ECS 289 / MAE 298, Lecture 15Mar 2, 2011

“Diffusion, Cascades and Influence, Part II”

Diffusion and cascades in networks(Nodes in one of two states)

• Viruses (human and computer)– contact processes– epidemic thresholds

• Technologies– Winner take all– Benefit of first to market– Benefit of second to market

• Political or social beliefs and societal norms

A long history of study, now trying to add impact of underlyingnetwork structure.

Diffusion, Cascade behaviors, and influential nodesPart I: Ensemble models (last time)

Generating functions / Master equations / giant componentsGenerating function approach: Watts PNAS (2002)

• Fractional threshold model (Φi).– Node “activated” once a fraction of it’s neighbors ≥ Φi areactive.– A vulnerable node is one that needs only a single neighborto be active before it flips (i.e., Φi ≤ 1/k).– Use generating functions to calculate the expected size ofclusters of vulnerable nodes.

• Results– Heterogeneity in thresholds (Φi) enhances global cascades.– Heterogeneity of degree (Pk) suppresses global cascades.

Diffusion, Cascade behaviors, and influential nodesPart I: Ensemble models (last time)

Generating functions / Master equations / giant components

Master equation approach: Pastor-Satorras and Vespignani

• Contact process, epidemic spreading– Probability of becoming “activated” is proportional to thenumber of active neighbors.

• Results– Heterogeneity of degree (Pk) enhances global spreading.

– For PLRG with 2 < γ < 3 the epidemic threshold λc→ 0.

Diffusion, Cascade behaviors, and influential nodesPart II: Contact processes with individual node preferences

• Long history of empirical / qualitative study in the socialsciences (Peyton Young, Granovetter, Martin Nowak ...;diffusion of innovation; societal norms)

• Recent theorems: “network coordination games” (biggerpayout if connected nodes in the same state)(Kleinberg, Kempe, Tardos, Dodds, Watts, Domingos)

• Finding the influential set of nodes, or the k most influentialOften NP-hard and not amenable to approximation algorithms

• Key distinction:– Thresholds of activation (leads to unpredictable behaviors)– Diminishing returns (submodular functions nicer)

Diffusion, Cascade behaviors, and influential nodesPart III: Markov chains and mixing times

• New game-theoretic approaches (general coordination games)– Results in an Ising model.– Montanari and Saberi PNAS 2010.

• Techniques from Parts I and II suggest:– Innovations spread quickly in highly connected networks.– Long-range links benefit spreading.– High-degree nodes quite influential (enhance spreading).

• Techniques from Part III suggest:– Innovations spread quickly in locally connected networks.– Local spatial coordination enhances spreading (having aspatial metric; graph embeddable in small dimension).– High-degree nodes slow down spreading.

Part II. Network Coordination Games

• The most basic model: Reviewed in Kleinberg “Cascading Behaviorin Networks: Algorithmic and Economic Issues”, Chap 24 of AlgorithmicGame Theory, (Cambridge University Press, 2007).

• Again each node in one of two states, say {−1,+1}.

• Play a game with each connected neighbor independently.Total payout is sum over all games.

• Assume neighbor(s) of j in fixed state while j updates.

• Positive payout if connected nodes i and j adopt the samestate. No payout if they differ. And -1 can have different payoutthat +1 coordinated behavior.

Payout matrix:q 00 (1-q)

How each node operates

• Again assume all other nodes fixed while node j updates.

• It has kAj nodes in state −1, and kBj nodes in state +1.

• If it chooses state −1, payout of qkAj .

• If it chooses state +1, payout of (1− q)kBj .

• Chooses −1 if qkAj > (1− q)kBj .

• Substitute in kj = kAj + kBj and rearrange:

Criteria: choose −1 if kBj < qkj and +1 if kBj > qkj.

• A threshold model! Adopt +1 if a fraction q of your neighborshave state +1.

Contagion threshold and cascades

• Start all nodes in −1. And all nodes update synchronously atdiscrete time steps.

• Key question: When is there a small set of nodes S, that whenset to +1 convert all (or almost all) of the population?

• A set S is contagious if every other node is converted by S.

• Easier for S to be contagious if the threshold q is small.

• Define the contagion threshold of a graph G to be themaximum q for which there exists a finite contagous set.

• (Like with generating functions, here no notion of how long ittakes for the full network to be activated. Just a final steady-state answer.)

Progressive vs. non-progressive processes

• The model thus far is non-progressive: nodes can flip from −1to +1 and back to −1.

• This makes the situation less stable. Consider a line of all −1at the start with a single +1 in the center, and q = 1/2. At nexttime steps neighbors of the +1 flip, but the +1 switches backto −1! And the whole system ends up “blinking”.

• Progressive: Once you flip, always stay in that state.

• The line above now all flips to +1 in a wavefront moving rightand left-wards.

Theorem:The Contagion Threshold for any Graph is at most 1/2.

• (Recall the contagion threshold is the maximum value of q forwhich a finite contagious set exists.)

• Independent of progressive vs non-progressive.

• “A behavior cant spread very far if it requires a strict majorityof your friends to convince you to adopt it.”

• This means if q > 1/2 on any graph, it cannot support acascade and the full graph will not be activated.

• This is for any graph: uniform degree, power law, etc.

Extending this simple model

So far all nodes have same fractional threshold q, and allneighbors contribute equally in calculation of fraction.

• The General Linear Threshold Model

– Directed graphs (not reciprocal influence necessarily).

– Each node has a threshold chosen uniformly at randombetween [0, 1].

– Each neighbor exerts a non-negative weight. The onlyconstraint is that sum over all the weights is less than or equalto 1.

– Note we now have diversity of influence (e.g., spouse/relativecan exert stronger weight than coworker/friend).

Finding the influential nodesMotivation

• Viral marketing – use word-of-mouth effects to sell product withminimal advertising cost.

• Design of search tools to track news, blogs, and other forms ofon-line discussion about current events

Finding the influential nodes: formally

• The minimum set S ∈ V that will lead to the whole networkbeing activated.

• The optimal set of a specified size k = |S| that will lead tolargest portion of the network being activated.

Due to thresholds/ critical mass

• In general NP-hard to find optimal set S.

• NP-hard to even find a approximate optimal set (optimal towithin factor η1−ε where n is network size and ε > 0.)(“inapproximability”)

• Due to thresholds (esp if each node can have its own) mighthave a tiny activated final set of nodes but it jumps abruptly ifjust a few more nodes or, moreover, the right nodes activated.

• Kleinberg calls this abrupt response the “Knife edge” property

Diminishing returns(No longer a threshold, but a concave function)

• Each additional friend who adopts the new behavior enhancesyour chance of adopting the new behevaior, but with lessinfluence for each additional friend

NATO Advanced Study Institute on Mining Massive DataSets for Security

Diffusion and Cascading Behaviour in Networks

author:Jure Leskovec, Computer Science Department, Stanford Universitypublished: Nov. 28, 2007, recorded: September 2007, views: 933

CategoriesTop » Computer Science » Network Analysis

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Slides

0:00 Diffusion and Cascading Behavior in Networks0:08 Networks –Social and Technological1:41 Examples of Networks2:11 Networks of the Real-world (1)2:40 Networks of the Real-world (2)3:07 Mining Social Network Data4:07 Networks as Phenomena4:27 Models and Laws of Networks5:11 Networks: Rich Data5:51 Networks: A Matter of Scale6:26 Networks: Scale Matters7:26 Structure vs. Process8:02 Diffusion in Social Networks8:23 Overview9:06 Diffusion in Social Networks

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Description

Diffusion is a process by which information, viruses, ideas and new behavior spread overthe network. For example, adoption of a new technology begins on a small scale with afew “early adopters”, then more and more people adopt it as they observe friends andneighbors using it. Eventually the adoption of the technology may spread through thesocial network as an epidemic “infecting” most of the network. As it spreads over thenetwork it creates a cascade. Cascades have been studied for many years by sociologistsconcerned with the diffusion of innovation; more recently, researchers have investigatedcascades for selecting trendsetters for viral marketing, finding inoculation targets inepidemiology, and explaining trends in blogspace.

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Reviews and comments:

1 Jacob Lee, October 13, 2009 at 5:30 a.m.:

I notice that the first slide is identical to the slide used by Jon Kleinberg here:http://uk.video.yahoo.com/watch/62057...

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Turn off the lights

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Diminishing returns(Submodular / concave function)

• The benefit of adding elements decreases as the set to whichthey are being added grows.

• So no longer get to have more influence from family orother special nodes. (Instead its the first nodes exert moreinfluence.)

• Since no longer have special nodes easy to build up optimalset S of k nodes.

• Hill climbing – add one at the time nodes to the set S thatcause maximum impact.

Hill climbing

(from Leskovec talk)

Submodular and hill climbing more formally:

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Empirical observations

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Joining Livejournal: on online bulletin board network

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Probability of joining a community when k friends are already members

• Diminishing returns only sets in once k > 3.

• Network effect not illustrated by curve: If the k friends are highlyclustered, the new user is more likely to join.

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For a wealth of additional information see Leskovec talk:http://videolectures.net/mmdss07 leskovec dcbn/

• The role product category plays (books, dvds, videos, animedvds)

• Predicting recommendation success with linear models.

• How do people actually get recommendations– Amazon recommendation of similar purchases– Personalized rec based on previous purchases/likes– 68% of people consult friends and family before purchasinghome electronics [Burke 2003]. (i.e. More influenced by friendsthan strangers.)– 94% of users make recommendations w/o having receivedone (they are the seed nodes)

Summary – Important distinctions for cascade processes

• Contagion (e.g. a virus) versus social behaviors.

• Threshold models / critical mass (abrupt changes as set Sincreased)

• Diminishing returns (submodular / concave)– KKT03, KKT05: If individual function for all nodes exhibitdiminishing returns, the resulting influence function for thegraph will be submodular (“local to global”).– Can approximate such sets (hill climbing)

Other interesting, related models on networks

• Voter models

• Synchronization (related to the spectral properties of A theadjacency matrix).

Finally: Part III: Markov chains and mixing times

• Montanari and Saberi, “The Spread of Innovations in SocialNetworks”, PNAS 2010

• Unlike any of the above models (which tell us about equilibriumsizes of activated populations), Markov Chain and mixing timestell us about the time it takes for innovations to be adopted!– sluggish– rapid fire spread