Statistical physics of human cooperation

19세기말 애국적 담론과 新小說의 정체성

Extensive cooperation among unrelated individuals is unique to humans, who often sacrifice personal benefits for the common good and work together to achieve what they are unable to execute alone. The evolutionary success of our species is indeed due, to a large degree, to our unparalleled other-regarding abilities. Yet, a comprehensive understanding of human cooperation remains a formidable challenge. Recent research in social science indicates that it is important to focus on the collective behavior that emerges as the result of the interactions among individuals, groups, and even societies. Non-equilibrium statistical physics, in particular Monte Carlo methods and the theory of collective behavior of interacting particles near phase transition points, has proven to be very valuable for understanding counterintuitive evolutionary outcomes. By studying models of human cooperation as classical spin models, a physicist can draw on familiar settings from statistical physics. However, unlike pairwise interactions among particles that typically govern solid-state physics systems, interactions among humans often involve group interactions, and they also involve a larger number of possible states even for the most simplified description of reality. The complexity of solutions therefore often surpasses that observed in physical systems. Here we review experimental and theoretical research that advances our understanding of human cooperation, focusing on spatial pattern formation, on the spatiotemporal dynamics of observed solutions, and on self-organization that may either promote or hinder socially favorable states.

Random Matrix Methods for Wireless Communications: Estimation

Rock is wrapped by paper, paper is cut by scissors and scissors are crushed by rock. This simple game is popular among children and adults to decide on trivial disputes that have no obvious winner, but cyclic dominance is also at the heart of predator–prey interactions, the mating strategy of side-blotched lizards, the overgrowth of marine sessile organisms and competition in microbial populations. Cyclical interactions also emerge spontaneously in evolutionary games entailing volunteering, reward, punishment, and in fact are common when the competing strategies are three or more, regardless of the particularities of the game. Here, we review recent advances on the rock–paper–scissors (RPS) and related evolutionary games, focusing, in particular, on pattern formation, the impact of mobility and the spontaneous emergence of cyclic dominance. We also review mean-field and zero-dimensional RPS models and the application of the complex Ginzburg–Landau equation, and we highlight the importance and usefulness of statistical physics for the successful study of large-scale ecological systems. Directions for future research, related, for example, to dynamical effects of coevolutionary rules and invasion reversals owing to multi-point interactions, are also outlined.

/pdf/cyclic-dominance-in-evolutionary-games-a-review-34pqhh53un.pdf

Cyclic dominance in evolutionary games: A review

We consider the local eigenvalue distribution of large self-adjoint $$N\times N$$
 random matrices $$\mathbf {H}=\mathbf {H}^*$$
 with centered independent entries. In contrast to previous works the matrix of variances $$s_{i j}=\mathbbm {E} |h_{i j}|^2$$
 is not assumed to be stochastic. Hence the density of states is not the Wigner semicircle law. Its possible shapes are described in the companion paper (Ajanki et al. in Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095
 ). We show that as N grows, the resolvent, $$\mathbf {G}(z)=(\mathbf {H}-z)^{-1}$$
 , converges to a diagonal matrix, $$\mathrm {diag}(\mathbf {m}(z))$$
 , where $$\mathbf {m}(z)=(m_1(z),\dots ,m_N(z))$$
 solves the vector equation $$-1/m_i(z) = z\,+\,\sum _j s_{i j} m_j(z) $$
 that has been analyzed in Ajanki et al. (Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095
 ). We prove a local law down to the smallest spectral resolution scale, and bulk universality for both real symmetric and complex hermitian symmetry classes.

/pdf/universality-for-general-wigner-type-matrices-1yqlzd6ycb.pdf

Universality for general Wigner-type matrices

Analyzing coexistence and survival scenarios of Lotka-Volterra (LV) networks in which the total biomass is conserved is of vital importance for the characterization of long-term dynamics of ecological communities. Here, we introduce a classification scheme for coexistence scenarios in these conservative LV models and quantify the extinction process by employing the Pfaffian of the network's interaction matrix. We illustrate our findings on global stability properties for general systems of four and five species and find a generalized scaling law for the extinction time.

/pdf/coexistence-and-survival-in-conservative-lotka-volterra-2i5e78jxp7.pdf

Coexistence and survival in conservative Lotka-Volterra networks.

We consider real symmetric or complex hermitian random matrices with correlated entries. We prove local laws for the resolvent and universality of the local eigenvalue statistics in the bulk of the spectrum. The correlations have fast decay but are otherwise of general form. The key novelty is the detailed stability analysis of the corresponding matrix valued Dyson equation whose solution is the deterministic limit of the resolvent.

/pdf/stability-of-the-matrix-dyson-equation-and-random-matrices-2far5bwotb.pdf

Stability of the matrix Dyson equation and random matrices with correlations

Let $ S $ be a positivity preserving symmetric linear operator acting on bounded functions. The nonlinear equation $ -\frac{1}{m}=z+Sm $ with a parameter $ z $ in the complex upper half-plane $ \mathbb{H} $ has a unique solution $ m $ with values in $ \mathbb{H} $. We show that the $ z $-dependence of this solution can be represented as the Stieltjes transforms of a family of probability measures $ v $ on $ \mathbb{R} $. Under suitable conditions on $ S $, we show that $ v $ has a real analytic density apart from finitely many algebraic singularities of degree at most three. 
Our motivation comes from large random matrices. The solution $ m $ determines the density of eigenvalues of two prominent matrix ensembles; (i) matrices with centered independent entries whose variances are given by $ S $ and (ii) matrices with correlated entries with a translation invariant correlation structure. Our analysis shows that the limiting eigenvalue density has only square root singularities or a cubic root cusps; no other singularities occur.

Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half‐Plane

Condensation phenomena arise through a collective behaviour of particles. They are observed in both classical and quantum systems, ranging from the formation of traffic jams in mass transport models to the macroscopic occupation of the energetic ground state in ultra-cold bosonic gases (Bose-Einstein condensation). Recently, it has been shown that a driven and dissipative system of bosons may form multiple condensates. Which states become the condensates has, however, remained elusive thus far. The dynamics of this condensation are described by coupled birth-death processes, which also occur in evolutionary game theory. Here we apply concepts from evolutionary game theory to explain the formation of multiple condensates in such driven-dissipative bosonic systems. We show that the vanishing of relative entropy production determines their selection. The condensation proceeds exponentially fast, but the system never comes to rest. Instead, the occupation numbers of condensates may oscillate, as we demonstrate for a rock-paper-scissors game of condensates.

/pdf/evolutionary-games-of-condensates-in-coupled-birth-death-4bs9us2iae.pdf

Torben Krüger

Papers

Universality for general Wigner-type matrices

Coexistence and survival in conservative Lotka-Volterra networks.

Stability of the matrix Dyson equation and random matrices with correlations

Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half‐Plane

Evolutionary games of condensates in coupled birth–death processes