Complexity

Volume 2017 (2017), Article ID 3548591, 16 pages

https://doi.org/10.1155/2017/3548591

## Self-Organized Societies: On the Sakoda Model of Social Interactions

^{1}Departamento de Ingeniería Industrial, Universidad de los Andes, Bogotá, Colombia^{2}CeiBA Complex Research Center, Bogotá, Colombia^{3}Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Avda. Diagonal las Torres 2640, Peñalolén, Santiago, Chile^{4}UAI Physics Center, Universidad Adolfo Ibáñez, Santiago, Chile

Correspondence should be addressed to Sergio Rica

Received 3 October 2016; Revised 7 December 2016; Accepted 14 December 2016; Published 23 January 2017

Academic Editor: Mattia Frasca

Copyright © 2017 Pablo Medina et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We characterize the behavior and the social structures appearing from a model of general social interaction proposed by Sakoda. The model consists of two interacting populations in a two-dimensional periodic lattice with empty sites. It contemplates a set of simple rules that combine attitudes, ranges of interactions, and movement decisions. We analyze the evolution of the 45 different interaction rules via a Potts-like energy function which drives the system irreversibly to an equilibrium or a steady state. We discuss the robustness of the social structures, dynamical behaviors, and the existence of spatial long range order in terms of the social interactions and the equilibrium energy.

#### 1. Introduction

Throughout the history, the society has self-organized in a myriad of social structures and behaviors which appears as a response to the attitudes, decisions, and expectations among individuals, that is to say, from local (usually simple) rules to global behavior [1]. This capacity appears in animals [2], insects [3, 4], among others, and it flourishes in human beings since the early life [5]. The basic underlying mechanism of this self-organization lies on the evaluation of the expectations resulting from the attitudes among individuals inducing, or not, a mobility decision.

Interestingly, despite the complex nature of social interactions, the social behavior shares many common features with a variety of physical systems. Segregation, inclusion, and aggregation are examples of a collective order arising from simple and local individual rules based on attitudes, decisions, and expectations among individuals [6–9].

In the late forties, Sakoda introduced in his Ph.D. thesis [10] a first general discrete dynamical model for social interactions and published later [11], at the same time of the well-known Schelling’s social segregation model [12, 13], which is, as we will show, only a special class already included in the original Sakoda model. The particular case of the Schelling segregation model and its variations have been studied widely through agent based models [9, 14–23]. However, the Sakoda dynamics and its underlying richness, which explain other different social phenomena far from segregation, are mostly unknown; thus they have not been, to the best of our knowledge, studied previously in a deeper manner.

The original Sakoda model roughly consists of social interactions among two groups of individuals evolving in a network according to specific attitudes of attraction, repulsion, and neutrality. An individual evaluates its social expectative at all possible available locations, preferring those near individuals associated with attractive (positive) attitudes and avoiding locations near individuals associated with repulsive (negative) ones. This procedure is repeated randomly among all possible individuals; henceforth Sakoda’s algorithm is iterated recursively driving the system, under some conditions, to a well-organized spatial pattern.

The dynamics is quite rich because of three aspects: () the large number of combinations of the possible attitudes, () the effect of the separation distance among individuals (i.e., the interaction could be of short or long range), and () the mobility of the individuals (the individuals may move into its own neighborhood or they may migrate far away).

This produces a myriad of possible patterns of the individuals’ arrangements in the space (social structures) to be analyzed. Some of them were already recognized by Sakoda in his early work [11].

In the present paper we discuss and complement the Sakoda original classification of social structures. More precisely, we characterize all independent individual interactions (45 different interactions) in the case of short range interaction and long range movement. We introduce a Potts energy [24, 25], which has been the natural -state generalization of the Ising model. As the former, the Potts model also displays a phase transition in statistical physics. In the context of the Sakoda model, the Potts energy happens to be a Lyapunov function in the case of symmetric interactions that governs the future dynamics and quantifies different attractors in terms of the parameters of the problem.

This Potts-like energy turns out to be a common principle for all possible interactions and describes the evolution of different social interacting systems. Finally, we discuss the role of empty sites and the existence of long range order in the social structures.

The present paper is organized as follows. Section 2 presents the Sakoda model and the numerical scheme. Section 3.1 shows the energy principle that mandates the dynamical evolution of the system. Next, Section 4 characterizes the various social structures arising from different interactions possibilities. A quantitative energy-based study is done in Section 5.1. The role of vacancies is studied in Section 5.2 and the existence of long range order is discussed in Section 5.3. Finally, we conclude in Section 6 and complement the analytics with a series of appendices.

#### 2. The Basic Model

The Sakoda model consists in a regular and periodic lattice with nodes in a two-dimensional space. Each node is associated with a discrete variable and may take the values . States denote an occupancy by an individual of one of the two types (+ or −); the state denotes an empty place or a vacancy. Only when a node is empty may an individual (+ or −) occupy the free spot and reassign its value with its current one.

The social interaction is mainly characterized by the possible attitudes among individuals which are summarized in a matrix. We call hereafter the “-matrix,” which for the sake of space we will symbolize in a vectorized form: . The entries take three possible values: . These indicate, respectively, an attractive , a neutral (0), or a repulsive attitude from members of the type to the individuals of the type . There exist different possible cases which are reduced, because of symmetry, to 45 independent social structures. In Appendix A, we discuss the symmetries of the “-matrices” and its properties in a deeper manner.

Next, we model the preferences of one individual over a particular place in the lattice based on the spatial locations and the attitudes of other individuals. Following the original ideas of Sakoda, we propose a function that quantifies the social expectations of the individual :where denotes a symmetric () interaction strength. We avoid self-interactions taking . On the other hand, the sign of the interaction is given by (do not confuse the variable that takes the values with the entries of the matrix that take the same values ):

In general, this expression is not symmetric; that is, . We underline that though is symmetric, the full interaction is not necessarily symmetric. The coefficients may include short and long range interactions. The mobility of an individual could be also of long or short movement. An individual located at the node would move towards an empty node , if . Here represents the potential value (1) evaluated at the empty node, , occupied by the individual . As a general rule, the individual chooses the place everywhere in the lattice that makes the social expectation function the highest. In case of degeneracy, that is, if there are nodes, , such that , one of these is selected randomly with the same probability. In a short movement case, the mobility of the individual is restricted to its Moore neighborhood (the eight closest neighbors).

To summarize, the movement of individuals occurs in the following manner: during an iteration step, an individual is selected randomly. Then, it evaluates the potential function at every available empty site in its range of movement for a potential new location. Next, the individual selects the site with the highest value of the potential expectation (1) and then it moves to this site. If there is not a site that improves the potential function, then the individual stays in place. The algorithm is iterated until the system reaches a fixed point; otherwise, it runs forever.

We point out that the rule preserves the initial number of individuals of all types. That is, if , , and are the number of individuals of the types +, −, and the number of vacancies, respectively, they keep their initial value during the evolution. We define the fractions by and (naturally ). Hereafter, as in Sakoda original paper [11], we limit our study to the case of .

##### 2.1. Numerics

For the following, we restrict our central results only to the case of short range interactions and long range movements: an individual evaluates its preferences considering the Moore neighborhood; that is, , if is in the 8 closest neighbors of the site , and elsewhere. Then, the individual may look forward the highest expectation at any empty node in the lattice. We discuss the cases of short and long range interactions/mobility at the end of the paper and in Appendix F.

The numerical simulations for all 45 possible -matrices were run in a two-dimensional periodic lattice of . The evolution usually runs for time steps, which happens to be sufficient to reach equilibrium or a steady state. We have also simulated the case of without significant dependence on the system size.

The initial condition consists of a random distribution of states characterized by a fraction of vacancy states. For all of our simulations, we have checked the robustness of the social structures considering different random initial configurations. We focus our attention on five levels () of different initial concentration of vacancies as a good representative sample (see Section 5.2). Other initial distributions may be studied using the same frame and tools developed in the present paper.

Figure 1 summarizes some social structures resulting from the afore-mentioned algorithm for four characteristic -matrices and for three different values of the vacancy fraction, . Over all the manuscript, the light gray cells (red cells online) represent individuals at the state and the dark gray cells (blue cells online) represent individuals, while the white ones represent the empty spaces, .