Influence theory

Abstract

Influence theory is a systematic study of formal models of the communicative influence of one person or group of people on another person or group. In that sense influence theory is an overarching philosophical discipline that includes aspects of decision theory and game theory as sub-disciplines as well as established models of de facto segregation, cultural change, opinion polarization, and epistemic networks. What we offer here is a structured outline of formal results that have been scattered across a range of disciplinary contexts from mathematics, physics and computer science to economics and political science, supplemented with a number of new models, emphasizing their place within the philosophical framework of a general theory of influence. What such an outline offers, we propose, is the prospect of new and important cross-fertilizations and expansions in formal attempts to model the diverse patterns of communicative influence.

Appendix

A compendium of examples above in terms of parameters outlined in Sect. 2:

Model	Agents	Traits	Network	Timing	Mechanism	Purpose	Dynamics
4.1		2 binary X,Y	1-Way influence	N/A	Influence on neighbor	formal	Immediate unanimity
4.2		2 binary X,Y	Mutual influence	Simultaneous	Influence on neighbor	formal	Oscillation from XY, YX
4.3		2 binary X,Y	Mutual influence	Sequential	Influence on neighbor	formal	First actor determines unanimity
4.4		2 binary X,Y	Mutual influence, Proba-bilistic	Simultaneous	Probability p of influence on neighbor	formal	Asymptotic approach to equilibrium
4.5		2 binary X,Y	Mutual influence, proba-bilistic	Sequential	Probability p of influence on neighbor	formal	Asymptotic approach to equilibrium, faster convergence than simultaneous
4.6		2 traits, adaptable to multiple	Mutual influence	Simultaneous & sequential variations	Influence on neighbor at particular trait, deterministic or probabilistic	formal	Allows a measure of degree of influence in terms of number of trait changes
4.7		Continuous scale on single trait	Mutual influence	Simultaneous & sequential variations	Each agent influences the other to move halfway to its position, or recalcitrant agents	formal	Various equilibria given differences in timing, rule and agent recalcitrance
4.8		2 binary, continuous, multiple	Mutual influence	Strategically handled influence	Probabilistic influence	formal	Altered probabilities with unbalanced influence events
5.1 Population model	Multiple, proportions of population	2 binary X,Y	Fully connected or random mixing	Simultaneous all with all	Probability of conversion of neighbor	formal	Markov model, unique equilibrium regardless of initial proportions, by Perron-Frobenius theorem
5.2 SIR, SIRS infection models	Multiple, proportions of population	Susceptible infected recovered	Random mixing	Simultaneous all with all	Infection of neighbor with R0 of pathogen, recovered immunity (SIR) or possible reinfection (SIRS)	explanatory predictive	Classical S-curve of infection for SIR model, endemic equilibrium possible in SIRS model
5.3 Deterministic cycle through states	Multiple, proportions of population	2 binary X,Y	Fully connected	Simultaneous all with all	100% probability of conversion of neighbor	formal	Violates conditionsof Markov model: oscillation
5.4 Variable transition probabilities, Grim Mar St. Denis (1998)	Multiple, proportions of population	2 binary X, Y and various	Fully connected	Simultaneous or sequential	Probabilistic conversion	formal	Equilibrium, oscillatory, periodic, and chaotic all possible dyanmics
5.5 Granovetter (1978)	Multiple, with hetero-geneous thresholds	2 binary: riot, not	Fully connected	Simultaneous	Agents with heterogeneous thresholds convert to ‘riot’ given specific proportions of the population rioting	explanatory	Rioting population depends not merely on average thresholds to riot but on distribution of thresholds
5.6 Lehrer and Wagner (1981)	Multiple	Multiple agents have opinion probability and reliability weights for other agents	Fully connected	Iterated simultaneous	Agents change opinion probabilities to weighted average of population	Formal, normative	Consensus in which all agents share same opinion probabilities develops as Markov process
6.1	Multiple	2 binary X, Y	Linear, 1-dimensional	Simultaneous	X changes to Y given a Y neighbor, or does so with probability p	formal	Total conversion to Y given any Y, speed dependent on probability
6.2	Multiple	2 binary X, Y	Linear, 1-dimensional	Simultaneous	Y changes to X if surrounded by X on both sides	formal	Results dependent on initial configuration
6.3	Multiple	2 binary X, Y	Linear, 1-dimensional	Simultaneous	Non-symmetrical: Y preceded by 2 X’s will convert to X	formal	Results dependent on initial configuration
6.4	Multiple	2 binary X, Y	Linear, 2-dimensional, finite alternation of X and Y	Simultaneous	Conversion to other view when surrounded on each side	formal	Results dependent on initial configuration
6.5	Multiple	2 binary X, Y	Linear loop	Simultaneous	Conversion to other view when surrounded on each side	formal	Oscillation with loop at one point; unanimity with loop at another
7.1 shared information	Multiple, with individual evidence as well as cumulative	Binary	Linear, 1-dimensional	Sequential along line	Agents share decisions and evidence down the line	formal	Given a small probability of correctness, individuals down the line will accumulate evidence toward the correct decision with probability 1
7.2 Information cascades Bikhchandani et al. (1998)	Multiple, with individual evidence as well as cumulative	Binary	Linear, 1-dimensional	Sequential along line	Agents share decisions but not individual evidence, make decisions on cumulative data	formal, explanatory	Cascades of decision, with calculable probabilities of misinformation
7.3 Information cascades in enclaves	Multiple in isolated enclaves, individual and cumulative evidence	Binary	Linear, 1-dimensional	Sequential along lines of each enclave	Agents share decisions but not individual evidence, make decisions on cumulative data	formal, explanatory	Increased probability of polarization, increased probability of misinformation in at least one enclave
7.4 Information cascades on trees	Multiple in isolated enclaves, individual and cumulative evidence	Binary	Tree network	Sequential from earlier on tree	Agents share decisions but not individual evidence, make decisions on cumulative data	formal, explanatory	Similar cascade phenomena
7.5 Fedderson Pesendorfer 1998; Austin-Smith Feddersen (2005, 2006)	Multiple agents	Private signals with ‘bias’ or standard of reasonable doubt	Fully connected	Simultaneous voting, or voting after straw poll, with either unanimous or majority decision rule	Agents may vote or share information on private signals strategically	Formal, explanatory normative	Incentives for strategic voting are different under unanimity and majority rule; error rates are higher under unanimity
7.6 Hegselmann Krause (2002, 2005, 2006)	Multiple agents	Continuous scale of opinion	Association network with similarity threshold on trait	Simultaneous	Agents’ opinions move toward average of those in their threshold range	formal, explanatory	With high threshold, convergence on opinion. With low threshold, polarization of opinion in population
8.1 Schelling, (1969, 1971, 1978)	Multiple agents	2 unchanging traits or colors X,Y	2-dimensional lattice with low density (empty cells)	Random	Agents do not change traits, but move on array when number of ‘like’ neighbors is below threshold t	formal, explanatory	With relatively low threshold (30%), clear clustered areas of residential segregation appear
8.2 Schelling with high, low density and heterogeneous thresholds	Multiple agents	2 unchanging traits or colors X,Y	2-dimensional lattice with low or high density	Random	Agents move when number of like neighbors is below threshold t, but one set of agents have a ‘don’t care’ low threshold	formal, explanatory	At low density, agents do not cluster. At high density, they are forced to
8.3 Cultural divfusion Axelrod (1997)	Multiple agents	5 traits, each with 1 of 10 values	Agents fixed in a 2-dimensional lattice	Neighbor pairs chosen at random, interact with a probability correlate to number of traits in common	Given interaction, the ‘active’ agent changes one of its non-matching traits to that of the other agent	formal, explanatory	Wide areas of convergence a rossall traits with islands of isolated ‘cultures’
8.4 Imitative game theory Nowak & May (1992, 1993)	Multiple agents	2 strategies: Cooperate, Defect	2-dimensional lattice	Simultaneous	Play rounds of Prisoner’s Dilemma, then imitate most successful neighbor	formal	Maintained proportion of cooperative strategies; symmetrical patterns
8.5 Imitative game theory Huberman & Glance (1993)	Multiple agents	2 strategies: Cooperate, Defect	2-dimensional lattice	Random	Play rounds of Prisoner’s Dilemma, then imitate most successful neighbor	formal	Symmetrical patterns destroyed; results for cooperation altered from simultaneous case
8.6 Game of Go	Multiple agents	2 traits, black or white	2-dimensional lattice	Sequential	Strategic moves attempting to occupy points or surround opponent	formal	Strategically complex
9.1 Small worlds Watts & Strogatz (1998)	Multiple agents	Various	Range of networks from ring with probability of ring rewiring to random	Simultaneous	Influence between nodes on links	formal, explanatory	With ring networks, clustering coefficient high and characteristic path length long. At random, path length short but clustering coefficient low. In ‘small worlds,’ high clustering with short path lengths
9.2 Preferential attachment networks Barabási & Albert (1999)	Multiple agents	Various	Networks formed with new nodes attaching with higher probability to nodes with many links	Sequential addition	Influence between nodes on links	formal, explanatory	Scale free networks form
9.3 Epistemic networks Zollman (2007, 2010a, 2010b)	multiple	Choice of theories as to the higher-paying of 2 ‘bandits’	Sample networks: ring, wheel, and complete	Simultaneous	Update chosen theory on the basis of individual testing of a chosen bandit and input from linked nodes	formal, normative	Ring formations resuilt in slower community convergence with higher group accuracy probability
9.4 Bayesian networks	multiple	Various	Directed acyclic graphs	Sequential	Bayesian updating of nodes from current links and previous values	formal	Various dynamics given differences in timing, rule, and agent recalcitrance
10.1 Wolfram rule 126	multiple	Black or white	Linear, 1-dimensional	Simultaneous	Cell is black just in case self and two neighbors not all blck or all white on previous generation	formal	Forms Sierpinski triangle
10.2 Wolfram rule 110	multiple	Black or white	Linear, 1-dimensional	Simultaneous	‘right-handed’: like 126 except a single black to the left is insufficient to make a cell black on the next generation	formal	Computationally universal, formally undecidasble (Cook, 2004)
11.1 Game of Life	multiple	Alive or dead	2-dimensional lattice	Simultaneous	If alive, stays alive on next iteration iff 2 or 3 neighbors are currently alive. If dead, stays dead unless 3 neighbors are currently alive	formal	Computationally universal, formally undecidasble (Berlekamp, Conway & Guy 1982)
11.2 Spatialized Prisoner’s Dilemma	Multiple	Many states to form electrons on wires, logic gates, memory units	2-dimensional lattice	Simultaneous	Cells play Prisoner’s Dilemma with neighbors, convert to most successful local strategy	formal	Computationally universal, formally undecidasble (Grim, Mar & St. Denis 1998)