/Users/batagelj/work/Python/temporal/violence')

with open("violence.p","rb") as p: V = pickle.load(p)
W = V['mat']
n = len(W); All = range(n)
names = V['nam']
times = V['tim']
# listMatrix(W,names,False)
for i,name in enumerate(names): print(i,name)

# All-All
ca = sumSyM(W,All,All)
# police - 3
cb = sumSyM(W,[3],All)
# fascists - 6
cc = sumSyM(W,[6],All)

comp = {}
comp['All-All']    = ca
comp['Pol-All']    = cb
comp['Fas-All']    = cc
with open('violComp.json',mode='w',encoding='utf-8') as f:
   json.dump(comp,f,indent=2)

Change to R for drawing.

> setwd("C:/Users/batagelj/work/Python/WoS/SN5/time")
> library(rjson)
> source("C:\\Users\\batagelj\\work\\Python\\WoS\\SN5\\time\\plotTime.R")
> C <- fromJSON(file="violComp.json")
> V <- fromJSON(file="violence.json")
> names(C)
[1] "Pol-All" "All-All" "Fas-All"
> names(V)
[1] "nam" "mat" "num" "tim" "tit"
> lTimes <- V$tim; lTimes <- c(lTimes,"Jan-23")
> tMin <- 1;  tMax <- length(lTimes)
> times <- c(1,7,13,19,25,31,37,43,49)
> pol <- C[[1]]; wMax <- 200
> plotTime(pol,tMin,tMax,times,lTimes,wMax,h=250,main="Police",
  file="violPol.pdf",PDF=TRUE)
> fas <- C[[3]]
> plotTime(fas,tMin,tMax,times,lTimes,wMax,h=250,main="Fascists",
  file="violFas.pdf",PDF=TRUE,col="blue") 
> all <- C[[2]]; wMax <- 700
> plotTime(all,tMin,tMax,times,lTimes,wMax,h=250,main="All-All",
  file="violAll.pdf",PDF=TRUE,col="orange")

violpol.pdf - police to all

violfas.pdf - fascists to all

violall.pdf - all to all

Collaboration networks

Transforming WoS2Pajek data into temporal network

See the pictures in Drawing the temporal quantities.

SN5

( COHEN_S , COHEN_S ) =
   [(16, 17, 0.4444), (28, 29, 0.1111), (29, 30, 1.0816), (33, 34, 1.0625), (34, 35, 0.1111),
    (35, 36, 1.0), (36, 37, 0.0556), (37, 38, 0.1927), (38, 39, 0.5)]
( BORGATTI_S , BORGATTI_S ) =
   [(19, 20, 1.1111), (20, 21, 0.6944), (21, 22, 0.6736), (22, 23, 0.1736), (23, 24, 2.8889),
    (24, 25, 0.5556), (25, 26, 0.6667), (27, 28, 0.25), (28, 29, 0.5069), (29, 30, 0.0400),
    (30, 31, 0.6667), (32, 33, 0.1025), (33, 34, 0.0625), (34, 35, 0.9914), (36, 37, 1.5556),
    (37, 38, 0.5), (38, 39, 0.1033)]
( BORGATTI_S , CARLEY_K ) =
   [(37, 38, 0.125)]
( BORGATTI_S , FREEMAN_L ) =
   [(22, 23, 0.125)]
( BORGATTI_S , WHITE_D ) =
   [(22, 23, 0.0625), (25, 26, 0.2222)]
( BORGATTI_S , KRACKHAR_D ) =
   [(37, 38, 0.125)]
( BORGATTI_S , JOHNSON_J ) =
   [(34, 35, 0.04)]
( BORGATTI_S , EVERETT_M ) =
   [(19, 20, 0.2222), (20, 21, 0.3472), (21, 22, 0.7222), (22, 23, 0.2222), (23, 26, 0.4444),
    (27, 28, 0.25), (28, 29, 0.2222), (30, 31, 0.6666), (34, 35, 0.04), (36, 37, 0.2222),
    (37, 38, 0.25)]
( BORGATTI_S , BOYD_J ) =
   [(20, 21, 0.125), (21, 22, 0.0625)]
( CARLEY_K , CARLEY_K ) =
   [(22, 23, 0.4444), (24, 25, 0.3611), (25, 26, 0.1267), (26, 27, 0.0625), (27, 28, 2.0),
    (28, 29, 0.4444), (30, 31, 1.3472), (31, 32, 0.04), (33, 34, 1.0), (34, 35, 0.0625),
    (35, 36, 0.3611), (37, 38, 0.4392), (38, 39, 0.5625)]
...
( HOLLAND_P , LEINHARD_S ) =
   [(7, 8, 0.2222), (8, 9, 0.4444), (9, 10, 0.2222), (12, 13, 0.4444), (14, 15, 0.125)]
...
( BERNARD_H , KILLWORT_P ) =
   [(7, 8, 0.2222), (9, 11, 0.2222), (11, 13, 0.125), (13, 14, 0.375), (18, 19, 0.125),
    (21, 22, 0.0964), (22, 23, 0.08), (26, 27, 0.0556), (28, 29, 0.0278), (29, 30, 0.1111),
    (32, 33, 0.0278), (34, 35, 0.0408), (37, 38, 0.2083)]
...
( BATAGELJ_V , BATAGELJ_V ) =
   [(13, 15, 0.1111), (23, 24, 0.75), (25, 26, 1.0625), (28, 30, 1.0), (30, 31, 0.3125),
    (31, 32, 0.5069), (32, 34, 0.4444), (35, 36, 0.1736), (36, 37, 0.4444), (38, 39, 0.2504)]
...
( BATAGELJ_V , FERLIGOJ_A ) =
   [(13, 15, 0.2222), (23, 24, 0.5), (25, 26, 0.0625), (31, 32, 0.0625), (35, 36, 0.1736),
    (38, 39, 0.0004)]
...
( DOREIAN_P , FERLIGOJ_A ) =
   [(23, 24, 0.125), (25, 26, 0.125), (31, 32, 0.125), (35, 36, 0.2361), (38, 39, 0.0004)]

---------------------------

18.3.2014
-------------------------------------------------------------------------------------
JSON

> library(rjson)
> T <- fromJSON(file="test.json")
> names(T)
[1] "Cn"  "tim" "num" "nam" "tit" "WAn"
> T$nam
[1] "A" "B" "C" "D" "E"
> T$num
[1] 5
>
> T$tit
[1] "small text example"
> T$tim
[1] "2000" "2001" "2002" "2003"
> T$Cn
[[1]]
[[1]][[1]]
[[1]][[1]][[1]]
[1] 1.00 2.00 0.25

[[1]][[1]][[2]]
[1] 2.0000 3.0000 0.2178

...

> T$Cn[[5]][[5]]
[[1]]
[1] 2.0000 3.0000 0.1089

[[2]]
[1] 3.00 4.00 0.25

>

# Batagelj, Doreian, Ferligoj
>>> BDF = [45,52,80]
>>> sumSyM(C,BDF,BDF)
[(1982, 1984, 0.4444), (1992, 1993, 1.75), (1994, 1995, 0.625), (2000, 2001, 0.625),
 (2004, 2005, 1.2917), (2007, 2008, 0.0025)]
>>> sumSyM(C,BDF,BDF,True)
[(1970, 1971, 2), (1974, 1975, 1.0), (1980, 1982, 1.0), (1982, 1983, 2), (1983, 1984, 1),
 (1984, 1985, 0.1111), (1985, 1986, 3.4444), (1986, 1987, 4.0), (1987, 1988, 2.0),
 (1988, 1989, 5.0), (1989, 1990, 1.1111), (1990, 1991, 0.1736), (1992, 1993, 3.4444),
 (1994, 1995, 4.5556), (1995, 1996, 0.1111), (1996, 1997, 1.8544), (1997, 1999, 1.0),
 (1999, 2000, 0.7569), (2000, 2001, 1.4444), (2001, 2002, 2), (2002, 2003, 1.4844),
 (2003, 2004, 0.1111), (2004, 2005, 2.3681), (2005, 2006, 0.5555), (2006, 2007, 2.0),
 (2007, 2008, 0.2537)]

# Borgatti, Everett
>>> BE = [1, 79]
>>> sumSyM(C,BE,BE)
[(1988, 1989, 0.4444), (1989, 1990, 0.6944), (1990, 1991, 1.4444), (1991, 1992, 0.4444),
 (1992, 1995, 0.8888), (1996, 1997, 0.5), (1997, 1998, 0.4444), (1999, 2000, 1.3333),
 (2003, 2004, 0.08), (2005, 2006, 0.4444), (2006, 2007, 0.5)]
# Physicist: Strogatz, Watts, Newman, Holme, Park, Barabasi, Albert
>>> Fiz = [46,19,10,18,42,12,17]
>>> sumSyM(C,Fiz,Fiz)
[(1999, 2000, 1.8333), (2000, 2001, 1.2744), (2001, 2002, 0.2222), (2002, 2003, 0.9444),
 (2003, 2005, 0.8889), (2005, 2006, 0.9444), (2006, 2007, 0.6667), (2007, 2008, 0.5)]

# Doreian with All
>>> n = len(C); All = range(n)
>>> sumSyM(C,[52],All)
[(1984, 1986, 0.2222), (1989, 1990, 0.2222), (1990, 1991, 0.4097), (1992, 1993, 0.375),
 (1994, 1995, 0.25), (1996, 1997, 0.33), (2000, 2001, 0.25), (2001, 2002, 0.2222),
 (2003, 2004, 0.2222), (2004, 2005, 0.4722), (2007, 2008, 0.0033)]
# All with All
>>> cf = sumSyM(C,All,All)
>>> cf
[(1976, 1977, 1.1389), (1977, 1978, 1.0139), (1978, 1979, 1.4583), (1979, 1980, 0.4444),
 (1980, 1981, 0.25), (1981, 1982, 1.5833), (1982, 1983, 1.1944), (1983, 1984, 1.1389),
 (1984, 1985, 1.3333), (1985, 1986, 0.5556), (1986, 1987, 0.8889), (1987, 1988, 0.6944),
 (1988, 1989, 1.3333), (1989, 1990, 1.5139), (1990, 1991, 4.0652), (1991, 1992, 2.4694),
 (1992, 1993, 4.0972), (1993, 1994, 5.0694), (1994, 1995, 2.9201), (1995, 1996, 2.7483),
 (1996, 1997, 3.1183), (1997, 1998, 2.0694), (1998, 1999, 2.4705), (1999, 2000, 8.6219),
 (2000, 2001, 4.4839), (2001, 2002, 3.3511), (2002, 2003, 5.0107), (2003, 2004, 3.2219),
 (2004, 2005, 3.7156), (2005, 2006, 2.4583), (2006, 2007, 3.9417), (2007, 2008, 2.9678)]
>>>