Download Assortativity and Mixing Complex Networks CSYS/MATH 303, Spring, 2011 Prof. Peter Dodds

Assortativity and
Mixing
Assortativity and Mixing
Definition
Complex Networks
CSYS/MATH 303, Spring, 2011
Assortativity by
degree
General mixing
Contagion
References
Prof. Peter Dodds
Department of Mathematics & Statistics
Center for Complex Systems
Vermont Advanced Computing Center
University of Vermont
Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.
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Outline
Assortativity and
Mixing
Definition
General mixing
Definition
Assortativity by
degree
Contagion
References
General mixing
Assortativity by degree
Contagion
References
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Basic idea:
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Random networks with arbitrary degree distributions
cover much territory but do not represent all
networks.
Moving away from pure random networks was a key
first step.
We can extend in many other directions and a
natural one is to introduce correlations between
different kinds of nodes.
Node attributes may be anything, e.g.:
Assortativity and
Mixing
Definition
General mixing
Assortativity by
degree
Contagion
References
1. degree
2. demographics (age, gender, etc.)
3. group affiliation
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We speak of mixing patterns, correlations, biases...
Networks are still random at base but now have more
global structure.
Build on work by Newman [4, 5] , and Boguñá and
Serano. [1] .
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General mixing between node categories
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Assume types of nodes are countable, and are
assigned numbers 1, 2, 3, . . . .
Consider networks with directed edges.
an edge connects a node of type µ
eµν = Pr
to a node of type ν
Assortativity and
Mixing
Definition
General mixing
Assortativity by
degree
Contagion
References
aµ = Pr(an edge comes from a node of type µ)
bν = Pr(an edge leads to a node of type ν)
I
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Write E = [eµν ], ~a = [aµ ], and ~b = [bν ].
Requirements:
X
X
X
eµν = 1,
eµν = aµ , and
eµν = bν .
µν
ν
µ
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Notes:
Assortativity and
Mixing
Definition
General mixing
I
Varying eµν allows us to move between the following:
1. Perfectly assortative networks where nodes only
connect to like nodes, and the network breaks into
subnetworks.
P
Requires eµν = 0 if µ 6= ν and µ eµµ = 1.
2. Uncorrelated networks (as we have studied so far)
For these we must have independence: eµν = aµ bν .
3. Disassortative networks where nodes connect to
nodes distinct from themselves.
I
Disassortative networks can be hard to build and
may require constraints on the eµν .
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Basic story: level of assortativity reflects the degree
to which nodes are connected to nodes within their
group.
Assortativity by
degree
Contagion
References
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Correlation coefficient:
Assortativity and
Mixing
Definition
General mixing
I
Quantify the level of assortativity with the following
assortativity coefficient [5] :
P
P
aµ bµ
Tr E − ||E 2 ||1
µ eµµ −
P µ
r=
=
1 − µ aµ bµ
1 − ||E 2 ||1
Assortativity by
degree
Contagion
References
where || · ||1 is the 1-norm = sum of a matrix’s entries.
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Tr E is the fraction of edges that are within groups.
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||E 2 ||1 is the fraction of edges that would be within
groups if connections were random.
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1 − ||E 2 ||1 is a normalization factor so rmax = 1.
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When Tr eµµ = 1, we have r = 1. X
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When eµµ = aµ bµ , we have r = 0. X
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Assortativity and
Mixing
Correlation coefficient:
Definition
General mixing
Notes:
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Assortativity by
degree
r = −1 is inaccessible if three or more types are
present.
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Disassortative networks simply have nodes
connected to unlike nodes—no measure of how
unlike nodes are.
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Minimum value of r occurs when all links between
non-like nodes: Tr eµµ = 0.
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rmin =
Contagion
References
−||E 2 ||1
1 − ||E 2 ||1
where −1 ≤ rmin < 0.
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Assortativity and
Mixing
Scalar quantities
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Now consider nodes defined by a scalar integer
quantity.
Examples: age in years, height in inches, number of
friends, ...
ejk = Pr (a randomly chosen edge connects a node
with value j to a node with value k ).
aj and bk are defined as before.
Can now measure correlations between nodes
based on this scalar quantity using standard Pearson
correlation coefficient ():
P
hjk i − hjia hk ib
j k j k (ejk − aj bk )
r=
=q
q
σa σb
hj 2 i − hji2 hk 2 i − hk i2
a
I
a
b
This is the observed normalized deviation from
randomness in the product jk .
Definition
General mixing
Assortativity by
degree
Contagion
References
b
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Assortativity and
Mixing
Degree-degree correlations
Definition
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Natural correlation is between the degrees of
connected nodes.
General mixing
Assortativity by
degree
Now define ejk with a slight twist:
Contagion
References
ejk = Pr
= Pr
an edge connects a degree j + 1 node
to a degree k + 1 node
an edge runs between a node of in-degree j
and a node of out-degree k
I
Useful for calculations (as per Rk )
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Important: Must separately define P0 as the {ejk }
contain no information about isolated nodes.
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Directed networks still fine but we will assume from
here on that ejk = ekj .
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Assortativity and
Mixing
Degree-degree correlations
Definition
General mixing
I
Notation reconciliation for undirected networks:
P
j k j k (ejk − Rj Rk )
r=
σR2
Assortativity by
degree
Contagion
References
where, as before, Rk is the probability that a
randomly chosen edge leads to a node of degree
k + 1, and
σR2 =
X
j

2
X
j 2 Rj − 
jRj  .
j
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Degree-degree correlations
Assortativity and
Mixing
Definition
General mixing
Assortativity by
degree
Error estimate for r :
Contagion
I
Remove edge i and recompute r to obtain ri .
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Repeat for all edges and compute using the jackknife
method () [2]
X
σr2 =
(ri − r )2 .
References
i
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Mildly sneaky as variables need to be independent
for us to be truly happy and edges are correlated...
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age A to package B indicates that A relies on components of B for its operation. Biological networks: "k#
protein-protein interaction network in the yeast S. Cerevisiae !53$; "l# metabolic network of the bacterium E.
Coli !54$; "m# neural network of the nematode worm C. Elegans !5,55$; tropic interactions between species
in the food webs of "n# Ythan Estuary, Scotland !56$ and "o# Little Rock Lake, Wisconsin !57$.
Measurements of degree-degree correlations
Group
Social
Technological
Biological
Network
Assortativity and
Mixing
Definition
Type
Size n
Assortativity r
Error & r
General mixing
Assortativity by
degree
a
a
b
c
d
e
f
Physics coauthorship
Biology coauthorship
Mathematics coauthorship
Film actor collaborations
Company directors
Student relationships
Email address books
undirected
undirected
undirected
undirected
undirected
undirected
directed
52 909
1 520 251
253 339
449 913
7 673
573
16 881
0.363
0.127
0.120
0.208
0.276
!0.029
0.092
0.002
0.0004
0.002
0.0002
0.004
0.037
0.004
g
h
i
j
Power grid
Internet
World Wide Web
Software dependencies
undirected
undirected
directed
directed
4 941
10 697
269 504
3 162
!0.003
!0.189
!0.067
!0.016
0.013
0.002
0.0002
0.020
k
l
m
n
o
Protein interactions
Metabolic network
Neural network
Marine food web
Freshwater food web
undirected
undirected
directed
directed
directed
2 115
765
307
134
92
!0.156
!0.240
!0.226
!0.263
!0.326
0.010
0.007
0.016
0.037
0.031
Contagion
References
!58,59$. The algorithm is as follows.
lmost never is. In this paper, therefore, we take an
I Social
"1# Given the desired
edge distribution e jk , we first caltend to be assortative
(homophily)
ive approach,
making usenetworks
of computer simulation.
culate the corresponding distribution of excess degrees q k
would like to generate on a computer a random netfrom networks
Eq. "23#, and then
invertto
Eq.be
"22# to find the degree
Technological
tend
aving, forIinstance,
a particular valueand
of thebiological
matrix
distribution:
his also fixes the degree distribution, via Eq. "23#.$ In
disassortative
2$ we discussed one possible way of doing this using
q k!1 /k
rithm similar to that of Sec. II C. One would draw
.
"27#
p k"
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rom the desired distribution e jk and then join the deq j!1 / j
%
Spreading on degree-correlated networks
Assortativity and
Mixing
Definition
General mixing
Assortativity by
degree
I
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Next: Generalize our work for random networks to
degree-correlated networks.
As before, by allowing that a node of degree k is
activated by one neighbor with probability Bk 1 , we
can handle various problems:
Contagion
References
1. find the giant component size.
2. find the probability and extent of spread for simple
disease models.
3. find the probability of spreading for simple threshold
models.
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Spreading on degree-correlated networks
Assortativity and
Mixing
Definition
General mixing
Assortativity by
degree
Contagion
I
Goal: Find fn,j = Pr an edge emanating from a
degree j + 1 node leads to a finite active
subcomponent of size n.
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Repeat: a node of degree k is in the game with
probability Bk 1 .
~ 1 = [Bk 1 ].
Define B
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References
Plan: Find the
generating function
~ 1 ) = P∞ fn,j x n .
Fj (x; B
n=0
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Spreading on degree-correlated networks
Assortativity and
Mixing
Definition
I
General mixing
Recursive relationship:
~ 1) = x 0
Fj (x; B
+x
∞
X
ejk
k =0
∞
X
k =0
Rj
Assortativity by
degree
(1 − Bk +1,1 )
Contagion
References
h
ik
ejk
~ 1) .
Bk +1,1 Fk (x; B
Rj
I
First term = Pr that the first node we reach is not in
the game.
I
Second term involves Pr we hit an active node which
has k outgoing edges.
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Next: find average size of active components
reached by following a link from a degree j + 1 node
~ 1 ).
= Fj0 (1; B
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Spreading on degree-correlated networks
Assortativity and
Mixing
Definition
General mixing
I
I
~ 1 ), set x = 1, and rearrange.
Differentiate Fj (x; B
~ 1 ) = 1 which is true when no giant
We use Fk (1; B
Assortativity by
degree
component exists. We find:
References
~ 1) =
Rj Fj0 (1; B
∞
X
k =0
I
ejk Bk +1,1 +
∞
X
Contagion
~ 1 ).
kejk Bk +1,1 Fk0 (1; B
k =0
Rearranging and introducing a sneaky δjk :
∞
X
k =0
∞
X
~ 1) =
ejk Bk +1,1 .
δjk Rk − kBk +1,1 ejk Fk0 (1; B
k =0
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Spreading on degree-correlated networks
Assortativity and
Mixing
Definition
General mixing
I
Assortativity by
degree
In matrix form, we have
Contagion
~ 0 (1; B
~ 1 ) = EB
~1
AE,B~ F
References
1
where
h
i
AE,B~
= δjk Rk − kBk +1,1 ejk ,
1 j+1,k +1
h
i
~ 0 (1; B
~ 1)
~ 1 ),
F
= Fk0 (1; B
k +1
h i
~1
[E]j+1,k +1 = ejk , and B
= Bk +1,1 .
k +1
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Spreading on degree-correlated networks
Assortativity and
Mixing
Definition
I
So, in principle at least:
General mixing
~ 0 (1; B
~ 1 ) = A−1 EB
~ 1.
F
~
E,B1
Assortativity by
degree
Contagion
References
I
~ 0 (1; B
~ 1 ), the average size of an active
Now: as F
component reached along an edge, increases, we
move towards a transition to a giant component.
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Right at the transition, the average component size
explodes.
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Exploding inverses of matrices occur when their
determinants are 0.
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The condition is therefore:
det AE,B~ = 0
1
.
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Spreading on degree-correlated networks
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General condition details:
det AE,B~ = det δjk Rk −1 − (k − 1)Bk ,1 ej−1,k −1 = 0.
1
I
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The above collapses to our standard contagion
condition when ejk = Rj Rk .
~ 1 = B~1, we have the condition for a simple
When B
Assortativity and
Mixing
Definition
General mixing
Assortativity by
degree
Contagion
References
disease model’s successful spread
det δjk Rk −1 − B(k − 1)ej−1,k −1 = 0.
I
~ 1 = ~1, we have the condition for the
When B
existence of a giant component:
det δjk Rk −1 − (k − 1)ej−1,k −1 = 0.
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Bonusville: We’ll find a much better version of this
set of conditions later...
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Spreading on degree-correlated networks
Assortativity and
Mixing
Definition
We’ll next find two more pieces:
1. Ptrig , the probability of starting a cascade
2. S, the expected extent of activation given a small
seed.
General mixing
Assortativity by
degree
Contagion
References
Triggering probability:
I
Generating function:
~ 1) = x
H(x; B
∞
X
h
ik
~ 1) .
Pk Fk −1 (x; B
k =0
I
Generating function for vulnerable component size is
more complicated.
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Spreading on degree-correlated networks
Assortativity and
Mixing
Definition
General mixing
I
Want probability of not reaching a finite component.
~ 1)
Ptrig = Strig =1 − H(1; B
∞
h
ik
X
~ 1) .
Pk Fk −1 (1; B
=1 −
Assortativity by
degree
Contagion
References
k =0
I
~ 1 ).
Last piece: we have to compute Fk −1 (1; B
I
Nastier (nonlinear)—we have to solve the recursive
expression we
started with when x = 1:
~ 1 ) = P∞ ejk (1 − Bk +1,1 )+
Fj (1; B
k =0 Rj
h
ik
P∞ ejk
~
B
F
(1;
B
)
.
k
1
k =0 Rj k +1,1
I
Iterative methods should work here.
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Spreading on degree-correlated networks
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Truly final piece: Find final size using approach of
Gleeson [3] , a generalization of that used for
uncorrelated random networks.
I
Need to compute θj,t , the probability that an edge
leading to a degree j node is infected at time t.
I
Evolution of edge activity probability:
Assortativity and
Mixing
Definition
General mixing
Assortativity by
degree
Contagion
References
θj,t+1 = Gj (θ~t ) = φ0 + (1 − φ0 )×
∞
k −1 X
ej−1,k −1 X k − 1 i
θk ,t (1 − θk ,t )k −1−i Bki .
Rj−1
i
k =1
I
i=0
Overall active fraction’s evolution:
φt+1 = φ0 +(1−φ0 )
∞
X
k =0
Pk
k X
k
i=0
i
θki ,t (1−θk ,t )k −i Bki .
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Spreading on degree-correlated networks
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As before, these equations give the actual evolution
of φt for synchronous updates.
~ θ~t ).
Contagion condition follows from θ~t+1 = G(
~ around θ~0 = ~0.
Expand G
θj,t+1 = Gj (~0)+
∞
X
∂Gj (~0)
k =1
I
I
I
∂θk ,t
∞
1 X ∂ 2 Gj (~0) 2
θk ,t +
θ +. . .
2!
∂θk2,t k ,t
Assortativity and
Mixing
Definition
General mixing
Assortativity by
degree
Contagion
References
k =1
If Gj (~0) 6= 0 for at least one j, always have some
infection.
∂Gj (~0)
~
If Gj (0) = 0 ∀ j, want largest eigenvalue
> 1.
∂θk ,t
Condition for spreading is therefore dependent on
eigenvalues of this matrix:
∂Gj (~0)
ej−1,k −1
=
(k − 1)Bk 1
∂θk ,t
Rj−1
Insert question from assignment 9 ()
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tively mixed network.
These findings are intuitively reasonable. If the network mixes assortatively, then the high-degree vertices
will tend to stick together in a subnetwork or core group
of higher mean degree than the network as a whole. It is
reasonable to suppose that percolation would occur earlier
within such a subnetwork. Conversely, since percolation
will be restricted to this subnetwork, it is not surprising
How the giant component changes with
assortativity:
General mixing
Contagion
0.8
giant component S
Definition
Assortativity by
degree
1.0
I
More assortative
networks
percolate for lower
average degrees
I
But disassortative
networks end up
with higher
extents of
spreading.
0.6
0.4
assortative
neutral
disassortative
0.2
0.0
Assortativity and
Mixing
1
10
100
exponential parameter κ
FIG. 1. Size of the [4]
giant component as a fraction of graph
from Newman, 2002
size
for graphs with the edge distribution given in Eq. (9). The
points are simulation results for graphs of N ! 100 000 vertices, while the solid lines are the numerical solution of Eq. (8).
Each point is an average over ten graphs; the resulting statistical errors are smaller than the symbols. The values of p are
0.5 (circles), p0 ! 0:146 . . . (squares), and 0:05 (triangles).
References
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References I
Assortativity and
Mixing
Definition
General mixing
[1] M. Boguñá and M. Ángeles Serrano.
Generalized percolation in random directed networks.
Phys. Rev. E, 72:016106, 2005. pdf ()
Assortativity by
degree
Contagion
References
[2] B. Efron and C. Stein.
The jackknife estimate of variance.
The Annals of Statistics, 9:586–596, 1981. pdf ()
[3] J. P. Gleeson.
Cascades on correlated and modular random
networks.
Phys. Rev. E, 77:046117, 2008. pdf ()
[4] M. Newman.
Assortative mixing in networks.
Phys. Rev. Lett., 89:208701, 2002. pdf ()
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References II
Assortativity and
Mixing
Definition
General mixing
Assortativity by
degree
Contagion
References
[5] M. E. J. Newman.
Mixing patterns in networks.
Phys. Rev. E, 67:026126, 2003. pdf ()
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