Statistical Mechanics of Complex Networks: Economy, Biology and Computer Networks

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Statistical Mechanics of Complex Networks
Economy, Biology and Computer Networks

• Albert Diaz-Guilera
• Universitat de Barcelona

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Outline

• Complex systems
• Topological properties of networks
• Complex networks in nature and society
• Tools
• Models
• Dynamics

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Physicist out their land

• Multidisciplinary research
• Reductionism simplicity
• Scaling properties
• Universality

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Multidisciplinary research

• Intricate web of researchers coming from very
  different fields
• Different formation and points of view
• Different languages in a common framework
• Complexity

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Complexity

• Challenge Accurate and complete description of
  complex systems
• Emergent properties out of very simple rules
• unit dynamics
• interactions

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Why is network anatomy important

• Structure always affects function
• The topology of social networks affects the
  spread of information
• Internet
• access to the information
• - electronic viruses

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Current interest on networks

• Internet access to huge databases
• Powerful computers that can process this
  information
• Real world structure
• regular lattice?
• random?
• all to all?

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Network complexity

• Structural complexity topology
• Network evolution change over time
• Connection diversity links can have directions,
  weights, or signs
• Dynamical complexity nodes can be complex
  nonlinear dynamical systems
• Node diversity different kinds of nodes

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Topological properties

• Degree distribution
• Clustering
• Shortest paths
• Betweenness
• Spectrum

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Degree

• Number of links that a node has
• It corresponds to the local centrality in social
  network analysis
• It measures how important is a node with respect
  to its nearest neighbors

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Degree distribution

• Gives an idea of the spread in the number of
  links the nodes have
• P(k) is the probability that a randomly selected
  node has k links

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What should we expect?

• In regular lattices all nodes are identical
• In random networks the majority of nodes have
  approximately the same degree
• Real-world networks this distribution has a
  power-law tail

scale-free networks
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Clustering

• Cycles in social network analysis language
• Circles of friends in which every member knows
  each other

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Clustering coefficient

• Clustering coefficient of a node
• Clustering coefficient of the network

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What happens in real networks?

• The clustering coefficient is much larger than it
  is in an equivalent random network

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Ego-centric vs. socio-centric

• Focus is on links surrounding particular agents
  (degree and clustering)
• Focus on the pattern of connections in the
  networks as a whole (paths and distances)
• Local centrality vs. global centrality

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Distance between two nodes

• Number of links that make up the path between two
  points
• Geodesic shortest path
• Global centrality points that are close to
  many other points in the network.
• Global centrality defined as the sum of minimum
  distances to any other point in the networks

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3
1
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Local vs global centrality
A,C B G,M J,K,L All other
Local 5 5 2 1 1
global 43 33 37 48 57
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Global centrality of the whole network?

• Mean shortest path average over all pairs of
  nodes in the network

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Betweenness

• Measures the intermediary role in the network
• It is a set of matrices, one for ach node
• Comments on Fig. 5.1

Ratio of shortest paths bewteen i and j that go
through k
There can be more than one geodesic between i
and j
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Pair dependency

• Pair dependency of point i on point k
• Sum of betweenness of k for all points that
  involve i
• Row-element on column-element

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Betweenness of a point

• Half the sum (count twice) of the values of the
  columns
• Ratio of geodesics that go through a point
• Distribution (histogram) of betweenness
• The node with the maximum betweenness plays a
  central role

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Spectrum of the adjancency matrix

• Set of eigenvalues of the adjacency matrix
• Spectral density (density of eigenvalues)

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• A symmetric and real gt eigenvalues are real and
  the largest is not degenerate
• Largest eigenvalue shows the density of links
• Second largest related to the conductance of the
  graph as a set of resistances
• Quantitatively compare different types of
  networks

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Tools

• Input of raw data
• Storing format with reduced disk space in a
  computer
• Analyzing translation from different formats
• Computer tools have an appropriate language
  (matrices, graphs, ...)
• Import and export data

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Complex networks in nature and society

• NOT regular lattices
• NOT random graphs
• Huge databases and computer power

simple mathematical analysis
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Networks of collaboration

• Through collaboration acts
• Examples
• movie actor
• board of directors
• scientific collaboration networks (MEDLINE,
  Mathematical, neuroscience, e-archives,..)
• gt Erdös number

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Coauthorship network
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Communication networks
Hyperlinks (directed)
Hosts, servers, routers through physical cables
(not directed)
Flow of information within a company employees
process information Phone call networks (?2)
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Internet
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Networks of citations of scientific papers

• Nodes papers
• Links (directed) citations
• ?3

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Social networks

• Friendship networks (exponential)
• Human sexual contacts (power-law)
• Linguistics words are connected if
• Next or one word apart in sentences
• Synonymous according to the Merrian-Webster
  Dictionary

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Biological networks

• Neural networks neurons synapses
• Metabolic reactions molecular compounds
  metabolic reactions
• Protein networks protein-protein interaction
• Protein folding two configurations are connected
  if they can be obtained from each other by an
  elementary move
• Food-webs predator-prey (directed)

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C. elegans neural network
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Food webs
East River, CO, USA
Little Rock Lake, WI, USA
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Engineering networks

• Power-grid networks generators, transformers,
  and substations through high-voltage
  transmission lines
• Electronic circuits electronic components
  (resistor, diodes, capacitors, logical gates)
  wires
• Software engineering

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Average path length
random graph
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Clustering
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Degree distribution
movie actors
internet
high energy coauthorship
neuroscience coauthorship
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Models

• Random graph (Erdös-Renyi)
• Small world (Watts-Strogatz)
• Scale-free networks (Barabasi-Albert)

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Random graph

• Binomial model start with N nodes, every pair of
  nodes being connected with probability p
• The total number of links, n, is a random
  variable
• E(n)pN(N-1)/2
• Probability of generating a graph, G0N,n

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Degree distribution

• The degree of a node follows a binomial
  distribution (in a random graph with p)
• Probability that a given node has a connectivity
  k
• For large N, Poisson distribution

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Mean short path

• Assume that the graph is homogeneous
• The number of nodes at distance l are ltkgtl
• How to reach the rest of the nodes?
• lrand to reach all nodes gt klN

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Clustering coefficient

• Probability that two nodes are connected (given
  that they are connected to a third)?

while it is constant for real networks
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Small world

• Crossover from regular lattices to random graphs
• Tunable
• Small world network with (simultaneously)
• Small average shortest path
• Large clustering coefficient (not obeyed by RG)

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Scale-free networks
Networks grow preferentially
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P(k)exp (-k2/A2)
P(k)k -g
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Dynamics

• Network dynamics
• global goal
• local goal
• Flow in complex networks
• ideas
• innovations
• computer viruses
• problems

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Global vs local optimization

• Design the goal is to optimize global quantity
  (distance, clustering, density, ...)
• Evolution decision taken at node level

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Virus spreading
fraction of infected nodes
prevalence in scale-free networks
infection rate
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Communication model

• Communicating agents computers, employees
• Communication channels cables, email, phone
• Information packets packets, problems
• Finite capacity of the agents to deliver
  information

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Summary