# Derived networks

⌘ ⊕ ⊙ ∈ ⋂ ⋃ ⋀ ⋁ ⊂ ⊆∖ ∑ ∏ ∃ ∀ ⇒ → ∞ ✕ ≤ ≥ ℝ ℕ ℤ ∙ ✶ ∅ ★ ⌻ ⊚ ⊛ √ ← ● ✿ Δ

## Temporal co-occurrence networks

Let the binary matrix A=[aep] describe a two-mode network on the set of events E and the set of of participants P:

aep = 1 if p participated in the event e ; and 0 otherwise.

The function d: E → T assigns to each event e the date d(e) when it happened. T = [first, last]. Using these data we can construct two temporal affiliation matrices:

• instantaneous: Ai=[aiep], where aiep = [(d(e),d(e)+1,1)] if aep = 1 ; and [] otherwise
• cumulative: Ac=[acep], where acep = [(d(e),last+1,1)] if aep = 1 ; and [] otherwise

Using the multiplication of temporal matrices over the combinatorial semiring we get the corresponding instantaneous and cumulative co-occurrence matrices Ci = AiTAi and Cc = AcTAc .

A typical example of such a matrix is the papers authorship matrix where E is the set of papers, P is the set of authors and d is the publication year \citep{bibnet}.

Temporal collaboration.

ci['IDI/B','HCL/B']              cc['IDI/B','HCL/B']
1 : (2003, 2004, 1)              1 : (2003, 2004, 1)
2 : (2004, 2005, 2)              2 : (2004, 2005, 3)
3 : (2005, 2006, 3)              3 : (2005, 2006, 6)
4 : (2006, 2007, 2)              4 : (2006, 2007, 8)
5 : (2007, 2008, 1)              5 : (2007, 2008, 9)
6 : (2008, 2009, 7)              6 : (2008, 2009, 16)
7 : (2009, 2010, 6)              7 : (2009, 2010, 22)
8 : (2010, 2011, 7)              8 : (2010, 2011, 29)
9 : (2011, 2013, 18)             9 : (2011, 2012, 47)
10 : (2012, 2013, 65)

ci['HCL/B','HCL/B']              cc['HCL/B','HCL/B']
1 : (1997, 1998, 2)              1 : (1997, 1998, 2)
2 : (1998, 1999, 5)              2 : (1998, 1999, 7)
3 : (1999, 2000, 8)              3 : (1999, 2000, 15)
4 : (2000, 2001, 7)              4 : (2000, 2001, 22)
5 : (2001, 2002, 5)              5 : (2001, 2002, 27)
6 : (2002, 2003, 6)              6 : (2002, 2003, 33)
7 : (2003, 2004, 14)             7 : (2003, 2004, 47)
8 : (2004, 2005, 20)             8 : (2004, 2005, 67)
9 : (2005, 2006, 10)             9 : (2005, 2006, 77)
10 : (2006, 2007, 14)            10 : (2006, 2007, 91)
11 : (2007, 2008, 20)            11 : (2007, 2008, 111)
12 : (2008, 2009, 28)            12 : (2008, 2009, 139)
13 : (2009, 2010, 56)            13 : (2009, 2010, 195)
14 : (2010, 2011, 78)            14 : (2010, 2011, 273)
15 : (2011, 2012, 84)            15 : (2011, 2012, 357)
16 : (2012, 2013, 112)           16 : (2012, 2013, 469)

The triple (s,f,v) in a temporal quantity cipq tells that in the time interval [s,f) there were v events in which both p and q took part.

The triple (s,f,v) in a temporal quantity ccpq tells that in the time interval [s,f) there were in total v accumulated events in which both p and q took part.

The diagonal matrix entries cipp and ccpp contain the temporal quantities counting the number of events in the time intervals in which the participant p took part.

For example, in a data set on the stem cell research during 1997–2012 in Spain collected by Gisela Cantos-Mateos \citep{stem} we get from the basic two-mode network, where E is the set of papers and P is the set of institutions, for selected two institutions (HCL/B = University Hospital Clínic de Barcelona, Barcelona and IDI/B = Institut d'Investigacions Biomèdiques August Pi i Sunyer, Barcelona) the collaboration temporal quantities presented in Table~\ref{tec}.

The first column in the table contains the yearly collaboration (co-authorship) data and the second column contains the cumulative collaboration data. Let's read the table:

ci[IDI/B, HCL/B](2005, 2006) = 3 - in the year 2005 researchers from both institutions published 3 joint papers;

ci[IDI/B, HCL/B](2011, 2013) = 18 - in the years 2011 and 2012 researchers from both institutions published 18 joint papers each year;

ci[HCL/B, HCL/B](2010, 2011) = 78 - in the year 2010 researchers from the institution HCL/B published 78 papers;

cc[IDI/B, HCL/B](2008, 2009) = 16 - till the year 2008 (included) researchers from both institutions published 16 joint papers.

Note that the violence network from Section~\ref{activ} is essentially a co-occurrence network that could be obtained from the more primitive instantaneous two-mode network about violent actions reported in journal articles and the involved political actors.

>>> print('start WIT:',datetime.datetime.now().ctime())
start WIT: Tue Jun 23 10:05:52 2015
>>> print('Network WIT')
Network WIT
>>> WIT = TQ.Ianus2Mat("./gisela/WIT.ten")
>>> print('input done:',datetime.datetime.now().ctime())
input done: Tue Jun 23 10:06:55 2015
>>> print('sum =',TQ.MatSummary(WIT['mat']))
sum = (1997, 2013, 1, 1)
>>> print(WIT.keys())
dict_keys(['met', 'tit', 'til', 'typ', 'tin', 'mat', 'dim', 'nam'])
>>> print('dim =',WIT['dim'])
dim = (5839, 577, 1997, 2013)
>>> print('met =',WIT['met'])
met = re
au Gisela Cantos Mateos
ti Spanish "Stem cell" research (1997-2012)
de Gisela collected from WoS data about Spanish “Stem cell” research (1997-2012).
She manually unified the names of institutions in Excel and produced the CSV
files with tables: (Documents,Institutions), (Documents,KeyWordsPlus) and
(Documents,PublicationYear).
dt October, 2014
er
re
ti transformation to Ianus format
dt October 28, 2014
de The data in CSV format were transformed into Ianus format.
er
>>> print('nam =',WIT['nam'][0:10])
nam = ['5', '8', '9', '15', '30', '43', '44', '45', '46', '47']
>>> print('tit =',WIT['tit'])
tit = papersXinstitutions
>>> print('tin =',WIT['tin'][0:10])
tin = [[(2007, 2013, 1)], [(2007, 2013, 1)], [(2007, 2013, 1)], [(2007, 2013, 1)], [(2007, 2013, 1)],
[(2007, 2013, 1)], [(2007, 2013, 1)], [(2007, 2013, 1)], [(2007, 2013, 1)], [(2007, 2013, 1)]]
>>> print('til =',WIT['til'][0:10])
til = []
>>> print('transpose:',datetime.datetime.now().ctime())
transpose: Tue Jun 23 10:09:50 2015
>>> IWTmat = TQ.MatTrans(WIT['mat'])
>>> print('done:',datetime.datetime.now().ctime())
done: Tue Jun 23 10:10:26 2015
>>> print('multiply:',datetime.datetime.now().ctime())
multiply: Tue Jun 23 10:11:48 2015

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