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The Structure and Function of Complex Networks
Part I
Jim Vallandingham
M. E. J. Newman
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Introduction
• Paper is a review of – Network types – Common network properties– Network models
• Examine large networks– Millions / Billions of nodes
• Statistical methods are an attempt to find something to “play the part of the eye” in current network analysis
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Organization
I. DefinitionsII. Types of NetworksIII. Properties of NetworksIV. Random GraphsV. Extensions to Random GraphsVI. Markov Graphs
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Definitions
• Network | Graph: – Composed of items : vertices / nodes– Connections between vertices : edges
• Directed edge:– One that runs in only one direction
• Degree:– Number of edges connected to a vertex– Directed graph has an in-degree and out-degree
for each vertex
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Definitions
Vertex Degree
1 2
2 3
3 2
4 3
5 3
6 1
Undirected Graph
Vertex In-Degree Out-Degree
1 0 2
2 2 0
3 2 2
4 1 1
Directed Graph
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Definitions
• Component:– Set of vertices connected together by edges
• Geodesic Path:– The shortest path through the network from one
vertex to another.– Can be multiple geodesic paths between two vertices
• Diameter:– Length of the longest geodesic path – In terms of edges
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DefinitionsThree components in a network
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Types of Networks
A. Social NetworksB. Information NetworksC. Technological NetworksD. Biological Networks
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Social Networks
• Definition:– Set of people or groups of people with some
interaction pattern between them• Early Work:
– “Southern Women Study”• Social circles of small southern town in 1936
– Social networks of factory workers in 1930’s• Current Work:
– Business communities– Sexual partner studies
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Social NetworksInternet Chat Relay (IRC) communications between individuals
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Social NetworksDating relationships between students in
a high school
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Social Networks
• “Small-World” experiments– Looked at the distribution of path lengths in
network– Participants were asked to pass letter around in an
attempt to reach a specific individual– Shown that there is usually short path between
any two vertices in a network– Later became the basis of the “6 degrees of
separation” concept.
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Social Networks
• Problems with traditional social networks– Based on questionnaires
• Labor intensive process which limits the size of network• Source of bias which skews results
– “Friend” might mean different thing to different people
• Presents need for other methods for probing social networks
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Social Networks
• Collaboration Networks– Affiliation networks in which vertices collaborate
in groups of some sort– Edges are created between pairs of nodes that
have a common group membership
– Classic Example : IMDB – Internet Movie Database• Vertices are actors• Edges indicate two actors have been in the same film
together
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Social Networks
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Social Networks
• Other social network data sources– Phone Calls– Email– Instant Messaging
• Produce Millions of pieces of data a day – Demonstrate the need for new analytical methods
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Information Networks
• Also known as “knowledge networks”• Definition:
– Representation of how information moves through a population or group
• Classic Example:– Network of citations between academic papers
• Directed edges• Mostly acyclic
– Papers can only cite other papers already written and not future papers. (not always true)
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Information NetworksCitation Network for Inferring network mechanisms: The Drosophila melanogaster protein interaction network
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Information Networks
• The World Wide Web– Network of information containing pages
• Vertices are the pages themselves• Edge is created when one page links to another
– No constraints as seen in the citation network• Cycles • Multiple edges between vertices
– Power-law in-degree and out-degree distributions
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Information Networks
Graph of Relationships between Facebook pages. Example of an Information Network with Social Network aspects.
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Information Networks
• Preference Networks– Includes two kinds of vertices
• Individuals • Objects of their preference
– Example: books or films
– Edges connect vertices of different types– Edges can be weighted
– Example of Bipartite Information Network
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Technological Networks
• Definition:– Man-made networks designed for the
transportation of a resource or commodity
• Examples– Power grid– Airline routes– The Internet
• Physical network of machines
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Technological Networks
Bandwidth transfer in Europe between countries
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Biological Networks
• Wide variety of biological systems can be represented as networks
• Metabolic Pathways– Vertices are metabolic substrates and products– Directed edges between known reaction exists
that produces product from substrate• Protein Interactions
– Mechanistic physical interactions between proteins
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Biological Networks
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Biological Networks
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Biological NetworksPortion of yeast protein interactions
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Biological Networks
• Gene Regulatory Networks– Expression of protein coded by particular genes– Controlled by other proteins
• Act as inducers and inhibitors
– Vertices represent proteins– Edges represent dependencies between proteins– One of the first networked dynamical systems for
which large-scale modeling attempts were made
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Biological Networks
• Food Webs– Vertices represent species– Directed edge indicates predatory relationship
• Could be the other way in terms of carbon movement
• Neural Networks– Actual biological neuron pathways
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Biological Networks
Reef fish food web
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Biological NetworksRat hippocampal neurons
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Properties of Networks
• Look at features that are common to many types of networks
• May or may not encode important or relevant information for any one graph
• Might be suggestive of the mechanisms in how real networks are formed
• Most involve how real networks are different than random graphs
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Properties of Networks
• Small-World Effect• Transitivity• Degree Distribution• Network Resilience• Mixing Patterns• Degree Correlation• Community Structure• Network Navigation
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Properties of Networks
• Small-World Effect• Transitivity• Degree Distribution• Network Resilience• Mixing Patterns• Degree Correlation• Community Structure• Network Navigation
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Properties: Small-World Effect
• Most pairs of vertices are connected by a relatively short path through the network
• Distance between any two vertices in a graph is usually much smaller than the total number of vertices
• Deals with the geodesic distance property– Uses Mean Geodesic Distance :
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Properties: Small-World Effect
• can be measured in O(mn) time where• m is the number of edges• n is the number of vertices
– Usually is much smaller than n • Can be problematic if there are multiple
components in the graph– Represented as ∞ edges and thus ∞ average
geodesic distance– Alternate way is to exclude any vertices that
connect multiple components
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Properties: Small-World Effect
• This property implies that spread of x through real networks occurs fast– Rumor– Information
• Mathematically obvious– If number of vertices within distance r grows
exponentially– Value of will increase as log n – “small-world” can refer to networks in which value of
l scales logarithmically or slower with network size
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Properties: Small-World Effect
• Biological example: protein-protein interactions in the yeast, S. cerevisiae
• Vertices: 1870• Edges: 2240
• : 6.80
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Properties of Networks• Small-World Effect
• Transitivity• Degree Distribution• Network Resilience• Mixing Patterns• Degree Correlation• Community Structure• Network Navigation
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Properties: Transitivity
• Probability that if vertex A is connected to vertex B, and vertex B is connected to vertex C, than vertex A will also be connected to vertex C
• In social network terms: the friend of your friend is likely also to be your friend
• Also known as clustering – This is confusing as it has another meaning– Quantified using the Clustering Coefficient
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Properties: Transitivity
C : Clustering coefficient
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Properties: Transitivity
1
8
2
1
3
4
567
8
Fraction of Transitive Triples
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Properties: Transitivity
Can also be defined locally for each vertex
With this value the definition of C becomes:
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Properties: TransitivityAlternative method for clustering coefficient
1
C1 = 1 / 1 = 1
2C2 = 1
C3 = 1/6
C4 = 0
C5 = 0
3
4
5
C = 1/5(1+1+(1/6))
C = 13/30
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Properties: Transitivity
• Two definitions labeled C(1) and C(2) in text• Effectively reverses the order of the operations:
– Taking the ratio of triangles to triples – Averaging over vertices
• C(2) calculates the mean of the ratio• C(1) calculates the ratio of the means• C(2) tends to weigh contributions of low-degree
vertices more heavily– Give significantly different results
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Properties: Transitivity
• Ci used often as well in sociological literature– Called “network density”
• Both C(1) and C(2) usually are significantly higher in real networks than random graphs
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Properties of Networks• Small-World Effect• Transitivity
• Degree Distribution• Network Resilience• Mixing Patterns• Degree Correlation• Community Structure• Network Navigation
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Properties: Degree Distributions
• Degree of a vertex is the number of edges connected to that vertex
• pk is the probability that a vertex chosen at random has a degree k
• Look at by creating a histogram of pk – Called the degree distribution for that network
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Properties: Degree Distributions
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Properties: Degree Distributions
• Real World networks are usually highly right-skewed– Long right tail of values above the mean
• Measuring of the tail is difficult– small sample size in that section– Usually noisy
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Properties: Degree DistributionsHistograms depicting the Noise and lack of measurements
indicative of the tail section of the degree distribution
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Properties: Degree Distributions• Many real world graph degree distributions
follow power laws in their tails– pk ~ k-α
• for some constant α• Others have exponential tails
– pk ~ e-k/κ
• Knowing this makes power-law and exponential distributions easy to find experimentally– Plot on logarithmic scales : power laws– Semi-logarithmic scales : exponentials
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Properties: Degree Distributions
Power law Exponential
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Properties: Degree Distributions
• Power-law degree distributions sometimes called scale-free networks
• Include networks of:– World wide web– Metabolic pathways– Telephone calls
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Properties of Networks• Small-World Effect• Transitivity• Degree Distribution
• Network Resilience• Mixing Patterns• Degree Correlation• Community Structure• Network Navigation
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Properties: Network Resilience
• How resilient is a network to the removal of its vertices– How the geodesic distance is affected by node
deletion
• Two main removal processes discussed1.Random removal of vertices2.Targeted removal
• Usually remove the vertices with highest degrees
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Properties: Network Resilience
• Two recent studies done on the resilience of the Internet and World Wide Web– One study found that these networks resilient to
random deletions but vulnerable to targeted ‘attacks’
– Other study found the opposite: WWW resilient to targeted attack as well as deletion of all vertices with degree greater than 5 would be needed
– Difference attributed to the high skew of degree distribution as only a very small fraction of nodes have degree greater than 5
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Properties: Network Resilience
• Biological Example:– Metabolic network of yeast
Diameter: total of all path lengths divided by total
number of paths
Targeted
Random
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Properties of Networks• Small-World Effect• Transitivity• Degree Distribution• Network Resilience
• Mixing Patterns• Degree Correlation• Community Structure• Network Navigation
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Properties: Mixing Patterns
• What types of vertices associate with other types of vertices
• Examples:– Food web:
• Many links between herbivores and carnivores• Few links between carnivores and plants
– Internet:• Many links between end-users and ISP’s• Few between end-users and backbone
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Properties: Mixing Patterns
• Quantified by assortativity coefficient
• Other ways to look at assortative mixing– By scalar characteristics
• Age, income
– Vector characteristics• Location : 2D vector
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Properties of Networks• Small-World Effect• Transitivity• Degree Distribution• Network Resilience• Mixing Patterns
• Degree Correlation• Community Structure• Network Navigation
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Properties: Degree Correlations
• Special case of assortative mixing– Based on a particular scalar vertex property :
degree
• Do high-degree vertices ‘prefer’ other high-degree?
• Do high-degree associate more with low-degree vertices?
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Properties: Degree Correlations
• Several different ways to quantify:– Two-dimensional histogram – One-parameter curve based on the degree– A single number
• Positive for assortatively mixed networks• Negative for disassortative networks
• Social networks tend to be assortative• All other networks discussed are disassortative
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Properties: Degree Correlations
Degree Increasing
Deg
ree
Incr
easi
ng
Highest degree correlation
Yeast protein interactions
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Properties of Networks• Small-World Effect• Transitivity• Degree Distribution• Network Resilience• Mixing Patterns• Degree Correlation
• Community Structure• Network Navigation
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Properties: Community Structure
• Structure and formation of groups in the network• Social Networks:
– People tend to divide into sub-sections based on common interests, occupations, etc.
• Cluster Analysis– Extracting community structure from a network– Assigns connection strength to vertex pairs of interest– Finished process of cluster analysis can be
represented by a tree or dendrogram
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Properties: Community StructureGroups in protein interactions
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Properties of Networks• Small-World Effect• Transitivity• Degree Distribution• Network Resilience• Mixing Patterns• Degree Correlation• Community Structure
• Network Navigation
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Properties: Network Navigation
• Finding paths in networks• Use some domain knowledge about the network
– Example: small-world experiments – people knew who to give the letter to so as to reach the destination quickly
• If it were possible to construct artificial networks that were easy to navigate in the same way social networks seem to be, then they could be used for databases or P2P networks
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Other Properties
• Largest Component Size– The “Giant component”
• Betweenness Centrality: – Number of geodesic paths between other vertices
that run through a particular vertex
• Recurrent Motifs:– Small sub-graphs that repeat in the network
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Random Graphs
• Poisson Random Graphs
• Configuration Model• Extensions to Random Graphs• Markov Graphs
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Poisson Random Graphs
• Developed by – Solomnoff and Rapoport (1951)– Erdős and Rényi (1959)
• Used as a “straw man” when discussing graph theory
• Most of the interesting work is in how real world graphs are not like random graphs
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Poisson Random Graphs
• Building Random Graphs:• Very simple process
– Take some number n of vertices – Connect each pair with a probability p
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Poisson Random Graphs
• Many properties of the random graph are exactly solvable in the limit of large graph size.
• Probability of a vertex having degree k :– (Degree Distribution)
Hence the name ‘Poisson’
Exact in large graph limit
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Poisson Random Graphs
• Expected structure varies with p.• Most important property: phase transition
– From low-density, low-p state • Containing few edges and all components are small
– To high-density, high-p state• Extensive fraction of all vertices are joined together in
single giant component• Giant component is main significant feature of random
graphs discussed in this paper
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Poisson Random Graphs
• Two properties in random graphs :– Giant component size
• Calculating the expected size of the giant component:
– Mean size of the non-giant components:
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Poisson Random Graphs
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Poisson Random Graphs
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Poisson Random Graphs
• Models– Small-world effect
• Typical distance through network log n / log z
• Does not Model – Clustering coefficient
• Lower than real world– Degree Distribution
• Poisson instead of power-law / exponential– Random Mixing Pattern– No community structure– Navigation is impossible using local algorithms
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Poisson Random Graphs
Linear graph Logarithmic graphScale-freerandom
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Poisson Random Graphs
• Still, it forms the basis of our basic intuition about how networks behave
• Giant component & phase transition are ideas that underlie much of graph theory
• Many future models started with this random graph as a springboard
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Random Graphs
• Poisson Random Graphs
• Configuration Model• Extensions to Random Graphs• Markov Graphs
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Configuration model• Trying to make random graphs more realistic• Configuration model incorporates idea of non-
Poisson degree distribution• Building configuration model:
– pk : degree distribution : the fraction of vertices having degree k
– Degree sequence a set of n values of the degrees ki of vertices i = 1 … n
• Visualized as giving each vertex ki spokes sticking out of it
– Choose pairs of spokes at random and connect them
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Configuration model
• Two important points on the configuration model 1. pk is the distribution of degrees of vertices
• But not the degree of the vertex reached by following a randomly chosen edge
• k edges that arrive at a vertex of degree k, we are k times as likely to arrive at that vertex as some other vertex of degree = 1.
• Thus degree distribution of a random vertex is proportional to k pk
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Configuration model
2. Chance of finding a loop in a small component of the graph goes as n-1
– Probability that there is more than one path between any pair of vertices is O(n-1)
– Not true of most real world networks
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Configuration model
• Example : power-law degree distribution
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Configuration model
• Gets rid of Poisson degree distribution
• Still no clustering (transitivity)• Explanation :
– Configuration model graphs are suitable for modeling the global network
– Clustering is a characteristic of the local network
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Random Graphs
• Poisson Random Graphs• Configuration Model
• Extensions to Random Graphs• Markov Graphs
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Extension to Random Graph: Directed Graphs
• Directed Graphs: Each vertex has – An in-degree : j– An out-degree: k
• Control both in creation of the random graph
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Extension to Random Graph: Directed Graphs
Use of extended random graph to model directed network: WWW
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Extension to Random Graph: Bipartite Graphs
• Have two types of nodes • Edges run only between two different types
• Work well for modeling some real world networks
• Fail to capture the complexity of others
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Extension to Random Graph: Bipartite Graphs
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Extension to Random Graph: Bipartite Graphs
Indication of shortcomings of modeled bipartite graphs
The theoretical predictions of the last two data sets show account for only half of the actual clustering
present.
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Random Graphs
• Poisson Random Graphs• Configuration Model• Extensions to Random Graphs
• Markov Graphs
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Markov Graphs
• Generalized random graph models have serious shortcoming: – Fail to show transitivity
• Look for completely different model– Add clustering to generated systems
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Markov Graphs• Looks at properties (edge configurations) of a
graph• Use properties to construct “conditional tie
variables” (Xij) – Signify a relationship between nodes i & j – Xij = 1 if there is an observed relational tie– Xij = 0 otherwise
• These tie variables are not independent– Need some way to reflect dependency– Markovian dependence structure: ties are
conditionally dependent when they share a node.
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Markov Graphs
• Social Network Example:– Work ties among lawyers
• Vertices : Lawyers in a law firm• Edges : Collaboration (work ties) among them
– How is work flow structured?• Discernable form of local structuring?
– “Social ties are not interdependent of each other but the dependence is expressed through any persons directly involved in the ties in question”
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Markov GraphsNetwork Ties Among Lawyers
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Markov Graphs
Significant Graph Features when considering Markovian Relational ties
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Markov GraphsResults indicate improved local clustering (transitivity) representation.
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Markov Graphs
• Problem :– Tend to “condense”
• Form regions of complete cliques– Subsets of vertices in which each vertex is connected to every
other vertex in that subset
– Networks in the real world do not share this “clumpy” transitivity
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Markov GraphsClumping effect indicative of
Markov Graph representation
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Summary
• Types of Real World NetworksA. Social NetworksB. Information NetworksC. Technological NetworksD. Biological Networks
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Summary
• Properties of networks– Small-World Effect– Transitivity– Degree Distribution– Network Resilience– Mixing Patterns– Degree Correlation– Community Structure– Network Navigation
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Summary
• Random Graphs and extensions– Model only some of the properties found in real
networks– Motivates the exploration of other models that
can represent these properties