Tipping Points in Opinion Spread in Social Networks Boleslaw Szymanski Casey Doyle, Sameet Sreenivasan, Jierui Xie, Andrew Thompson, Chjan Lim and G. Korniss Social Cognitive Networks Academic Research Center Rensselaer Polytechnic Institute, Troy, NY, USA

Spread of Opinions Social Networks Nodes: individuals Links: social relationship (family, work, friendship) Many individuals with diverse social interactions between them. Big Question How does existence of information networks (internet with its social networking tools, cell phone networks, and so on) change dynamics of human interaction?

The Naming Game Features Rules Nodes contain a list of words Speaker and listener chosen A word is chosen from the speakers vocabulary If the listener has that word in their vocabulary, both delete all other words If the listener does not have that word, then the listener adds the word to their list with no deletion

A B AB A A B B B B B B B B speaker listener There are two opinions A,B, three states A, B, AB. A B A A AB B B B B Rule 1: Accept new opinion (for listener) The listener does not have this opinion, it adds the sent opinion to his list. Rule 2: Reinforce common opinion (for both) The listener already has this opinion, both retain only this opinion. A AB B Binary Agreement Model (BAM)

Classical BAM Model Neutral speaker selects one opinion from its state with equal probability. Neutral listener changes state to speaker’s opinion AB B A ½:½: ½:½: A A J. Xie S. Sreenivasan, G. Korniss and B.K. Szymanski, PRE (2011)

Variations on Classical BAM Propensity, p: allows for neutral speaker to select an opinion with unequal probability. Bias based on media or other social influence. Stickiness, s: allows for neutral listener to reject listener’s opinion. Neutral listener makes thoughtful decision AB B A p:p: (1-p): AAB A s: (1-s): A. Thompson, B.K. Szymanski, C. Lim, PRE (2014)

Propensity Model Propensity, p, causes a bias on neutral speakers. Saddle node fixed point migrates from one consensus to the other with p. Propensity, p, between [0, ½ ): Bias on the opinion BPropensity, p, between [½,1 ): Bias on the opinion A

Stickiness Model Drastic behavioral changes occur. Low stickiness, s ½ Saddle points moves Center manifold disappears Stable & saddle points to point (¼, ¼) Separate lines of equilibria switch places CINET Workshop – UAlbany, NY, August 11, 2015

Propensity and Stickiness Model This model holds similar characteristics to both the p and s model. Stickiness inverts the system Propensity biases neutrals towards one of the two opinions. Domain of values of p and s for which fixed points exist shrinks as s tends to ½. Outside of the hour-glass region for p and s a new stable solution with persistent neutrals arises Plot of p-s plane Probability s Probability p CINET Workshop – UAlbany, NY, August 11, 2015

Viral marketing by word of mouth Egyptian revolution Mr. 郭明义 ’s Charity Team in China How many people do we need to convince to make a difference? The value of committed agents has been recognized in sociology. “Never doubt that a small group of thoughtful, committed, citizens can change the world. Indeed, it is the only thing that ever has.“ - Margaret Mead Committed agents: people who stick to an opinion and continuously influence other. Limited budget -> rely on local interaction + commitment The Value of Commitment

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Committed Agents Start – Committed agents are all state A, population p – The rest are all state B Finite networks always reach consensus Look at consensus time Rate Equations

Tipping point: flows in opinion space p : fraction of agents committed to opinion A A non-absorbing (B-dominated, mixed) stable fixed point exists; All trajectories starting from initial condition flow to the non-absorbing fixed point Only all-A consensus fixed point exists All trajectories flow to consensus fixed point.

Tipping Points in Real Networks High school network, N=1127,  k  =9 p c =4.8% network, N=1133,  k  =9.6 p c =5% Facebook, N=1893,  k  =15 p c =7.8% Tipping point in empirical social networks is weakly impacted by the average degree  k  and is typically bounded by 10% [Xie (2012)]. Theory/prediction: Xie et al., PLoS One (2012);PLoS One (2012) Zhang, et al., PRE 2013

Competing Commitments Region I Three critical points, one saddle and two boundary Region II Only one stable point Divided into stable for A or stable for B

Traversing an off diagonal trajectory in parameter space Fraction committed to A is twice the fraction commited to B A dominates B dominates Discontinuous transitions occur along curves bounding the bistable region finite complete graphs

Commitment of initial nodes can be removed by repeated exposure to opposite opinion Committed node enters mixed state after w consecutive 'hits' Regains absorbing states for A and B p_c is now a function of w Waning Commitment Above critical population the majority absorbing point vanishes Below critical population the system retains both absorbing points but is biased toward majority Probability of minority consensus decreases exponentially with p

Beak narrows with greater commitment Stable points inside beak become absorbing states Competition with Waning Commitment

Stickiness with Committed Agents Stickiness, s Low stickiness A dominant AB dominant B dominant High stickiness

Idea Some ideas are more pleasing, leading to confirmation bias Can be modeled with inherent idea 'inertia' Opinion Inertia Opinion A is the minority Opinion B has inertia of 2 Rules There are two possible states, A and B. Each state has an inertia (how many times a node in that state must hear the opposite opinion before it switches) If a node listens to an opinion that confirms it's own, it's count toward switching goes to zero

Critical Populations

Lattice System

Public Relations and New Product Campaigns To promote a new product, do not try to reach all, instead select a segment of population that you can convince to become committed to a new product and they will convince the others: make the network work for you When to start actively defending reputation in case of events bring negative attention to a company?

Summary There are many different variants on the Naming Game Largely based on the binary agreement (2 state) version Stubbornness is a driving force behind many modifications Commitment leads to invariant nodes Stickiness forces nodes to avoid extreme opinions Opinion inertia makes nodes hold onto some opinions longer than others Bias also drive some variants Propensity parameter favors one state when exiting neutral