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Lecture 27: Information diffusion CS 765 Complex Networks Slides are modified from Lada Adamic, David Kempe, Bill Hackbor outline factors influencing information diffusion network structure: which nodes are connected? strength of ties: how strong are the connections? studies in information diffusion: Granovetter: the strength of weak ties J-P Onnela et al: strength of intermediate ties Kossinets et al: strength of backbone ties Davis: board interlocks and adoption of practices network position and access to information Burt: Structural holes and good ideas Aral and van Alstyne: networks and information advantage networks and innovation Lazer and Friedman: innovation factors influencing diffusion network structure (unweighted) density degree distribution clustering connected components community structure strength of ties (weighted) frequency of communication strength of influence spreading agent attractiveness and specificity of information Strong tie defined A strong tie frequent contact affinity many mutual contacts “forbidden triad”: strong ties are likely to “close” Less likely to be a bridge (or a local bridge) Source: Granovetter, M. (1973). "The Strength of Weak Ties", edge embeddeness embeddeness: number of common neighbors the two endpoints have neighborhood overlap: school kids and 1st through 8th choices of friends snowball sampling: will you reach more different kids by asking each kid to name their 2 best friends, or their 7th & 8th closest friend? Source: M. van Alstyne, S. Aral. Networks, Information & Social Capital is it good to be embedded? What are the advantages of occupying an embedded position in the network? What are the disadvantages of being embedded? Advantages of being a broker (spanning structural holes)? Disadvantages of being a broker? outline factors influencing information diffusion network structure: which nodes are connected? strength of ties: how strong are the connections? studies in information diffusion: Granovetter: the strength of weak ties J-P Onnela et al: strength of intermediate ties Kossinets et al: strength of backbone ties Davis: board interlocks and adoption of practices network position and access to information Burt: Structural holes and good ideas Aral and van Alstyne: networks and information advantage networks and innovation Lazer and Friedman: innovation how does strength of a tie influence diffusion? M. S. Granovetter: The Strength of Weak Ties, AJS, 1973: finding a job through a contact that one saw frequently (2+ times/week) 16.7% occasionally (more than once a year but < 2x week) 55.6% rarely 27.8% but… length of path is short contact directly works for/is the employer or is connected directly to employer strength of tie: frequency of communication Kossinets, Watts, Kleinberg, KDD 2008: which paths yield the most up to date info? how many of the edges form the “backbone”? source: Kossinets et al. “The structure of information pathways in a social communication network” the strength of intermediate ties strong ties frequent communication, but ties are redundant due to high clustering weak ties reach far across network, but communication is infrequent… “Structure and tie strengths in mobile communication networks” use nation-wide cellphone call records and simulate diffusion using actual call timing Localized strong ties slow infection spread. source: Onnela J. et.al. Structure and tie strengths in mobile communication networks how can information diffusion be different from simple contagion (e.g. a virus)? simple contagion: infected individual infects neighbors with information at some rate threshold contagion: individuals must hear information (or observe behavior) from a number or fraction of friends before adopting in lab: complex contagion (Centola & Macy, AJS, 2007) how do you pick individuals to “infect” such that your opinion prevails http://www.ladamic.com/netlearn/NetLogo4/DiffusionCompetition.html Framework The network consists of nodes (individual) and edges (links between nodes) Each node is in one of two states Susceptible (in other words, healthy) Infected Susceptible-Infected-Susceptible (SIS) model Cured nodes immediately become susceptible Infected by neighbor Susceptible Infected Cured internally Framework (Continued) Homogeneous birth rate β on all edges between infected and susceptible nodes Homogeneous death rate δ for infected nodes Healthy Prob. δ N2 Prob. β N1 Infected X N3 SIR and SIS Models An SIS model consists of two group Susceptible: Those who may contract the disease Infected: Those infected An SIR model consists of three group Susceptible: Those who may contract the disease Infected: Those infected Recovered: Those with natural immunity or those that have died. Important Parameters α is the transmission coefficient, which determines the rate at which the disease travels from one population to another. γ is the recovery rate: (I persons)/(days required to recover) R0 is the basic reproduction number. R0 So (So )1 / (Number of new cases arising from one infective) x (Average duration of infection) If R0 > 1 then ∆I > 0 and an epidemic occurs SIR and SIS Models SIS Model: S SI I I SI I SIR Model: S SI I SI I R I Diffusion in networks: ER graphs review: diffusion in ER graphs http://www.ladamic.com/netlearn/NetLogo501/ERDiffusion.html Quiz Q: When the density of the network increases, diffusion in the network is faster slower unaffected ER graphs: connectivity and density nodes infected after 10 steps, infection rate = 0.15 average degree = 2.5 average degree = 10 Quiz Q: When nodes preferentially attach to high degree nodes, the diffusion over the network is faster slower unaffected Diffusion in “grown networks” nodes infected after 4 steps, infection rate = 1 preferential attachment non-preferential growth http://www.ladamic.com/netlearn/NetLogo501/BADiffusion.html Diffusion in small worlds What is the role of the long-range links in diffusion over small world topologies? http://www.ladamic.com/netlearn/NetLogo4/SmallWorldDiffusionSIS.html diffusion of innovation surveys: farmers adopting new varieties of hybrid corn by observing what their neighbors were planting (Ryan and Gross, 1943) doctors prescribing new medication (Coleman et al. 1957) spread of obesity & happiness in social networks (Christakis and Fowler, 2008) online behavioral data: Spread of Flickr photos & Digg stories (Lerman, 2007) joining LiveJournal groups & CS conferences (Backstrom et al. 2006) + others e.g. Anagnostopoulos et al. 2008 Open question: how do we tell influence from correlation? approaches: time resolved data: if adoption time is shuffled, does it yield the same patterns? if edges are directed: does reversing the edge direction yield less predictive power? 27 Example: adopting new practices poison pills diffused through interlocks geography had little to do with it more likely to be influenced by tie to firm doing something similar & having similar centrality golden parachutes did not diffuse through interlocks geography was a significant factor more likely to follow “central” firms why did one diffuse through the “network” while the other did not? Source: Corporate Elite Networks and Governance Changes in the 1980s. Social Network and Spread of Influence Social network plays a fundamental role as a medium for the spread of INFLUENCE among its members Opinions, ideas, information, innovation… Direct Marketing takes the “word-of-mouth” effects to significantly increase profits Gmail, Tupperware popularization, Microsoft Origami … Problem Setting Given a limited budget B for initial advertising e.g. give away free samples of product estimates for influence between individuals Goal trigger a large cascade of influence e.g. further adoptions of a product Question Which set of individuals should B target at? Application besides product marketing spread an innovation detect stories in blogs Models of Influence First mathematical models [Schelling '70/'78, Granovetter '78] Large body of subsequent work: [Rogers '95, Valente '95, Wasserman/Faust '94] Two basic classes of diffusion models: threshold and cascade General operational view: A social network is represented as a directed graph, with each person (customer) as a node Nodes start either active or inactive An active node may trigger activation of neighboring nodes Monotonicity assumption: active nodes never deactivate Threshold dynamics The network: • aij is the adjacency matrix (N ×N) • un-weighted • undirected aij {0,1} aij a ji The nodes: • are labelled i , i from 1 to N; • have a state ; vi (t ) {0,1} • and a threshold ri from some distribution. Threshold dynamics Node i has state vi (t ) {0,1} and threshold ri Neighbourhood average: r i 1 ki a v ij j j Updating: if ri ri 1 vi unchanged otherwise The fraction of nodes in state vi=1 is r(t): Example Inactive Node 0.6 0.3 Active Node 0.2 X Threshold 0.2 Active neighbors 0.1 0.4 U 0.5 w 0.3 0.5 Stop! 0.2 v Independent Cascade Model When node v becomes active, it has a single chance of activating each currently inactive neighbor w. The activation attempt succeeds with probability pvw . Example 0.6 Inactive Node 0.3 0.2 X 0.4 0.5 w 0.2 U 0.1 0.3 0.2 0.5 v Stop! Active Node Newly active node Successful attempt Unsuccessful attempt outline factors influencing information diffusion network structure: which nodes are connected? strength of ties: how strong are the connections? studies in information diffusion: Granovetter: the strength of weak ties J-P Onnela et al: strength of intermediate ties Kossinets et al: strength of backbone ties Davis: board interlocks and adoption of practices network position and access to information Burt: Structural holes and good ideas Aral and van Alstyne: networks and information advantage networks and innovation Lazer and Friedman: innovation Burt: structural holes and good ideas Managers asked to come up with an idea to improve the supply chain Then asked: whom did you discuss the idea with? whom do you discuss supply-chain issues with in general do those contacts discuss ideas with one another? 673 managers (455 (68%) completed the survey) ~ 4000 relationships (edges) results people whose networks bridge structural holes have higher compensation positive performance evaluations more promotions more good ideas these brokers are more likely to express ideas less likely to have their ideas dismissed by judges more likely to have their ideas evaluated as valuable Aral & Alstyne: Study of a head hunter firm Three firms initially Unusually measurable inputs and outputs 1300 projects over 5 yrs and 125,000 email messages over 10 months (avg 20% of time!) Metrics (i) Revenues per person and per project, (ii) number of completed projects, (iii) duration of projects, (iv) number of simultaneous projects, (v) compensation per person Main firm 71 people in executive search (+2 firms partial data) 27 Partners, 29 Consultants, 13 Research, 2 IT staff Four Data Sets per firm 52 Question Survey (86% response rate) E-Mail Accounting 15 Semi-structured interviews Email structure matters Coefficientsa New Contract Revenue Unstandardized Coefficients B Ave. E-Mail Size Colleagues’ Ave. Response Time a. b. Sig. F B Std. Error 0.40 (Base Model) Best structural pred. Unstandardized Coefficients Adj. R2 Std. Error Contract Execution Revenue Coefficientsa 12604.0*** -10.7** -198947.0 Adj. R2 Sig. F 0.19 4454.0 0.52 .006 1544.0** 639.0 0.30 .021 4.9 0.56 .042 -9.3* 4.7 0.34 .095 168968.0 0.56 .248 157789.0 0.42 .026 Dependent Variable: Bookings02 Base Model: YRS_EXP, PARTDUM, %_CEO_SRCH, SECTOR(dummies), %_SOLO. -368924.0** a. Dependent Variable: Billings02 b. N=39. *** p<.01, ** p<.05, * p<.1 Sending shorter e-mail helps get contracts and finish them. Faster response from colleagues helps finish them. diverse networks drive performance by providing access to novel information network structure (having high degree) correlates with receiving novel information sooner (as deduced from hashed versions of their email) getting information sooner correlates with $$ brought in controlling for # of years worked job level …. Network Structure Matters Coefficientsa New Contract Revenue Unstandardized Coefficients B Std. Error Betweenness a. b. Unstandardized Coefficients Sig. F B Std. Error 0.40 (Base Model) Size Struct. Holes Adj. R2 Contract Execution Revenue Coefficientsa Adj. R2 Sig. F 0.19 13770*** 4647 0.52 .006 7890* 1297* 773 0.47 .040 1696** Dependent Variable: Bookings02 Base Model: YRS_EXP, PARTDUM, %_CEO_SRCH, SECTOR(dummies), %_SOLO. 4656 0.24 .100 697 0.30 .021 a. Dependent Variable: Billings02 b. N=39. *** p<.01, ** p<.05, * p<.1 Bridging diverse communities is significant. Being in the thick of information flows is significant. outline factors influencing information diffusion network structure: which nodes are connected? strength of ties: how strong are the connections? studies in information diffusion: Granovetter: the strength of weak ties J-P Onnela et al: strength of intermediate ties Kossinets et al: strength of backbone ties Davis: board interlocks and adoption of practices network position and access to information Burt: Structural holes and good ideas Aral and van Alstyne: networks and information advantage networks and innovation Lazer and Friedman: innovation networked coordination game choice between two things, A and B e.g. basketball and soccer if friends choose A, they get payoff a if friends choose B, they get payoff b if one chooses A while the other chooses B, their payoff is 0 coordinating with one’s friends Let A = basketball, B = soccer. Which one should you learn to play? fraction p = 3/5 play basketball fraction p = 2/5 play soccer which choice has higher payoff? d neighbors p fraction play basketball (A) (1-p) fraction play soccer (B) if choose A, get payoff p * d *a if choose B, get payoff (1-p) * d * b so should choose A if p d a ≥ (1-p) d b or p ≥ b / (a + b) two equilibria everyone adopts A everyone adopts B what happens in between? What if two nodes switch at random? Will a cascade occur? example: a = 3, b = 2 payoff for nodes interaction using behavior A is 3/2 as large as what they get if they both choose B nodes will switch from B to A if at least q = 2/(3+2) = 2/5 of their neighbors are using A how does a cascade occur suppose 2 nodes start playing basketball due to external factors e.g. they are bribed with a free pair of shoes by some devious corporation Quiz Q: Which node(s) will switch to playing basketball next? the complete cascade you pick the initial 2 nodes A larger example (Easley/Kleinberg Ch. 19) does the cascade spread throughout the network? http://www.ladamic.com/netlearn/NetLogo412/CascadeModel.html implications for viral marketing if you could pay a small number of individuals to use your product, which individuals would you pick? Quiz question: What is the role of communities in complex contagion enabling ideas to spread in the presence of thresholds creating isolated pockets impervious to outside ideas allowing different opinions to take hold in different parts of the network bilingual nodes so far nodes could only choose between A and B what if you can play both A and B, but pay an additional cost c? try it on a line Increase the cost of being bilingual so that no node chooses to do so. Let the cascade run Now lower the cost. What happens? Quiz Q: The presence of bilingual nodes helps the superior solution to spread throughout the network helps inferior options to persist in the network causes everyone in the network to become bilingual knowledge, thresholds, and collective action nodes need to coordinate across a network, but have limited horizons can individuals coordinate? each node will act if at least x people (including itself) mobilize nodes will not mobilize mobilization there will be some turnout innovation in networks network topology influences who talks to whom who talks to whom has important implications for innovation and learning better to innovate or imitate? brainstorming: more minds together, but also danger of groupthink working in isolation: more independence slower progress in a network context modeling the problem space Kauffman’s NK model N dimensional problem space N bits, each can be 0 or 1 K describes the smoothness of the fitness landscape how similar is the fitness of sequences with only 1-2 bits flipped (K = 0, no similarity, K large, smooth fitness) Kauffman’s NK model fitness K large distance K medium K small Update rules As a node, you start out with a random bit string At each iteration If one of your neighbors has a solution that is more fit than yours, imitate (copy their solution) Otherwise innovate by flipping one of your bits Quiz Q: Relative to the regular lattice, the network with many additional, random connections has on average: slower convergence to a local optimum smaller improvement in the best solution relative to the initial maximum more oscillations between solutions networks and innovation: is more information diffusion always better? linear network fully connected network Nodes can innovate on their own (slowly) or adopt their neighbor’s solution Best solutions propagate through the network source: Lazer and Friedman, The Parable of the Hare and the Tortoise: Small Worlds, Diversity, and System Performance networks and innovation fully connected network converges more quickly on a solution, but if there are lots of local maxima in the solution space, it may get stuck without finding optimum. linear network (fewer edges) arrives at better solution eventually because individuals innovate longer Coordination: graph coloring Application: coloring a map: limited set of colors, no two adjacent countries should have the same color graph coloring on a network Each node is a human subject. Different experimental conditions: knowledge of neighbors’ color knowledge of entire network Compare: regular ring lattice small-world topology scale-free networks Kearns et al., ‘An Experimental Study of the Coloring Problem on Human Subject Networks’, Science, 313(5788), pp. 824-827, 2006 simulation Kearns et al. “An Experimental Study of the Coloring Problem on Human Subject Networks” network structure affects convergence in coordination games, e.g. graph coloring http://www.ladamic.com/netlearn/NetLogo4/GraphColoring.html Quiz Q: As the rewiring probability is increased from 0 to 1 the following happens: the solution time decreases the solution time increases the solution time initially decreases then increases again to sum up network topology influences processes occurring on networks what state the nodes converge to how quickly they get there process mechanism matters: simple vs. complex contagion coordination learning diffusion can be simple (person to person) or complex (individuals having thresholds) people in special network positions (the brokers) have an advantage in receiving novel info & coming up with “novel” ideas in some scenarios, information diffusion may hinder innovation