Assortativity and Mixing. Outline. Definition. General mixing. Definition. Assortativity by degree. Contagion. References. Contagion.
|
|
- Dale Simpson
- 6 years ago
- Views:
Transcription
1 Outline Complex Networks, Course 303A, Spring, 2009 Prof Peter Dodds Department of Mathematics & Statistics University of Vermont Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 30 License Frame 1/25 Frame 2/25 Basic idea: between node categories Random networks with arbitrary distributions cover much territory but do not represent all networks Moving away from pure random networks was a key first step We can extend in many other directions and a natural one is to introduce correlations between different kinds of nodes Node attributes may be anything, eg: 1 2 demographics (age, gender, etc 3 group affiliation We speak of mixing patterns, correlations, biases Networks are still random at base but now have more global structure Build on work by Newman [3, 4] Frame 3/25 Assume types of nodes are countable, and are assigned numbers 1, 2, 3, Consider networks with directed edges ( an edge connects a node of type µ e µν = Pr to a node of type ν a µ = Pr(an edge comes from a node of type ν b µ = Pr(an edge leads to a node of type ν Write E = [e µν ], a = [a µ ], and b = [b µ ] Requirements: e µν = 1, µ ν ν e µν = a µ, and µ e µν = b ν Frame 4/25
2 Notes: Correlation coefficient: Varying e µν allows us to move between the following: 1 Perfectly assortative networks where nodes only connect to like nodes, and the network breaks into subnetworks Requires e µν = 0 if µ ν and µ e µµ = 1 2 Uncorrelated networks (as we have studied so far For these we must have independence: e µν = a µ b ν 3 Disassortative networks where nodes connect to nodes distinct from themselves Disassortative networks can be hard to build and may require constraints on the e µν Basic story: level of assortativity reflects the to which nodes are connected to nodes within their group Quantify the level of assortativity with the following assortativity coefficient [4] : µ r = e µµ µ a µb µ 1 µ a = Tr E E 2 1 µb µ 1 E 2 1 where 1 is the 1-norm = sum of a matrix s entries Tr E is the fraction of edges that are within groups E 2 1 is the fraction of edges that would be within groups if connections were random 1 E 2 1 is a normalization factor so r max = 1 When Tr e µµ = 1, we have r = 1 When e µµ = a µ b µ, we have r = 0 Frame 5/25 Frame 6/25 Correlation coefficient: Scalar quantities Notes: r = 1 is inaccessible if three or more types are presents Disassortative networks simply have nodes connected to unlike nodes no measure of how unlike nodes are Minimum value of r occurs when all links between non-like nodes: Tr e µµ = 1 where 1 r min < 0 r min = E E 2 1 Now consider nodes defined by a scalar integer quantity Examples: age in years, height in inches, number of friends, e jk = Pr a randomly chosen edge connects a node with value j to a node with value k a j and b k are defined as before Can now measure correlations between nodes based on this scalar quantity using standard Pearson correlation coefficient ( : r = j k j k(e jk a j b k σ a σ b = j 2 a j 2 a jk j a k b k 2 b k 2 b Frame 7/25 This is the observed normalized deviation from randomness in the product jk Frame 8/25
3 Degree- correlations Degree- correlations Natural correlation is between the s of connected nodes Now define e jk with a slight twist: ( an edge connects a j + 1 node e jk = Pr to a k + 1 node ( an edge runs between a node of in- j = Pr and a node of out- k Useful for calculations (as per R k Important: Must separately define P 0 as the {e jk } contain no information about isolated nodes Directed networks still fine but we will assume from here on that e jk = e kj Degree- correlations Frame 9/25 Notation reconciliation for undirected networks: j k r = j k(e jk R j R k MIXING PATTERNS IN NETWORKS where, as before, R k is the probability that a randomly chosen edge leads to a node of k + 1, and σ 2 R = j σ 2 R j 2 R j j jr j TABLE II Size n, assortativity coefficient r, and expected error r on the assortativity, for a number of social, technological, and biological networks, both directed and undirected Social networks: coauthorship networks of a physicists and biologists 46 and b mathematicians 47, in which authors are connected if they have coauthored one or more articles in learned journals; c collaborations of film actors in which actors are connected if they have appeared together in one or more movies 5,7 ; d directors of fortune 1000 companies for 1999, in which two directors are connected if they sit on the board of directors of the same company 48 ; e romantic not necessarily sexual relationships between students at a US high school 49 ; f network of address books of computer users on a large computer system, in which an edge from user A to user B indicates that B appears in A s address book 50 Technological networks: g network of high voltage transmission lines in the Western States Power Grid of the United States 5 ; h network of direct peering relationships between autonomous systems on the Internet, April ; i network of hyperlinks between pages in the World Wide Web domain ndedu, circa ; j network of dependencies between software packages in the GNU/Linux operating system, in which an edge from package A to package B indicates that A relies on components of B for its operation Biological networks: k protein-protein interaction network in the yeast S Cerevisiae 53 ; l metabolic network of the bacterium E Measurements Coli 54 ; m neural network of theof nematode - worm C Elegans 5,55 ; tropic interactions correlations between species in the food webs of n Ythan Estuary, Scotland 56 and o Little Rock Lake, Wisconsin 57 2 PHYSICAL REVIEW E 67, Frame 10/25 Error estimate for r: Remove edge i and recompute r to obtain r i Repeat for all edges and compute using the jackknife method ( [1] σ 2 r = i (r i r 2 Mildly sneaky as variables need to be independent for us to be truly happy and edges are correlated Frame 11/25 Group Network Type Size n Assortativity r Error r a Physics coauthorship undirected a Biology coauthorship undirected b Mathematics coauthorship undirected Social c Film actor collaborations undirected d Company directors undirected e Student relationships undirected f address books directed g Power grid undirected Technological h Internet undirected i World Wide Web directed j Software dependencies directed k Protein interactions undirected l Metabolic network undirected Biological m Neural network directed n Marine food web directed o Freshwater food web directed m almost never is In this paper, therefore, we take an alternative approach, making use of computer simulation We would like to generate on a computer a random network having, for instance, a particular value of the matrix e jk This also fixes the distribution, via Eq 23 In Ref 22 we discussed disassortative one possible way of doing this using an algorithm similar to that of Sec II C One would draw edges from the desired distribution e jk and then join the k ends randomly in groups of k to create the network This algorithm has also been discussed recently by 58,59 The algorithm is as follows 1 Given the desired edge distribution e jk, we first calculate the corresponding distribution of excess s q k from Eq 23, and then invert Eq 22 to find the distribution: Frame 12/25 p k q k 1 /k 27 q j 1 / j j Social networks tend to be assortative (homophily Technological and biological networks tend to be
4 Spreading on -correlated networks Spreading on -correlated networks Next: Generalize our work for random networks to -correlated networks As before, by allowing that a node of k is activated by one neighbor with probability β k1, we can handle various problems: 1 find the giant component size 2 find the probability and extent of spread for simple disease models 3 find the probability of spreading for simple threshold models Goal: Find f n,j = Pr an edge emanating from a j + 1 node leads to a finite active subcomponent of size n Repeat: a node of k is in the game with probability β k1 Define β 1 = [β k1 ] Plan: Find the generating function F j (x; β 1 = n=0 f n,jx n Frame 13/25 Frame 14/25 Spreading on -correlated networks Spreading on -correlated networks Recursive relationship: F j (x; β 1 = x 0 + x e jk R j (1 β k+1,1 e [ jk β k+1,1 F k (x; R k β 1 ] j First term = Pr that the first node we reach is not in the game Second term involves Pr we hit an active node which has k outgoing edges Next: find average size of active components reached by following a link from a j node = F j (1; β 1 Frame 15/25 Differentiate F j (x; β 1, set x = 1, and rearrange We use F k (1; β 1 = 1 which is true when no giant component exists We find: R j F j (1; β 1 = e jk β k+1,1 + ke jk β k+1,1 F k (1; β 1 Rearranging and introducing a sneaky δ jk : ( δjk R k kβ k+1,1 e jk F k (1; β 1 = e jk β k+1,1 Frame 16/25
5 Spreading on -correlated networks Spreading on -correlated networks So, in principle at least: In matrix form, we have F (1; β 1 = A 1 E, β 1 E β 1 where [A E, β1 ] A E, β1 F (1; β 1 = E β 1 = δ jkr k kβ k+1,1 e jk, j+1,k+1 [ F (1; β 1 ] = F k (1; β 1, k+1 [ ] [E] j+1,k+1 = e jk, and β1 = β k+1,1 k+1 Now: as F (1; β 1, the average size of an active component reached along an edge, increases, we move towards a transition to a giant component Right at the transition, the average component size explodes Exploding inverses of matrices occur when their determinants are 0 The condition is therefore: det A E, β1 = 0 Frame 17/25 Frame 18/25 Spreading on -correlated networks Spreading on -correlated networks General condition details: det A E, β1 = det ( δ jk R k 1 (k 1β k,1 e j 1,k 1 = 0 The above collapses to our standard contagion condition when e jk = R j R k When β 1 = β 1, we have the condition for a simple disease model s successful spread We ll next find two more pieces: 1 P trig, the probability of starting a cascade 2 S, the expected extent of activation given a small seed Triggering probability: det ( δ jk R k 1 β(k 1e j 1,k 1 = 0 When β 1 = 1, we have the condition for the existence of a giant component: Generating function: H(x; β 1 = x P k [F k 1 (x; β k 1 ] det ( δ jk R k 1 (k 1e j 1,k 1 = 0 Bonusville: We ll find another (possibly better version of this set of conditions later Frame 19/25 Generating function for vulnerable component size is more complicated Frame 20/25
6 Spreading on -correlated networks Want probability of not reaching a finite component P trig = S trig =1 H(1; β 1 =1 P k [F k 1 (1; β k 1 ] Last piece: we have to compute F k 1 (1; β 1 Nastier (nonlinear we have to solve the recursive expression we started with when x = 1: F j (1; β 1 = e jk R j (1 β k+1,1 + Iterative methods should work here e jk VOLUME 89, NUMBER 20 P H Y S I C A L R E V I E W L E T T E R S 11 NOVEMBER 2002 Equation (7 diverges at the point at which the determinant of A is zero This point marks the phase transition at which a giant component forms in our graph By considering the behavior of Eq (7 close to the transition, where hsi must be large and positive in the absence of a giant component, we deduce that a giant component exists in the network when deta > 0 This is the appropriate generalization for a network with assortative mixing of the criterion of Molloy and Reed [16] for the existence of a giant component To calculate the size S of the giant component, we define u k to be the probability that an edge connected to a vertex of remaining k leads to another vertex that does not belong to the giant component Then P S 1 p 0 X1 p k u k k 1 ; u k e jk u k k j : (8 k 1 Pk e jk R j β k+1,1 [F k (1; β 1 ] k Spreading on -correlated networks As before, these equations give the actual evolution of φ t for synchronous updates condition follows from θ t+1 = G( θ t equation Need small θ 0 to take off so we linearize G around θ 0 = 0 θ j,t+1 = G j ( 0 + k=1 G j ( 0 θ k,t θ k,t + k=1 Largest eigenvalue of G j ( 0 θ k,t must exceed 1 Condition for spreading is therefore dependent on eigenvalues of this matrix: G j ( 0 θ k,t = e j 1,k 1 (k 1(β k1 β k0 R j 1 Insert question from assignment 5 ( To test these results and to help Frame form21/25 a more complete picture of the properties of assortatively mixed networks, we have also performed computer simulations, generating networks with given values of e jk and measuring their properties directly Generating such networks is not entirely trivial One cannot simply draw a set of pairs j i ;k i for edges i from the distribution e jk, since such a set would almost certainly fail to satisfy the basic topological requirement that the number Assortativity of edges andending at vertices of k must be a multiple of k Instead, therefore we propose the following Monte Carlo algorithm for generating graphs First, we generate a random graph with the desired distribution according to the prescription given in Ref [16] Then we apply a Metropolis dynamics to the graph in which on each step we Assortativity choose at byrandom two edges, denoted by the vertex pairs, v 1 ;w 1 and v 2 ;w 2, that they connect We measure the remaining s j 1 ;k 1 and j 2 ;k 2 for these vertex pairs, and then replace the edges with two new ones v 1 ;v 2 and w 1 ;w 2 with probability 2 G j ( min 1; e j1 j 2 e k1 k 2 = e j1 k 1 e j2 k 2 This dynamics conserves the sequence, is ergodic on the set of graphs having 0 θk,t 2 θk,t 2 that + sequence, and, with the choice of acceptance probability above, satisfies detailed balance for state probabilities Q i e ji k i, and hence has the required edge distribution e jk as its fixed point As an example, consider the symmetric binomial form j k j k e jk N e j k = p j q k p k q j ; (9 j k where p q 1, > 0, and N e 1= is a normalizing constant (The binomial probabilities p and q should not be confused with the quantities p k and q k introduced earlier This distribution is chosen for analytic tractability, although its behavior is also quite natural: the distribution of the sum j k of the s at the ends of an edge falls off as a simple Frame exponential, 23/25 while that sum is distributed between the two ends binomially, Spreading on -correlated networks Truly final piece: Find final size using approach of Gleeson [2], a generalization of that used for uncorrelated random networks Need to compute θ j,t, the probability that an edge leading to a j node is infected at time t the parameter p controlling the assortative mixing From Eq (3, the value of r is Evolution of edge activity probability: 8pq 1 r 2e 1= 1 2 p q ; (10 2 θ j,t+1 = G j ( θ t = φ 0 + (1 φ 0 which can take both positive and negative values, passing through zero when p p k 1 e p j 1,k 1 ( 0:1464 In Fig 1 we show the size of the giant component k 1 for graphs of this type as a function of the scale parameter, from bothrour j 1 numerical simulations i and the exact solution k=1 above As the figure i=0 shows, the two are in good agreement The three curves in the figure are for p 0:05, where the graph is disassortative, p p 0, where it is neutral (neither assortative nor disassortative, and p 0:5, where it is assortative k As becomes large we see the expected phase transition at which φ t+1 a giant = φcomponent 0 +(1 φ forms 0 ThereP are k two important points to notice about the figure First, i=0 the position of the phase transition moves lower as the graph becomes more assortative That is, the graph percolates more easily, creating a giant component, if the high vertices preferentially associate with other high ones Second, notice that, by contrast, the size of the giant component for large is smaller in the assortatively mixed network Overall active fraction s evolution: θ i k,t (1 θ k,t k 1 i β ki ( k θk,t i i (1 θ k,t k i β ki These findings are intuitively reasonable If the net- How the giant component changes with assortativity work mixes assortatively, then the high- vertices will tend to stick together in a subnetwork or core group of higher mean than the network as a whole It is reasonable to suppose that percolation would occur earlier within such a subnetwork Conversely, since percolation will be restricted to this subnetwork, it is not surprising giant component S exponential parameter κ assortative neutral disassortative FIG 1 Size of the giant component as a fraction of graph from Newman, 2002 [3] size for graphs with the edge distribution given in Eq (9 The points are simulation results for graphs of N vertices, while the solid lines are the numerical solution of Eq (8 Each point is an average over ten graphs; the resulting statistical errors are smaller than the symbols The values of p are 05 (circles, p 0 0:146 (squares, and 0:05 (triangles More assortative networks percolate for lower average s But disassortative networks end up with higher extents of spreading Frame 22/25 Frame 24/
7 I B Efron and C Stein The jackknife estimate of variance The Annals of Statistics, 9: , 1981 pdf ( J P Gleeson Cascades on correlated and modular random networks Phys Rev E, 77:046117, 2008 pdf ( M Newman Assortative mixing in networks Phys Rev Lett, 89:208701, 2002 pdf ( M E J Newman patterns in networks Phys Rev E, 67:026126, 2003 pdf ( Frame 25/25
Complex Networks, Course 303A, Spring, Prof. Peter Dodds
Complex Networks, Course 303A, Spring, 2009 Prof. Peter Dodds Department of Mathematics & Statistics University of Vermont Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.
More informationRandom Networks. Complex Networks, CSYS/MATH 303, Spring, Prof. Peter Dodds
Complex Networks, CSYS/MATH 303, Spring, 2010 Prof. Peter Dodds Department of Mathematics & Statistics Center for Complex Systems Vermont Advanced Computing Center University of Vermont Licensed under
More informationRandom Networks. Complex Networks CSYS/MATH 303, Spring, Prof. Peter Dodds
Complex Networks CSYS/MATH 303, Spring, 2011 Prof. Peter Dodds Department of Mathematics & Statistics Center for Complex Systems Vermont Advanced Computing Center University of Vermont Licensed under the
More informationarxiv: v4 [cond-mat.dis-nn] 16 May 2011
Direct, physically-motivated derivation of the contagion condition for spreading processes on generalized random networks arxiv:1101.5591v4 [cond-mat.dis-nn] 16 May 2011 Peter Sheridan Dodds, 1,2, Kameron
More informationQuantitative Metrics: II
Network Observational Methods and Topics Quantitative Metrics: II Degree correlation Exploring whether the sign of degree correlation can be predicted from network type or similarity to regular structures,
More informationNetwork Observational Methods and. Quantitative Metrics: II
Network Observational Methods and Whitney topics Quantitative Metrics: II Community structure (some done already in Constraints - I) The Zachary Karate club story Degree correlation Calculating degree
More informationContagion. Complex Networks CSYS/MATH 303, Spring, Prof. Peter Dodds
Complex Networks CSYS/MATH 33, Spring, 211 Basic Social Prof. Peter Dodds Department of Mathematics & Statistics Center for Complex Systems Vermont Advanced Computing Center University of Vermont Licensed
More informationMini course on Complex Networks
Mini course on Complex Networks Massimo Ostilli 1 1 UFSC, Florianopolis, Brazil September 2017 Dep. de Fisica Organization of The Mini Course Day 1: Basic Topology of Equilibrium Networks Day 2: Percolation
More informationComplex Networks CSYS/MATH 303, Spring, Prof. Peter Dodds
Complex Networks CSYS/MATH 303, Spring, 2011 Prof. Peter Dodds Department of Mathematics & Statistics Center for Complex Systems Vermont Advanced Computing Center University of Vermont Licensed under the
More informationComplex networks: an introduction
Alain Barrat Complex networks: an introduction CPT, Marseille, France ISI, Turin, Italy http://www.cpt.univ-mrs.fr/~barrat http://cxnets.googlepages.com Plan of the lecture I. INTRODUCTION II. I. Networks:
More informationDesign and characterization of chemical space networks
Design and characterization of chemical space networks Martin Vogt B-IT Life Science Informatics Rheinische Friedrich-Wilhelms-University Bonn 16 August 2015 Network representations of chemical spaces
More information6.207/14.15: Networks Lecture 12: Generalized Random Graphs
6.207/14.15: Networks Lecture 12: Generalized Random Graphs 1 Outline Small-world model Growing random networks Power-law degree distributions: Rich-Get-Richer effects Models: Uniform attachment model
More informationSelf Similar (Scale Free, Power Law) Networks (I)
Self Similar (Scale Free, Power Law) Networks (I) E6083: lecture 4 Prof. Predrag R. Jelenković Dept. of Electrical Engineering Columbia University, NY 10027, USA {predrag}@ee.columbia.edu February 7, 2007
More informationECS 289 F / MAE 298, Lecture 15 May 20, Diffusion, Cascades and Influence
ECS 289 F / MAE 298, Lecture 15 May 20, 2014 Diffusion, Cascades and Influence Diffusion and cascades in networks (Nodes in one of two states) Viruses (human and computer) contact processes epidemic thresholds
More informationGenerating Functions for Random Networks
for Random Networks Complex Networks CSYS/MATH 303, Spring, 2011 Prof. Peter Dodds Department of Mathematics & Statistics Center for Complex Systems Vermont Advanced Computing Center University of Vermont
More informationNetwork Analysis and Modeling
lecture 0: what are networks and how do we talk about them? 2017 Aaron Clauset 003 052 002 001 Aaron Clauset @aaronclauset Assistant Professor of Computer Science University of Colorado Boulder External
More information0.20 N=000 N=5000 0.30 0.5 0.25 P 0.0 P 0.20 0.5 0.05 0.0 0.05 0.00 0.0 0.2 0.4 0.6 0.8.0 n r 0.00 0.0 0.2 0.4 0.6 0.8.0 n r 0.4 N=0000 0.6 0.5 N=50000 P 0.3 0.2 0. P 0.4 0.3 0.2 0. 0.0 0.0 0.2 0.4 0.6
More informationWeb Structure Mining Nodes, Links and Influence
Web Structure Mining Nodes, Links and Influence 1 Outline 1. Importance of nodes 1. Centrality 2. Prestige 3. Page Rank 4. Hubs and Authority 5. Metrics comparison 2. Link analysis 3. Influence model 1.
More informationLiu et al.: Controllability of complex networks
complex networks Sandbox slides. Peter Sheridan Dodds Department of Mathematics & Statistics Center for Complex Systems Vermont Advanced Computing Center University of Vermont of 2 Outline 2 of 2 RESEARCH
More informationNetwork Science: Principles and Applications
Network Science: Principles and Applications CS 695 - Fall 2016 Amarda Shehu,Fei Li [amarda, lifei](at)gmu.edu Department of Computer Science George Mason University 1 Outline of Today s Class 2 Robustness
More informationTELCOM2125: Network Science and Analysis
School of Information Sciences University of Pittsburgh TELCOM2125: Network Science and Analysis Konstantinos Pelechrinis Spring 2015 Figures are taken from: M.E.J. Newman, Networks: An Introduction 2
More informationSocial Networks- Stanley Milgram (1967)
Complex Networs Networ is a structure of N nodes and 2M lins (or M edges) Called also graph in Mathematics Many examples of networs Internet: nodes represent computers lins the connecting cables Social
More informationShannon Entropy and Degree Correlations in Complex Networks
Shannon Entropy and Degree Correlations in Complex etworks SAMUEL JOHSO, JOAQUÍ J. TORRES, J. MARRO, and MIGUEL A. MUÑOZ Instituto Carlos I de Física Teórica y Computacional, Departamento de Electromagnetismo
More information1 Complex Networks - A Brief Overview
Power-law Degree Distributions 1 Complex Networks - A Brief Overview Complex networks occur in many social, technological and scientific settings. Examples of complex networks include World Wide Web, Internet,
More informationCommunities Via Laplacian Matrices. Degree, Adjacency, and Laplacian Matrices Eigenvectors of Laplacian Matrices
Communities Via Laplacian Matrices Degree, Adjacency, and Laplacian Matrices Eigenvectors of Laplacian Matrices The Laplacian Approach As with betweenness approach, we want to divide a social graph into
More informationCSI 445/660 Part 6 (Centrality Measures for Networks) 6 1 / 68
CSI 445/660 Part 6 (Centrality Measures for Networks) 6 1 / 68 References 1 L. Freeman, Centrality in Social Networks: Conceptual Clarification, Social Networks, Vol. 1, 1978/1979, pp. 215 239. 2 S. Wasserman
More informationVirgili, Tarragona (Spain) Roma (Italy) Zaragoza, Zaragoza (Spain)
Int.J.Complex Systems in Science vol. 1 (2011), pp. 47 54 Probabilistic framework for epidemic spreading in complex networks Sergio Gómez 1,, Alex Arenas 1, Javier Borge-Holthoefer 1, Sandro Meloni 2,3
More informationNetworks as a tool for Complex systems
Complex Networs Networ is a structure of N nodes and 2M lins (or M edges) Called also graph in Mathematics Many examples of networs Internet: nodes represent computers lins the connecting cables Social
More informationA framework for analyzing contagion in banking networks
A framework for analyzing contagion in banking networks arxiv:.432v [q-fin.gn] 9 Oct 2 T. R. Hurd, James P. Gleeson 2 Department of Mathematics, McMaster University, Canada 2 MACSI, Department of Mathematics
More informationarxiv:cond-mat/ v1 3 Sep 2004
Effects of Community Structure on Search and Ranking in Information Networks Huafeng Xie 1,3, Koon-Kiu Yan 2,3, Sergei Maslov 3 1 New Media Lab, The Graduate Center, CUNY New York, NY 10016, USA 2 Department
More informationContagion. Basic Contagion Models. Social Contagion Models. Network version All-to-all networks Theory. References. 1 of 58.
Complex Networks CSYS/MATH 33, Spring, 2 Prof. Peter Dodds Department of Mathematics & Statistics Center for Complex Systems Vermont Advanced Computing Center University of Vermont Basic Social Spreading
More informationKINETICS OF COMPLEX SOCIAL CONTAGION. János Kertész Central European University. Pohang, May 27, 2016
KINETICS OF COMPLEX SOCIAL CONTAGION János Kertész Central European University Pohang, May 27, 2016 Theory: Zhongyuan Ruan, Gerardo Iniguez, Marton Karsai, JK: Kinetics of social contagion Phys. Rev. Lett.
More informationarxiv:cond-mat/ v2 [cond-mat.stat-mech] 7 Jan 2000
Epidemics and percolation in small-world networks Cristopher Moore 1,2 and M. E. J. Newman 1 1 Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501 2 Departments of Computer Science and
More informationDegree Distribution: The case of Citation Networks
Network Analysis Degree Distribution: The case of Citation Networks Papers (in almost all fields) refer to works done earlier on same/related topics Citations A network can be defined as Each node is
More informationEigenvalues by row operations
Eigenvalues by row operations Barton L. Willis Department of Mathematics University of Nebraska at Kearney Kearney, Nebraska 68849 May, 5 Introduction There is a wonderful article, Down with Determinants!,
More informationKINETICS OF SOCIAL CONTAGION. János Kertész Central European University. SNU, June
KINETICS OF SOCIAL CONTAGION János Kertész Central European University SNU, June 1 2016 Theory: Zhongyuan Ruan, Gerardo Iniguez, Marton Karsai, JK: Kinetics of social contagion Phys. Rev. Lett. 115, 218702
More informationECS 253 / MAE 253, Lecture 15 May 17, I. Probability generating function recap
ECS 253 / MAE 253, Lecture 15 May 17, 2016 I. Probability generating function recap Part I. Ensemble approaches A. Master equations (Random graph evolution, cluster aggregation) B. Network configuration
More informationLecture VI Introduction to complex networks. Santo Fortunato
Lecture VI Introduction to complex networks Santo Fortunato Plan of the course I. Networks: definitions, characteristics, basic concepts in graph theory II. III. IV. Real world networks: basic properties
More informationOnline Social Networks and Media. Link Analysis and Web Search
Online Social Networks and Media Link Analysis and Web Search How to Organize the Web First try: Human curated Web directories Yahoo, DMOZ, LookSmart How to organize the web Second try: Web Search Information
More informationAnalytically tractable processes on networks
University of California San Diego CERTH, 25 May 2011 Outline Motivation 1 Motivation Networks Random walk and Consensus Epidemic models Spreading processes on networks 2 Networks Motivation Networks Random
More informationComplex Networks, Course 303A, Spring, Prof. Peter Dodds
Complex Networks, Course 303A, Spring, 2009 Prof. Peter Dodds Department of Mathematics & Statistics University of Vermont Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.
More informationScale-Free Networks. Outline. Redner & Krapivisky s model. Superlinear attachment kernels. References. References. Scale-Free Networks
Outline Complex, CSYS/MATH 303, Spring, 200 Prof. Peter Dodds Department of Mathematics & Statistics Center for Complex Systems Vermont Advanced Computing Center University of Vermont Licensed under the
More informationople are co-authors if there edge to each of them. derived from the five largest PH, HEP TH, HEP PH, place a time-stamp on each h paper, and for each perission to the arxiv. The the period from April 1992
More informationBayesian Inference for Contact Networks Given Epidemic Data
Bayesian Inference for Contact Networks Given Epidemic Data Chris Groendyke, David Welch, Shweta Bansal, David Hunter Departments of Statistics and Biology Pennsylvania State University SAMSI, April 17,
More informationGeneralized Exponential Random Graph Models: Inference for Weighted Graphs
Generalized Exponential Random Graph Models: Inference for Weighted Graphs James D. Wilson University of North Carolina at Chapel Hill June 18th, 2015 Political Networks, 2015 James D. Wilson GERGMs for
More informationMarkov Chains Handout for Stat 110
Markov Chains Handout for Stat 0 Prof. Joe Blitzstein (Harvard Statistics Department) Introduction Markov chains were first introduced in 906 by Andrey Markov, with the goal of showing that the Law of
More informationAttacks on Correlated Peer-to-Peer Networks: An Analytical Study
The First International Workshop on Security in Computers, Networking and Communications Attacks on Correlated Peer-to-Peer Networks: An Analytical Study Animesh Srivastava Department of CSE IIT Kharagpur,
More informationErzsébet Ravasz Advisor: Albert-László Barabási
Hierarchical Networks Erzsébet Ravasz Advisor: Albert-László Barabási Introduction to networks How to model complex networks? Clustering and hierarchy Hierarchical organization of cellular metabolism The
More informationSpectral Graph Theory for. Dynamic Processes on Networks
Spectral Graph Theory for Dynamic Processes on etworks Piet Van Mieghem in collaboration with Huijuan Wang, Dragan Stevanovic, Fernando Kuipers, Stojan Trajanovski, Dajie Liu, Cong Li, Javier Martin-Hernandez,
More informationTopic 4 Randomized algorithms
CSE 103: Probability and statistics Winter 010 Topic 4 Randomized algorithms 4.1 Finding percentiles 4.1.1 The mean as a summary statistic Suppose UCSD tracks this year s graduating class in computer science
More informationCHAPTER 1. Relations. 1. Relations and Their Properties. Discussion
CHAPTER 1 Relations 1. Relations and Their Properties 1.1. Definition of a Relation. Definition 1.1.1. A binary relation from a set A to a set B is a subset R A B. If (a, b) R we say a is Related to b
More informationCSI 445/660 Part 3 (Networks and their Surrounding Contexts)
CSI 445/660 Part 3 (Networks and their Surrounding Contexts) Ref: Chapter 4 of [Easley & Kleinberg]. 3 1 / 33 External Factors ffecting Network Evolution Homophily: basic principle: We tend to be similar
More informationCh. 2: Lec. 1. Basics: Outline. Importance. Usages. Key problems. Three ways of looking... Colbert on Equations. References. Ch. 2: Lec. 1.
Basics: Chapter 2: Lecture 1 Linear Algebra, Course 124C, Spring, 2009 Prof. Peter Dodds Department of Mathematics & Statistics University of Vermont Instructor: Prof. Peter Dodds Lecture room and meeting
More informationLearning latent structure in complex networks
Learning latent structure in complex networks Lars Kai Hansen www.imm.dtu.dk/~lkh Current network research issues: Social Media Neuroinformatics Machine learning Joint work with Morten Mørup, Sune Lehmann
More informationModeling, Analysis, and Control of Information Propagation in Multi-layer and Multiplex Networks. Osman Yağan
Modeling, Analysis, and Control of Information Propagation in Multi-layer and Multiplex Networks Osman Yağan Department of ECE Carnegie Mellon University Joint work with Y. Zhuang and V. Gligor (CMU) Alex
More informationSTA 4273H: Statistical Machine Learning
STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Computer Science! Department of Statistical Sciences! rsalakhu@cs.toronto.edu! h0p://www.cs.utoronto.ca/~rsalakhu/ Lecture 7 Approximate
More informationSupplementary Information
Supplementary Information For the article"comparable system-level organization of Archaea and ukaryotes" by J. Podani, Z. N. Oltvai, H. Jeong, B. Tombor, A.-L. Barabási, and. Szathmáry (reference numbers
More informationLecture 10. Under Attack!
Lecture 10 Under Attack! Science of Complex Systems Tuesday Wednesday Thursday 11.15 am 12.15 pm 11.15 am 12.15 pm Feb. 26 Feb. 27 Feb. 28 Mar.4 Mar.5 Mar.6 Mar.11 Mar.12 Mar.13 Mar.18 Mar.19 Mar.20 Mar.25
More informationarxiv: v1 [physics.soc-ph] 15 Dec 2009
Power laws of the in-degree and out-degree distributions of complex networks arxiv:0912.2793v1 [physics.soc-ph] 15 Dec 2009 Shinji Tanimoto Department of Mathematics, Kochi Joshi University, Kochi 780-8515,
More informationMachine Learning and Modeling for Social Networks
Machine Learning and Modeling for Social Networks Olivia Woolley Meza, Izabela Moise, Nino Antulov-Fatulin, Lloyd Sanders 1 Spreading and Influence on social networks Computational Social Science D-GESS
More informationExact solution of site and bond percolation. on small-world networks. Abstract
Exact solution of site and bond percolation on small-world networks Cristopher Moore 1,2 and M. E. J. Newman 2 1 Departments of Computer Science and Physics, University of New Mexico, Albuquerque, New
More informationSupplementary Information Activity driven modeling of time varying networks
Supplementary Information Activity driven modeling of time varying networks. Perra, B. Gonçalves, R. Pastor-Satorras, A. Vespignani May 11, 2012 Contents 1 The Model 1 1.1 Integrated network......................................
More informationarxiv: v1 [physics.soc-ph] 13 Oct 2016
A framewor for analyzing contagion in assortative baning networs T. R. Hurd, J. P. Gleeson, 2 and S. Melni 2 ) Department of Mathematics, McMaster University, Canada 2) MACSI, Department of Mathematics
More informationEE595A Submodular functions, their optimization and applications Spring 2011
EE595A Submodular functions, their optimization and applications Spring 2011 Prof. Jeff Bilmes University of Washington, Seattle Department of Electrical Engineering Winter Quarter, 2011 http://ee.washington.edu/class/235/2011wtr/index.html
More informationModularity and anti-modularity in networks with arbitrary degree distribution
RESEARCH Open Access Research Modularity and anti-modularity in networks with arbitrary degree distribution Arend Hintze* and Christoph Adami* Abstract Background: Much work in systems biology, but also
More informationnetworks in molecular biology Wolfgang Huber
networks in molecular biology Wolfgang Huber networks in molecular biology Regulatory networks: components = gene products interactions = regulation of transcription, translation, phosphorylation... Metabolic
More information6.207/14.15: Networks Lectures 4, 5 & 6: Linear Dynamics, Markov Chains, Centralities
6.207/14.15: Networks Lectures 4, 5 & 6: Linear Dynamics, Markov Chains, Centralities 1 Outline Outline Dynamical systems. Linear and Non-linear. Convergence. Linear algebra and Lyapunov functions. Markov
More informationSpectral Methods for Subgraph Detection
Spectral Methods for Subgraph Detection Nadya T. Bliss & Benjamin A. Miller Embedded and High Performance Computing Patrick J. Wolfe Statistics and Information Laboratory Harvard University 12 July 2010
More informationFlip Your Grade Book With Concept-Based Grading CMC 3 South Presentation, Pomona, California, 2016 March 05
Flip Your Grade Book With Concept-Based Grading CMC South Presentation, Pomona, California, 016 March 05 Phil Alexander Smith, American River College (smithp@arc.losrios.edu) Additional Resources and Information
More informationCS224W: Social and Information Network Analysis
CS224W: Social and Information Network Analysis Reaction Paper Adithya Rao, Gautam Kumar Parai, Sandeep Sripada Keywords: Self-similar networks, fractality, scale invariance, modularity, Kronecker graphs.
More information12. LOCAL SEARCH. gradient descent Metropolis algorithm Hopfield neural networks maximum cut Nash equilibria
12. LOCAL SEARCH gradient descent Metropolis algorithm Hopfield neural networks maximum cut Nash equilibria Lecture slides by Kevin Wayne Copyright 2005 Pearson-Addison Wesley h ttp://www.cs.princeton.edu/~wayne/kleinberg-tardos
More informationComplex contagions and hybrid phase transitions
Journal of Complex Networks Advance Access published August 14, 2015 Journal of Complex Networks (2014) Page 1 of 23 doi:10.1093/comnet/cnv021 Complex contagions and hybrid phase transitions Joel C. Miller
More information6.842 Randomness and Computation March 3, Lecture 8
6.84 Randomness and Computation March 3, 04 Lecture 8 Lecturer: Ronitt Rubinfeld Scribe: Daniel Grier Useful Linear Algebra Let v = (v, v,..., v n ) be a non-zero n-dimensional row vector and P an n n
More informationCS 246 Review of Linear Algebra 01/17/19
1 Linear algebra In this section we will discuss vectors and matrices. We denote the (i, j)th entry of a matrix A as A ij, and the ith entry of a vector as v i. 1.1 Vectors and vector operations A vector
More informationMatrices and Vectors
Matrices and Vectors James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 11, 2013 Outline 1 Matrices and Vectors 2 Vector Details 3 Matrix
More informationSpectral Graph Theory Tools. Analysis of Complex Networks
Spectral Graph Theory Tools for the Department of Mathematics and Computer Science Emory University Atlanta, GA 30322, USA Acknowledgments Christine Klymko (Emory) Ernesto Estrada (Strathclyde, UK) Support:
More informationClass President: A Network Approach to Popularity. Due July 18, 2014
Class President: A Network Approach to Popularity Due July 8, 24 Instructions. Due Fri, July 8 at :59 PM 2. Work in groups of up to 3 3. Type up the report, and submit as a pdf on D2L 4. Attach the code
More informationCLASS NOTES FOR APRIL 14, 2000
CLASS NOTES FOR APRIL 14, 2000 Announcement: Section 1.2, Questions 3,5 have been deferred from Assignment 1 to Assignment 2. Section 1.4, Question 5 has been dropped entirely. 1. Review of Wednesday class
More informationCS 781 Lecture 9 March 10, 2011 Topics: Local Search and Optimization Metropolis Algorithm Greedy Optimization Hopfield Networks Max Cut Problem Nash
CS 781 Lecture 9 March 10, 2011 Topics: Local Search and Optimization Metropolis Algorithm Greedy Optimization Hopfield Networks Max Cut Problem Nash Equilibrium Price of Stability Coping With NP-Hardness
More informationThere are two things that are particularly nice about the first basis
Orthogonality and the Gram-Schmidt Process In Chapter 4, we spent a great deal of time studying the problem of finding a basis for a vector space We know that a basis for a vector space can potentially
More informationECS 289 / MAE 298, Lecture 7 April 22, Percolation and Epidemiology on Networks, Part 2 Searching on networks
ECS 289 / MAE 298, Lecture 7 April 22, 2014 Percolation and Epidemiology on Networks, Part 2 Searching on networks 28 project pitches turned in Announcements We are compiling them into one file to share
More informationNonparametric Bayesian Matrix Factorization for Assortative Networks
Nonparametric Bayesian Matrix Factorization for Assortative Networks Mingyuan Zhou IROM Department, McCombs School of Business Department of Statistics and Data Sciences The University of Texas at Austin
More informationAny live cell with less than 2 live neighbours dies. Any live cell with 2 or 3 live neighbours lives on to the next step.
2. Cellular automata, and the SIRS model In this Section we consider an important set of models used in computer simulations, which are called cellular automata (these are very similar to the so-called
More informationData Mining and Analysis: Fundamental Concepts and Algorithms
Data Mining and Analysis: Fundamental Concepts and Algorithms dataminingbook.info Mohammed J. Zaki 1 Wagner Meira Jr. 2 1 Department of Computer Science Rensselaer Polytechnic Institute, Troy, NY, USA
More informationAlgebra & Trigonometry for College Readiness Media Update, 2016
A Correlation of Algebra & Trigonometry for To the Utah Core Standards for Mathematics to the Resource Title: Media Update Publisher: Pearson publishing as Prentice Hall ISBN: SE: 9780134007762 TE: 9780133994032
More information12. LOCAL SEARCH. gradient descent Metropolis algorithm Hopfield neural networks maximum cut Nash equilibria
Coping With NP-hardness Q. Suppose I need to solve an NP-hard problem. What should I do? A. Theory says you re unlikely to find poly-time algorithm. Must sacrifice one of three desired features. Solve
More informationLink Prediction. Eman Badr Mohammed Saquib Akmal Khan
Link Prediction Eman Badr Mohammed Saquib Akmal Khan 11-06-2013 Link Prediction Which pair of nodes should be connected? Applications Facebook friend suggestion Recommendation systems Monitoring and controlling
More informationGraph Theory and Networks in Biology arxiv:q-bio/ v1 [q-bio.mn] 6 Apr 2006
Graph Theory and Networks in Biology arxiv:q-bio/0604006v1 [q-bio.mn] 6 Apr 2006 Oliver Mason and Mark Verwoerd February 4, 2008 Abstract In this paper, we present a survey of the use of graph theoretical
More informationSynchronization of a General Delayed Complex Dynamical Network via Adaptive Feedback
Synchronization of a General Delayed Complex Dynamical Network via Adaptive Feedback Qunjiao Zhang and Junan Lu College of Mathematics and Statistics State Key Laboratory of Software Engineering Wuhan
More informationEffciency of Local Majority-Rule Dynamics on Scale-Free Networks
AAPPS Bulletin June 2006 11 Effciency of Local Majority-Rule Dynamics on Scale-Free Networks Haijun Zhou Dynamic processes occurring on a network may be greatly influenced by the particular structure and
More informationOverlapping Community Structure & Percolation Transition of Community Network. Yong-Yeol Ahn CCNR, Northeastern University NetSci 2010
Overlapping Community Structure & Percolation Transition of Community Network Yong-Yeol Ahn CCNR, Northeastern University NetSci 2010 Sune Lehmann James P. Bagrow Communities Communities Overlap G. Palla,
More informationDS504/CS586: Big Data Analytics Graph Mining II
Welcome to DS504/CS586: Big Data Analytics Graph Mining II Prof. Yanhua Li Time: 6-8:50PM Thursday Location: AK233 Spring 2018 v Course Project I has been graded. Grading was based on v 1. Project report
More informationCS 277: Data Mining. Mining Web Link Structure. CS 277: Data Mining Lectures Analyzing Web Link Structure Padhraic Smyth, UC Irvine
CS 277: Data Mining Mining Web Link Structure Class Presentations In-class, Tuesday and Thursday next week 2-person teams: 6 minutes, up to 6 slides, 3 minutes/slides each person 1-person teams 4 minutes,
More informationRandomized Algorithms
Randomized Algorithms Prof. Tapio Elomaa tapio.elomaa@tut.fi Course Basics A new 4 credit unit course Part of Theoretical Computer Science courses at the Department of Mathematics There will be 4 hours
More informationLower Bounds for Testing Bipartiteness in Dense Graphs
Lower Bounds for Testing Bipartiteness in Dense Graphs Andrej Bogdanov Luca Trevisan Abstract We consider the problem of testing bipartiteness in the adjacency matrix model. The best known algorithm, due
More informationMAE 298, Lecture 8 Feb 4, Web search and decentralized search on small-worlds
MAE 298, Lecture 8 Feb 4, 2008 Web search and decentralized search on small-worlds Search for information Assume some resource of interest is stored at the vertices of a network: Web pages Files in a file-sharing
More informationSpectral Analysis of Directed Complex Networks. Tetsuro Murai
MASTER THESIS Spectral Analysis of Directed Complex Networks Tetsuro Murai Department of Physics, Graduate School of Science and Engineering, Aoyama Gakuin University Supervisors: Naomichi Hatano and Kenn
More informationarxiv:cond-mat/ v1 [cond-mat.dis-nn] 4 May 2000
Topology of evolving networks: local events and universality arxiv:cond-mat/0005085v1 [cond-mat.dis-nn] 4 May 2000 Réka Albert and Albert-László Barabási Department of Physics, University of Notre-Dame,
More informationECEN 689 Special Topics in Data Science for Communications Networks
ECEN 689 Special Topics in Data Science for Communications Networks Nick Duffield Department of Electrical & Computer Engineering Texas A&M University Lecture 8 Random Walks, Matrices and PageRank Graphs
More informationCS224W: Analysis of Networks Jure Leskovec, Stanford University
CS224W: Analysis of Networks Jure Leskovec, Stanford University http://cs224w.stanford.edu 10/30/17 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 2
More information