Networks and Discrete Mathematics
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1 Aristotle University, School of Mathematics Master in Web Science Networks and Discrete Mathematics Small Words-Scale-Free- Model Chronis Moyssiadis Vassilis Karagiannis 7/12/2012 WS.04 Webscience: lecture 5 Regular Networks The simplest networks are the Regular Networks (Periodic repetition of the Unit Cell) Each node has constant degree. Discrete Geometry Classification by Symmetry [Barlow, Fedorov 1890, Conway,ea 1999] Models of crystals For example the nodes in the cubic crystal of NaCl has degree 6 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 2
2 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 3 Random Networks The PSTN (Public Switched Telephone Network) is not Regular Distributions of Calls and Connections obey the Poisson Law Need for Random Networks 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 4
3 Random Graphs G(n,p) (Erdös Rényi 1959 model): every possible edge occurs independently with probability p G(n,M) assigns equal probability to all graphs with exactly M edges (this can be viewed as a snapshot at a particular time (M) of the random graph process, which is a stochastic process that starts with n vertices and no edges and at each step adds one new edge chosen uniformly from the set of missing edges) 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 5 Natural Complex Networks The Complex networks of Life are neither completely regular (like crystals) nor completely random (like PSTN). Life manifests between order and chaos [Ilya Prigogine, 1980] Discovery of 2 new classes of Communication Networks: -Small World Networks [Strogats, Watts 1998] -Self - Similar (Scale-Free) Networks [Barabasi,Albert 1999] 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 6
4 6 degree of separation [Milgram 1967] The Death of Distance The experiment: Random people from Nebraska were to send a letter (via intermediaries) to a stock broker in Boston. Could only send to someone with whom they were on a first-name basis. Among the letters that found the target, the average number of links was six. Stanley Milgram ( ) 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 7 Real-World Networks Small World Model While random graphs exhibit some futures of real world networks (small diameters or average distances relative to growing average degree), they lack other characteristics. Stanley Milgram (1967), Harvard, what is the probability that two randomly selected people would know each other? On average 5.5 hops (in a sparse network) Six degrees of separation Why? In social networks (which are sparse networks) it has observed that the friends of us are usually also friends between them, or in other words people tend to group into relative small (or large clusters) in a way that in some cases does not depend to the place of living. 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 8
5 Six Degrees of Separation Milgram (1967) Allan Wagner? Robert Sternberg Mike Tarr Kentaro Toyama John Guare wrote the play Six Degrees of Separation, based on this concept. Everybody on this planet is separated by only six other people. Six degrees of separation. Between us and everybody else on this planet. The president of the United States. A gondolier in Venice It s not just the big names. It s anyone. A native in a rain forest. A Tierra del Fuegan. An Eskimo. I am bound to everyone on this planet by a trail of six people 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 9 Real World Networks Small World Model Suppose that we have a Poisson random social network. Then: P(i and j know each other) = P(link {i, j} exists) = p. Moreover P(i and j know each other when both are friends of k) = P(i and j know each and both are friends of k) / P(i and j are friends of k) = p 3 / p 2 = p k j i The existence of any acquaintance between i and j is independent of any other acquaintance between iand k or between j and k. 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 10
6 Real World Networks Small World Model Usually a friendship network having ourselves as a root may have a picture like the one drawn below 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 11 Real World Networks Small World Model A way to define the previous probability is by the enumeration of the connected triples and the triangles in the network. A connected triple centered at node i is defined as a path of length 2 having node ias the intermediate node. The number of all possible connected triples at node i having degree d i is: d () ( 1) i d i i Ti d 2 2 The transitivity of a node i as also the clustering coefficient of a node i is defined as: 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 12
7 Real World Networks Small World Model This network has one triangle and eight connected triples. The individual vertices have local clustering coefficients 1, 1, 1/6, 0 and 0, hence the clustering coefficient is equal to C(G) = 13/30 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 13 Real World Networks Small World Model Watts and Strogatz (1998). A one dimensional lattice is graph on n nodes such that if we order them into a ring each node is connected to the first k left hand neighbors and to the first k right hand neighbors. Such a lattice with n=20 nodes and k = 2 is presented below (2k regular graph). Then with probability p, we decide independently for each link {u,v} if we replace it with a link {u,w} where w is chosen uniformly at random from the set of nodes. The importance of Watts and Strogatz model is due to the fact that it started the active and important field of modeling largescale networks by random graphs defined by simple rules 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 14
8 Real-World Networks Small World Model Using n = 20, k = 2, and rewiring probability p=0.1 the diameter from 5 goes to 4, while theoretically it was expected near log20 = /12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 15 Real-World Networks Small World Model For a network G we compute the distance from each node ito each other and then we take the average distance for node i. Doing the same for each node in the network we get a set of V(G) average distances. The characteristic path length of the network G is the median of the all the average distances and sometimes is used instead of average path length (which is the average distance of the graph). 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 16
9 Από τη σελίδα 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 17 Dynamic Growth and Preferential Attachment Unfortunately, however, the statistics resulting from the Watts/Strogatz model does not match the observed small worlds. These networks, such as power grids, hollywood actors, web pages, and peer networks, exhibit power-law distribution of edges among the nodes while the W/S model does not. Barabási and his team found that two techniques resulted in power-law distributions: Dynamic Construction Preferential Attachment The first constructs small worlds graphs dynamically, rather than rewiring a graph in place. Secondly, the rewiring of nodes occurs not randomly, but preferentially attaching to the most connected nodes. They introduced a family of such graphs which start with a small number of disconnected nodes. They then dynamically wire into the graph new nodes of fixed order K, with each edge being wired to existing nodes preferentially according to the number of edges the existing node has. 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 18
10 To the left, the example starts with 3 unwired nodes. It starts adding nodes one at a time, with two edges per new node. The edges are wired to destination nodes preferentially, with probability being proportional to edges already attached to the target node. The sequence continues until all nodes are added to the graph. The resulting distribution of edges to nodes is power-law. 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 19 SW Networks Protein Networks Road maps, food chains, electric power grids, metabolite processing networks, telephone call graphs (after 1995) social networks: influence Networks. Erdos Networks Neural Networks, Short term Memory Networks WWW 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 20
11 Scale free networks A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. That is, the fraction P(k) of nodes in the network having k connections to other nodes goes for large values of k as ~ where c is a normalization constant and is a parameter whose value is typically in the range 2 3, although occasionally it may lie outside these bounds. Many networks are conjectured to be scale-free, including: World Wide Web links, biological networks, and social networks, although the scientific community is still discussing these claims as more sophisticated data analysis techniques become available. Preferential attachment and the fitness model have been proposed as mechanisms to explain conjectured power law degree distributions in real networks. 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 21 History In studies of the networks of citations between scientific papers, Derek de Solla Price showed in 1965 that the number of links to papers i.e., the number of citations they receive had a heavytailed distribution following a Pareto distribution or power law, and thus that the citation network was scale-free. Price also proposed a mechanism to explain the occurrence of power laws in citation networks, which he called "cumulative advantage" but which is today more commonly known under the name preferential attachment. 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 22
12 History Albert-László Barabási and colleagues at the University of Notre Dame studying scale-free networks in 1999 found that --- some nodes, which they called "hubs", had many more connections than others and that - the network as a whole had a power-law distribution of the number of links connecting to a node. Barabási and Albert called the mechanism who explains the appearance of the power-law distribution, "preferential attachment" while analytic solutions for this mechanism were presented in 2000 by Dorogovtsev, Mendes and Samukhin and independently by Krapivsky, Redner, and Leyvraz, and later rigorously proved by mathematician Béla Bollobás. Fault Tolerance. Immunity to Random attacks. 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 23 example s( G) For a graph G let s( G) d SG ( ) = id j and s (, i j) ÎE max This gives a metric between 0 and 1, such that graphs with low S(G) are "scale-rich", and graphs with S(G) close to 1 are "scale-free". This definition includes the notion of self-similarity implied in the name "scale-free". = å 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 24
13 SS Networks Collaboration Networks: -Citations between Scientific Papers Protein interaction Networks Sexual Networks, Disease Spread Genetic Disease Networks, Word Rank Networks www [Barabasi,ea 1999] 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 25 Application of Self-Similar Networks Combating Aids NATURE 2003 Vol 423, 685 Living with the AIDS Virus: New approaches may be needed if AIDS is to be controlled. The Epidemic and the Response In India 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 26
14 Power Law Distribution Power-Law Distribution ( υναμοκατανομή) It is scale invariant The main attribute of power laws that makes them interesting is their scale invariance. Given a relation, scaling the argument by a constant factor causes only a proportionate scaling of the function itself. That is, Universality In physics, phase transitions in thermodynamic systems are associated with the emergence of power-law distributions of certain quantities, whose exponents are referred to as the critical exponents of the system. Diverse systems with the same critical exponents that is, which display identical scaling behaviour as they approach criticality can be shown, via renormalization group theory, to share the same fundamental dynamics. For instance, the behavior of water and CO2 at their boiling points fall in the same universality class because they have identical critical exponents. In fact, almost all material phase transitions are described by a small set of universality classes. 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 27 Preferential Attachment BA model 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 28
15 Preferential Attachment BA model No cycle is created under the BA model, so other models are proposed. 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 29 Scientific Collaboration Network 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 30
16 Degree centrality 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 31 Closeness centrality 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 32
17 Betweenness centrality Betweenness is a centrality measure of a vertex within a graph (there is also edge betweenness, which is not discussed here). Betweenness centrality quantifies the number of times a node acts as a bridge along the shortest path between two other nodes. It was introduced as a measure for quantifying the control of a human on the communication between other humans in a social network by Linton Freeman. In his conception, vertices that have a high probability to occur on a randomly chosen shortest path between two randomly chosen vertices have a high betweenness. The betweenness of a vertex in a graph, with vertices is computed as follows: 1. For each pair of vertices,, compute the shortest paths between them. 2. For each pair of vertices (s,t), determine the fraction of shortest paths that pass through the vertex in question (here, vertex ). 3. Sum this fraction over all pairs of vertices (s,t). More compactly the betweenness can be represented as where, is total number of shortest paths from node to node and, is the number of those paths that pass through. 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 33 Hue (from red=0 to blue=max) shows the node betweenness. 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 34
18 Examples of A) Degree centrality, B) Closeness centrality, C) Betweenness centrality, D) Eigenvector centrality, E) Katz centrality and F) Alpha centrality of the same graph 7/12/2012 WS.04 lecture 5: Small Words-Scale-Free- Models. C. Moyssiadis V. Karagiannis 35
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