Classes will begin shortly Networks Complexity and their Applications Classes will begin shortly Cesar A. Hidalgo PhD

Complexity Complex Systems: Components: -Large number of parts -Properties of parts are heterogeneously distributed -Parts interact through a host of non-trivial interactions -Adaptability -Evolvability

EMERGENCE 1+1>2 “The Whole is Greater than the sum of its parts” Aristotle An aggregate system is not equivalent to the sum of its parts. People’s action can contribute to ends which are no part of their intentions. (Smith)* Local rules can produce emergent global behavior For example: The global match between supply and demand More is different (Anderson)** There is emerging behavior in systems that escape local explanations. (Anderson) **Murray Gell-Mann “You do not need Something more to Get something more” TED Talk (2007)” **Phillip Anderson “More is Different” Science 177:393–396 (1972) *Adam Smith “The Wealth of Nations” (1776)

20 billion neurons 60 trillion synapses In addition to the neocortex, the cortical region of the human brain contains more primitive components called the olfactory cortex and the hippocampus that occur in reptiles as well as mammals. The mammalian versions of these structures, however, are associated with other regions of the cortex, hypothalamus, and thalamus in a ring-like assembly centered around the brainstem known as the limbic system. The emotional responses or feelings that mammals experience are produced by the limbic system, which closely interacts with other parts of the brain. The limbic system also is a functional center for long-term memory. 20 billion neurons 60 trillion synapses

WHY NETWORKS?

NETWORKS = ARCHITECTURE OF COMPLEXITY

Networks

Networks? We all had some academic experience with networks at some point in our lives

Types of Networks Simple Graph. Symmetric, Binary. Example: Countries that share a border in South America

Types of Networks Bi-Partite Graph, and projections

Types of Networks Directed Graphs

Types of Networks Weighted Graphs 2 years 4 years 1 year 7 years

Representing Networks ADJACENCY MATRIX Simple Graph: Symmetric, Binary. Directed Graph: Non-Symmetric, Binary. Directed and Weighted Graph: Any Matrix

Networks are usually sparser than matrices List of Edges or Links A B D c F G S A B B D A C A F B G G F A S Example: The World Social Network Nodes = 6x109 Links=103 x 6x109/2 = 3x1012 Possible Links= (6x109-1)x 6x109/2 ~= 2x1019 Number of Zeros= 6x1018 - 3x1012 ~2x1019 Density ~ 10-6 or 10-7

A network is a “space”. Networks? 1 2 3 4 5 6 7 What if we start making neighbors of non-consecutive numbers? Cartesian Space (Lattice) 2-d 1 3 4 5 6 7 2 Now we have different paths between One number and another Cartesian Space (Lattice) 1-d 1 2 3 4 5 6 7

Networks now and then

Konigsberg bridge problem, Euler (1736) Eulerian path: is a route from one vertex to another in a graph, using up all the edges in the graph Eulerian circuit: is a Eulerian path, where the start and end points are the same A graph can only be Eulerian if all vertices(nodes), except two (the beginning and the end), have an even number of edges Euler's solution to the Königsberg Bridge problem leads to a result that applies to all graphs. Before we see this however, we need some more terminology. Firstly, a Eulerian path is a route from one vertex to another in a graph, using up all the edges in the graph. A Eulerian circuit is a Eulerian path, where the start and end points are the same. This is equivalent to what would be required in the problem. Given these terms a graph is Eulerian if there exists an Eulerian circuit, and Semi-Eulerian if there exists a Eulerian path that is not a circuit. Finally, the degree of a vertex is the number of edges that lead from it. The result above showed that a graph can only be Eulerian if all vertices have an even number of edges from them. In other words, each vertex must have an even degree. It was later proven that any graph with all vertices of even degree will be Eulerian. Similarly if and only if a graph has only 2 vertices with odd order, it will be Semi-Eulerian. Leonhard Euler

La Firma del Diablo (The Signature of the Devil)

La Firma del Diablo (Cheat) (The Signature of the Devil)

Connect each house to the three utilities without crossing the lines  Kazimierz Kuratowski  (February 2, 1896 – June 18, 1980) A finite graph is planar if and only if it does not contain a subgraph that is a subdivision of K5 (the complete graph on five vertices) or K3,3 (complete bipartite graph on six vertices, three of which connect to each of the other three).

Planar Graph Non-Planar Graphs

Some Examples of Networks in “Nature”

We have taken the data from National Institute on Money in State Politics a nonpartisan organization dedicated to the documentation and research on campaign financing at the state level. For now we have used only data about Governors elected in 2006 and the top 100 donors. Currently we show only donors who donated to more than one candidate or a single donation bigger than 100000 $. http://www.weshowthemoney.com/

PNAS 2005

The Political Blogosphere and the 2004 U. S The Political Blogosphere and the 2004 U.S. Election: Divided They Blog Lada A. Adamic and Natalie Glance, LinkKDD-2005

http://presidentialwatch08.com/index.php/map/

http://www.blogopole.fr/

http://prezoilmoney.oilchangeusa.org/

John Maynard Keynes Charles Darwin William Thomson (Lord Kelvin) Adam Smith Charles Lyell Thomas Henry Huxley Joel Levy, Scientific Feuds, New Holland Publishers (2010) ISBN 978 1 84773 717 5

Marriage Offspring Friend Relationship

CA Hidalgo, B Klinger, A-L Barabasi, R Hausmann. Science (2007)

Protein-Protein Interactions (Rual, Nature (2005) + Stelzl Cell (2005)+ Literature Curated Interactions)

Metabolic Networks

IN THIS FIRST PART OF THE COURSE: We will be asking the questions… -How do we characterize the structure of networks? -Why networks have the structure that they do? -What are the implications of network structure?

RANDOM GRAPH THEORY

Paul Erdos Alfred Renyi Random Graph Theory Erdos-Renyi Model (1959) Original Formulation: N nodes, n links chosen randomly from the N(N-1)/2 possible links. Alternative Formulation: N nodes. Each pair is connected with probability p. Average number of links =p(N(N-1))/2; Random Graph Theory Works on the limit N-> and studies when do properties on a graph emerge as function of p. Alfred Renyi

Random Graph Theory: Erdos-Renyi (1959) Subgraphs Trees Cycles k Cliques k k(k-1)/2 Nodes: Links: k k-1

GN,p F(k,l) CNk  Nk pl /a k! a pl E= Random Graph Theory: Erdos-Renyi (1959), Bollobas (1985) CNk Can choose the k nodes in N choose k ways GN,p  Nk pl /a Which for large N goes like F(k,l) We can permute the nodes we choose in k! ways, but have to remember not to double count isomorphisms (a) k! a pl Each link occurs with Probability p E=

E(k,l)  Nk pl /a p(N)~ cN-k/l Random Graph Theory: Erdos-Renyi (1959), Bollobas (1985) E(k,l)  Nk pl /a In the threshold (i.e. E(k,l) is finite): p(N)~ cN-k/l Which implies a number of subgraphs that does not depend on N or k. Probability of a 5 clique? (5 nodes all connected to each other). Number of links =~ 10 Probability =~ N-1/2 Average Degree =~ N1/2 If N=106, then Average Degree =~ 1,000 R. Albert, A.-L. Barabasi, Rev. Mod. Phys (2002)

Question for the class: Subgraphs appear suddenly (percolation threshold) Probability of having a property Question for the class: Given that the critical connectivity is p(N)~ cN-k/l When does a random graph become connected? p

Degree Distribution Random Graph Theory: Erdos-Renyi (1959) K=8 Binomial distribution For large N approaches a poison distribution K=4

Clustering Random Graph Theory: Erdos-Renyi (1959) Ci=triangles/possible triangles Clustering Coefficient = <C>

Distance Between A and B?

Random Graph Theory: Erdos-Renyi (1959) Average Path Length Number of nodes at distance m from a randomly chosen node Hence the average path length is <k>4 <k>3 <k>2 <k> m

Structure of Random Networks SUMMARY: Structure of Random Networks Degree Distribution: is Binomial, Poisson, Gaussian (Bell Shaped) Clustering: Decays as 1/N. Is Equal to the Average degree over the number of nodes in the system. Percolation Threshold: at a probability of 1/N. Average Path Length: Grows as the log of N. Is it a small world?

GRADING SYSTEM 4 Short Homeworks (10% grade each) 1 Class Project (50% of grade) Assistance & participation (10% of grade) CLASSES Thursdays 3:00pm-5:30pm (E14-525) (from 4:30pm to 5:30pm the class will work on projects). COURSE PAGE macro.media.mit.edu TA’s Daniel Smilkov and Amy Zhao Yu

CLASS PLAN Feb 5 Intro & Random Networks Feb 12 Small World and Scale Free Networks (HW1) Feb 19 Implications of Scale Free Networks Feb 26 Structural Measures in Networks (HW2) March 5 Structural Measures in Networks 2 March 12 Preliminary Findings of Research Team March 19 Social Networks March 26 SPRING BREAK MIT (No Class) CLASS PLAN April 2 Measurements and Information in Complex Systems April 9 Economic Complexity 1(HW4) April 16 Economic Complexity (Formal Models) April 23 MIT Media Lab Member’s Week (No Class) April 30 Economic Complexity and class wrap up. May 7 and 14 Project Presentation

LEARNER TRACK RESARCHER TRACK