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"Centrality
Bavelas  Leavitt experiments
effects of communication structure on performance
one factor: distance from integrator
Intuitive notions of centrality and centralization
Freeman's review & distillation
center of star is the ultimate in centrality
Centralization
extent to which graph revolves around a single individual
sum of differences between centrality of most central node and centrality of all others, divided by maximum possible.
Degree
definition:
number of nodes adjacent to given node
number of edges incident upon
ci = sumj(aij)
normalization
probability of transmission
indegree and outdegree
Bonacich Eigenvector centrality
trouble with degree: two people could have same degree, but one person knows only people who know hardly anyone, while the other knows people who know lots of people
solution is iterated degree: a person's centrality is equal to the sum of the degrees of the people they are connected to:
ci = (xijdj
iterative version: let ci(0) = 1; ci(1) = (xijcj(0); ci(2) = (xijcj(1);
but why stop with two iterations?
ci(k) = (xijcj(k1)
ci(k) increases as k increases, but ratios of the values tend to stabilize, so that ci(100) = ci(99)
so at some point ci = (xijcj or in matrix form c = Xc
c is an eigenvector of X and lambda is an eigenvalue
cc' = X or xij = cicj : the principal eigenvectorof a matrix is that vector which when multiplied by itself, gives the best possible estimate of the matrix values: its a model of the matrix.
valued data
directed data
righteigenvector: c = Xc
lefteigenvector: c = cX (prestige)
in a symmetric matrix cleft = transpose of cright
to sum up, degree is how many people you know; eigenvector is how many people you know who are in the know
Closeness
definition:
total distance of given node from all others (same as average
sumj(dij)
inverse measure
need connected graph
speed of transmission
incloseness and outcloseness
Betweenness
definition:
loosely: # of times someone needs given node to get to someone else. measured as the number of paths that pass through a given point.
more accurately: # of shortest paths, and we make an adjustment for the existence of alternative shortest paths between pairs of points. if two paths, we count each as 1
gij = # of geodesics from i to j; gikj # of geodesics from i to j via k; ck = sumi sumj {gijk/gij}
can interpret inner term as the probability that signal passes through k on way from i to j
control of transmission: broker, gatekeeper, liaison
Pitts article
based on geodesics only.
Flow betweenness
proportion reduction in maximum flow if a given node is removed from the network.
Influence
katz
a person influences those they are connected to. these in turn influence those whom they are connected to, including the first person, who reinfluences those he is connected to. positive feedback loop.
chains of influence, long or short, are just walks.
total influence on others is the total number walks of any length from a given node to all others.
assume we can measure i's influence on j. maybe 1/0 scale: does or doesn't. One way to do it is ask "who knows what he's talking about in this organization?" then take the transpose of that, because if i mentions j as being knowledgeable, then j has influence over i. So R = A'
# of walks for length k from each person to each other is given by Rk, so total influence of i on j is W = R + R2 + R3 + ... + RK + ....R( and the total influence of i on all others then, is the row sums of W. (C = W1)
two problems:
weights long walks as much as short walks (in fact more since there are more long walks than short ones).
the series doesn't converge. numbers go to infinity.
solution:
introduce an attenuation factor : assume that influence declines with distance. i.e. weight the walks inversely by their length. W = R + 2R2 + 3R3 + ... + kRk
if < 1 then as k approaches infinity, k approaches 0.
Conveniently, it happens that (I  R)1 = 0R0 + 1R1 + 2R2 + 3R3 + ... + kRk as long as abs(1/) > (dominant eigenvalue of R) or < 1/
So W = (I  R)1  I and C is again the row sums of W
[(I  R)1  I]e = (I  R)1e  Ie = (I  R)1e  1 = row sums of (I  R)1 minus 1.
think of as probability that i persuades j
in the example, simple degree gives (40, 20, 20, 60, 20, 80) but katz gives (47, 4, 4, 41, 22, 45). node one has low degree but is chosen by the two nodes that everyone else chooses.
node 1 is the expert that is not visible to the masses but is known by the cognoscenti
if we choose = 1/, then we have 1/1.6837716 which is 0.5939048. Result is {61, 0, 0, 50, 29, 54}  which is the column eigenvector of matrix
if we choose
hard to pick without empirical evidence. and if we choose arbitrarily, can get numbers that don't work: can't invert the matrix.
helpful fact: <= greatest row or column sum. and is only equal to that bound when all rows (or columns) have equal sums.
so can always pick equal to 1/greatest_row_sum
incidentally, if we were to make the rows or columns stochastic (add to 1), the eigenvalue would be 1, and this series would converge if is just less than 1.
status versus influence: rowsums or column sums?
hubbell
Hubbell wanted a measure in which the status of an actor is a function of the status of the people it is connected to, plus a factor that was just due to the person herself ci = xijcj + ei or in matrix terms, c = Xc + e
we can solve for C as follows: c = Xc + e; c  Xc = e; (I  X)c = e; c = (I  X)1e
suppose e = column vector of 1s. then Hubbell's C is just row marginals of (I  X)1. Let X = R. Then C = row marginals of (I  R)1. almost like katz. in fact Hubbell = Katz + 1.
But if E is not vector of 1s, then Katz and Hubbell can be different.
if E is degree of X (e = X1) then katz = hubbell:
h = Xh + X1; h  Xh = X1; (I  X)h = X1; h = (I  X)1X1 = [X0 + X1 ...]X1 = X0X1 + X1X1 + X2X1 ... = X11 + X21 + X31.. = [X1 + X2 + X3...]1 = KATZ
Bonacich revisited
suppose e is vector of zeros. then ci = xijcj or c = Xc. when can this occur? when lambda equal 1, or when X = R and < 1/. c = 1/Rc or c = Rc. so in this case hubbell = bonacich.
Power
in general, intuitively, we expect centrality to be related to power. Lots of empirical evidence. but the relationship clearly differs by context
Cook et al
created series of lab experiments in which pairs of actors must strike a deal with each other. have to divide up 24 points. can only make deals with people they are connected to, which is controlled by experimenter.
objective is to reveal the structural determinants of power in exchange networks. had two basic theoretical ideas going in: emerson dependency theory and network centrality
emerson: a has power over b if b depends on a more than a depends on b
b depends on a to the extent that (1) there are few alternatives to A available to B, and (2) B really needs whatever A has got
in a star graph, center has many options, periphery has few
in a 5pt line, center three have two options, but only B and D connected to those with no options, so C is weak
in exhcnage experiments, (2) is a constant and can be disregarded
centrality: degree relates to dependency; closeness: hard to relate to power in these experiments; betweenness works to the extent that we consider only paths of length two.
in star graph, everything predicts same, and agrees with experimental results
in 5pt line, dependency predicts middle is weak, but centrality predicts middle is strong. empirical result is middle is weak.
in order to save graphtheoretic orientation, try to develop new centrality measure called vulnerability = reduction in maximum flow.
is glimmerings of notion of excludability, developed by markovsky
is based on specific process (power in exchange networks) rather than general principles.
Markovsky et al
gpi = # of vertexdisjoint paths of odd length minus # of even length paths
they add on some additional assumptions: i seeks exchange with j only if p(i) > p(j) or p(i)  p(j) > p(i)  p(k) for all k
excludability
Bonacich power
Purely graphtheoretic measures versus theoretical measures tied to specific substantive arena
DISPLAY
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Input dataset: C:\DATA\KATZ
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1 0 0 0 0 0 1
2 0 0 1 0 0 1
3 0 1 0 1 0 1
4 1 0 0 0 1 0
5 0 0 0 1 0 1
6 1 0 0 1 0 0
Indegree:
2 1 1 3 1 4
INFLUENCE
(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((
Attenuation factor: 0.5000000
Method: KATZ
Input dataset: C:\DATA\KATZ
1 2 3 4 5 6
1 1.00 0.00 0.00 0.80 0.40 1.20
2 2.67 0.33 0.67 2.40 1.20 2.93
3 3.33 0.67 0.33 3.20 1.60 3.47
4 2.00 0.00 0.00 1.40 1.20 1.60
5 2.00 0.00 0.00 2.00 1.00 2.00
6 2.00 0.00 0.00 1.60 0.80 1.40
Descriptive Statistics
1 2 3 4 5 6
1 Mean 2.17 0.17 0.17 1.90 1.03 2.10
2 Std Dev 0.71 0.25 0.25 0.76 0.37 0.83
3 Sum 13.00 1.00 1.00 11.40 6.20 12.60
4 Variance 0.51 0.06 0.06 0.58 0.14 0.69
5 Euc Norm 5.59 0.75 0.75 5.02 2.69 5.53
6 Minimum 1.00 0.00 0.00 0.80 0.40 1.20
7 Maximum 3.33 0.67 0.67 3.20 1.60 3.47
8 N of Obs 6.00 6.00 6.00 6.00 6.00 6.00
Descriptive Statistics
1 2 3 4 5 6 7 8
MeanStd De SumVarianEuc NoMinimuMaximuN of O
1 0.57 0.47 3.40 0.22 1.80 0.00 1.20 6.00
2 1.70 1.01 10.20 1.02 4.84 0.33 2.93 6.00
3 2.10 1.29 12.60 1.67 6.04 0.33 3.47 6.00
4 1.03 0.77 6.20 0.59 3.16 0.00 2.00 6.00
5 1.17 0.90 7.00 0.81 3.61 0.00 2.00 6.00
6 0.97 0.77 5.80 0.59 3.03 0.00 2.00 6.00
EIGENVECTOR ANALYSIS
Eigenvalue = 1.6837716
Column Eigenvector:
1 .61439642
2 0.00000000
3 0.00000000
4 .49500917
5 .29398832
6 .53949405
Row Eigenvector:
1 .30276
2 1.04830
3 1.25532
4 0.55559
5 0.63273
6 0.50978
INFLUENCE
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Attenuation factor: 0.5939048
Method: KATZ
Input dataset: C:\DATA\KATZ
1 2 3 4 5 6
1 112660288.00 0.00 0.00 90768560.00 53907884.00 98925640.00
2 390083616.00 0.54 0.92 314284000.00 186654784.00 342527712.00
3 467117504.00 0.92 0.54 376348960.00 223515456.00 410170240.00
4 206741392.00 0.00 0.00 166568176.00 98925632.00 181537120.00
5 235444992.00 0.00 0.00 189694192.00 112660288.00 206741392.00
6 189694192.00 0.00 0.00 152833520.00 90768560.00 166568176.00
Descriptive Statistics COLUMN MEANS ONLY
266956992 0.24 0.24 215082896 127738768 234411712
Descriptive Statistics ROW MEANS ONLY
1 59377064.00
2 205591680.00
3 246192032.00
4 108962056.00
5 124090144.00
6 99977408.00
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