The structure and function of complex networks
The structure and function of complex networks(2003)Author:
M.E.J. NewmanPresenter: Guoliang LiuDate:5/4/2012OutlineNetworks in
the real worldProperties of networksRandom graphsExponential random
graphs and Markov graphsThe small-world modelModels of network
growthProcesses happening or happenedNetworks in the real
worldOutlineNetworks in the real worldProperties of networksRandom
graphsExponential random graphs and Markov graphsThe small-world
modelModels of network growthProcesses happening or
happenedProperties of networksThe small-world effectTransitivity or
clusteringDegree distributionsNetwork resilienceMixing
patternsDegree correlationsCommunity structureNetwork
navigationOther network propertiesThe small-world effectStarted
from 1960s by Stanley Milgram6 steps between two nodes.Undirected
networks, define l to be the mean geodesic distance between vertex
pairs in a network:
In case that networks have more than one component
Small-world effect: the value l scales logarithmically or slower
with network size or fixed mean degree.
6Transitivity or clusteringIn many networks it is found that if
vertex A is connected to vertex B and vertex B to vertex C, then
there is a heightened probability that A is connected to C.
Cluster coefficient:Global valueLocal value
Transitivity or clusteringAn example to calculate clustering
coefficient C
Transitivity or clusteringDegree distributions
Degree distributionsDegree distributionsDegree distributions
Degree distributions
Network resilienceResilience to the removal of vertices(or
edges.). Popular in Epidemiology, in CS, the attack in complex
networks.How to remove?Remove vertices at randomTarget some
specific class of vertices, such as those with the highest
degree.How to measure resilience?Distances l increasement on
average.Thorough study by Holme et al. attack vulnerability of
complex networksMixing patternsIn most kinds of networks, and the
probabilities of connection between vertices often depends on
types.Food webSocial network of couples(In social also called
assortative mixing or homophily)
Mixing patternsHow to measure mixing patterns?Eij: the number of
edges in a network that connect vertices of types i and j.
eij measure the fraction of edges that fall between vertices of
types i and j.
Assortative mixing coefficient:
Degree correlationsA special case of assortative mixing
according to a scalar vertex property is mixing according to vertex
degree, referred as degree correlation.Community structureNetwork
navigationIdeas from Stanley Milgrams experiment.Not only
small-world effect.But also people are good at finding them.Target:
build efficient database structures or better peer-to-peer computer
networks. E.g. local search in unstructured networksOther
propertiesGiant componentBetweenness
centralitySelf-similarityOutlineNetworks in the real
worldProperties of networksRandom graphsExponential random graphs
and Markov graphsThe small-world modelModels of network
growthProcesses happening or happenedRandom graphsPoisson random
graphGeneralized random graphsPoisson random graphsWhy we talk
about Poisson random graphs?Basic intuition about the networks
behaviors from the study of random graph.Poisson distribution is a
classic one and later ideas all started from Poisson random
graphPoisson random graphsPoisson random graphsEarly work, simple
model of a networkGn,p, each pair connects with probability pLater,
consider the mean degree z = p(n-1)
When we focus on low-density, low-p state and high-density,
high-p state.A single giant componentThe remainder of vertices
occupying smaller components with exponential size distribution and
finite mean size,
Poisson random graphsLet u be the fraction of vertices on the
graph that do not belong to the giant component.
The fraction S of the graph occupied by the giant component is S
= 1-u and
Poisson random graphs
Poisson random graphsTalk about again about Small-world effect
in Poisson random graphs.The mean number of neighbors a distance l
away from a vertex in a random graph is zd , and hence the value of
d needed to encompass the entire network zl ~n. Thus l =
logn/logz.Shortcomings of PoissonLow clustering coefficient C = p,
when n is large, p~ n-1 ~0Unlike the real-world distribution.
Generalized random graphsConfiguration modelThe vertex at the
end of a randomly chosen edge is proportional to kpk .Excess
degree(how many edges there are leaving such a vertex other than
the one we arrived along) qk .
Define two generating functions
Generalized random graphsConsider the giant component:
Consider the clustering coefficient
Generalized random graphs
Generalized random graphsDirected graphsBipartite graphsDegree
correlationsOutlineNetworks in the real worldProperties of
networksRandom graphsExponential random graphs and Markov graphsThe
small-world modelModels of network growthProcesses happening or
happenedExponential random graphs and Markov graphsBackground:
properties such as transitivity is not incorporated into random
graph models.Exponential random graphs and Markov graphsExponential
random graphs (P* models) is a set of measurable properties of a
single graph. is a set of inverse-temperature of field
parametersEach graph G appears with probability:
Partition function Z is
Exponential random graphs and Markov graphsThe calculation of
the ensemble average of a graph observable is then found by taking
a suitable derivative of free energy f=-logZ.There are ways to
express f in closed form. But carrying through the entire
field-theoretic program is not easyThe question of how to carry out
calculations in exponential random graph ensembles is open
OutlineNetworks in the real worldProperties of networksRandom
graphsExponential random graphs and Markov graphsThe small-world
modelModels of network growthProcesses happening or happenedThe
small-world modelLess sophisticated but more tractable model of a
network with high transitivity.Built on lattices of any dimension
or topology( Take one dimension with L vertices, each vertex has k
neighbors with fewer spacing away)Each edge Rewires randomly with
possibility p (create shortcuts)
The small-world modelExtreme 1, P=0, clustering coefficient
C=(3k-3)/(4k-2), Mean geodesic distance tend to L/4k for large
LExtreme 2, P=1,C~2k/LMean geodesic distance logL/logKNumerical
simulation by Watts and Strogatz showed there exists a sizable
region in between there two extremes for which the model has both
low path lengths and high transitivity.
The small-world modelModification of the rewiring methodsBoth
ends of edges can be rewiredAllow double edges and self
edges(maintain original edges)
The small-world modelA. Clustering coefficientBy Barrat and
Weigt
By Newman
B. Degree distribution(Since it is not the goal, behaves badly
compared with real-world networks)
The small-world modelC. Average path lengthP=0, l~ L/4k ,
large-world P=1, l~logL, small-world0
