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Introduction of Network Science Prof. Cheng-Shang Chang ( 張正尚教授 ) Institute of Communications Engineering National Tsing Hua University Hsinchu Taiwan

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Outline What is network science? A brief history of network science Review of the mathematics of networks Diffusion, distributed averaging, random gossip, synchronization Network formation Structure of networks (Community detection) Conclusion

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What is network science? 2005 National Research Council of the National Academies “Organized knowledge of networks based on their study using the scientific method” Social networks, biological networks, communication networks, power grids, …

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A visualization of the network structure of the Internet at the level of “autonomous systems” (Newman, 2003)

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A social network (Newman, 2003)

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A food web of predator-prey interactions between species in a freshwater lake (Newman, 2003)

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Power grid map power-grid-map-shows-whos-got-the-power.php

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Citation networks ementIHDP.htm

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Two key ingredients The study of a collections of nodes and links (graphs) that represent something real The study of dynamic behavior of the aggregation of nodes and links

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Definition of Network G(t)={V(t), E(t), f(t): J(t)} t: time V: node (vertex, actor) E: link (edge) f: NxN topology (adjacency matrix) J: algorithm for the evolution of the network (microrule)

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Definition of Network Science (by Ted G. Lewis) The study of the theoretical foundation of network structure/dynamic behaviors and the application of network to many subfields Social network analysis (SNA) Collaboration networks (citations, online social networks) Emergent systems (power grids, the Internet) Physical science systems (phase transition, percolation theory) Life science systems (epidemics, metabolic processes)

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A brief history: The pre-network period ( ) 1736 Leonhard Euler: seven bridge of Konigsberg problem 1925 Yule: preferential attachment An explanation for the evolution of the Internet and WWW 1927 Kermack and McKendrick: epidemic model (diffusion of innovation, the spread of information) Erdos and Renyi: random graph model

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The meso-network period ( ) 1967 Stanley Milgram “Six degree of separation” Communication project If you do not know the target person, forward the request to a personal acquaintance Small-world effect: the diameter of a network increases as ln(n)

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The meso-network period ( ) 1972 Bonacich :influence network Distributed consensus Kirchhoff’s network: the value of a node is equal to the difference between the sum of values from input and output links States and differential equations Fixed point (steady state) 1984 Kuramoto: synchronization in coupled linear systems

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The modern period (1998- present) 1998 Holland: emergence as the final state (of the fixed point problem) 1998 Watts and Strogatz: a generative procedure of rewiring the links in a regular graph The small-world model Crossover point and phase transition

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The modern period (1998- present) 1999 M. Faloutsos, P. Faloutsos and C. Faloutsos: observed a power law in their graph of the Internet 1999 Barabasi: math model for scale-free networks 2000 Dorogovtsev: power law in many biological systems 1999 Kleinberg: power law in webgraph 2002 Girvan and Newman: community structure

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The modern period (1998- present) Atay network (a generalization of the Kirchhoff network) Emergence and synchronization: Heart beating The chirping of crickets Distributed consensus Propagation of influence

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Review of the mathematics of networks

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Networks and their representations A networks is a graph Vertices (nodes, sites, actors) Edges (links, bonds, ties) n: number of nodes m: number of edges Multiedges Self-edges (self-loops) Simple network (simple graph): a network that has neither self-edges nor multiedges Multigraph: a network with multiedges Multiedge Self-edge

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Adjacency matrix A: an n × n matrix A ij =1 if there is an edge between vertices i and j. A ij =0 otherwise. For a network with no self-edges, the diagonal elements are all zero. It is symmetric A=

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Directed networks Adjacency matrix: A ij =1 if there is an edge from j to i. With self-edges: A ii =1 for a single edge from vertex i to itself in a directed network A=

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Degree The degree of a vertex is the number of edges connected to it. k i : the degree of vertex i m: number of edges 2m ends of edges (every edge has two ends)

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Mean degree c: the mean degree of a vertex in an undirected graph The maximum possible number of edges is (n-1)n/2.

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Density Density (connectance): the fraction of the maximum number of edges that actually present For large network (n is very large)

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Density A (large) network is said to be dense if the density ρ tends to a constant as On the other hand, it is said to be sparse if ρ tends to 0 as

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Regular graphs A regular graph is a graph in which all the vertices have the same degree. k-regular graph: every vertex has degree k 2-regular: ring 4-regular: square lattice

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Path Path: a sequence of connected vertices Self-avoiding path: a path that does not intersect itself Length of a path: the number of edges in the path If there is a path of length 2 from j to i via k, then A ik A kj =1.

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Paths and adjacency matrix : the number of paths of length 2 from j to i : the number of paths of length 3 from j to i

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Geodesic paths A geodesic path (shortest path) is a path between two vertices that no shorter path exists Geodesic distance (shortest distance): the length of a geodesic path The smallest value r such that Geodesic paths are self-avoiding (Why?) Geodesic paths are not necessarily unique

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Diameter The diameter d of a graph is the length of the longest geodesic paths between any pairs of vertices in a network. Suppose that is the geodesic distance between vertices i and j

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Components A network is connected if there is a path from every vertex to any other vertex. Disconnected networks can be separated into several components. Components: There is a path from every vertex in the subnetwork to any other vertex in the same subnetwork. No other vextex can be added while preserving this property.

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Diffusion Diffusion is the process by which gas moves from regions of high density to regions of low, driven by the relative pressure of the different regions. Diffusion in a network (Influence network): The spread of an idea The spread of a disease

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Diffusion in a network Suppose that we have some commodity on the vertices. Let be the amount of the commodity at vertex i at time t Suppose that community moves from vertex j to an adjacent vertex i at rate C is called the diffusion constant.

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Governing equation for diffusion in a network is the degree of vertex i is the Kronecker delta, which is 1 if i=j and 0 otherwise.

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Governing equation for diffusion in a network Let D be the diagonal matrix with vertex degrees along its diagonal. Graph Laplacian: L=D-A In matrix form, A system of linear differential equations

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Solving the system of linear differential equations Suppose v i and λ i are the i th eigenvector and eigenvalue. Guess the solution has the form

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Eigenvalues of the graph Laplacian The Laplacian is symmetric. It has real eigenvalues. The Laplacian is positive- semidefinite. All its eigenvalues are nonnegative. The vector (1,1,…,1) is an eigenvector with eigenvalue 0.

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Algebraic connectivity The number of zero eigenvalues of the Laplacian is the number of components. The Laplacian can be written in a block form. The network is connected if and only if the second smallest eigenvalue of the Laplacian is nonzero. Algebraic connectivity: the second smallest eigenvalue of the Laplacian

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Distributed averaging consensus Lin Xiao and Stephen Boyd, “Systems & Control Letters,” 53 (2004) 65 – 78. Consider a network (connected graph) G=(V,E) Each vertex i holds an initial scalar value x i (0) in R, and x(0)=(x 1 (0),…, x n (0)) Two vertices can communicate with each other, if and only if they are neighbors. The problem is to compute the average of the initial values,,via a distributed algorithm

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Motivation Sensor networks (measuring temperature) A flock of flying birds

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Distributed linear iterations Constant edge weights In matrix form L=D-A is the Laplacian of the graph

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Distributed linear iterations W=I- L The vector (1,1,…,1) is an eigenvector with eigenvalue 0 of the Laplacain L. L is symmetric for an undirected graph W is a doubly stochastic matrix, i.e., all the row sums and column sums are all equal to 1. If W is a nonnegative matrix, then W can be viewed as the probability transition matrix of a Markov chain and where is a matrix with all its elements being 1.

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Condition for convergence As The condition for W to be a nonnegative matrix, k i is the degree of vertex i Distributed linear iteration is guaranteed to converge if

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Randomized gossip algorithms Stephen Boyd, Arpita Ghosh, Balaji Prabhakar, and Devavrat Shah, IEEE Transactions on Information Theory, VOL. 52, NO. 6, pp , JUNE Gossip algorithm: an algorithm in which each node can communicate with no more than one neighbor in each time slot. Consider a network (connected graph) G=(V,E) Each vertex i holds an initial scalar value x i (0) in R, and x(0)=(x 1 (0),…, x n (0)) The problem is to compute the average of the initial values,,via a gossip algorithm

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Asynchronous time model Each vertex has a clock which ticks at the times of a rate 1 Poisson process. Superposition of independent Poisson processes is also a Poisson process with the rate equal to the sum of the rates of the original Poisson processes. Uniformization: consider a Poisson process with rate n for clock ticks (as there are n vertices). With probability 1/n, a clock tick is chosen for vertex i.

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Asynchronous time model In the kth time slot, let node i’s clock tick and let it contact some neighboring node j with probability P ij. Both vertices set their values equal to the average of their current values. With probability, the random matrix W(k) is where Q is the permutation matrix that interchange the ith and jth coordinates.

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Spread of information Other objective functions, e.g., max, min. How fast is information distributed over a network via a randomized gossip algorithm? Start from the initial state x(0)=(1,0,…,0), i.e., only the first vertex has the information. If x i (t)>0, then vertex i must have been “visited” (at least once) by time t via the randomized gossip algorithm. can be used to bound the probability that all the vertices received the information.

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Influence network x i (t): the degree of influence (power) of vertex i at time t : the influence from j to i We still have But the weight matrix W is much more complicated. It may not be nonnegative, or doubly stochastic. Convergence might be a problem.

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Command hierarchies

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Emergent power Define power as the degree of influence in a social network How does one increase his/her power? Acquisition of weight influence: increase the influence to others and reduce the influence from others Acquisition of link influence: rewiring links (by knowing more important people) is less effective.

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Synchronization and desynchronization Desynchronization has many applications Fair resource scheduling as Time Division Multiple Access. Resource scheduling in wireless sensor networks. Phenomenon of mutual synchronization The flashing of fireflies in south Asia. Spreading identical oscillators into a round-robin schedule. 52

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Desynchronization algorithms A general framework for distributed algorithm to achieve desynchronization needed in TDMA. All the nodes can communicate with each other. Each node is modelled by an oscillator with the same fundamental frequency. There is no clock drift in every oscillator. J.Degesys, I. Rose, A. Patel, R. Nagpal, “Desync: Self-Organizing desynchronization and TDMA on wireless sensor networks,” IPSN, 2007 Degesys, Rose, Patel, Nagpal (2007) 55

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Desynchronization algorithms The DESYNC-STALE algorithm 56 Fire!

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Desynchronization algorithms The DESYNC-STALE algorithm When a node reaches the end of the cycle, it fires and resets its phase back to 0. It waits for the next node to fire and jump to a new phase according to a certain function. The jumping function only uses the firing information of the node fires before it and the node fires after it. 57

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Desynchronization algorithms When a node reaches the end of the cycle, it fires and resets its phase back to Fire!

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Desynchronization algorithms It waits for the next node to fire and jump to a new phase according to a certain function. 59 Fire!

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Desynchronization algorithms The jumping function only uses the firing information of the node fires before it and the node fires after it. 60 Fire!

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Desynchronization algorithms An extension of the fair scheduling scheme to the GPS. Every node is assigned a weight and the amount of bandwidth received by a node is proportional to its weight. They proposed an algorithm with two oscillators in each node and showed the convergence in the ideal case. Generalized process sharing scheme (GPS) Pagliari, Hong, Scaglione (2010) R. Pagliari, Y.-W. Hong, and A Scaglione “Bio-inspired algorithms for decentralized round-robin and proportional fair scheduling,” IEEE Journal on Selected Areas in Communications: Special Issue on Bio-Inspired Networking,

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Desynchronization algorithms Both the DESYNC-STALE and the extension of the GPS scheme are shown to work properly by various computer simulations. Lack of rigorous theoretical proofs in many aspects The rate of convergence of the DESYNC-STALE algorithm. The convergence of the stale GPS scheme. 62

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Desynchronization algorithms All the node are not likely to be identical A particular node need to interact with the “outside” world, and might not have the freedom to adjust its local clock. The master node in Bluetooth. The collector node in a wireless sensor network. The master clock in parallel analog-to-digital converters. 63

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Desynchronization algorithms Consider the desynchronization problem with an anchored node The anchored node never adjusts its phase. Except the anchored node, all the other nodes are identical and they do not know which node the anchored node is. 64

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Dynamics in Anchored Desynchronization 65 Fire!

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Network formation Erdos-Renyi random graph Configuration model Preferential attachment Small world Formation of social networks by random triad connections

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Random Graphs

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The G(n,p) model There are n vertices. With probability p, we place an edge independently between each distinct pair. First studied by Solomonoff and Rapoport in Erdos and Renyi published a series of papers on this model. For a sample G in G(n,p) with m edges, The probability of drawing a graph with m edges is then

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The mean degree in the G(n,p) model The mean value of m is This is a direct result of the binomial distribution as there are n(n-1)/2 independent Bernoulli random variables with parameter p. The mean degree in a graph with m edges is 2m/n. Thus the mean degree in G(n,p) is

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Degree distribution A given vertex in G(n,p) is connected with probability p to each of the n-1 other vertices. The degree of of a given vertex is thus a random variable with the binominal distribution B(n-1,p), i.e., Let the mean degree c=(n-1)p be fixed as n goes to Then the binomial distribution B(n-1,p) converges to the Poisson distribution with mean c, i.e., This is called Poisson random graph (as the limit of the Erdos and Renyi random graph)

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Clustering coefficient The clustering coefficient C is a measure of transitivity. It is defined as the probability that two network neighbors of a vertex are also neighbors of each other. As each edge is connected independently with probability p, This is one of several aspects that the random graph differs sharply most from real-world networks.

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Giant component Consider the Poisson random graph For the case p=0, there are no edges in the network at all. Each vertex is completely isolated. For the case p=1, every possible edge in the network is present and the network is an n- clique. Phase transition: an interesting question is how the transition between the two extremes occurs if we increase p from 0 to 1. Giant component: a network component whose size grows in proportion to n.

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The size of the giant component Let u be the average fraction of the vertices in the random graph that do not belong to the giant component. Suppose that a randomly chosen vertex i does not belong to the giant component (with probability u). For any other vertex j Either i is not connected to j (with probability 1-p), or i is connected to j but j itself is not a member of the giant component (with probability pu).

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A sample of an Erdos-Renyi graph hots2.html

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The configuration model Given a specific degree sequence Give vertex i k i “stubs” Choose two of the stubs uniformly at random and create an edge between the two vertices of the two chosen stubs. (they might be the same vertices (self edges)) Repeat the process until all the stubs are chosen.

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From degree sequence to degree distribution Specify the degree distribution p k Draw a degree sequence k 1,k 2, … with probability Then use the degree sequence to generate a random graph via using the configuration model. Scale free networks: the degree distribution obeys the power law (Pareto distribution).

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Preferential attachment Richer-get-richer effect Cumulative advantage Experience of shopping ( 品牌效應 )

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Price’s model Every newly appearing paper cites previous ones chosen at random proportional to the number of citations these previous papers already have. Degree distributions obey the power law. A special case is the Barabasi and Albert model

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The small-world model

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Regular graphs vs. random graphs Random graphs have low transitivity (clustering coefficient). Random graphs have small diameter. On the other hand, regular graphs have large diameter and some of them have large transitivity.

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A simple one-dimensional network model (ring) Every vertex is connected to c nearest neighbors in a line (a) or in a ring (b). Here c=6.

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Clustering coefficient Count the number of triangles Two steps forward and one step back The length of the last step can be chosen between 1,2,…c/2. The number of ways to choose the first two steps is the number of positive integer solutions to, which is

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Clustering coefficient The total number of triangles is Each vertex has degree c and the number of connected triples centered at a vertex is This clustering coefficient varies from 0 for c=2 up to a maximum of ¾ when c

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The mean shortest path The farthest one can move around the ring in a single step is c/2. So two vertices m lattice spacing apart are connected by a shortest path of 2m/c steps. Averaging over the complete range of m from 0 to n/2 gives a mean shortest path of n/2c. By contrast, the mean shortest path in the ER random graph is

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The small-world model Watts and Strogatz 1998 Interpolate between the ring model and the random graph by moving or rewiring edges from the ring to random positions. Start with a ring model with n vertices in which every vertex has degree c. Go through each edge in turn and with probability p we remove that edge and replace it with one (shortcut) that joins two vertices chosen uniformly at random.

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The small-world model

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The crucial point about the Watts-Strogatz small-world model is that as p is increased from 0 the clustering coefficient is maintained up to quite large values of p while the small-world behavior, meaning short average path lengths, already appears for quite modest values of p. As a result, there is a substantial range of intermediate values for which the model shows both effects simultaneously, i.e., large clustering coefficient and short average path length.

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Clustering coefficient (solid line) and average path length (dashed line)

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Scaling function for the small-world model

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Formation of Social Networks by Random Triad Connections Join work with Prof. Duan-Shin Lee Director of the Institute of Communications Engineering National Tsing Hua University

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A Network Formation Model for Social Networks At time zero, the network consists of a clique with m 0 vertices. At time t, which is a non-negative integer, a new vertex is attached to one of the existing vertices in the network. – The attached existing vertex is selected with equal probability. – This step is called the uniform attachment step. Each neighbor of the attached existing vertex is attached to the new vertex with probability a and not attached with probability 1-a. – This step is called the triad formation step. – Friends’ friends are more likely to be friends. 91 Institute of Communications Engineering National Tsing-Hua University

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Uniform Attachment and Triad Formation when 92 Institute of Communications Engineering National Tsing-Hua University t = 0

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Uniform Attachment and Triad Formation when 93 Institute of Communications Engineering National Tsing-Hua University t = 1 uniform attachment triad formation with probability a do nothing with probability 1-a

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Uniform Attachment and Triad Formation when 94 Institute of Communications Engineering National Tsing-Hua University t = 2

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Community detection INFOCOM 2011 Cheng-Shang Chang, Chin-Yi Hsu, Jay Cheng, and Duan-Shin Lee Institute of Communications Engineering National Tsing Hua University Taiwan, R.O.C.

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96 Detecting Community Community : It is the appearance of densely connected groups of vertices, with only sparser connections between groups. Modularity (Girman and Newman 2002) : It is a property of a network and a specifically proposed division of that network into communities. It measures when the division is a good one, in the sense that there are fewer than expected edges between communities.

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97 Detecting Community Example :

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98 Algorithms for Detecting Community Many algorithms have been proposed in the literature. Basically, they can be classified into four categories: (1) divisive algorithms (2) agglomerative algorithms (3) graph partitioning and clustering algorithms (4) data compression algorithms Newman’s fast algorithm also belongs to this class Our algorithm belongs to this class

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99 Agglomerative algorithms Features of agglomerative algorithms Clusters are iteratively merged if their similarity is sufficiently high. Our distribution based clustering algorithms starts by viewing each vertex as a community with only one member. The algorithm repeatedly merges the two most positively correlated communities into a new community until all the remaining communities are negatively correlated.

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100 Agglomerative Algorithms Example :

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Our Contributions In spite of all the efforts in developing community detection algorithms, there are still many questions that we do not have satisfactory answers. What is a community in a network? Even with a definition of a community, what would be the right index for measuring the performance of a graph partition? We will provide a general probabilistic framework for these questions. 101

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Our Contributions In spite of all the efforts in developing community detection algorithms, there are still many questions that we do not have satisfactory answers. What is a community in a network? Even with a definition of a community, what would be the right index for measuring the performance of a graph partition? We will provide a general probabilistic framework for these questions. 102

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Our Contributions Characterization of a graph : the key idea of our framework is to characterize a graph by a bivariate distribution that specifies the probability of the two vertices appearing at both ends of a “randomly” selected path in the graph. Definition of a community: With such a bivariate distribution, we can then define a community as a set of vertices with the property that it is more likely to find the other end in the same community given one of the two ends in a randomly selected path is already in the community. Correlation measures: To detect communities, we define a class of correlation measures that can be used for measuring how two vertices (and two communities) are related. Two communities are positively (resp. negatively) correlated if the value of a correlation measure for these two communities is positive (resp. negative). 103

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Our Contributions A class of distribution-based clustering algorithms : as a generalization of Newman’s fast algorithm, we propose a class of distribution-based clustering algorithms for community detection. Two theoretic results that can be proved for a distribution-based clustering algorithm: (i) it guarantees that every community detected by the algorithm satisfies the definition of a community under certain technical conditions for the bivariate distribution, (ii) the algorithm increases the “modularity” index in each merge of two positively correlated communities. 104

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105 Correlation Measures Definition: For any two indicator random variables X and Y, is called a correlation measure in this paper if (1) is solely determined by the bivariate distribution of X and Y, (2) if and only if X and Y are independent, (3) if and only if X and Y are positively correlated, i.e., From (2) and (3), we also know that if and only if X and Y are negatively correlated.

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106 Covariance: For two indicator random variables X and Y, we have Examples of Correlation Measures

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107 Correlation: Note that the correlation of two random variables X and Y, denoted by, can be computed as follows:,where Examples of Correlation Measures

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108 Mutual information: The mutual information of two random variables X and Y, denoted by, can be computed as follows: Examples of Correlation Measures

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109 Instead of characterizing a network by a graph, we characterize a network by a bivariate distribution. As mentioned before, Now, let be the sum of all the elements in a matrix A, i.e., Then, we can rewrite the bivariate distribution: { 1/2m, if vertices v and w are connected, 0, otherwise. Probabilistic Framework

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110 Recall that the bivariate distribution above is the probability for the two ends of a randomly selected edge in a graph. Now, our idea is to generate the needed bivariate distribution by randomly selecting the two ends of a path. We first consider a function f that maps an adjacency matrix A to another matrix f(A). Then we define a bivariate distribution from f(A) by Probabilistic Framework

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111 A random selection of a path with length not greater than 2: Consider a graph with an n n adjacency matrix A and A path with length l is selected with probability for l = 0, 1, and 2 Probabilistic Framework

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112 A random walk on a graph: It can be characterized by a Markov chain with the n n transition probability matrix, where is the transition probability from vertex v to vertex w. The stationary probability that the Markov chain is in vertex v, denoted by, is. is the probability that we select a path with length l, l = 1, 2, …. The probability of selecting a random walk (path) with vertices is Probabilistic Framework

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113 We then have the bivariate distribution We can simply let l = 0 for all l > 2 and this leads to Probabilistic Framework

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114 (1) Input a bivariate distribution, v,w = 1, 2, …, n that characterizes the two randomly selected nodes V and W, and a correlation measure for two indicator random variables. (2) Initially, there are n communities, indexed from 1 to n, with each community containing exactly one node. Specifically, let be the set of nodes in community i. Then, i = 1, 2, …, n. Distribution-based Clustering Algorithm

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115 (3) Let (resp. ) be the indicator random variable for the event that V is in community i (resp. W is in community j). Then Compute for all i and j. Distribution-based Clustering Algorithm

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116 (4) Find the two (distinct) communities that have the largest correlation measure. Group these two communities into a new community. Suppose that community i and community j are grouped into a new community k. Then and update Distribution-based Clustering Algorithm

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117 (5) For all, compute and. (6) Repeat (4) until either there is only one community left or all the remaining pairs of communities have negative measures, i.e., for all. Distribution-based Clustering Algorithm

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118 Definition: A set of nodes S is a community in a probabilistic sense if If, then this is equivalent to It is more likely to find the other node in the same community given that one of a randomly selected pair of two nodes is already in the community. Definition of a Community

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119 Definition of a Community Theorem 1: Suppose that is symmetric and, for all v = 1, 2, …, n. Then every community detected by any distribution-based clustering algorithm is a community in the probabilistic sense.

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120 Definition: Consider a bivariate distribution with v, w = 1, 2, …, n. Let, c = 1, 2, …,C, be a partition of {1, 2, …, n}, i.e., is an empty set for and. The modularity index Q with respect to the partition where c = 1, 2, …,C, is Theorem 2: Suppose that is symmetric. Then for any distribution-based clustering algorithm, the modularity index is non-decreasing in every iteration. Definition of the Modularity Index

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121 Each point in these figures is an average over 100 random graphs. In these figures, we also show 95% confidence intervals for all data points. In our simulation results, we will consider three distribution-based clustering algorithms: (1) covariance algorithm (2) correlation algorithm (3) mutual information algorithm Simulation Result

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122 To map a graph with an adjacency matrix A to a bivariate distribution. Recall that,, we will consider the following three types of functions: (1), i.e., (2), i.e., (3), i.e., Simulation Result

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123 Simulation Result

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124 Simulation Result

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125 Simulation Result

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126 Simulation Result

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127 Simulation Result W. W. Zachary, J. Anthropol, Res. 33, 452, 1977.

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128 Simulation Result

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129 Simulation Result

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130 Simulation Result

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Conclusion In 2005, the National Science Foundation in the U.S. realized that there is a need for “organized knowledge of networks” based on the scientific method. This will require the integration of the knowledge in various fields, including the Internet, power grids, social networks, physical networks, and biological networks. The main mathematical tool for network science is the study of the dynamics of graphs.

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Research problems How is life formed? Is the emergence of life through random rewiring of DNAs according a certain microrule? How powerful is a person in a community? How much is he/she worth? Can these be evaluated by the people he/she knows? How can one bring down the Internet? What is the best strategy to defend one’s network from malicious attacks? How are these related to the topology of a network? Why is there a phase change from water to ice? Can this be explained by using the percolation theory? Does the large deviation theory play a role here?

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