Contagion Outline Contagion Contagion Basic Contagion Basic Contagion Models Models Complex Networks, Course 295A, Spring, 2008 Social Contagion Social Contagion Models Models Basic Contagion Models Granovetter’s model Granovetter’s model Network version Network version Theory Theory Prof. Peter Dodds Groups Groups References References Social Contagion Models Department of Mathematics & Statistics Granovetter’s model University of Vermont Network version Theory Groups References Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License . Frame 1/66 Frame 2/66 Contagion models Contagion Spreading mechanisms Contagion Basic Contagion Basic Contagion Some large questions concerning network Models Models Social Contagion Social Contagion contagion: Models ◮ General spreading Models Granovetter’s model Granovetter’s model mechanism: Network version Network version 1. For a given spreading mechanism on a given Theory Theory Groups State of node i depends Groups network, what’s the probability that there will be References References on history of i and i ’s global spreading? neighbors’ states. 2. If spreading does take off, how far will it go? ◮ Doses of entity may be 3. How do the details of the network affect the stochastic and outcome? history-dependent. 4. How do the details of the spreading mechanism ◮ May have multiple, affect the outcome? uninfected interacting entities infected spreading at once. ◮ Next up: We’ll look at some fundamental kinds of spreading on generalized random networks. Frame 3/66 Frame 4/66
Spreading on Random Networks Contagion Contagion condition Contagion ◮ We need to find: Basic Contagion Basic Contagion Models Models r = the average # of infected edges that one random ◮ For random networks, we know local structure is Social Contagion Social Contagion infected edge brings about. Models Models pure branching. ◮ Define β k as the probability that a node of degree k Granovetter’s model Granovetter’s model Network version Network version ◮ Successful spreading is ∴ contingent on single Theory Theory is infected by a single infected edge. Groups Groups edges infecting nodes. References ◮ References Success Failure: ∞ kP k � r = · β k · ( k − 1 ) � k � ���� � �� � k = 0 ���� Prob. of # outgoing prob. of infection infected connecting to edges a degree k node ���� ∞ kP k � + � k � · ( 1 − β k ) · 0 ◮ Focus on binary case with edges and nodes either ◮ ���� � �� � infected or not. k = 0 # outgoing Prob. of infected no infection Frame 5/66 Frame 6/66 edges Contagion condition Contagion Contagion condition Contagion Basic Contagion ◮ Case 2: If β k = β < 1 Basic Contagion then Models Models ◮ Our contagion condition is then: Social Contagion Social Contagion r = β � k ( k − 1 ) � Models Models > 1 . Granovetter’s model Granovetter’s model � k � Network version Network version ∞ ( k − 1 ) kP k Theory Theory � r = β k > 1 . Groups Groups � k � References References k = 0 ◮ A fraction (1- β ) edges do not transmit the infection. ◮ Analogous phase transition to giant component case ◮ Case 1: If β k = 1 then but critical value of � k � is increased. r = � k ( k − 1 ) � ◮ Aka bond percolation. > 1 . � k � ◮ Resulting degree distribution P ′ k : ∞ � i � ◮ Good: This is just our giant component condition � P ′ k = β k ( 1 − β ) i − k P i . k again. i = k ◮ We can show F P ′ ( x ) = F P ( β x + 1 − β ) . Frame 7/66 Frame 8/66
Contagion condition Contagion Contagion condition Contagion Basic Contagion Basic Contagion Models Models ◮ Example: β k = 1 / k . Social Contagion Social Contagion ◮ Cases 3, 4, 5, ...: Now allow β k to depend on k Models Models Granovetter’s model Granovetter’s model ◮ ◮ Asymmetry: Transmission along an edge depends ∞ ∞ Network version Network version ( k − 1 ) kP k ( k − 1 ) kP k � � Theory Theory r = β k = Groups Groups on node’s degree at other end. � k � � k � k References References k = 1 k = 1 ◮ Possibility: β k increases with k ... unlikely. ∞ ( k − 1 ) P k = � k � − 1 = 1 − 1 ◮ Possibility: β k is not monotonic in k ... unlikely. � = � k � � k � � k � ◮ Possibility: β k decreases with k ... hmmm. k = 1 ◮ β k ց is a plausible representation of a simple kind of ◮ Since r is always less than 1, no spreading can social contagion. occur for this mechanism. ◮ The story: ◮ Decay of β k is too fast. More well connected people are harder to influence. ◮ Result is independent of degree distribution. Frame 9/66 Frame 10/66 Contagion condition Contagion Contagion condition Contagion Basic Contagion Basic Contagion ◮ Example: β k = H ( 1 k − φ ) ◮ The contagion condition: Models Models where 0 < φ ≤ 1 is a threshold and H is the Social Contagion Social Contagion Models Models ⌊ 1 φ ⌋ Heaviside function. Granovetter’s model Granovetter’s model ( k − 1 ) kP k � Network version Network version r = > 1 . ◮ Infection only occurs for nodes with low degree. Theory Theory � k � Groups Groups k = 1 ◮ Call these nodes vulnerables: References References they flip when only one of their friends flips. ◮ As φ → 1, all nodes become resilient and r → 0. ◮ As φ → 0, all nodes become vulnerable and the ◮ contagion condition matches up with the giant ∞ ∞ ( k − 1 ) kP k ( k − 1 ) kP k H ( 1 � � component condition. r = β k = k − φ ) � k � � k � ◮ Key: If we fix φ and then vary � k � , we may see two k = 1 k = 1 phase transitions. ⌊ 1 φ ⌋ ◮ Added to our standard giant component transition, ( k − 1 ) kP k � = where ⌊·⌋ means floor. we will see a cut off in spreading as nodes become � k � k = 1 more connected. Frame 11/66 Frame 12/66
Thresholds Contagion Social Contagion Contagion Basic Contagion Basic Contagion Models Models Social Contagion Social Contagion Models Models Granovetter’s model Granovetter’s model Network version Network version Theory Theory Groups Groups References References ◮ What if we now allow thresholds to vary? ◮ We need to backtrack a little... Frame 13/66 Frame 14/66 Social Contagion Contagion Social Contagion Contagion Basic Contagion Basic Contagion Models Models Social Contagion Social Contagion Examples abound Models Models Granovetter’s model Granovetter’s model We need to understand influence ◮ Harry Potter ◮ being polite/rude Network version Network version Theory Theory Groups ◮ Who influences whom? Very hard to measure... Groups ◮ voting ◮ strikes References References ◮ What kinds of influence response functions are ◮ gossip ◮ innovation there? ◮ residential segregation ◮ Rubik’s cube ◮ Are some individuals super influencers? ◮ ipods ◮ religious beliefs Highly popularized by Gladwell [5] as ‘connectors’ ◮ obesity ◮ leaving lectures ◮ The infectious idea of opinion leaders (Katz and Lazarsfeld) [8] SIR and SIRS contagion possible ◮ Classes of behavior versus specific behavior: dieting Frame 15/66 Frame 16/66
One perspective Contagion The hypodermic model of influence Contagion Basic Contagion Basic Contagion Models Models Social Contagion Social Contagion Models Models Granovetter’s model Granovetter’s model “In historical events great men—so-called—are but labels Network version Network version Theory Theory serving to give a name to the event, and like labels they Groups Groups have the least possible connection with the event itself. References References Every action of theirs, that seems to them an act of their own free will, is in an historical sense not free at all, but in bondage to the whole course of previous history, and predestined from all eternity.” —Leo Tolstoy, War and Peace. Frame 17/66 Frame 18/66 The two step model of influence [8] Contagion The general model of influence Contagion Basic Contagion Basic Contagion Models Models Social Contagion Social Contagion Models Models Granovetter’s model Granovetter’s model Network version Network version Theory Theory Groups Groups References References Frame 19/66 Frame 20/66
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