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CS109/Stat121/AC209/E-109 Data ScienceNetwork Models
Hanspeter Pfister & Joe [email protected] /
[email protected]
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mailto:[email protected]:[email protected]?subject=
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This Week HW4 due tonight at 11:59 pm Friday lab 10-11:30 am in
MD G115
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Examples from Newman (2003)I Introduction 3
FIG. 2 Three examples of the kinds of networks that are the
topic of this review. (a) A food web of predator-prey
interactionsbetween species in a freshwater lake [272]. Picture
courtesy of Neo Martinez and Richard Williams. (b) The network
ofcollaborations between scientists at a private research
institution [171]. (c) A network of sexual contacts between
individualsin the study by Potterat et al. [342].
A. Types of networks
A set of vertices joined by edges is only the simplesttype of
network; there are many ways in which networksmay be more complex
than this (Fig. 3). For instance,there may be more than one
different type of vertex in anetwork, or more than one different
type of edge. Andvertices or edges may have a variety of
properties, nu-merical or otherwise, associated with them. Taking
theexample of a social network of people, the vertices mayrepresent
men or women, people of different nationalities,locations, ages,
incomes, or many other things. Edgesmay represent friendship, but
they could also representanimosity, or professional acquaintance,
or geographicalproximity. They can carry weights, representing,
say,how well two people know each other. They can also bedirected,
pointing in only one direction. Graphs com-posed of directed edges
are themselves called directed
graphs or sometimes digraphs, for short. A graph rep-resenting
telephone calls or email messages between in-dividuals would be
directed, since each message goes inonly one direction. Directed
graphs can be either cyclic,meaning they contain closed loops of
edges, or acyclicmeaning they do not. Some networks, such as food
webs,are approximately but not perfectly acyclic.
One can also have hyperedgesedges that join morethan two
vertices together. Graphs containing such edgesare called
hypergraphs. Hyperedges could be used to in-dicate family ties in a
social network for examplen in-dividuals connected to each other by
virtue of belongingto the same immediate family could be
represented byan n-edge joining them. Graphs may also be
naturallypartitioned in various ways. We will see a number
ofexamples in this review of bipartite graphs: graphs thatcontain
vertices of two distinct types, with edges runningonly between
unlike types. So-called affiliation networks
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GraphsA graph G=(V,E) consists of a vertex set V and an
edge set E containing unordered pairs {i,j} of vertices.
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graph multigraph
The degree of vertex v is the number of edges attached to
it.
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A Plea for Clarity: What is a Network?
graph vs. multigraph (are loops, multiple edges ok? What is a
simple graph?)
directed vs. undirected weighted vs. unweighted dynamics of vs.
dynamics on labeled vs. unlabeled network as quantity of interest
vs. quantities of
interest on networks
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Why model networks?
Hard to interpret hairballs. We can define some interesting
features
(statistics) of a network, such as measures of clustering, and
compare the observed values against those of a model
Warning: much of the network literature carelessly ignores the
way in which the network data were gathered (sampling) and whether
there are missing/unknown nodes or edges!
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Erdos-Renyi Random Graph Model
Independently flip coins with prob. p of heads Let n get large
and p get small, with the average
degree c = (n-1)p held constant.
What happens for c < 1? What happens for c > 1? What
happens for c = 1?
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Degree Sequences
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Take V = { 1, . . . , n} and let di be the degreeof vertex i
.The degree sequence of G is d = (d1, . . . , dn ).
n = 9, d = (3, 4, 3, 3, 4, 3, 3, 3, 2)
A sequence d is graphical if there is a graph G with degree
sequenced.G is a realization of d.
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MCMC on Networks
mixing times, burn-in, bottlenecks, autocorrelation,...
Switchings Chain
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Power Laws
Power-law (a.k.a. scale-free) networks: the number of vertices
of degree k is proportional to k-
Stumpf et al (2005): Subnets of scale-free networks are not
scale-free, especially for large
Their subnets are i.i.d. node-based. What about features other
than degree distributions?
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p1 Model (Holland-Leinhardt 1981)
3.4 The p1 Model for Social Networks
A conceptually separate thread of research developed in parallel
in the statistics and socialsciences literature, starting with the
introduction of the p1 model. Consider a directed graphon the set
of n nodes. Holland and Leinhardts p1 model focuses on dyadic
pairings andkeeps track of whether node i links to j, j to i,
neither, or both. It contains the followingparameters:
! : a base rate for edge propagation,
" i (expansiveness): the eect of an outgoing edge from i,
#j (popularity): the eect of an incoming edge into j,
$ij (reciprocation/mutuality): the added eect of reciprocated
edges.
Let P (0, 0) be the probability for the absence of an edge
between i and j, Pij(1, 0) theprobability of i linking to j (1
indicates the outgoing node of the edge), Pij(1, 1) theprobability
of i linking to j and j linking to i. The p1 model posits the
following probabilities(see [149]):
log Pij(0, 0) = %ij, (3.1)
log Pij(1, 0) = %ij + " i + #j + ! , (3.2)
log Pij(0, 1) = %ij + " j + #i + ! , (3.3)
log Pij(1, 1) = %ij + " i + #j + " j + #i + 2! + $ij. (3.4)
In this representation of p1, %ij is a normalizing constant to
ensure that the probabilitiesfor each dyad (i, j) add to 1. For our
present purposes, assume that the dyad is in oneand only one of the
four possible states. The reciprocation eect, $ij, implies that the
oddsof observing a mutual dyad, with an edge from node i to node j
and one from j to i, isenhanced by a factor of exp($ij) over and
above what we would expect if the edges occuredindependently of one
another.
The problem with this general p1 representation is that there is
a lack of identification ofthe reciprocation parameters. The
following special cases of p1 are identifiable and of
specialinterest:
1. " i = 0, #j = 0, and $ij = 0. This is basically an
Erdos-Renyi-Gilbert model fordirected graphs: each directed edge
has the same probability of appearance.
2. $ij = 0, no reciprocal e! ect. This model eectively focuses
solely on the degree distri-butions into and out of nodes.
3. $ij = $, constant reciprocation. This was the version of p1
studied in depth by Hollandand Leinhardt using maximum likelihood
estimation.
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ERGMs (Exponential Random Graph Models)
26 J. BLIT ZSTE IN AN D P. DIA CONIS
More formally, den e a probabilit y measure P! on the space of
all graphs on n verticesby
P! (G) = Z ! 1 exp
!
!n"
i =1
! i di (G)
#
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where Z is a normalizing constant. The real parameters ! 1, . .
. , ! n are chosen to achievegiven expected degrees. Th is model
appearsexplicitly in Park and Newman [59], using thetools and
languageof stati sti cal mechanics.
Holland and Leinhardt [35] give iterati ve algorithms for the
maximum likelihood es-timators of the parameters, and Snijders [65]
considers MCMC methods. Techniques ofHaberman [31] can be used to
prove that the maximum likelihood estimates of the ! i
areconsistent and asymptotically normal as n " # , provided that
there is a constant B suchthat |! i | $ B for all i .
Such exponential modelsare standard fare in statist ics, stati
sti cal mechanics, and socialnetworking (where they are called p"
models). They are used for directed graphs in Hollandand Leinhardt
[35] and for graphs in Frank and Strauss [27, 70] and Snijders [65,
66],with a variety of su! cient stati sti cs (see the surveys in
[3], [56], and [66]). One standardmotivation for using the
probabilit y measure P! when the degree sequence is the mainfeature
of interest is that thi s model gives the maximum entr opy distrib
ution on graphswith a given expected degreesequence(see Laurit zen
[43] for furth er discussion of th is).Unl ike most other
exponential modelson graphs, the normalizing constant Z is
available inclosed form. Furth ermore, there is an easy method of
sampling exact ly from P! , as shownby the following. The same
formulas are given in [59], but for completeness we provide abrief
proof.
Lemma 1. Fix real parameters ! 1, . . . , ! n . Let Yij be
independent binary random vari-ablesfor 1 $ i < j $ n, with
P (Yij = 1) =e! (! i + ! j )
1 + e! (! i + ! j )= 1 ! P (Yij = 0).
Form a random graph G by creating an edgebetween i and j if and
only i f Yij = 1. ThenG is distributed according to P! , with
Z =$
1# i
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Pseudolikelihood (Strauss-Ikeda 80)
Fix a pair of nodes {i,j}, and consider the indicator r.v. of
whether an edge {i,j} is present in G.
Conditioning on the rest of G yields great simplification:
P (edge{ i, j }| rest)P (no edge{ i, j }| rest)
= e!! (x(G+ )! x(G" ))
So use logistic regression? Be careful of variance
estimates!
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MCMCMLE (Geyer-Thompson 92)
But why and not = ?
Dont know the normalizing constant!
From now on we write
P(G) =exp( x(G))
c()
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Write
Fix some baseline ! 0 and estimate log-likelihood ratio.
l(! ) ! l(! 0) = ( ! ! ! 0)!x(G) ! logc(! )c(! 0)
c(! )c(! 0)
= E ! 0q! (G)q! 0 (G)
Ratio of normalizing constants is:
= q! (G)/c (! )
So can approximate the MLE via MCMC.
What about the choice of ! 0 though?
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i.i.d. node i.i.d. edge snowball RDS short paths
Erdos
DyadIndep.
ERGM
Fixed degree
Geom
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Latent Space Models
Reproduced with permission of the copyright owner. Further
reproduction prohibited without permission.
Hoff et al (2002) model:
Reproduced with permission of the copyright owner. Further
reproduction prohibited without permission.
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Degrees
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Normalization?
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Closenessuses the reciprocal of the average shortest distance to
other nodes
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Betweenness
many variations: shortest paths
vs. flow maximization vs. all paths vs. random paths
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Eigenvector Centralityuse eigenvector of A corresponding to the
largest
eigenvalue (Bonacich); more generally, power centrality
