Final Report Summary - STATPHYSNETFORMGAME (The statistical physics of network formation games)
During his visit the researcher learned the basic concepts and relevant results of evolutionary game theory that is considered as a general theoretical background for mathematical consideration of living systems. This new field of sciences adopts the concepts and methods of statistical physics for investigating fundamental problems in social and biological systems consisting of many participants. He studied several models with using different approaches and methods.
First he studied the spreading of species/strategies/opinions in a spatial system where the preference is varied linearly along one of the directions and found a set of critical exponents characterizing the inhomogeneous spatial patterns. The spreading mechanism via the so-called voter model is also studied in a one-dimensional system that exhibits strong analogy to the Ising model ensuring the applicability of the transfer matrix method. The evolutionary matching pennies game with a synchronized stochastic strategy update have indicated the emergence and growing of domains with oscillating spatio-temporal patterns. This system also exhibits an Ising-type critical transition when increasing the level of noise.
In collaboration with other groups the researcher studied the evolution of the mobile phone network in Cote d'Ivory and the history of connections provided by worldwide cargo ship movements.
The above-mentioned results were presented in different conferences and group seminars, were published (or the publishing is in progress) in scientific journals, and expanded the knowledge and experience of the candidate that can be utilized in his future research and teaching activity. The last statement is justified by recognizing the applicability of his cartogram method to the investigation of the networks of neurons (in human brain). Some other results can be utilized in the near future both in further development of the networks of internet or mobile phone, and in managing the investments in trade via cargo ships. See more attached.
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The focus of this project was to understand the formation and dynamics of complex networks and spatio-temporal patterns with the methods of statistical physics. Some of our work was motivated by fundamental mathematical questions while others were driven by the analysis of real data.
I. THE GEOGRAPHY AND CARBON FOOTPRINT OF MOBILE PHONE USE IN COTE D'IVOIRE
In 2013, the mobile phone provider Orange released a data base of mobile phone use in Cote d'Ivoire. These data provided a rare insight into the use of mobile technology
in a developing country. We performed a series of spatial data analyses that reveal important geographic aspects of the country's mobile phone coverage. We first mapped the locations of base stations, which house the antennas, with respect to the population distribution and the number and duration of calls at each base station. We used a cartographic technique called “density-equalizing map or cartogram that was
developed by the researcher in 2004. This method allowed us to scale all subprefectures in Cote d'Ivoire such that the areas are proportional to the number of inhabitants.
Although the geographic distribution of the base stations is largely explained by the heterogeneous population distribution, the point pattern on the cartogram still identifies regions of significantly higher per-capita base station density. Especially
in Abidjan the dots are noticeably aggregated. On the contrary, sparser point densities on the cartogram highlight those regions on which expansion of the network should be focused in the future. The resultant cartogram helps the investor to improve the efficiency by providing a roughly equal percapita number of base stations for the entire population of Cote d'Ivoire.
In further analyses, we estimated the energy consumed by the mobile phone network. It is found that mobile networks in Cote d'Ivoire are likely to contribute a greater proportion to the national greenhouse gas emissions than those in industrialized
countries. However, the mobile network still only consumes 0.95% of Cote d'Ivoire's electrical energy and is therefore only of minor concern for the nation's CO2 carbon footprint.
II. THE DEVELOPMENT OF THE GLOBAL NETWORK OF CARGO SHIPPING
Cargo shipping is estimated to transport 90% of world trade so that, in economic terms, the routes of cargo ships form arguably the world's most important transport network. Peviously the researcher had performed an analysis of the network based on data from 2007 exclusively. Now, due to the collaboration with Cesar Ducruet (CNRS Paris) the researcher can access to cargo ship records for a longer period, going as far back as 1890. In the knowledge of these data, the distributions of vessel calls and
the ports' degrees (i.e. the number of other ports that a port is directly linked with) become possible. Much previous work on economic and social networks has claimed that such centrality distributions in economic and social networks have a characteristic mathematical form: a power law, also sometimes called a “scalefree
distribution. Power laws have characteristically heavy tails, which means that some nodes are significantly better connected than the average. Our work, however, demonstrates that a power law is inadequate to fit the cargo ship data. Instead we propose alternatives, such as lognormal or Weibull distributions, that perform consistently better for all years for which we have data. In short, Cargo ship traffic has thus for the entire study period been heavy-tailed, but the distribution is not scale-free.
The Gini coefficient is a measure of inequality in economics. The Gini coefficient of port traffic has slightly, but statistically significantly, decreased over the study period, highlighting a tendency towards a more polycentric distribution in cargo
shipping. These results are informative for economic forecasts and provide important lessons for successful planning of port expansions.
III. OPINION FORMATION ON A GRADIENT
The voter model, as a simple example of a collective game-theoretic social behaviour, was introduced to describe how individuals change their inherently preferred opinion if their friends disagree. The early investigations were focused on systems with homogeneous preferences. Real preferences, however, often depend on regional and cultural differences. The resultant effects are investigated by assuming spatial gradients in the environment that affect real election results, especially between rural and urban populations, as it has been pointed out by many political scientists. However, the quantitative consequences for opinion formation have so far been unknown.
We approach opinion formation with the tools of statistical physics to analyze the percolation pattern (i.e. the geometry of connected clusters of like-minded opinions). We show that opinion clusters are typically in the standard (i.e. independent) percolation universality class. As a consequence, influencing
each other's opinion usually only creates consensus on a local scale, but over long distances the opinions remain uncorrelated. Thus, opinions remain mixed if averaged over long length scales. Additionally we have also presented an alternative model where a sharp spatial division between two opinions occurs.
The geometrical features of the interfaces, separating the large homogeneous domains, are investigated numerically and analytically on a two-dimensional lattice. It is demonstrated by numerical simulations how the interface widths, fractal dimensions, and cluster size distributions differ/vary between these two scenarios. Quantitatively, the interface width exhibits a power law behaviour in the same way as it is described in many physical systems. For small gradients we could quantify the fractal dimension (7/4) of the interface. In the opposite case it is justified that the interface is not a fractal despite its roughness. The cluster size distribution (for weak gradients) exhibits the same asymptotic power law behaviour as it is found for the traditional percolation theory, while the results are consistent with
first-order transition in the percolation for large gradients. Our results settle a controversial debate in the literature about the universality class of percolation
in “non-consensus opinion models the model for a small gradient was previously claimed to differ from standard percolation, but our data convincingly demonstrates the opposite. For large gradients the model proves that with alternative rules for opinion formation non-standard percolation is conceivable.
IV. THE ISING CHAIN CONSTRAINED TO AN EVEN OR ODD NUMBER OF POSITIVE SPINS
The above work on opinion formation raises the intriguing question of a general equation-based solution for those models. It turns out that, after some mathematical
transformations, the one-dimensional models can indeed be mapped onto a classic problem in statistical physics: the Ising model. In the Ising model, sites are occupied by so-called spins that can point either up or down. The total energy
in the Ising model is determined by the number of neighbours pointing in opposite directions. The mapping from opinion formation to Ising model required one additional twist that had not been previously studied: the number of positive spins is constrained to be either odd or even, depending on whether the total number of individuals in the opinion formation model is an odd or even number. This observation called for a thorough investigation of the problem. In a recent publication I calculated the partition function using a generalization of the transfer matrix
method. On this basis, I derived the exact magnetization, susceptibility, internal energy, heat capacity and correlation function. I showed that in general the constraints substantially slow down convergence to the thermodynamic limit (i.e. the
limit of infinitely many nodes in the network). By taking the thermodynamic limit together with the limit of zero temperature and zero magnetic field, the constraints lead to new scaling functions and different probability distributions for the magnetization. This work is a starting point for investigating the more general case of opinions on two-dimensional or even more complex social networks.
V. MATCHING-PENNIES GAME ON LATTICE WITH SYNCHRONIZED UPDATES
Preliminary results show that the spatial matching-pennies game, a textbook example of a two-player game with no pure Nash equilibrium, exhibits a behaviour that is similar to those described by the Ising model when increasing the strength of stochasticity. We have studied an evolutionary game where the players were located on the sites of a square lattice, they played matching-pennies games with their four neighbours, and in discrete time steps they could modify their strategy simultaneously to have higher income. In this systems four types growing domains are observed. These domains represent the four phases of a cyclic oscillation (from white to chessboard to black to anti-chessboard to white, etc.) Similar pattern evolution characterizes the so-called chimera states studied progressively in the last years. The publication of these results is in progress. Finally we mention that we have observed more complex chimera states in the above system if rock-paper-scissors game is substituted for the matching-pennies game.
VI. CONTINUING WORK
At the end of Marie Curie Fellowship the researcher began collaborating with Geza Odor (a researcher in the same group) on analyzing data from the Open Connectome project. From the website (www.openconnectomeproject.org) we obtained network data for the human brain where nodes are voxels in MRI scans and edges are fibre tracts. The same statistical techniques that he developed for the analysis of cargo ship routes can be successfully applied to the brain. Our results may open the path towards more realistic models of brain functions and dynamics.
First he studied the spreading of species/strategies/opinions in a spatial system where the preference is varied linearly along one of the directions and found a set of critical exponents characterizing the inhomogeneous spatial patterns. The spreading mechanism via the so-called voter model is also studied in a one-dimensional system that exhibits strong analogy to the Ising model ensuring the applicability of the transfer matrix method. The evolutionary matching pennies game with a synchronized stochastic strategy update have indicated the emergence and growing of domains with oscillating spatio-temporal patterns. This system also exhibits an Ising-type critical transition when increasing the level of noise.
In collaboration with other groups the researcher studied the evolution of the mobile phone network in Cote d'Ivory and the history of connections provided by worldwide cargo ship movements.
The above-mentioned results were presented in different conferences and group seminars, were published (or the publishing is in progress) in scientific journals, and expanded the knowledge and experience of the candidate that can be utilized in his future research and teaching activity. The last statement is justified by recognizing the applicability of his cartogram method to the investigation of the networks of neurons (in human brain). Some other results can be utilized in the near future both in further development of the networks of internet or mobile phone, and in managing the investments in trade via cargo ships. See more attached.
************************************************************
The focus of this project was to understand the formation and dynamics of complex networks and spatio-temporal patterns with the methods of statistical physics. Some of our work was motivated by fundamental mathematical questions while others were driven by the analysis of real data.
I. THE GEOGRAPHY AND CARBON FOOTPRINT OF MOBILE PHONE USE IN COTE D'IVOIRE
In 2013, the mobile phone provider Orange released a data base of mobile phone use in Cote d'Ivoire. These data provided a rare insight into the use of mobile technology
in a developing country. We performed a series of spatial data analyses that reveal important geographic aspects of the country's mobile phone coverage. We first mapped the locations of base stations, which house the antennas, with respect to the population distribution and the number and duration of calls at each base station. We used a cartographic technique called “density-equalizing map or cartogram that was
developed by the researcher in 2004. This method allowed us to scale all subprefectures in Cote d'Ivoire such that the areas are proportional to the number of inhabitants.
Although the geographic distribution of the base stations is largely explained by the heterogeneous population distribution, the point pattern on the cartogram still identifies regions of significantly higher per-capita base station density. Especially
in Abidjan the dots are noticeably aggregated. On the contrary, sparser point densities on the cartogram highlight those regions on which expansion of the network should be focused in the future. The resultant cartogram helps the investor to improve the efficiency by providing a roughly equal percapita number of base stations for the entire population of Cote d'Ivoire.
In further analyses, we estimated the energy consumed by the mobile phone network. It is found that mobile networks in Cote d'Ivoire are likely to contribute a greater proportion to the national greenhouse gas emissions than those in industrialized
countries. However, the mobile network still only consumes 0.95% of Cote d'Ivoire's electrical energy and is therefore only of minor concern for the nation's CO2 carbon footprint.
II. THE DEVELOPMENT OF THE GLOBAL NETWORK OF CARGO SHIPPING
Cargo shipping is estimated to transport 90% of world trade so that, in economic terms, the routes of cargo ships form arguably the world's most important transport network. Peviously the researcher had performed an analysis of the network based on data from 2007 exclusively. Now, due to the collaboration with Cesar Ducruet (CNRS Paris) the researcher can access to cargo ship records for a longer period, going as far back as 1890. In the knowledge of these data, the distributions of vessel calls and
the ports' degrees (i.e. the number of other ports that a port is directly linked with) become possible. Much previous work on economic and social networks has claimed that such centrality distributions in economic and social networks have a characteristic mathematical form: a power law, also sometimes called a “scalefree
distribution. Power laws have characteristically heavy tails, which means that some nodes are significantly better connected than the average. Our work, however, demonstrates that a power law is inadequate to fit the cargo ship data. Instead we propose alternatives, such as lognormal or Weibull distributions, that perform consistently better for all years for which we have data. In short, Cargo ship traffic has thus for the entire study period been heavy-tailed, but the distribution is not scale-free.
The Gini coefficient is a measure of inequality in economics. The Gini coefficient of port traffic has slightly, but statistically significantly, decreased over the study period, highlighting a tendency towards a more polycentric distribution in cargo
shipping. These results are informative for economic forecasts and provide important lessons for successful planning of port expansions.
III. OPINION FORMATION ON A GRADIENT
The voter model, as a simple example of a collective game-theoretic social behaviour, was introduced to describe how individuals change their inherently preferred opinion if their friends disagree. The early investigations were focused on systems with homogeneous preferences. Real preferences, however, often depend on regional and cultural differences. The resultant effects are investigated by assuming spatial gradients in the environment that affect real election results, especially between rural and urban populations, as it has been pointed out by many political scientists. However, the quantitative consequences for opinion formation have so far been unknown.
We approach opinion formation with the tools of statistical physics to analyze the percolation pattern (i.e. the geometry of connected clusters of like-minded opinions). We show that opinion clusters are typically in the standard (i.e. independent) percolation universality class. As a consequence, influencing
each other's opinion usually only creates consensus on a local scale, but over long distances the opinions remain uncorrelated. Thus, opinions remain mixed if averaged over long length scales. Additionally we have also presented an alternative model where a sharp spatial division between two opinions occurs.
The geometrical features of the interfaces, separating the large homogeneous domains, are investigated numerically and analytically on a two-dimensional lattice. It is demonstrated by numerical simulations how the interface widths, fractal dimensions, and cluster size distributions differ/vary between these two scenarios. Quantitatively, the interface width exhibits a power law behaviour in the same way as it is described in many physical systems. For small gradients we could quantify the fractal dimension (7/4) of the interface. In the opposite case it is justified that the interface is not a fractal despite its roughness. The cluster size distribution (for weak gradients) exhibits the same asymptotic power law behaviour as it is found for the traditional percolation theory, while the results are consistent with
first-order transition in the percolation for large gradients. Our results settle a controversial debate in the literature about the universality class of percolation
in “non-consensus opinion models the model for a small gradient was previously claimed to differ from standard percolation, but our data convincingly demonstrates the opposite. For large gradients the model proves that with alternative rules for opinion formation non-standard percolation is conceivable.
IV. THE ISING CHAIN CONSTRAINED TO AN EVEN OR ODD NUMBER OF POSITIVE SPINS
The above work on opinion formation raises the intriguing question of a general equation-based solution for those models. It turns out that, after some mathematical
transformations, the one-dimensional models can indeed be mapped onto a classic problem in statistical physics: the Ising model. In the Ising model, sites are occupied by so-called spins that can point either up or down. The total energy
in the Ising model is determined by the number of neighbours pointing in opposite directions. The mapping from opinion formation to Ising model required one additional twist that had not been previously studied: the number of positive spins is constrained to be either odd or even, depending on whether the total number of individuals in the opinion formation model is an odd or even number. This observation called for a thorough investigation of the problem. In a recent publication I calculated the partition function using a generalization of the transfer matrix
method. On this basis, I derived the exact magnetization, susceptibility, internal energy, heat capacity and correlation function. I showed that in general the constraints substantially slow down convergence to the thermodynamic limit (i.e. the
limit of infinitely many nodes in the network). By taking the thermodynamic limit together with the limit of zero temperature and zero magnetic field, the constraints lead to new scaling functions and different probability distributions for the magnetization. This work is a starting point for investigating the more general case of opinions on two-dimensional or even more complex social networks.
V. MATCHING-PENNIES GAME ON LATTICE WITH SYNCHRONIZED UPDATES
Preliminary results show that the spatial matching-pennies game, a textbook example of a two-player game with no pure Nash equilibrium, exhibits a behaviour that is similar to those described by the Ising model when increasing the strength of stochasticity. We have studied an evolutionary game where the players were located on the sites of a square lattice, they played matching-pennies games with their four neighbours, and in discrete time steps they could modify their strategy simultaneously to have higher income. In this systems four types growing domains are observed. These domains represent the four phases of a cyclic oscillation (from white to chessboard to black to anti-chessboard to white, etc.) Similar pattern evolution characterizes the so-called chimera states studied progressively in the last years. The publication of these results is in progress. Finally we mention that we have observed more complex chimera states in the above system if rock-paper-scissors game is substituted for the matching-pennies game.
VI. CONTINUING WORK
At the end of Marie Curie Fellowship the researcher began collaborating with Geza Odor (a researcher in the same group) on analyzing data from the Open Connectome project. From the website (www.openconnectomeproject.org) we obtained network data for the human brain where nodes are voxels in MRI scans and edges are fibre tracts. The same statistical techniques that he developed for the analysis of cargo ship routes can be successfully applied to the brain. Our results may open the path towards more realistic models of brain functions and dynamics.
