Analytical solution of the voter model on uncorrelated networks

Vázquez, Federico; Eguı́luz, Vı́ctor M.

doi:10.1088/1367-2630/10/6/063011

Cited by 178 publications

(336 citation statements)

References 46 publications

Supporting

Mentioning

313

Contrasting

Unclassified

Order By: Relevance

“…The system dynamics is traced in form of an aggregate order parameter and the system is reformulated on the macro-scale as a differential equation which describes the temporal evolution of that parameter. In many cases, the average opinion (due to the analogy to spin systems often called »magnetization«) has proven to be an adequate choice, but sometimes the number of (re)active interfaces yields a more handable transformation (e.g., Frachebourg and Krapivsky, 1996;Krapivsky and Redner, 2003;Vazquez and Eguíluz, 2008). A mean-field analysis for the VM on the complete graph was presented by Slanina and Lavicka (2003), and naturally, we come across the same results using our method (Sec.…”

mentioning

confidence: 57%

“…Slanina and Lavicka (2003) derive expressions for the asymptotic exit probabilities and the mean time needed to converge, but the partial differential equations that describe the full probability distribution for the time to reach the stationary state is too difficult to be solved analytically (Slanina and Lavicka, 2003, 4). Further analytical results have been obtained for the VM on d-dimensional lattices (Cox, 1989;Frachebourg and Krapivsky, 1996;Liggett, 1999;Krapivsky and Redner, 2003) as well as for networks with uncorrelated degree distributions (Sood and Redner, 2005;Vazquez and Eguíluz, 2008). It is noteworthy, that the analysis of the VM (and more generally, of binary-state dynamics) on networks has inspired a series of solution techniques such as refined mean-field descriptions (e.g., Sood and Redner, 2005;Moretti et al, 2012), pairwise approximation (e.g., De Oliveira et al, 1993;Vazquez and Eguíluz, 2008;Schweitzer and Behera, 2008;Pugliese and Castellano, 2009) and approximate master equations (e.g., Gleeson, 2011Gleeson, , 2013.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Markov Chain Aggregation for Agent-Based Models

Banisch

2016

Understanding Complex Systems

View full text Add to dashboard Cite

AcknowledgementsScience needs freedom. It needs free thought, it needs free time, it needs free talk. Philippe Blanchard is among those teachers who are deeply committed to such an understanding of science. I am very proud and grateful to be among his students.I am also very grateful to Ricardo Lima. He is probably the person who engaged most in the details of this project and should really be an honorary member of the reading committee. Thank you for always inspiring discussions and the critical reading of all parts of this work.Tanya Araújo has given me encouragement since the first day we met. She also read through all the thesis and her advises (especially during the last turbulent months) helped a lot to finalize the writing.I was once told that those people who are most busy (those that really are and do not just pretend to be) are also the people who always have time when you approach them with some question or meet them on the floor. I don't know whether this is a general rule, but it certainly applies to Dima Volchenkov. Thank you for an open ear whenever I knocked on your door.A special thanks goes to Hanne Litschewsky, our great secretary in E5. Due to Hanne, I could experience how comfortable, how helpful, yet sometimes essential it is to be supported in all the bureaucratic aspects of science.All of this would have been a lot more difficult without the unconditional support of my family. I am very thankful to my parents, my parents-in-law, and especially to my wife Nannette.Financial support of the German Federal Ministry of Education and Research (BMBF) through the project Linguistic Networks is also gratefully acknowledged (http://project.linguistic-networks.net). AbstractThis thesis introduces a Markov chain approach that allows a rigorous analysis of a class of agent-based models (ABMs). It provides a general framework of aggregation in agent-based and related computational models by making use of Markov chain aggregation and lumpability theory in order to link between the micro and the macro level of observation.The starting point is a microscopic Markov chain description of the dynamical process in complete correspondence with the dynamical behavior of the agent model, which is obtained by considering the set of all possible agent configurations as the state space of a huge Markov chain. This is referred to as micro chain, and an explicit formal representation including microscopic transition rates can be derived for a class of models by using the random mapping representation of a Markov process. The explicit micro formulation enables the application of the theory of Markov chain aggregation -namely, lumpability -in order to reduce the state space of the micro chain and relate microscopic descriptions to a macroscopic formulation of interest. Well-known conditions for lumpability make it possible to establish the cases where the macro model is still Markov, and in this case we obtain a complete picture of the dynamics including the transient stage, the most interesting phase in applications.F...

show abstract

mentioning

confidence: 57%

mentioning

confidence: 99%

Markov Chain Aggregation for Agent-Based Models

Banisch

2016

Understanding Complex Systems

View full text Add to dashboard Cite

show abstract

“…We assume that each individual interacts with people living in her own location (family, friends, neighbors) with a probability α, while with probability 1 − α she does so with individuals of her work place. Once an individual interacts with others, its opinion is updated following a noisy voter model [6,7,8,9,10]: an interaction partner is chosen and the original agent copies her opinion imperfectly (with a certain probability of making mistakes). A more detailed description of the SIRM model can be found in [26].…”

Section: Model Definition and Analytical Descriptionmentioning

confidence: 99%

Persistence in Voting Behavior: Stronghold Dynamics in Elections

Pérez

Fernández-Gracia

Ramasco

et al. 2015

Social Computing, Behavioral-Cultural Modeling, and Prediction

View full text Add to dashboard Cite

Abstract. Influence among individuals is at the core of collective social phenomena such as the dissemination of ideas, beliefs or behaviors, social learning and the diffusion of innovations. Different mechanisms have been proposed to implement inter-agent influence in social models from the voter model, to majority rules, to the Granoveter model. Here we advance in this direction by confronting the recently introduced Social Influence and Recurrent Mobility (SIRM) model, that reproduces generic features of vote-shares at different geographical levels, with data in the US presidential elections. Our approach incorporates spatial and population diversity as inputs for the opinion dynamics while individuals' mobility provides a proxy for social context, and peer imitation accounts for social influence. The model captures the observed stationary background fluctuations in the vote-shares across counties. We study the so-called political strongholds, i.e., locations where the votes-shares for a party are systematically higher than average. A quantitative definition of a stronghold by means of persistence in time of fluctuations in the voting spatial distribution is introduced, and results from the US Presidential Elections during the period 1980-2012 are analyzed within this framework. We compare electoral results with simulations obtained with the SIRM model finding a good agreement both in terms of the number and the location of strongholds. The strongholds duration is also systematically characterized in the SIRM model. The results compare well with the electoral results data revealing an exponential decay in the persistence of the strongholds with time.

show abstract

“…Fragmentation transitions have been frequently observed in simulations [73,100,101,105]. However, common analytical approaches that faithfully capture other phase transitions [72,81,84] yield only rough approximations for the fragmentation threshold [114,115]. In this chapter we present a novel approach, which allows for a precise analytical estimation of fragmentation thresholds.…”

Section: Fragmentation Transitions In Models For Opinion Formationmentioning

confidence: 99%

Emergence and persistence of diversity in complex networks

Böhme

2013

Eur. Phys. J. Spec. Top.

View full text Add to dashboard Cite

Complex networks are employed as a mathematical description of complex systems in many different fields, ranging from biology to sociology, economy and ecology. Dynamical processes in these systems often display phase transitions, where the dynamics of the system changes qualitatively. In combination with these phase transitions certain components of the system might irretrievably go extinct. In this case, we talk about absorbing transitions. Developing mathematical tools, which allow for an analysis and prediction of the observed phase transitions is crucial for the investigation of complex networks.In this thesis, we investigate absorbing transitions in dynamical networks, where a certain amount of diversity is lost. In some real-world examples, e.g. in the evolution of human societies or of ecological systems, it is desirable to maintain a high degree of diversity, whereas in others, e.g. in epidemic spreading, the diversity of diseases is worthwhile to confine. An understanding of the underlying mechanisms for emergence and persistence of diversity in complex systems is therefore essential. Within the scope of two different network models, we develop an analytical approach, which can be used to estimate the prerequisites for diversity.In the first part, we study a model for opinion formation in human societies. In this model, regimes of low diversity and regimes of high diversity are separated by a fragmentation transition, where the network breaks into disconnected components, corresponding to different opinions. We propose an approach for the estimation of the fragmentation point. The approach is based on a linear stability analysis of the fragmented state close to the phase transition and yields much more accurate results compared to conventional methods.In the second part, we study a model for the formation of complex food webs. We calculate and analyze coexistence conditions for several types of species in ecological communities. To this aim, we employ an approach which involves an iterative stability analysis of the equilibrium with respect to the arrival of a new species. The proposed formalism allows for a direct calculation of coexistence ranges and thus facilitates a systematic analysis of persistence conditions for food webs.In summary, we present a general mathematical framework for the calculation of absorbing phase transitions in complex networks, which is based on concepts from percolation theory. While the specific implementation of the formalism differs from model to model, the basic principle remains applicable to a wide range of different models.

show abstract

Analytical solution of the voter model on uncorrelated networks

Cited by 178 publications

References 46 publications

Markov Chain Aggregation for Agent-Based Models

Markov Chain Aggregation for Agent-Based Models

Persistence in Voting Behavior: Stronghold Dynamics in Elections

Emergence and persistence of diversity in complex networks

Contact Info

Product

Resources

About