The Structure and Function of Complex Networks Part I Jim Vallandingham M. E. J. Newman.

The Structure and Function of Complex Networks

Part I

Jim Vallandingham

M. E. J. Newman

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

Organization

I. DefinitionsII. Types of NetworksIII. Properties of NetworksIV. Random GraphsV. Extensions to Random GraphsVI. Markov Graphs

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

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

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

DefinitionsThree components in a network

Types of Networks

A. Social NetworksB. Information NetworksC. Technological NetworksD. Biological Networks

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

Social NetworksInternet Chat Relay (IRC) communications between individuals

Social NetworksDating relationships between students in

a high school

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.

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

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

Social Networks

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

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)

Information NetworksCitation Network for Inferring network mechanisms: The Drosophila melanogaster protein interaction network


• 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


Graph of Relationships between Facebook pages. Example of an Information Network with Social Network aspects.


• 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

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

Technological Networks

Bandwidth transfer in Europe between countries

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

Biological Networks

Biological NetworksPortion of yeast protein interactions

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

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

Biological Networks

Reef fish food web

Biological NetworksRat hippocampal neurons

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

Properties of Networks

• Small-World Effect• Transitivity• Degree Distribution• Network Resilience• Mixing Patterns• Degree Correlation• Community Structure• Network Navigation

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 :


• 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


• 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


• Biological example: protein-protein interactions in the yeast, S. cerevisiae

• Vertices: 1870• Edges: 2240

• : 6.80

Properties of Networks• Small-World Effect

• Transitivity• Degree Distribution• Network Resilience• Mixing Patterns• Degree Correlation• Community Structure• Network Navigation

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


C : Clustering coefficient


1

8

2

1

3

4

567

8

Fraction of Transitive Triples


Can also be defined locally for each vertex

With this value the definition of C becomes:

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


• 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


• 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

Properties of Networks• Small-World Effect• Transitivity

• Degree Distribution• Network Resilience• Mixing Patterns• Degree Correlation• Community Structure• Network Navigation

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


• 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

Properties: Degree DistributionsHistograms depicting the Noise and lack of measurements

indicative of the tail section of the degree distribution

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


Power law Exponential


• Power-law degree distributions sometimes called scale-free networks

• Include networks of:– World wide web– Metabolic pathways– Telephone calls

Properties of Networks• Small-World Effect• Transitivity• Degree Distribution

• Network Resilience• Mixing Patterns• Degree Correlation• Community Structure• Network Navigation

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


• 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


• Biological Example:– Metabolic network of yeast

Diameter: total of all path lengths divided by total

number of paths

Targeted

Random

Properties of Networks• Small-World Effect• Transitivity• Degree Distribution• Network Resilience

• Mixing Patterns• Degree Correlation• Community Structure• Network Navigation

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

Properties: Mixing Patterns

• Quantified by assortativity coefficient

• Other ways to look at assortative mixing– By scalar characteristics

• Age, income

– Vector characteristics• Location : 2D vector

Properties of Networks• Small-World Effect• Transitivity• Degree Distribution• Network Resilience• Mixing Patterns

• Degree Correlation• Community Structure• Network Navigation

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?


• 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


Degree Increasing

Deg

ree

Incr

easi

ng

Highest degree correlation

Yeast protein interactions

Properties of Networks• Small-World Effect• Transitivity• Degree Distribution• Network Resilience• Mixing Patterns• Degree Correlation

• Community Structure• Network Navigation

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

Properties: Community StructureGroups in protein interactions

Properties of Networks• Small-World Effect• Transitivity• Degree Distribution• Network Resilience• Mixing Patterns• Degree Correlation• Community Structure

• Network Navigation

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

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

Random Graphs

• Poisson Random Graphs

• Configuration Model• Extensions to Random Graphs• Markov Graphs

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


• Building Random Graphs:• Very simple process

– Take some number n of vertices – Connect each pair with a probability p


• 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


• 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


• Two properties in random graphs :– Giant component size

• Calculating the expected size of the giant component:

– Mean size of the non-giant components:


• 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


Linear graph Logarithmic graphScale-freerandom


• 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

Random Graphs

• Poisson Random Graphs

• Configuration Model• Extensions to Random Graphs• Markov Graphs

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

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

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

Configuration model

• Example : power-law degree distribution

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

Random Graphs

• Poisson Random Graphs• Configuration Model

• Extensions to Random Graphs• Markov Graphs

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

Extension to Random Graph: Directed Graphs

Use of extended random graph to model directed network: WWW

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


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.

Random Graphs

• Poisson Random Graphs• Configuration Model• Extensions to Random Graphs

• Markov Graphs

Markov Graphs

• Generalized random graph models have serious shortcoming: – Fail to show transitivity

• Look for completely different model– Add clustering to generated systems

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.

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”

Markov GraphsNetwork Ties Among Lawyers

Markov Graphs

Significant Graph Features when considering Markovian Relational ties

Markov GraphsResults indicate improved local clustering (transitivity) representation.

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

Markov GraphsClumping effect indicative of

Markov Graph representation

Summary

• Types of Real World NetworksA. Social NetworksB. Information NetworksC. Technological NetworksD. Biological Networks

Summary

• Properties of networks– Small-World Effect– Transitivity– Degree Distribution– Network Resilience– Mixing Patterns– Degree Correlation– Community Structure– Network Navigation

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

The Structure and Function of Complex Networks Part I Jim Vallandingham M. E. J. Newman.

Documents

Transcript of The Structure and Function of Complex Networks Part I Jim Vallandingham M. E. J. Newman.