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Statistical Inference for NetworksSystems Biology Doctoral Training Centre
Theoretical Systems Biology Module
HILARY TERM 2008PROF. GESINE REINERT
http://www.stats.ox.ac.uk/ reinertWITH PAO-YANG CHEN
http://www.stats.ox.ac.uk/ chenAND WAQAR ALI
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Overview Lecture 1: Network summaries.What are networks? Some examples fromsocial science and from biology. The need to summarise networks. Clustering coefficient,degree distribution, shortest path length, motifs, between-ness, second-order summaries. Rolesin networks, derived from these summary statistics, and modules in networks. Directed andweighted networks. The choice of summary should depend on the research question.
Lecture 2:Models of random networks.Models would provide further insight into the net-work structure. Classical Erds-Renyi (Bernoulli) random graphs and their random mixtures,Watts-Strogatz small worlds and the modification by Newman, Barabasi-Albert scale-free net-works, exponential random graph models.
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Lecture 3:Fitting a model: parametric methods.Deriving the distribution of summarystatistics. Parametric tests based on the theoretical distribution of the summary statistics (onlyavailable for some of the models).
Lecture 4:Statistical tests for model fit: nonparametric methods.Quantile-quantile plotsand other visual methods. Monte-Carlo tests based on shuffling edges with the number of edgesfixed, or fixing the node degree distribution, or fixing some other summary. The particularissue of testing for power-law dependence. Subsampling issues. Tests carried out on the samenetwork are not independent.
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Lecture 5:Statistical inference for networks: local properties.Inferring characteristics fora missing node from the existing network. Log-linear regression models. Inferring missingedges and identifying false-positive edges. Logistic regression models.
Lecture 6:Statistical inference for networks: modules, motifs and roles.Identifying sim-ilar edges in networks. Clustering algorithms. Comparison of networks: two networks on thesame set of nodes. Regression models.
Lecture 7:Further topics.Hierarchical networks. Dynamics on networks.
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Suggested reading
1. U. Alon: An Introduction to Systems Biology Design Principles of Biological Circuits.Chapman and Hall 2007.
2. S.N. Dorogovtsev and J.F.F. Mendes:Evolution of Networks. Oxford University Press2003.
3. R. Durrett:Random Graph Dynamics. Cambridge University Press 2007.
4. R. Gentleman, V. Carey, W. Huber, R. Irizarry, S. Dutoit (eds).Bioinformatics andComputational Biology Solutions using R and Bioconductor. Springer 2005.
5. W. de Nooy, A. Mrvar and V. Bagatelj.Exploratory Social Network Analysis with Pajek.Cambridge University Press 2005.
6. S. Wasserman and K. Faust:Social Network Analysis. Cambridge University Press1994.
7. D. Watts:Small Worlds. Princeton University Press 1999.
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This part of the module will take place Wednesday 27 February, Monday 10 March, andFriday 14 March, from 9:30 - 12 and 2-5 in the DTC.
The teaching will be a mixture of lectures, worked examples, and computer exercises.We shall use theR language in connection withBioconductor. Both of these are open
source.
Lecture notes will be published athttp://www.stats.ox.ac.uk/ reinert/dtc/networks.html.The notes may cover more material than the lectures. The notes may be updated through-
out the course.
The statistical analysis of networks is a very complex topic, far beyond what could becovered in 3-day course. Hence the goal of the class is to give a brief overview of the basics,highlighting some of the issues to be addressed.
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1 Network summaries
1.1 What are networks?Networks are just graphs. Often one would think of a network as a connected graph, but notalways. In this lecture course we shall usenetworkandgraph interchangeably.
Here are some examples of networks (graphs).
MAPK: Pancreatic pathway
KRAS
PIK3R4 RALGDS
ARHGEF6 AKT3
ARAF
RALA
RAC1 CHUK BAD CASP9
MAP2K1
RALBP1 PLD1
NFKB1 BCL2L1
MAPK1 MAPK8
(suppress apopt.)(cytosk. remod.)
(anti−apopt.)
(DNA; prolif. gn.)
(cell surv.)
This graph shows part of the KEGG pancreatic cancer model, surrounding the MAPK signalingpathway.
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Flo: Florentine Families
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Marriage relations between Florentine families, in the order0 ACCIAIUOL,1 ALBIZZI,2 BARBADORI,3 BISCHERI,4 CASTELLAN,5 GINORI,6 GUADAGNI,7 LAMBERTES,8 MEDICI,9 PAZZI,10 PERUZZI,11 PUCCI,12 RIDOLFI,13 SALVIATI,14 STROZZI,15 TORNABUON.The Medici beat their arch-rivals, the Strozzi.
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Marriage relations between Florentine families: different drawing program.
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Yeast: A plot of a connected subset of Yeast protein interactions.
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0-10
Networks arise in a multitude of contexts, such as
- metabolic networks- protein-protein interaction networks- spread of epidemics- neural network ofC. elegans- social networks- collaboration networks (Erdos numbers ... )- Membership of management boards- World Wide Web- power grid of the Western US
The study of networks has a long tradition in social science, where it is calledSocial Net-work Analysis. The networks under consideration are typically fairly small. In contrast, start-ing at around 1997, statistical physicists have turned their attention to large-scale properties ofnetworks. Our lectures will try to get a glimpse on both approaches.
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Typical networks in systems biology are
• Metabolic networks
• Gene interaction networks
• Protein interaction networks.
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Research questions include
• How do these networks work? Where could we best manipulate a network in order toprevent, say, tumor growth?
• How did these biological networks evolve? Could mutation affect whole parts of thenetwork at once?
• How similar are these networks? If we study some organisms very well, how muchdoes that tell us about other organisms?
• How are these networks interlinked? Can we infer information from gene interactionnetworks that would be helpful for protein interaction networks?
• What are the building principles of these networks? How is resilience achieved, andhow is flexibility achieved? Could we learn from biological networks to build man-made efficient networks?
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From a statistical viewpoint, questions include
• How to best describe networks?
• How to infer characteristics of nodes in the network?
• How to infer missing links, and how to check whether existing links are not false posi-tives
• How to compare networks from related organisms?
• How to predict functions from networks?
• How to find relevant sub-structures of a network?
Statistical inference relies on the assumption that there is some randomness in the data. Beforewe turn our attention to modelling such randomness, let’s look at how to describe networks, orgraphs, in general.
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1.2 What are graphs?A graphconsists ofnodes(sometimes also calledvertices) andedges(sometimes also calledlinks). We typically think of the nodes as actors, or proteins, or genes, or metabolites, and wethink of an edge as an interaction between the two nodes at either end of the edge. Sometimesnodes may possess characteristics which are of interest (such as structure of a protein, orfunction of a protein). Edges may possess different weights, depending on the strength of theinteraction. For now we just assume that all edges have the same weight, which we set as 1.
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Mathematically, we abbreviate a graphG asG = (V, E), whereV is the set of nodes andE is the set of edges. We use the notation|S| to denote the number of elements in the setS.Then|V | is the number of nodes, and|E| is the number of edges in the graphG. If u andvare two nodes and there is an edge fromu to v, then we write that(u, v) ∈ E, and we say thatv is aneighbourof u.
If both endpoints of an edge are the same, then the edge is aloop. For now we excludeself-loops, as well as multiple edges between two nodes.
Edges may bedirectedor undirected. A directed graph, or digraph, is a graph where alledges are directed. Theunderlyinggraph of a digraph is the graph that results from turning alldirected edges into undirected edges. Here we shall mainly deal with undirected graphs.
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Two nodes are calledadjacentif they are joined by an edge. A graph can be described byits adjacency matrixA = (au,v). This is a square|V | × |V | matrix. Each entry is either 0 or1;
au,v = 1 if and only if (u, v) ∈ E.
As we assume that there are no self-loops, all elements on the diagonal of the adjacency matrixare 0. If the edges of the graph are undirected, then the adjacency matrix will be symmetric.
The adjacency matrix entries tell us for every nodev which nodes are within distance 1 ofv. If we take the matrix produceA2 = A×A, the entry for(u, v) with u 6= v would be
a(2)(u, v) =∑w∈V
au,waw,v.
If a(2)(u, v) 6= 0 thenu can be reached fromv within two steps;u is within distance 2 ofv.Higher powers can be interpreted similarly.
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Example: Adjacency matrix for marital relation between Florentine families (seeWasserman+Faust, p.744).
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 00 0 0 0 0 1 1 0 1 0 0 0 0 0 0 00 0 0 0 1 0 0 0 1 0 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 1 0 0 0 1 00 0 1 0 0 0 0 0 0 0 1 0 0 0 1 00 1 0 0 0 0 0 0 0 0 0 0 0 0 0 00 1 0 1 0 0 0 1 0 0 0 0 0 0 0 10 0 0 0 0 0 1 0 0 0 0 0 0 0 0 01 1 1 0 0 0 0 0 0 0 0 0 1 1 0 10 0 0 0 0 0 0 0 0 0 0 0 0 1 0 00 0 0 1 1 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 1 0 0 0 0 0 1 10 0 0 0 0 0 0 0 1 1 0 0 0 0 0 00 0 0 1 1 0 0 0 0 0 1 0 1 0 0 00 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0
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A completegraph is a graph such that every pair of nodes is joined by an edge. Theadjacency matrix has entry 0 on the diagonal, and 1 everywhere else.
A bipartitegraph is a graph where the node setV is decomposed into two disjoint subsets,U andW , say, such that there are no edges between any two nodes inU , and also there are noedges between any two nodes inW ; all edges have one endpoint inU and the other endpointin W . An example is a network of co-authorship and articles;U could be the set of authors,W the set of articles, and an author is connected to an article by an edge if the author is aco-author of that article. The adjacency matrixA can then be arranged such that it is of theform [
0 A1
A2 0
].
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1.3 Network summariesThedegreedeg(v) of a nodev is the number of edges which involvev as an endpoint. Thedegree is easily calculated from the adjacency matrixA;
deg(v) =∑
u
au,v.
Theaverage degreeof a graph is then the average of its node degrees.
(For directed graphs we would define thein-degreeas the number of edges directed at thenode, and theout-degreeas the number of edges that go out from that node.)
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The clustering coefficientof a nodev is, intuitively, the proportion of its ”friends” whoare friends themselves. Mathematically, it is the proportion of neighbours ofv which areneighbours themselves. In adjacency matrix notation,
C(v) =
∑u,w∈V au,vaw,vau,w∑
u,w∈V au,vaw,v.
The(average) clustering coefficientis defined as
C =1
|V |∑v∈V
C(v).
0-21
Note that ∑u,w∈V
au,vaw,vau,w
is the number of triangles involvingv in the graph. Similarly,∑u,w∈V
au,vaw,v
is the number of2-starscentred aroundv in the graph. The clustering coefficient is thus theratio between the number of triangles and the number of 2-stars. The clustering coefficientdescribes how ”locally dense” a graph is. Sometimes the clustering coefficient is also calledthetransitivity.
The clustering coefficient in the Florentine family example is 0.1914894; the average clus-tering coefficient in the Yeast data is 0.1023149.
0-22
Exercise 1:
• Draw an undirected complete graph on 6 nodes, and write down its adjacency matrix.Determine the degrees of the 6 nodes. What is the clustering coeffient?
• Draw two different undirected graphs on 6 nodes where each node has degree 2, andwrite down their adjacency matrices. What are their clustering coefficients?Hint: agraph does not have to be connected.
0-23
In a graph apath from nodev0 to nodevn is an alternating sequence of nodes andedges,(v0, e1, v1, e2, . . . , vn−1, en, vn) such that the endpoints ofei are vi−1 and vi, fori = 1, . . . , n. A graph is calledconnectedif there is a walk between any pair of nodes in thegraph, otherwise it is calleddisconnected. Thedistance (u, v) between two nodesu andv isthe length of the shortest path joining them. This path does not have to be unique.
We can calculate the distance`(u, v) from the adjacency matrixA as the smallest powerp of A such that the(u, v)-element ofAp is not zero.
In a connected graph, theaverage shortest path lengthis defined as
` =1
|V |(|V | − 1)
∑u 6=v∈V
`(u, v).
The average shortest path length describes how ”globally connected” a graph is.
0-24
Example: H. Pylori and Yeast protein interaction network comparison:
n ` CH.Pylori 686 4.137637 0.016
Yeast 2361 4.376182 0.1023149
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Node degree, clustering coefficient, and shortest path length are the most common sum-maries of networks. Other popular summaries, to name but a few, are: thebetween-ness of anedgecounts the proportion of shortest paths between any two nodes which pass through thisedge. Similarly, thebetween-ness of a nodeis the proportion of shortest paths between anytwo nodes which pass through this node. Theconnectivityof a connected graph is the smallestnumber of edges whose removal results in a disconnected graph.
In addition to considering these general summary statistics, it has proven fruitful to de-scribe networks in terms ofmotifs; these are building- block patterns of networks such as afeed-forward loop, see the book byAlon. Here we think of a motif as a subgraph with a fixednumber of nodes and with a given topology. In biological networks, it turns out that motifsseem to be conserved across species. They seem to reflect functional units which combine toregulate the cellular behaviour as a whole.
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The decomposition ofcommunitiesin networks, small subgraphs which are highly con-nected but not so highly connected to the remaining graph, can reveal some structure of thenetwork. Identifyingroles in networks singles out specific nodes with special properties, suchas hub nodes, which are nodes with high degree.
Looking at the ”spectral decomposition”, i.e. at eigenvectors and eigenvalues, of the adja-cency matrix, provides another set of summaries, such ascentrality.
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The above network summaries provide an initial go at networks. Specific networks mayrequire specific concepts. In protein interaction networks, for example, there is a differencewhether a protein can interact with two other proteins simultaneously (party hub) or sequen-tially (date hub). In addition, the research question may suggest other summaries. For example,in fungal networks, there are hardly any triangles, so the clustering coefficient does not makemuch sense for these networks.
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Excursion:Milgram and thesmall world effect.
In 1967 the American sociologist Milgram reported a series of experiments of the follow-ing type. A number of people from a remote US state (Nebraska, say) are asked to havea letter (or package) delivered to a certain person in Boston, Massachusetts (such as thewife of a divinity student). The catch is that the letter can only be sent to someone whomthe current holder knew on a first-name basis. Milgram kept track of how many interme-diaries were required until the letters arrived; he reported a median of six; see for examplehttp : //www.uaf.edu/northern/bigworld.html. This made him coin the notion ofsixdegrees of separation, often interpreted as everyone being six handshakes away from the Pres-ident. While the experiments were somewhat flawed (in the first experiment only 3 lettersarrived), the concept ofsix degrees of separationhas stuck.
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2 Models of random networksIn order to judge whether a network summary is ”unusual” or whether a motif is ”frequent”,there is an underlying assumption of randomness in the network.
Network data are subject to various errors, which can create randomness, such as
• There may be missing edges in the network. Perhaps a node was absent (social network)or has not been studied yet (protein interaction network).
• Some edges may be reported to be present, but that recording is a mistake. Dependingon the method of determining protein interactions, the number of suchfalse positiveinteractions can be substantial, of around 1/3 of all interactions.
• There may be transcription errors in the data.
• There may be bias in the data, some part of the network may have received higherattention than another part of the network.
Often network data are snapshots in time, while the network might undergo dynamical changes.In order to understand mechanisms which could explain the formation of networks, math-
ematical models have been suggested.
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2.1 Bernoulli (Erdos-Renyi) random graphsThe most standard random graph model is that of Erdos and Renyi (1959). The (finite) nodesetV is given, say|V | = n, and an edge between two nodes is present with probabilityp,independently of all other edges. As there are(
n
2
)=
n(n− 1)
2
potential edges, the expected number of edges is then(n
2
)p.
Each node hasn − 1 potential neighbours, and each of thesen − 1 edges is present withprobabilityp, and so the expected degree of a node is(n − 1)p. As the expected degree of anode is the same for all nodes, the average degree is(n− 1)p.
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Similarly, the average number of triangles in the graph is(n
3
)p3 =
n(n− 1)(n− 2)
6p3,
and the average number of 2-stars is (n
3
)p2.
Thus, with a bit of handwaving, we would expect an average clustering coefficient of about(n3
)p3(
n3
)p2
= p.
0-32
In a Bernoulli random graphs, your friends are no more likely to be friends themselves thanwould be a two complete strangers. This model is clearly not a good one for social networks.Below is an example from scientific collaboration networks (N. Boccara, Modeling ComplexSystems, Springer 2004, p.283. We can estimatep as the fraction of average node degreeandn − 1; this estimate would also be an estimate of the clustering coefficient in a Bernoullirandom graph.
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Network n ave degree C CBernoulli
Los Alamos archive 52,909 9.7 0.43 0.00018MEDLINE 1,520,251 18.1 0.066 0.000011NCSTRL 11,994 3.59 0.496 0.0003
Also in real-world graphs often the shortest path length is much shorter than expectedfrom a Bernoulli random graph with the same average node degree. The phenomenon of shortpaths, often coupled with high clustering coefficient, is called thesmall world phenomenon.Remember the Milgram experiments!
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2.2 The Watts-Strogatz modelWatts and Strogatz (1998)published a ground-breaking paper with a new model for smallworlds; the version currently most used is as follows. Arrange then nodes ofV on a lattice.Then hard-wire each node to itsk nearest neighbours on each side on the lattice, wherek issmall. Thus there arenk edges in this hard-wired lattice. Now introduce random shortcutsbetween nodes which are not hard-wired; the shortcuts are chosen independently, all with thesame probability.
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If there are no shortcuts, then the average distance between two randomly chosen nodesis of the ordern, the number of nodes. But as soon as there are just a few shortcuts, thenthe average distance between two randomly chosen nodes has an expectation of orderlog n.Thinking of an epidemic on a graph - just a few shortcuts dramatically increase the speed atwhich the disease is spread.
It is possible to approximate the node degree distribution, the clustering coefficient, andthe shortest path length reasonably well mathematically; we may come back to these approxi-mations later.
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While the Watts-Strogatz model is able to replicate a wide range of clustering coefficientand shortest path length simultaneously, it falls short of producing the observed types of nodedegree distributions. It is often observed that nodes tend to attach to ”popular” nodes; popu-larity is attractive.
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2.3 ”The” Barabasi-Albert modelIn 1999, Barabasi and Albert noticed that the actor collaboration graph adn the World WideWeb had degree distributions that were of the type
Prob(degree = k) ∼ Ck−γ
for k →∞. Such behaviour is calledpower-law behaviour; the constantγ is called thepower-law exponent. Subsequently a number of networks have been identified which show this type ofbehaviour. They are also calledscale-free random graphs. To explain this behaviour, Barabasiand Albert introduced thepreferential attachmentmodel for network growth. Suppose that theprocess starts at time 1 with 2 nodes linked bym (parallel) edges. At every timet ≥ 2 we adda new node withm edges that link the new node to nodes already present in the network. Weassume that the probabilityπi that the new node will be connected to a nodei depends on thedegreedeg(i) of i so that
πi =deg(i)∑j deg(j)
.
To be precise, when we add a new node we will add edges one at a time, with the second andsubsequent edges doing preferential attachment using the updated degrees.
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This model has indeed the property that the degree distribution is approximately powerlaw with exponentγ = 3. Other exponents can be achieved by varying the probability forchoosing a given node.
Unfortunately the above construction will not result in any triangles at all. It is possibleto modify the construction, adding more than one edge at a time, so thatany distribution oftriangles can be achieved.
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2.4 Erdos-Renyi Mixture GraphsAn intermediate model with not quite so many degrees of freedom is given by the Erdos-Renyimixture model, also known aslatent block modelsin social science (Nowicky and Snijders(2001)). Here we assume that nodes are of different types, say, there areL different types.Then edges are constructed independently, such that the probability for an edge varies onlydepending on the type of the nodes at the endpoints of the edge.Robin et alhave shown thatthis model is very flexible and is able to fit many real-world networks reasonably well. It doesnot produce a power-law degree distribution however.
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Exercise 2:Consider an Erdos-Renyi mixture model with two types of nodes. Type 1,of which there aren1 nodes, has edge probabilityp1; whereas Type 2, of which there aren2 nodes, has edge probabilityp2; the edge probability for an edge between a Type-1 nodeand a Type-2 node isp1,2. We are interested in the average node degree and the clusteringcoefficient.
• What should average node degree and the clustering coefficient be ifp1,2 = 0? What ifp1,2 6= 0 butp1 = p2 = 0?
• What would the average node degree be in general?
• For the clustering coefficient, which types of triangles would you need to consider?What would you expect for the casep1,2 = 0? What ifp1,2 6= 0 butp1 = p2 = 0?
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2.5 Exponential random graph (p∗) modelsNetworks have been analysed for ”ages” in the social science literature, see for example thebook byWasserman and Faust. Here usually digraphs are studied, and the research questionsare different from biological networks. Typical research questions could be
• Is there a tendency in friendship towards transitivity; are friends of friends my friends?
• What is the role of explanatory variables such as income on the position in the network?
• What is the role of friendship in creating behaviour (such as smoking)?
• Is there a hierarchy in teh network?
• Is the network influenced by other networks for which the membership overlaps?
0-42
Exponential random graph (p∗) modelsmodel the whole adjacency matrix of a graphsimultaneously, making it easy to incorporate dependence. Suppose thatX is our randomadjacency matrix. The general form of the model is
Prob(X = x) =1
κexp{
∑B
λBzB(x)},
where the summation is over all subsetsB of the set of potential edges,
zB(x) =∏
(i,j)∈B
xi,j
is the network statistic corresponding to the subsetB, κ is a normalising quantity so that theprobabilities sum to 1, and the parameterλB = 0 for all x unless all the the variables inB aremutually dependent.
The simplest such model is that the probability of any edge is constant across all possibleedges, i.e. the Bernoulli graph, for twhich
Prob(X = x) =1
κexp{λL(x)},
whereL(x) is the number of edges in the networkx andλ is a parameter.
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For social networks, Frank and Strauss (1986) introducedMarkov dependence, wherebytwo possible edges are assumed to be conditionally dependent if they share a node. For non-directed networks, the resulting model has parameters relating only to the configurationsstarsof various types, and triangles. If the numberL(x) of edges, the numberS2(x) of two-stars,the numberS3(x) of three-stars, and the numberT (x) of triangles are included, then the modelreads
Prob(X = x) =1
κexp{λ1L(x) + λ2S2(x) + λ3S3(x) + λ4T (x)}.
By setting the parameters to particular values and then simulating the distribution, we canexamine global properties of the network.
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2.6 Specific models for specific networksDepending on the research question, it may make sense to build a specific network model.For example, a gene duplication model has been suggested which would result in a power-lawlike node degree distribution. For metabolic pathways, a number of Markov models have beenintroduced. When thinking of flows through networks, it may be a good idea to use weightednetworks; the weights could themselves be random.
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Further referencesJ.-J. Daudin, F. Picard, S. Robin (2006). A mixture model for random graphs. Preprint.
O. Frank and D. Strauss (1986). Markov graphs.Journal of the American StatisticalAssociation81, 832-842.
K. Nowicky and T. Snijders (2001). Estimation and Prediction for Stochastic Blockstruc-tures.Journal of the American Statistical Association455, Vol. 96, pp. 1077-1087.
S. Milgram (1967). The small world problem.Psychology Today2, 60–67.
0-46
Recap and additionsNetworks are complex, hence the need to find good summaries. The most common ones arenode degrees, clustering coefficient, andshortest path length.
Addendum: summaries based on spectral properties of the adjacency matrix.
If λi are the eigenvalues of the adjacency matrixA, then the spectral density of the graphis defined as
ρ(λ) =1
n
∑i
δ(λ− λi),
whereδ(x) is the delta function. For Bernoulli random graphs, ifp is constant asn → ∞,thenρ(λ) converges to a semicircle.
The eigenvalues can be used to compute thekth moments,
Mk =1
n
∑i
(λi)k =
1
n
∑i1,i2,...,ik
ai1,i2ai2,i3 · · · aik−1,ik .
0-47
The quantitynMk is the number of paths returning to the same node in the graph, passingthroughk edges, where these paths may contain nodes that were already visited.
Because in a tree-like graph a return path is only possible going back through alreadyvisited nodes, the presence of odd moments is an indicator for the presence of cycles in thegraph.
Thesubgraph centrality
Sci =
∞∑k=0
(Ak)i,i
k!
measures the ”centrality” of a node based on the number of subgraphs in which the node takespart. It can be computed as
Sci =
n∑j=i
vj(i)2eλi ,
wherevj(i) is theith element of thejth eigenvector.
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Addendum: entropy-type summaries
The structure of a network is related to its reliability and speed of information propagation.If a random walk starts on nodei going to nodej, the probability that it goes through a givenshortest pathπ(i, j) between these vertices is
P(π(i, j)) =1
d(i)
∑b∈N (π(i,j))
1
d(b)− 1,
whered(i) is the degree of nodei, andN (π(i, j)) is the set of nodes in the pathπ(i, j)excludingi andj. Thesearch informationis the total information needed to identify one of allthe shortest paths betweeni andj and is given by
S(i, j) = − log2
∑π(i,j)
P(π(i, j)).
Similarly, an entropy can be defined based on the predictability of a message flow.
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Further reading:L. da F. Costa, F.A. Rodrigues, P.R. Villas Boas, G. Travieso (2007). Characterization of
complex networks: a survey of measurements. Advances in Physics 56, Issue 1 January 2007,167 - 242.
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Recap: network models
We looked at Bernoulli random graphs and their mixtures, Watts-Strogatz small worlds,Barabasi-Albert scale-free networks, and exponential random graphs. We saw that in thesemodels the summaries are dependent. As an extreme case, knowing the degree sequence mayalready completely specify the network.
Specific networks may allow for specific modelling, and summaries may be chosen to bestreflect the main features of the network.
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3 Fitting a model: parametric methodsParametricjust means that we have a finite set of parameters which fully specify the model.For example:
Bernoulli (Erdos-Renyi) random graphs
In the random graph model of Erdos and Renyi (1959), the (finite) node setV is given, say|V | = n. We denote the set of all potential edges byE; thus|E| =
(n2
). An edge between two
nodes is present with probabilityp, independently of all other edges. Herep is an unknownparameter.
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3.1 Parameter estimationIn classical (frequentist) statistics we often estimate unknown parameters via the method ofmaximum likelihood.
The likelihoodof the parameter given the data is just the probability of seeing the data wesee, given the parameter.
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Example:Bernoulli random graphs.
Our data is the network we see. We describe the data using the adjacency matrix, denoteit by x here because it is the realisation of a random adjacency matrixX. Recall that theadjacency matrix is the square|V | × |V |matrix where each entry is either 0 or 1;
xu,v = 1 if and only if there is an edge betweenu andv.
The likelihood of p being the true value of the edge probability if we seex is
L(p;x) =∏
(i,j)∈E
{pxi,j (1− p)1−xi,j
}.
For example,
L(0.5;x) =∏
(i,j)∈E
{(0.5)xi,j (1− 0.5)1−xi,j
}=
∏(i,j)∈E
0.5 = 0.5|E|.
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In general we can simplify
L(p;x) = (1− p)|E|∏
(i,j)∈E
(p
1− p
)xi,j
= (1− p)|E|(
p
1− p
)∑(i,j)∈E xi,j
.
Note thatt =∑
(i,j)∈E xi,j is the total number of edges in the random graph.
To maximise the likelihood, we often take logs, and then differentiate. Here this wouldgive
`(p;x) = logL(p;x)
= |E| log(1− p) + t log p− t log(1− p);
and
∂`(p;x)
∂p= − 1
1− p(|E| − t) +
t
p.
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To find a maximum we equate this to zero and solve forp,
t
p=
1
1− p(|E| − t) ⇐⇒ t(1− p) = p(|E| − t)
⇐⇒ t = p|E| ⇐⇒ p =t
|E| .
We can check that the second derivative of` is less than zero, so the fraction of edges that arepresent in the network,
p =t
|E| ,
is our maximum-likelihood estimator.
Maximum-likelihood estimators have attractive properties; under some regularity condi-tions they would not only converge to the true parameter as the sample size tends to infinity,but it would also be approximately normally distributed if suitably standardized, and we canapproximate the asymptotic variance.
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Maximum-likelihood estimation also works well in Erdos-Renyi Mixture graphs when thenumber of types is known, and it works well in Watts-Strogatz small world networks when thenumberk of nearest neighbours we connect to is known. When the number of types, or thenumber of nearest neighbours, is unknown, then things become messy.
In Barabasi-Albert models, the parameter would be the power exponent for the node de-gree, as occurring in the probability for an incoming node to connect to some nodei alreadyin the network.
In exponential random graphs, unless the network is very small, maximum-likelihood es-timation quickly becomes numerically unfeasible. Even in a simple model like
Prob(X = x) =1
κexp{λ1L(x) + λ2S2(x) + λ3S3(x) + λ4T (x)}
the calculation of the normalising constantκ becomes numerically impossible very quickly.
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3.2 Markov Chain Monte Carlo estimationA Markov chain is a stochastic process where the state at timen only depends on the stateat timen − 1, plus some independent randomness; a random walk is an example. A Markovchain isirreducible if any set of states can be reached from any other state in a finite numberof moves. The Markov chain isreversibleif you cannot tell whether it is running forwards intime or backwards in time. A distribution isstationaryfor the Markov chain if, when you startin the stationary distribution, one step after you cannot tell whether you made any step or not;the distribution of the chain looks just the same.
There are mathematical definitions for these concepts, but we only need the main resulthere:
If a Markov chain is irreducible and reversible, then it will have a unique stationary dis-tribution, and no matter in which state you start the chain, it will eventually converge to thisstationary distribution.
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We make use of this fact by looking at our target distribution, such as the distribution forX in an exponential random graph model, as the stationary distribution of a Markov chain.
This Markov chain lives on graphs, and moves are adding or deleting edges, as well asadding types or reducing types. Finding suitable Markov chains is an active area of research.
The ergm package has MCMC implemented for parameter estimation. We need to beaware that there is no guarantee that the Markov chain has reached its stationary distribution.Also, if the stationary distribution is not unique, then the results can be misleading. Unfortu-nately in exponential random graph models it is known that in some small parameter regionsthe stationary distribution is not unique. Another very active area of research.
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3.3 Assessing the model fitSuppose that we have estimated our parameters in our model of interest. We can now use thismodel to see whether it does actually fit the data.
To that purpose we study the (asymptotic) distributions of our summary statisticsnodedegree, clustering coefficient, andshortest path length. Then we see whether our observedvalues are plausible under the estimated model.
Often, secretly we would like to find that they are not plausible! Because then we canreject, say, the simple random graph model, and conclude that something more complicated isgoing on.
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3.4 A quick review of distributionsJust a quick reminder of some classical distributions which often appear as limiting distribu-tions.
The normal distributionN (µ, σ2)
This distribution has meanµ and varianceσ2. Its shape is given by the Bell curve. Itsdensity is awkward to write down, but probabilities can be calculated numerically.
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-4 -2 0 2 4
0.0
0.1
0.2
0.3
0.4
Normal Probability Densities
x
dens
ity
N(0,1)N(0,2)N(1,1)
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For a normally distributed random variable, around 2/3 of the time it will be withinσ (thestandard deviation, square root of the variance) of the mean.
Around 95% of the time it will be within2σ of the mean.
Around 99% of the time it will be within3σ of the mean.
Thus if an observed value is further than3σ away fromµ, we would find that ratherunusual; we would reject the null hypothesis that the data is normallyN (µ, σ2) distributed atthe level 1%.
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The Central Limit Theoremtells us that, in a sequence of independent identically dis-tributed observations with finite variance, the sample mean will converge to a normal distribu-tion, and the standardised sample mean will be approximately standard normal.
Fact: If the observations are dependent, but only ”weakly” dependent, then the CentralLimit Theorem still holds. (Another area of research.)
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Binomial Probability distibution with n=10 and p=0.6
x
P(X
=x)
0.00
0.10
0.20
Binomial Probability distibution with n=50 and p=0.6
x
P(X
=x)
0.00
0.04
0.08
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The Poisson distribution
When considering the occurrence of ”rare” events, an approximation with a Poisson dis-tribution is often more appropriate than a normal approximation.
The Poisson distributionlives on the non-negative integers; it has a parameterλ andXhas Poisson distribution with parameterλ if
P (X = k) = e−λ λk
k!, k = 0, 1, 2, . . .
It is relatively easy to calculate that mean and variance both equal toλ.
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Poisson Probability distibution with lambda=5
x
P(X
=x)
0.00
0.05
0.10
0.15
Poisson Probability distibution with lambda=10
x
P(X
=x)
0.00
0.04
0.08
0.12
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A famous result result states that if the expected numbernp of successes inn independentBernoulli trials with probability of successp each is such thatnp → λ asn → ∞ , then thenumber of successes in these trials is approximately Poisson distributed with parameterλ.
Note:np → λ asn →∞means that in each trial the probability of success is very small.A Poisson approximation is used forrare events.
The Poisson approximation is also good if the trials are only ”weakly” dependent.
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Binomial Probability distibution with n=50 and p=0.1
x
P(X
=x)
0.00
0.05
0.10
0.15
Poisson Probability distibution with lambda=5
x
P(X
=x)
0.00
0.05
0.10
0.15
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3.5 The distribution of summary statistics in Bernoulli randomgraphs
In a Bernoulli random graph onn nodes, with edge probabilityp, the network summaries arepretty well understood.
3.5.1 The degree of a random node
Pick a nodev, and denote its degree byD(v), say.
The degree is calculated as the number of neighbours of this node. Each of the other(n− 1) nodes is connected to our nodev with probabilityp, independently of all other nodes.Thus the distribution ofD(v) is Binomial with parametersn andp, for each nodev.
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Typically we look at relatively sparse graphs, and so a Poisson approximation applies. IfX denotes the random adjacency matrix, then, in distribution,
D(v) =∑
u:u 6=v
Xu,v ≈ Poisson((n− 1)p).
Note that the node degrees in a graph are not independent. We have seen last time thatthere isno graph on 6 nodes which has 5 nodes of degree 5 and 1 node of degree 1. SoD(v)doesnot stand for the average node degree.
Computer exercise:Simulate many Bernoulli random graphs and look at their averagenode degree distribution. Also: pick node at random in each of the simulated graph, find itsdegree, and look at the distribution of these degrees.
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How about the average degree of a node? Denote it byD. Note that the average does notonly take integer values, so would certainly not be Poisson distributed. But
D =1
n
n∑v=1
D(v) =2
n
n∑v=1
∑u<v
Xu,v,
noting that each edge gets counted twice. As theXu,v are independent, we can use a Poissonapproximation again, giving that
n∑v=1
∑u<v
Xu,v ≈ Poisson
(n(n− 1)
2p
)and so, in distribution,
D ≈ 2
nZ,
whereZ ∼ Poisson(
n(n−1)2
p)
.
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3.5.2 The clustering coefficient of a random node
Here it gets a little tricky already. Recall that theclustering coefficientof a nodev is,
C(v) =
∑u,w∈V Xu,vXw,vXu,w∑
u,w∈V Xu,vXw,v.
The ratio of two random sums is not easy to evaluate. If we just look at∑u,w∈V
Xu,vXw,vXu,w
then we see that we have a sum of dependent random variables.
Exercise:Calculate the covariance ofXu,vXw,vXu,w andXu,vXt,vXu,t, wherew 6= t.Recall: for random variablesX andY , the covariance is defined as
Cov(X, Y ) = E(XY )− E(X)E(Y ).
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Most 3-tuples(u, w, v) and(r, s, t), though, will not share an index, and henceXu,vXw,vXu,w
andXr,sXs,tXr,t will be independent. The dependence among the random variables overallis hence weak, so that a Poisson approximation applies. As
E∑
u,w,v∈V
Xu,vXw,vXu,w =
(n
3
)p3,
we obtain that, in distribution,
∑u,w.v∈V
Xu,vXw,vXu,w ≈ Poisson
((n
3
)p3
).
Similarly,
E∑
u,w∈V
Xu,vXw,v =
(n
2
)p2.
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K. Lin (2007) showed that, for the average clustering coefficient
C =1
n
∑v
C(v)
it is also true that, in distribution,
C ≈ 1
n(
n2
)p2
Z,
whereZ ∼ Poisson((
n3
)p3).
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Example.In the Florentine family data, we observe a total number of 20 edges, an averagenode degree of 2.5, and an average clustering coefficient of 0.1914894, with 16 nodes in total.Assess the null hypothesis that the data come from a Bernoulli random graph.
Let us assume that the null hypothesis is true, the data come from a Bernoulli randomgraph. Then we estimate
p =20(162
) =20× 2
16× 15=
1
6.
As in a Bernoulli random graphD ≈ 2nZ, whereZ ∼ Poisson
(n(n−1)
2p)
, under the null
hypothesis the average node degree would be
D ≈ 1
8Z,
whereZ ∼ Poisson(20). The probability under the null hypothesis thatD ≥ 2.5 would thenbe
P (Z ≥ 2.5× 8) = P (Z ≥ 20) ≈ 0.55,
so no reason to reject the null hypothesis.
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Exercise: Test the null hypothesis that the Florentine family data come from a Bernoullirandom graph using a test based on the average clustering coefficient.
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3.5.3 Shortest paths: Connectivity in Bernoulli random graph
Erdos and Renyi (1960)showed the following ”phase transition” for the connectedness of aBernoulli random graph.
If p = p(n) = log nn
+ cn
+ 0(
1n
)then the probability that a Bernoulli graph, denoted by
G(n, p) onn nodes with edge probabilityp is connected converges toe−e−c
.
Recall theO ando notation:f(n) = O(g(n)) asn →∞ if the fraction f(n)g(n)
is bounded
away from∞. If f(n) = o(g(n)) asn →∞ then the fractionf(n)g(n)
tends to zero asn →∞.
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Thediameterof a graph is the maximum diameter of its connected components; the diam-eter of a connected component is the longest shortest path length in that component.
Chung and Lu (2001)showed that, ifnp ≥ 1 then, asympotically, the ratio between thediameter and log n
log(np)is at least 1, and remains bounded above asn →∞.
If np → ∞ then the diameter of the graph is(1 + o(1)) log nlog(np)
. If nplog n
→ ∞, then thediameter is concentrated on at most two values.
In the Physics literature, the valuelog nlog(np)
is used for the average shortest path length in aBernoulli random graph. This has hence to be taken with a lot of grains of salt.
While we have some idea about how the diameter (and, relatedly, the shortest path length)behaves, it is an inconvenient statistics for Bernoulli random graphs, because the graph neednot be connected.
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3.6 The distribution of summary statistics in Watts-Strogatz smallworlds
Recall that in this model we arrange then nodes ofV on a lattice. Then hard-wire each nodeto its k nearest neighbours on each side on the lattice, wherek is small. Thus there arenkedges in this hard-wired lattice. Now introduce random shortcuts between nodes which are nothard-wired; the shortcuts are chosen independently, all with the same probabilityφ.
Thus the shortcuts behave like a Bernoulli random graph, but the graph will necessarily beconnected. The degreeD(v) of a nodev in the Watts-Strogatz small world is hence distributedas
D(v) = 2k + Binomial(n− 2k − 1, φ),
taking the fixed lattice into account. Again we can derive a Poisson approximation whenp issmall; see K.Lin (2007) for the details.
For the clustering coefficient there is a problem - triangles in the graph may now appear inclusters. Each shortcut between nodesu andv which are a distance ofk + a ≤ 2k apart onthe circle createsk − a− 1 triangles automatically.
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Thus a Poisson approximation will not be suitable; instead we use acompound Poissondistribution. A compound Poisson distribution arises as the distribution of a Poisson numberof clusters, where the cluster sizes are independent and have some distribution themselves. Ingeneral there is no closed form for a compound Poisson distribution.
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The compound Poisson distribution also has to be used when approximating the numberof 4-cycles in the graph, or the number of other small subgraphs which have the clumpingproperty.
It is also worth noting that when counting the joint distribution of the number of trianglesand the number of 4-cycles, these counts are not independent, not even in the limit; a bivariatecompound Poisson approximation with dependent components is required. See Lin (2007) fordetails.
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3.6.1 The shortest path length
LetD denote shortest distance between two randomly chosen points, and abbreviateρ = 2kφ.Then (Barbour + Reinert) show that uniformly in|x| ≤ 1
4log(nρ),
P
(D >
1
ρ
(1
2log(nρ) + x
))=
∫ ∞
0
e−y
1 + e2xydy + O
((nρ)−
15 log2(nρ)
)if the probability of shortcuts is small. If the probability of shortcuts is relatively large, thenDwill be concentrated on one or two points.
Note thatD is the shortest distance between two randomly chosen points,not the averageshortest path. Again the difference can be considerable (Computer exercise).
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Dependent sampling
Our data are usually just one graph, and we calculate all shortest paths. But there is muchoverlap between shortest paths possible, creating dependence
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Simulation:n = 500, k = 1, φ = 0.01
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Simulation: dependent vs independent
We simulate 100 replicas, and calculate the average shortest path length in each network.We compare this distribution to the theoretical approximate distribution; we carry out 100chi-square tests:
n k φ E.no mean p-value max p-value300 1 0.01 3 1.74 E-09 8.97 E-08
0.167 50 0.1978 0.89132 0.01 6 0 0
1000 1 0.003 3 1.65E-13 3.30 E-120.05 50 0.0101 0.1124
2 0.03 60 0.0146 0.2840
Thus the two statistics are close if the expected numberE.no of shortcuts is large (or verysmall); otherwise they are significantly different.
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Recall: chi-square test of goodness-of-fit
We want to test whether a data set comes from a conjectured (null) distribution. We groupour data into cells such that under the null distribution, the count in each cell is expected to beat least 5.
Then we take the sum
X2 =∑cellsi
(Observedi − Expectedi)2
Expectedi.
If the data come indeed from the null distribution, thenX2 will be approximately chi-squareddistributed, with degrees of freedom ”number(cells) - number(fitted parameters) - 1”.
Thep-value is the probability of seeingX2 as least as large as the observed value if thenull distribution is the correct distribution.
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Aside: When comparing continuous distributions, theKolmogorov-Smirnov testis anothernonparametric alternative, as areWilcoxon tests.
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3.7 The distribution of summary statistics in Barabasi-Albert mod-els
The node degree distribution is given by the model directly, as that is how it is designed.
The clustering coefficient depends highly on the chosen model. In the original Barabasi-Albert model, when only one new edge is created at any single time, there will be no triangles(beyond those from the initial graph). The model can be extended to match any clusteringcoefficient, but even if only two edges are attached at the same time, the distribution of thenumber of the clustering coefficient is unknown to date.
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The expected value, however, can be approximated. Fronczaket al. (2003) studied themodels where the network starts to grwo from an initial cluster ofm fully connected nodes.Each new node that is added to the network createdm edges which connect it to previouslyadded nodes. The probability of a new edge to be connected to a nodev is proportional to thedegreed(v) of this node. If both the number of nodes,n, andm are large, then the expectedaverage clustering coefficient is
EC =m− 1
8
(log n)2
n.
The average pathlenghincreases approximately logarithmically with network size. Ifγ = 0.5772 denotes the Euler constant, then Fronczaket al. (2004) show for the mean averageshortest path length that
E` ∼ log n− log(m/2)− 1− γ
log log n + log(m/2)+
3
2.
The asymptotic distribution is not understood.
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3.8 The distribution of summary statistics in exponential randomgraph models
The distribution of the node degree, clustering coefficient, and the shortest path length is poorlystudied in these models. One reason is that these models are designed to predict missing edges,and to infer characteristics of nodes, but their topology itself has not often been of interest.
The summary statistics appearing in the model try to push the random networks towardscertain behaviour with respect to these statistics, depending on the sign and the size of theirfactorsθ.
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When only the average node degree and the clustering coefficient are included in themodel, then a strange phenomenon happens. For many combinations of parameter valuesthe model produces networks that are either full (every edge exists) or empty (no edge exists)with probability close to 1. Even for parameters which do not produce this phenomenon, thedistribution of networks produced by the model is often bimodal: one mode is sparsely con-nected and has a high number of triangles, while the other mode is densely connected but witha low number of triangles. Again: active research.
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4 Statistical tests for model fit: nonparametric meth-ods
What if we do not have a suitable test statistic for which we know the distribution? We needsome handle on the distribution, so here we assume that we can simulate random samples fromour null distribution. There are a number of methods available.
4.1 Quantile-quantile plotsIt is often a good idea to use plots to visually assess the fit. A much used plot in Statistics aquantile-quantile plot.
The quantilesof a distribution are its ”percent points”; for example the 0.5 quantile isthe 50 % point, i.e. the median. Mathematically, the(sample) quantilesqα, are defined for0 ≤ α ≤ 1 so that a proportion of at leastα of the data are less or equal toqα and a proportionof at least1−α is greater or equal toqα. There are many (at least 8) definitions ofqα if αn isnot an integer.
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We plot the quantiles of our observed (empirical) distribution against the quantiles of ourhypothesised (null) distribution; if the two distributions agree, then the plot should result in aroughly diagonal line.
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Example: Simulate 1,000 random variables from a normal distribution. Firstly: mean zero,variance 1; secondly: mean 1, variance 3. Both QQ-plots are satisfactory.
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We can also use a quantile-quantile plot for two sets of simulated data, or for one set ofsimulated data and one set of observed data. The interpretation is always the same: if the datacome from the same distribution, then we should see a diagonal line; otherwise not. Here wecompare 1000 Normal (0,1) variables with 1000 Poisson (1) variables - clearly not a good fit.
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4.2 Monte-Carlo testsThe Monte Carlo test, attributed to Dwass (1957) and Barnard (1963), is an exact procedure ofvirtually universal application and correspondingly widely used.
Suppose that we would like to base our test on the statisticT0. We only need to be able tosimulate a random sampleT01, T02, . . . from the distribution, call itF0, determined by the nullhypothesis. We assume thatF0 is continuous, and, without loss of generality, that we rejectthe null hypothesisH0 for large values ofT0. Then, provided thatα = m
n+1is rational, we
can proceed as follows.
1. Observe the actual valuet∗ for T0, calculated from the data
2. Simulate a random sample of sizen from F0
3. Order the set{t∗, t01, . . . , t0n}4. RejectH0 if the rankof t∗ in this set (in decreasing order) is> m.
The basis of this test is that, underH0, the random variableT ∗ has the same distribution as theremainder of the set and so, by symmetry,
P(t∗ is among the largestm values) =m
n + 1.
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The procedure is exact however smalln might be. However, increasingn increases thepower of the test. The question of how largen should be is discussed by Marriott (1979), seealso Hall and Titterington (1989). A reasonable rule is to choosen such thatm > 5. Note thatwe will need more simulations to test at smaller values ofα.
An alternative view of the procedure is to count the numberM of simulated values> t∗.ThenP = M
nestimates the true significance levelP achieved by the data, i.e.
P = P(T0 > t∗|H0).
In discrete data, we will typically observe ties. We can break ties randomly, then the aboveprocedure will still be valid.
Unfortunately this test does not lead directly to confidence intervals.
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For random graphs, Monte Carlo tests often use shuffling edges with the number of edgesfixed, or fixing the node degree distribution, or fixing some other summary.
Suppose we want to see whether our observed clustering coefficient is ”unusual” for thetype of network we would like to consider. Then we may draw many networks uniformly atrandom from all networks having the same node degree sequence, say. We count how often aclustering coefficient at least as extreme as ours occurs, and we use that to test the hypothesis.
In practice these types of test are the most used tests in network analysis. They are calledconditional uniform graph tests.
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Some caveats:
In Bernoulli random graphs, the number of edges asymptotically determines the numberof triangles when the number of edges is moderately large. Thus conditioning on the numberof edges (or the node degrees, which determine the number of edges) gives degenerate results.More generally, we have seen that node degrees and clustering coefficient (and other subgraphcounts) are not independent, nor are they independent of the shortest path length. By fixingone summary we may not know exactly what we are testing against.
”Drawing uniformly at random” from complex networks is not as easy as it sounds. Algo-rithms may not explore the whole data set.
”Drawing uniformly at random”, conditional on some summaries being fixed, is related tosampling from exponential random graphs. We have seen already that in exponential randomgraphs there may be more than one stationary distribution for the Markov chain Monte Carloalgorithm; this algorithm is similar to the one used for drawing at random, and so we may haveto expect similar phenomena.
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4.3 Scale-free networksBarabasi and Albert introduced networks such that the distribution of node degrees the type
Prob(degree = k) ∼ Ck−γ
for k → ∞. Such behaviour is calledpower-law behaviour; the constantγ is called thepower-law exponent. The networks are also calledscale-free:
If α > 0 is a constant, then
Prob(degree = αk) ∼ C(αk)−γ ∼ C′k−γ ,
whereC′ is just a new constant. That is, scaling the argument in the distribution changes theconstant of proportionality as a function of the scale change, but preserves the shape of thedistribution itself. If we take logarithms on both sides:
log Prob(degree = k) ∼ log C − γ log k
log Prob(degree = αk) ∼ log C − γ log α− γ log k;
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scaling the argument results in a linear shift of thelog probabilities only. This equation alsoleads to the suggestion to plot thelog relfreq(degree = αk) of the empirical relative degreefrequencies againstlog k. Such a plot is called alog-log plot. If the model is correct, then weshould see a straight line; the slope would be our estimate ofγ.
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Example: Yeast data.
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These plots have a lot of noise in the tails. As an alternative,Newman (2005)suggeststo plot thelog of the empirical cumulative distribution function instead, or, equivalently, ourestimate for
log Prob(degree ≥ k).
If the model is correct, then one can calculate that
log Prob(degree ≥ k) ∼ C′′ − (γ − 1) log k.
Thus a log-log plot should again give a straight line, but with a shallower slope. The tails aresomewhat less noisy in this plot.
In both cases, the slope is estimated by least-squares regression: for our observations,y(k)(which could be log probabilities or log cumulative probabilities, for example) we find the linea + bk such that ∑
(y(k)− a− bk)2
is as small as possible.
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As a measure of fit, the sample correlationR2 is computed. For general observationsy(k)andx(k), for k = 0, 1, . . . , n, with averagesy andx, it is defined as
R =
∑k(x(k)− x)(y(k)− y)√
(∑
k(x(k)− x)2)(∑
k(y(k)− y)2).
It measures the strength of the linear relationship.
In linear regression,R2 > 0.9 would be rather impressive. However, the rule of thumb forlog-log plots is that1. R2 > 0.992. The observed data (degrees) should cover at least 3 orders of magnitude.
Examples include the World Wide Web at some stage, when it had around109 nodes. Thecriteria are not often matched.
Computer exercise:Generate random samples from your favourite probability distribution,make a log-log plot.
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A final issue for scale-free networks: It has been shown (Stumpf et al. (2005)that whenthe underlying real network is scale-free, then a subsample on fewer nodes from the networkwill not be scale-free. Thus if our subsample looks scale-free, the underlying real network willnot be scale-free.
In biological network analysis, is is debated how useful the concept of ”scale-free” be-haviour is, as many biological networks contain relatively few nodes.
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Further referencesA.D. Barbour and G. Reinert (2001). Small Worlds. Random Structures and algorithms 19, 54- 74.
A.D. Barbour and G. Reinert (2006). Discrete small world networks. Electronic Journal ofProbability 11, 12341283.
G. Barnard (1963). Contribution to the discussion of Bartlett’s paper. J. Roy. Statist. Soc. B,294.
F. Chung and L. Lu (2001). The diameter of sparse random graphs. Advances in AppliedMath. 26, 257–279.
M. Dwass (1957). Modified randomization tests for nonparametric hypotheses.Ann. Math.Stat.28, 181–187.
A. Fronczak, P. Fronczak and Janusz A. Holyst (2003). Mean-field theory for clustering coef-
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ficients in Barabasi-Albert networks Phys. Rev. E 68, 046126.
A. Fronczak, P. Fronczak, Janusz A. Holyst (2004). Average path length in random networksPhys. Rev. E 70, 056110.
P. Hall and D.M. Titterington (1989). The effect of simulation order on level accuracy andpower of Monte Carlo tests. J. Roy. Statist. Soc. B, 459.
K. Lin (2007). Motif counts, clustering coefficients, and vertex degrees in models of randomnetworks. D.Phil. dissertation, Oxford.
F. Marriott (1979). Barnard’s Monte Carlo tests: how many simulations? Appl. Statist.28, 75–77.
M.E.J. Newman (2005). Power laws, Pareto distributions and Zipf’s law. ContemporaryPhysics 46, 323351.
M. P. H. Stumpf, C.Wiuf and R. M. May (2005). Subnets of scale-free networks are not scale-
0-108
free: Sampling properties of networks. Proc Natl Acad Sci U S A. 2005 March 22; 102(12):4221 - 4224.
0-109
0-110
0-111
0-112
-3 -2 -1 0 1 2 3
-3-2
-10
12
3
Normal Q-Q Plot
Theoretical Quantiles
N(0
,1) r
ando
m s
ampl
es: s
ampl
e qu
antil
es
-3 -2 -1 0 1 2 3
-50
510
Normal Q-Q Plot
Theoretical Quantiles
N(1
,3) r
ando
m s
ampl
es: s
ampl
e qu
antil
es
0-113
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−3 −2 −1 0 1 2 3
01
23
45
x
y
0-114
01
23
4
-8-7-6-5-4-3-2
log
k
log P(k)
01
23
4
-8-6-4-2
log
k
log P(greater than k)
0-115
