Download Assortativity and Mixing Complex Networks, Course 303A, Spring, 2009 Prof. Peter Dodds .

Assortativity and
Mixing
Assortativity and Mixing
Definition
General mixing
Complex Networks, Course 303A, Spring, 2009
Assortativity by
degree
Contagion
Prof. Peter Dodds
References
Department of Mathematics & Statistics
University of Vermont
Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License
.
Frame 1/26
Outline
Assortativity and
Mixing
Definition
General mixing
Definition
Assortativity by
degree
General mixing
References
Contagion
Assortativity by degree
Contagion
References
Frame 2/26
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.
Assortativity and
Mixing
Definition
General mixing
Assortativity by
degree
Contagion
References
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.:
1. degree
2. demographics (age, gender, etc.)
3. group affiliation
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We speak of mixing patterns, correlations, biases...
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Networks are still random at base but now have more
global structure.
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Build on work by Newman [3, 4] .
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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)
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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
Frame 4/26
Connection to degree distribution:
Assortativity and
Mixing
Definition
General mixing
Assortativity by
degree
Contagion
References
Frame 5/26
Notes:
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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.
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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 and
Mixing
Definition
General mixing
Assortativity by
degree
Contagion
References
Frame 6/26
Correlation coefficient:
Assortativity and
Mixing
Definition
I
Quantify the level of assortativity with the following
assortativity coefficient [4] :
P
P
aµ bµ
Tr E − ||E 2 ||1
µ eµµ −
P µ
r=
=
1 − µ aµ bµ
1 − ||E 2 ||1
General mixing
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
Notes:
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General mixing
r = −1 is inaccessible if three or more types are
presents.
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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 =
Assortativity by
degree
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
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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
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Natural correlation is between the degrees of
connected nodes.
Definition
General mixing
Assortativity by
degree
Now define ejk with a slight twist:
Contagion
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
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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 .
References
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Assortativity and
Mixing
Degree-degree correlations
Definition
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Notation reconciliation for undirected networks:
P
j k j k (ejk − Rj Rk )
r=
σR2
General mixing
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 :
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Remove edge i and recompute r to obtain ri .
Contagion
References
Repeat for all edges and compute using the
jackknife method () [1]
σr2 =
X
(ri − r )2 .
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$.
Measurements of degree-degree correlations
Type
Size n
Assortativity r
Error & r
a
a
b
c
d
e
f
Physics coauthorship
Biology coauthorship
Mathematics coauthorship
Film actor collaborations
Company directors
Student relationships
Email address books
undirected
undirected
undirected
undirected
undirected
undirected
directed
52 909
1 520 251
253 339
449 913
7 673
573
16 881
0.363
0.127
0.120
0.208
0.276
!0.029
0.092
0.002
0.0004
0.002
0.0002
0.004
0.037
0.004
g
h
i
j
Power grid
Internet
World Wide Web
Software dependencies
undirected
undirected
directed
directed
4 941
10 697
269 504
3 162
!0.003
!0.189
!0.067
!0.016
0.013
0.002
0.0002
0.020
k
l
m
n
o
Protein interactions
Metabolic network
Neural network
Marine food web
Freshwater food web
undirected
undirected
directed
directed
directed
2 115
765
307
134
92
!0.156
!0.240
!0.226
!0.263
!0.326
0.010
0.007
0.016
0.037
0.031
Group
Social
Technological
Biological
Network
Assortativity and
Mixing
Definition
General mixing
Assortativity by
degree
Contagion
References
!58,59$. 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
Frame 13/26
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"
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
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 βk 1 , we
can handle various problems:
Assortativity by
degree
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
I
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Goal: Find fn,j = Pr an edge emanating from a
degree j + 1 node leads to a finite active
subcomponent of size n.
Assortativity by
degree
Contagion
References
Repeat: a node of degree k is in the game with
probability βk 1 .
~1 = [βk 1 ].
Define β
Plan: Find the
generating function
~1 ) = P∞ fn,j x n .
Fj (x; β
n=0
Frame 15/26
Spreading on degree-correlated networks
I
Assortativity and
Mixing
Definition
Recursive relationship:
General mixing
~1 ) = x 0
Fj (x; β
+x
∞
X
ejk
k =0
∞
X
k =0
Rj
(1 − βk +1,1 )
h
ik
ejk
~1 ) .
βk +1,1 Fk (x; β
Rj
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First term = Pr that the first node we reach is not in
the game.
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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; β
Assortativity by
degree
Contagion
References
Frame 16/26
Spreading on degree-correlated networks
Assortativity and
Mixing
Definition
I
I
~1 ), set x = 1, and rearrange.
Differentiate Fj (x; β
~1 ) = 1 which is true when no giant
We use Fk (1; β
component exists. We find:
General mixing
Assortativity by
degree
Contagion
References
~1 ) =
Rj Fj0 (1; β
∞
X
k =0
I
ejk βk +1,1 +
∞
X
~1 ).
kejk βk +1,1 Fk0 (1; β
k =0
Rearranging and introducing a sneaky δjk :
∞
X
k =0
∞
X
~1 ) =
ejk βk +1,1 .
δjk Rk − k βk +1,1 ejk Fk0 (1; β
k =0
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Spreading on degree-correlated networks
Assortativity and
Mixing
Definition
General mixing
I
In matrix form, we have
Assortativity by
degree
~ 0 (1; β
~1 ) = Eβ
~1
AE,β~ F
1
Contagion
References
where
h
i
AE,β~
= δjk Rk − k βk +1,1 ejk ,
1 j+1,k +1
h
i
~ 0 (1; β
~1 )
~1 ),
F
= Fk0 (1; β
k +1
h i
~1
[E]j+1,k +1 = ejk , and β
= βk +1,1 .
k +1
Frame 18/26
Spreading on degree-correlated networks
I
So, in principle at least:
Assortativity and
Mixing
Definition
~ 0 (1; β
~1 ) = A−1 Eβ
~1 .
F
~
E,β1
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~ 0 (1; β
~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:
General mixing
Assortativity by
degree
Contagion
References
det AE,β~ = 0
1
.
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Spreading on degree-correlated networks
I
General condition details:
det AE,β~ = det δjk Rk −1 − (k − 1)βk ,1 ej−1,k −1 = 0.
1
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The above collapses to our standard contagion
condition when ejk = Rj Rk .
~1 = β~1, we have the condition for a simple
When β
Assortativity and
Mixing
Definition
General mixing
Assortativity by
degree
Contagion
References
disease model’s successful spread
det δjk Rk −1 − β(k − 1)ej−1,k −1 = 0.
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~1 = ~1, we have the condition for the existence
When β
of a giant component:
det δjk Rk −1 − (k − 1)ej−1,k −1 = 0.
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Bonusville: We’ll find another (possibly better)
version of this set of conditions later...
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Spreading on degree-correlated networks
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.
Assortativity and
Mixing
Definition
General mixing
Assortativity by
degree
Contagion
References
Triggering probability:
I
Generating function:
~1 ) = x
H(x; β
∞
X
h
ik
~1 ) .
Pk Fk −1 (x; β
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
I
Want probability of not reaching a finite component.
Ptrig = Strig
~1 )
=1 − H(1; β
∞
h
ik
X
~1 ) .
Pk Fk −1 (1; β
=1 −
General mixing
Assortativity by
degree
Contagion
References
k =0
I
~1 ).
Last piece: we have to compute Fk −1 (1; β
I
Nastier (nonlinear)—we have to solve the recursive
expression P
we started with when x = 1:
~1 ) = ∞ ejk (1 − βk +1,1 )+
Fj (1; β
k =0 Rj
h
ik
P∞ ejk
~1 ) .
β
F
(1;
β
k
+1,1
k
k =0 Rj
I
Iterative methods should work here.
Frame 22/26
Spreading on degree-correlated networks
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Truly final piece: Find final size using approach of
Gleeson [2] , a generalization of that used for
uncorrelated random networks.
Need to compute θj,t , the probability that an edge
leading to a degree j node is infected at time t.
Assortativity and
Mixing
Definition
General mixing
Assortativity by
degree
Contagion
References
Evolution of edge activity probability:
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 βki .
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 βki .
Frame 23/26
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(
~
~
~
Linearize G around θ0 = 0.
∞
∞
X
X
∂Gj (~0)
∂ 2 Gj (~0) 2
~
θ +...
θj,t+1 = Gj (0) +
θk ,t +
∂θk ,t
∂θk2,t k ,t
k =1
k =1
Assortativity and
Mixing
Definition
General mixing
Assortativity by
degree
Contagion
References
If Gj (~0) 6= 0 for at least one j, always have some
infection.
∂Gj (~0)
~
If Gj (0) = 0 ∀ j, largest eigenvalue of
must
∂θk ,t
exceed 1.
Condition for spreading is therefore dependent on
eigenvalues of this matrix:
∂Gj (~0)
ej−1,k −1
=
(k − 1)βk 1
∂θk ,t
Rj−1
Insert question from assignment 5 ()
Frame 24/26
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
Definition
General mixing
Assortativity by
degree
1.0
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giant component S
0.8
0.6
0.4
assortative
neutral
disassortative
0.2
0.0
Assortativity and
Mixing
1
10
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100
exponential parameter κ
FIG. 1. Size of the [3]
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).
More assortative
networks
percolate for lower
average degrees
Contagion
References
But disassortative
networks end up
with higher
extents of
spreading.
Frame 25/26
References I
[1] B. Efron and C. Stein.
The jackknife estimate of variance.
The Annals of Statistics, 9:586–596, 1981. pdf ()
Assortativity and
Mixing
Definition
General mixing
Assortativity by
degree
Contagion
[2] J. P. Gleeson.
Cascades on correlated and modular random
networks.
Phys. Rev. E, 77:046117, 2008. pdf ()
References
[3] M. Newman.
Assortative mixing in networks.
Phys. Rev. Lett., 89:208701, 2002. pdf ()
[4] M. E. J. Newman.
Mixing patterns in networks.
Phys. Rev. E, 67:026126, 2003. pdf ()
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