Complex Networks Albert Diaz Guilera Universitat de Barcelona

Complex Networks 0Presentation 1Introduction –Topological properties –Complex networks in nature and society 2Random graphs: the Erdos-Rényi model 3Small worlds 4Preferential linking 5Dynamical properties –Network dynamics –Flow in complex networks

Presentation 2 hours per session approx homework –short exercises: analytical calculations –computer simulations –graphic representations What to do with the homework? –BSCW: collaborative network tool

BSCW Upload and download documents (files, graphics, computer code,...) Pointing to web addresses Adding notes as comments Discussions Information about access bscw.ppt

1. INTRODUCTION Complex systems Representations –Graphs –Matrices Topological properties of networks Complex networks in nature and society Tools

Physicist out their land Multidisciplinary research Reductionism = simplicity Scaling properties Universality

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

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

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

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?

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

Scaling and universality Magnetism Ising model: spin-spin interaction in a regular lattice Experimental models: they can be collapsed into a single curve Universality classes: different values of exponents

Representations From a socioeconomic point of view: representation of relational data How data is collected, stored, and prepared for analysis Collecting: reading the raw data (data mining)

Example People that participate in social events Incidence matrix:

Adjacence matrix: event by event Adjacence matrix: person by person

Graphs (graphic packages: list of vertices and edges) Persons Events

Bipartite graph Board of directors

Directed relationships Sometimes relational data has a direction The adjacency matrix is not symmetric Examples: –links to web pages –information –cash flow

Topological properties Degree distribution Clustering Shortest paths Betweenness Spectrum

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

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

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 tail scale-free networks

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

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

What happens in real networks? The clustering coefficient is much larger than it is in an equivalent random network

Directedness The flow of resources depends on direction Degree –In-degree –Out-degree Careful definition of magnitudes like clustering

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

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. (Fig. 5.1 SNA) Global centrality defined as the sum of minimum distances to any other point in the networks

Local vs global centrality A,CBG,MJ,K,LAll other Local55211 global

Global centrality of the whole network? Mean shortest path = average over all pairs of nodes in the network

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

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

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

Spectrum of the adjancency matrix Set of eigenvalues of the adjacency matrix Spectral density (density of eigenvalues)

Relation with graph topology k-th moment N*M = number of loops of the graph that return to their starting node after k steps k=3 related to clustering

A symmetric and real => 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

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

UCINET General purpose Compute basic concepts Exercises: –How to compute the quantities we have defined so far –Other measures (cores, cliques,...)

PAJEK Drawing package with some computations Exercises: –Draw the networks we have used –Check what can be computed –Displaying procedures

Complex networks in nature and society NOT regular lattices NOT random graphs Huge databases and computer power simple mathematical analysis

Networks of collaboration Through collaboration acts Examples: –movie actor –board of directors –scientific collaboration networks (MEDLINE, Mathematical, neuroscience, e-archives,..) => Erdös number

Communication networks Hyperlinks (directed) Hosts, servers, routers through physical cables (directed) Flow of information within a company: employees process information Phone call networks ( =2)

Networks of citations of scientific papers Nodes: papers Links (directed): citations =3

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

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)

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

Average path length

Clustering

Degree distribution