Volatility of Weak Ties Co-evolution of Selection and Influence in Social Networks Fang-Yi Yu
Volatility of Weak Ties Co-evolution of Selection and Influence in Social Networks Jie Gao, Grant Schoenebeck, Fang-Yi Yu
Co-evolution of Selection and Influence in Social Networks VOLATILITY OF WEAK TIES
An Experiment by Granovetter [1970] • Weak Ties and Changing Jobs Tie strength Frequency Found jobs friend 1/week 16.7% acquaintance 1/year 55.6% stranger less 27.8%
Volatility of Weak Ties Changing Jobs Bubble Filters • bring fresh information to a social • unfriending disproportionately affect group weak ties as compared to strong ties.
Outline • Model – Opinion formation: Influence and Selection – Network: Strong and Weak Ties • Simulation Results
Opinion Formation • Influence • Selection
Influence • Influence – agents changing their opinions to match their neighbors
Influence • Influence – agents changing their opinions to match their neighbors
Influence • Influence – agents changing their opinions to match their neighbors
Influence 𝒈 𝒋𝒐𝒈 • Influence – agents changing their opinions to match their neighbors – 𝜓 𝑢+1 𝑤 = 1 w.p. 𝑔 𝑗𝑜𝑔 𝑆 𝑢 (𝑤)
Influence 𝒈 𝒋𝒐𝒈 • Influence Voter Majority 3-Majority 1 – agents changing their opinions to match their neighbors 0.8 – 𝜓 𝑢+1 𝑤 = 1 w.p. 𝑔 𝑗𝑜𝑔 𝑆 𝑢 (𝑤) 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1
Selection • Influence – agents changing their opinions to match their neighbors • Selection – agents re-wiring to connect to new agents when the existing neighbor has a different opinion
Selection • Influence – agents changing their opinions to match their neighbors • Selection – agents re-wiring to connect to new agents when the existing neighbor has a different opinion
Selection • Influence – agents changing their opinions to match their neighbors • Selection – agents re-wiring to connect to new agents when the existing neighbor has a different opinion
Co-evolution of Selection and Influence • Influence – agents changing their opinions to match their neighbors Selection – bring new information through weak ties Opinion Network Influence
Co-evolution of Selection and Influence • Influence – agents changing their opinions to match their neighbors Selection – bring new information through weak ties Opinion Network • Selection, 𝑞 𝑡𝑓𝑚 – agents re-wiring to connect to new Influence agents when the existing neighbor has a different opinion – unfriend weak ties
Model of Network 𝑯 𝟏 = (𝑾, 𝑭 𝑻 , 𝑭 𝑿 ) • Strong ties 𝐹 𝑇 – grid edge – Not affected by selection
Model of Network 𝑯 𝟏 = (𝑾, 𝑭 𝑻 , 𝑭 𝑿 ) • Strong ties 𝐹 𝑇 – grid edge – Not affected by selection • Weak ties 𝐹 𝑋 – random edge – affected by selection
Model of Network 𝑯 𝟏 = (𝑾, 𝑭 𝑻 , 𝑭 𝑿 ) 1 4 𝑆 𝑢 𝑤 = 𝑟 𝑡𝑢𝑠𝑝𝑜 4 + 1 − 𝑟 𝑡𝑢𝑠𝑝𝑜 • Strong ties 𝐹 𝑇 6 – grid edge – Not affected by selection • Weak ties 𝐹 𝑋 – random edge – affected by selection • Strength of strong ties, 𝑟 𝑡𝑢𝑠𝑝𝑜 – Relative frequency of communication through strong ties
Sel-Inf 𝑯 𝟏 , 𝒈 𝒋𝒐𝒈 , 𝒒 𝒕𝒇𝒎𝒇𝒅𝒖 , 𝒓 𝒕𝒖𝒔𝒑𝒐𝒉 • Dynamic over binary opinion 𝜓 𝑢 – Agent 𝑤 has a random opinion 𝜓 𝑢 (𝑤)~{0.1} – At round 𝑢 + 1 , a random node 𝑤 is picked • Selection w.p. 𝑞 𝑡𝑓𝑚𝑓𝑑𝑢 – Pick an incident weak tie (𝑤, 𝑣) and rewire if 𝜓 𝑢 (𝑤) ≠ 𝜓 𝑢 (𝑣) • Influence w.p. 1 − 𝑞 𝑡𝑓𝑚𝑓𝑑𝑢 𝑢 (𝑤) fraction of opinion 1 in 𝑢 (𝑤)/𝑆 𝑋 – 𝑆 𝑇 strong/weak neighborhood of 𝑤 𝑗𝑜𝑔 𝑆 𝑢 (𝑤) – Update to 1 w.p. 𝑔
Sel-Inf 𝑯 𝟏 , 𝒈 𝒋𝒐𝒈 , 𝒒 𝒕𝒇𝒎𝒇𝒅𝒖 , 𝒓 𝒕𝒖𝒔𝒑𝒐𝒉 • Dynamic over binary opinion 𝜓 𝑢 – Agent 𝑤 has a random opinion 𝜓 𝑢 (𝑤)~{0.1} – At round 𝑢 + 1 , a random node 𝑤 is picked • Selection w.p. 𝑞 𝑡𝑓𝑚𝑓𝑑𝑢 – Pick an incident weak tie (𝑤, 𝑣) and rewire if 𝜓 𝑢 (𝑤) ≠ 𝜓 𝑢 (𝑣) • Influence w.p. 1 − 𝑞 𝑡𝑓𝑚𝑓𝑑𝑢 𝑢 (𝑤) fraction of opinion 1 in 𝑢 (𝑤)/𝑆 𝑋 – 𝑆 𝑇 strong/weak neighborhood of 𝑤 𝑗𝑜𝑔 𝑆 𝑢 (𝑤) – Update to 1 w.p. 𝑔
Sel-Inf 𝑯 𝟏 , 𝒈 𝒋𝒐𝒈 , 𝒒 𝒕𝒇𝒎𝒇𝒅𝒖 , 𝒓 𝒕𝒖𝒔𝒑𝒐𝒉 1 4 𝑆 𝑢 𝑤 = 𝑟 𝑡𝑢𝑠𝑝𝑜 • Dynamic over binary opinion 𝜓 𝑢 4 + 1 − 𝑟 𝑡𝑢𝑠𝑝𝑜 6 – Agent 𝑤 has a random opinion 𝜓 𝑢 (𝑤)~{0.1} – At round 𝑢 + 1 , a random node 𝑤 is picked • Selection w.p. 𝑞 𝑡𝑓𝑚𝑓𝑑𝑢 – Pick an incident weak tie (𝑤, 𝑣) and rewire if 𝜓 𝑢 (𝑤) ≠ 𝜓 𝑢 (𝑣) • Influence w.p. 1 − 𝑞 𝑡𝑓𝑚𝑓𝑑𝑢 𝑢 (𝑤) fraction of opinion 1 in 𝑢 (𝑤)/𝑆 𝑋 – 𝑆 𝑇 strong/weak neighborhood of 𝑤 𝑗𝑜𝑔 𝑆 𝑢 (𝑤) – Update to 1 w.p. 𝑔
Sel-Inf 𝑯 𝟏 , 𝒈 𝒋𝒐𝒈 , 𝒒 𝒕𝒇𝒎𝒇𝒅𝒖 , 𝒓 𝒕𝒖𝒔𝒑𝒐𝒉 • Dynamic over binary opinion 𝜓 𝑢 Small 𝑞 𝑡𝑓𝑚𝑓𝑑𝑢 – Agent 𝑤 has a random opinion 𝜓 𝑢 (𝑤)~{0.1} Influence Influence through weak through strong – At round 𝑢 + 1 , a random node 𝑤 is ties ties picked • Selection w.p. 𝑞 𝑡𝑓𝑚𝑓𝑑𝑢 Small 𝑟 𝑡𝑢𝑠𝑝𝑜 Strong 𝑟 𝑡𝑢𝑠𝑝𝑜 – Pick an incident weak tie (𝑤, 𝑣) and rewire if 𝜓 𝑢 (𝑤) ≠ 𝜓 𝑢 (𝑣) • Influence w.p. 1 − 𝑞 𝑡𝑓𝑚𝑓𝑑𝑢 Selection on None 𝑢 (𝑤) fraction of opinion 1 in 𝑢 (𝑤)/𝑆 𝑋 – 𝑆 𝑇 weak ties strong/weak neighborhood of 𝑤 𝑗𝑜𝑔 𝑆 𝑢 (𝑤) – Update to 1 w.p. 𝑔 Large 𝑞 𝑡𝑓𝑚𝑓𝑑𝑢
Outline • Model – Opinion formation: Influence and Selection – Network: Strong and Weak Ties • Simulation Results
Consensus Time of Voter Model Influence Influence through weak through strong ties ties Selection on weak ties
Consensus Time of Iterative Majority Influence Influence through weak through strong ties ties Selection on weak ties
Consensus Time Voter model 13-majority Majority
Low selection->Spread Voter model 13-majority Majority
High Selection->Bubble Filter Voter model 13-majority Majority
Strong Ties Voter model 13-majority Majority Fast Slow Stuck
Fast, Slow, and Stuck Consensus Time Number of Switches
Take-home Message • In influence dynamics, the strength of weak ties is to get new information and fresh ideas into the comfort zone created by strong ties. • In selection dynamics, the role of strong ties and weak ties, in terms of spreading fresh ideas, are swapped.
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