6 THE CLASSES~1 hour Networks~20 min Other Topics on Complexity (Bonus Section)NETWORKSClass 1: Random networks, simple graphs and basic network characteristics.Class 2: Scale-Free Networks. Class 3: Characterizing Network Topology. Class 4: Community Structure. Class 5: Network Dynamics. Class 6: Networks in Biology. Class 7: Networks in Economy.BONUS SECTIONClass 1: Chaos. Class 2: Fractals. Power-Laws. Self-Organized Criticality. Class 3: Drawing your own Networks using Cytoscape. Class 4: Community finding software. Class 5: Crowd-sourcing. Class 6: Synthetic Biology. Class 7: TBA.
8 Complex Systems: -Large number of parts Components:-Large number of parts-Properties of parts are heterogeneously distributed-Parts interact through a host of non-trivial interactions
9 EMERGENCEAn 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 behaviorFor example: The global match between supply and demandMore is different (Anderson)**There is emerging behavior in systems that escape local explanations. (Anderson)**Murray Gell-Mann“You do not needSomething more toGet something more”TED Talk (2007)”**Phillip Anderson“More is Different” Science 177:393–396 (1972)*Adam Smith“The Wealth of Nations” (1776)
10 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 neurons60 trillion synapses
17 NetworksEconomicsEmergence of Scaling in Random Networks - R Albert, AL Barabási - Science, 1999Cited by Innovation and Growth in the Global Economy GM Grossman, E Helpman Cited by 4542Statistical mechanics of complex networks - R Albert, AL Barabási - Reviews of Modern Physics, Cited by 3132Technical Change, Inequality, and the Labor Market - D Acemoglu - Journal of Economic Literature, Cited by 911Collective dynamics of'small-world' networks - Find Harvard DJ WATTS, SH STROGATZ - Nature, Cited by 6595The Market for Lemons: Quality Uncertainty and the Market Mechanism GA Akerlof - Cited by 4561The structure and function of complex networks - MEJ Newman - Arxiv preprint cond-mat/ , Cited by 2451The Pricing of Options and Corporate Liabilities F Black, M Scholes - Journal of Political Economy, Cited by 9870
18 Networks?We all had some academic experience with networks at some point in our lives
19 Types of Networks Simple Graph. Symmetric, Binary. Example: Countries that share a border in South America
22 Types of Networks Weighted Graphs 2 years 4 years 1 year 7 years
23 Simple Graph:Symmetric, Binary.Directed Graph:Non-Symmetric, Binary.Directed and Weighted Graph:Any Matrix
24 Networks are usually sparser than matrices List of Edges or LinksA BB DA CA FB GG FA SABDcFGSExample: The World Social NetworkNodes = 6x109Links=103 x 6x109/2 = 3x1012 Possible Links= (6x109-1)x 6x109/2 = 6x1018Number of Zeros= 6x x1012 ~5.9x1018
25 A network is a “space”. Networks? 1 2 3 4 5 6 7 What if we start making neighborsof non-consecutive numbers?Cartesian Space (Lattice) 2-d1345672Now we have different paths betweenOne number and anotherCartesian Space (Lattice)1-d1234567
27 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 graphEulerian circuit: is a Eulerian path, where the start and end points are the sameA graph can only be Eulerian if all vertices have an even number of edgesEuler'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
28 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 $.
37 Random Graph Theory Erdos-Renyi Model (1959) Paul Erdos 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
38 Random Graph Theory: Erdos-Renyi (1959) SubgraphsTreesCycleskCliqueskk(k-1)/2Nodes:Links:kk-1
39 GN,p F(k,l) CNk Nk pl /a k! a pl E= Random Graph Theory: Erdos-Renyi (1959), Bollobas (1985)CNkCan choose the k nodes in N choose k waysGN,p Nk pl /aWhich in the large Ngoes likeF(k,l)We can permute the nodes we choosein k! ways, but have to remember not to double count isomorphisms (a)k!aplEach link occurs withProbability pE=
40 E Nk pl /a p(N)~ cN-k/l E=cl/a=l Random Graph Theory: Erdos-Renyi (1959), Bollobas (1985)E Nk pl /aIn the threshold:p(N)~ cN-k/lWhich implies a number of subgraphs:E=cl/a=lBollobas (1985)R. Albert, A.-L. Barabasi, Rev. Mod. Phys (2002)
41 Subgraphs appear suddenly (percolation threshold) Probability of having a propertyQuestion for the class:Given that the critical connectivity is p(N)~ cN-k/lWhen does a random graph become connected?p
58 Total number of linkable actors: Weighted total of linkable actors: Average Kevin Bacon number: 2.946Kevin Bacon Number# of People121082204188360174741361785865668397111812Kevin Bacon
59 Average Connery number: 2.731 # of people12272221856033807214402635353765357668Sean Connery
60 Hollywood Revolves Around Click on a name to see that person's table.Steiger, Rod ( )Lee, Christopher (I) ( )Hopper, Dennis ( )Sutherland, Donald (I) ( )Keitel, Harvey ( )Pleasence, Donald ( )von Sydow, Max ( )Caine, Michael (I) ( )Sheen, Martin ( )Quinn, Anthony ( )Heston, Charlton ( )Hackman, Gene ( )Connery, Sean ( )Stanton, Harry Dean ( )Welles, Orson ( )Mitchum, Robert ( )Gould, Elliott ( )Plummer, Christopher (I) ( )Coburn, James ( )Borgnine, Ernest ( )Rod Steiger
61 Is there a "worst" center (or most obscure actor) in the Hollywood universe? Of course. I won't tell you the name of the person who produces the highest average number in the IMDb, but his/her table looks like this (as of June 29, 2004):XXXXXX Number# of people127342152861511584470094400471014890011107641211581318314
62 Kevin Bacon has +2000 co-workers, so does Sean Connery, while the worst connected actor in Hollywood has just 1.Are networks random?TO BE CONTINUED…….
67 Xn+1=Xn+9XnThe circle hikerOriginX1Xn+1=significand(10Xn)9x0X0X0=X1=X2=X3=….
68 Remember Not to Always Blame the Butterfly David Orrell
69 Take Home Messages NETWORKS -Networks can be used to represent a wide set of systems-The properties of random networks emerge suddenly as a function of connectivity.-The distance between nodes in random networks is small compared to network size L~ log(N)Networks can exhibit simultaneously: short average path length and high clustering(SMALL WORLD PROPERTY)The coexistence of these last two properties cannot be explained by random networksThe small world property of networks is not exclusive of “social” networks.BONUSDeterministic Systems are not necessarily predictable.But you shouldn’t always blame the butterfly.