ISSN 20790597, Russian Journal of Genetics: Applied Research, 2015, Vol. 5, No. 6, pp. 672–678. © Pleiades Publishing, Ltd., 2015. Original Russian Text © S.A. Lashin, E.A. Mamontova, Yu.G. Matushkin, N.A. Kolchanov, 2014, published in Vavilovskii Zhurnal Genetiki i Selektsii, 2014, Vol. 18, No. 4/3, pp. 1289–1298.
Mechanisms of the Formation and Propagation of Sociobiological Interactions: a Computer Simulation Study S. A. Lashina,b, E. A. Mamontovab, Yu. G. Matushkina,b, and N. A. Kolchanova,b a
Institute of Cytology and Genetics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, 630090 Russia b Novosibirsk State University, Novosibirsk, Russia email:
[email protected] Received October 15, 2014; in final form, October 31, 2014
Abstract—Over 10000 scientific studies have been dedicated to mathematical modeling and simulations of biologic systems at all levels of their organization. At the Institute of Cytology and Genetics, Novosibirsk, software tools are being developed for the analysis of the propagation of ideas/opinions in social systems with regard not only to social but also biological (genetic, physiological, and psychological) factors. The consid eration of more factors in these models allows better analysis of social processes, more precise predictions, and smarter strategies for process control. Agentbased modeling is an approach that is widely used for these tasks. In the present study, the simple model introduced by Bonabeau (2002) is considered and extended. In the original model, the complex behavior of the system arises on the basis of two different strategies of agent behavior. The extended sociobiological model describes more sophisticated agents with additional character istics. It allows social and biological submodels to be combined within a common software tool. As the num ber of agent parameters (dissatisfaction, enmity, and mobility) increases, the dynamic variability of the model increases manifold times. The addition of the dissatisfaction parameter enables an agent to compare his suc cess with other agents, which may be interpreted as selfesteem. The model can be further improved by the addition of various mechanisms of idea and opinion transmission, which will make it possible to model the dissemination of ideas within populations (and, in addition, the spread of diseases, rumors, and any sort of information), to model a heterogeneous distribution of parameters within population, and to consider differ ent topologies of social networks. Nevertheless, even now the “aggressor–defender” model can simulate many social and biological phenomena. Keywords: social system, agentbased model, parallel programming, “aggressor–defender” model DOI: 10.1134/S2079059715060040
INTRODUCTION Systems approaches to social epidemiology cur rently attract the increasing attention of scientists (El Sayed et al., 2012). The most important tools used in this field are social network analysis and agentbased modeling (ABM, or AB modeling). Social network analysis answers questions concerning the relation ships between network topology and the rate of spread (and extinction) of various sorts of information, whereas AB models often support direct simulation of a population with explicitly specified rules of informa tion generation and processing by agents. Although network analysis is suitable for social infection studies (Social Epidemiology, 2000), it demands huge vol umes of experimental data, which are seldom available in practice. These are the constraints of the methods. Thus, the employment of AB models to assess factors influencing idea propagation appears to be the most natural, especially in consideration of factors belong ing to different levels of the biologic or social organi zation of a community or its members (ElSayed et al., 2012). Here we consider the simple model proposed
by Bonabeau (2000), where phenomena arise on the basis of two different strategies of agent behavior. Our task was to extend this model into a sociobiological model where agents would have more parameters. This would allow variation of social and biological submodels within a single software tool. OUTLINE OF THE PROBLEM AgentBased Modeling Methods Agentbased modeling systems present particles as individual objects. They include Monte Carlo meth ods, cellular automata, neural networks, etc. The AB approach is commonly considered to be an alternative to describing a system in terms of differential equa tions (DEs). The complexity of DEs describing a sys tem increases exponentially with the behavior com plexity of an individual agent. Agentbased models not only solve this problem but have additional advantages. First, they allow con sideration of the process of phenomenon arise, whereas DEbased models describe the final state of
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the system and deal mainly with averaged parameter values. Second, AB models are flexible and adjustable. The heterogeneous topology of the interaction net work, agent memory, learning, and adaptation provide unpredictable effects that cannot be tracked in DEbased models. These features can be readily implemented in ABM. Finally, AB models can con sider a system at various levels of detail: individual agents, groups of agents, or the entire population. For each of these levels, individual strategies are tuned and specific results are obtained (Bonabeau, 2002). Originally, ABM was applied to the description of molecular dynamics processes (Metropolis et al., 1953; Alder and Wainwright, 1957, 1959). They later became popular in population biology (Matsuda et al., 1992) for description of the movement of interacting reasonable agents (McCarthy, 1959; Minsky, 1961; Hewitt, 1970). Interaction refers to the exchange of messages and a change of the internal states of the agents. This approach is presently applied worldwide to the modeling of opinion dynamics, culture propa gation, language dynamics, crowd behavior, financial flows, traffic, and various physical processes (Castell ano et al., 2000). Mathematical Modeling of Sociobiological Processes The emergence of fundamentally new social enti ties such as the Internet and the availability of large databases affected the application of numerical analy sis to sociology, which involves mathematical models borrowed from physics. A common feature of such models is that they can be employed to investigate the emergence of order in an originally disordered system. By order we imply concordance and uniformity (e.g., making a group decision or formation of a hierarchy, common language, and culture.) By disorder we mean fragmentation and discordance. The key points in the formation of a common opinion, culture, or language are the interaction of agents and their desire to be sim ilar to each other. Here are some known social dynamics models that use notions of physics and ABM. The voter model (Clifford and Sudbury, 1973; Holley and Liggett, 1755) is known as an example of a nonequilibrium sto chastic process that is analytically soluble for any dimensionality. The model with the majority rule (Galam, 2002) simulates the social inertia rule, which states that people unwillingly accept changes not approved by a broad majority. Models of social impact (Latané, 1981; Nowak et al., 1990; Lewenstein et al., 1992) consider the dependence of interaction inten sity on distance (both physical distance and distance in the abstract space of opinions). The Sznajd model (SznaidWeron and Sznajd, 2000) was originally designed to describe the propagation of ideas; how ever, it followed the behavior pattern of the voter model. The Deffuant (Deffuant et al., 2000) and Heg selmann–Krause (Hegselmann and Krause, 2002)
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models consider continuous opinions, that is, opin ions with a continuum of variants described by real numbers. The Axelrod model (Axelrod, 1997) is based on two fundamental mechanisms of cultural assimila tion: the urge of agents to become similar to each other and the tendency for similar agents to interact more frequently. In the ABM methodology, a system is modeled with a set of autonomous objects, each of which takes decisions separately. These objects are termed agents. Each agent estimates a situation independently of oth ers and alters his state (takes a decision) on the base of a certain set of rules. Agent models can be recalcu lated, which allows a broader assessment of statistical parameters of processes to be modeled in comparison to classical mathematical models. Even those models in which agent behavior is determined by very simple rules can generate a complex pattern of system behav ior. Naturally, more intricate models, in which agents can evolve by acquiring new properties, getting rid of old ones, or changing their behavior, can generate even more sophisticated system behavior patterns. The Aggressor–Defender Model This model was proposed by Bonabeau (2002). It is an AB model of the development of a community whose members (agents, or players) move according to certain rules. In particular, strategies of “cowards” and “defenders” are recognized. In the first case, agent A (coward) tries to hide from agent B (aggressor) behind agent C (defender). In the second case, agent A plays a “defender,” or peacemaker, occupying a position between agents B and C. This is a classic example of model showing the emergent nature of social pro cesses: (1) simple rules of individual behavior can pro duce unexpected results at the system/community level, and (2) minor changes in the rules can cause sig nificant changes in system behavior. We reproduced and considerably extended Bon abeau’s aggressor–defender model. Two alternative strategies are proposed for cowards and defenders. We introduce a new parameter, enmity. It is the closest distance to which players approach each other. In case of zero enmity, their locations can match exactly, and in case of enmity = 2r, where r (radius) is the radius of the circle designating a player, the agents avoid overly ing each other. The speed of an agent is introduced. In addition, all parameters of the model can be set as ran dom values from a specified distribution. Presently, we use the beta distribution. The Extended Aggressor–Defender Model The Bonabeau model was extended by complicat ing the behavior of individual agents and introducing parameters related to the entire population. First, in addition to the two options (coward or defender), each agent in the extended model could choose a move
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Agent A
X'
Active agent X
Agent B
Fig. 1. Example of location chart for three agents.
ment strategy: normal or along the median/antime dian line. Let A and B be a pair of agents considered by agent X. Normal movement implies the following algorithm: It is checked whether movement normal to the AB line is in agreement with the chosen mode. The location desired in the defender mode should be between A and B, and agent X' should hide behind B in the coward mode. In the case of disagreement, X as a coward tries to hide behind B at the minimum acceptable distance; as a defender, X tries to occupy the position exactly between A and B. If X in Fig. 1 is a coward preferring normal movement, point X' does not satisfy the desire of the coward to hide from the aggressor, and X makes
his way to a point behind B. With the assumption that X is a defender, point X' meets his intention. The movement along median implies that the player tries to occupy the position exactly halfway between A and B in the defender mode or the position symmetri cal to aggressor A about defender B in the coward mode. We term the latter direction of movement along antimedian (see Fig. 2). Let X(t) be the radius vector of the active agent at time t; A(t) and B(t) are radius vectors of the aggressor and defender (defended), respectively. Then the movement of the agent is described, depending on the mode and strategy, by equations (1)–(9). PM (protectoralongmedian): The agent goes to the position exactly between the aggressor and defended, moving along the median in triangle ABX: PM ( t ) = X ( t ) + min ( speed,| ( A ( t )/2 + B ( t )/2 )| ) × normalize ( A ( t )/2 + B ( t )/2 ); (2) X ( t + 1 ) = PM ( t ). PN (protectoralongnormal): The agent checks whether the movement towards line AB along the shortest path (i.e., normal) corresponds to the defender mode. If not, he moves along the median. PN ( t ) = X ( t ) + min ( speed,normal ( B ( t ) – A ( t ) ) ) × normalize ( normal ( B ( t ) – A ( t ) ) );
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Fig. 2. Decision rules for choice of movement direction by agents. RUSSIAN JOURNAL OF GENETICS: APPLIED RESEARCH
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desired position. The discontentment of the ith agent is calculated by Equation (10):
X(t + 1) ⎧ ⎪ PN ( t ), if |B ( t ) – A ( t )| = max ( |B ( t ) – A ( t ) |, (4) ⎪ = ⎨ |PN ( t ) – A ( t )|,|PN ( t + 1 ) – A ( t )|; ⎪ ⎪ ⎩ PM ( t ) otherwise. CM (cowardalongmedian): The agent tries to occupy the position symmetrical to the aggressor about the defender (antimedian movement). CM ( t ) = X ( t ) + min { speed, |2B ( t ) – A ( t )| }
(5)
× normalize ( 2B ( t ) – A ( t ) ); X ( t + 1 ) = CM ( t ).
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CN (cowardalongnormal): The agent checks whether the movement towards line AB along the shortest path (i.e., normal) corresponds to the coward mode. If not, he moves along the antimedian. CN ( t ) = X ( t ) + min ( speed, | normal ( B ( t ) – A ( t ) )| ) (7) × normalize ( normal ( B ( t ) – A ( t ) ) ); (8) hide ( t ) = B ( t ) + enmity × normalize ( B ( t ) – A ( t ) ); X(t + 1) ⎧ ⎪ ANP ( t ), if |CN ( t ) – A ( t )| = max{|CN ( t ) – A ( t )|, ⎪ (9) ⎪ |CN ( t ) – B ( t )|,|B ( t ) – A ( t )}; = ⎨ ⎪ X ( t ) + min{speed, |hide ( t )|} ⎪ ⎪ ⎩ × normalize (hide ( t )) otherwise, where speed is agent’s speed; normalize is the vector normalization operator, normal is the normal operator (with regard to the vector direction), and enmity is the enmity parameter.} The following new parameters, speed and enmity, have been used in Equations (1)–(9). Enmity implies the minimum acceptable distance between two agents. In the case of zero enmity, their locations can match exactly, and in case of enmity = 2r, where r is the radius of the circle designating a player, the agents avoid over lying each other. Speed is the distance for which an agent can move in a unit of time. We also introduce an output parameter, discontent ment, as a measure of relative distance from the
discontentment i = dis tan ce i /maxDis tan ce , (10) where distance is the distance to desired coordinates, and maxDistance is the maximum distance among agents in the specified neighborhood. The discontentment of an agent can be calculated in two ways. (1) The maximum value maxDistance is calculated for all agents of the model. We will term this value absolute. (2) Alternatively, the maximum value maxDistance is calculated within a neighborhood of specified radius. This discontentment value is termed relative. In this method, the neighborhood radius is an additional parameter of the model. The mean discontentment is calculated by averag ing the discontentments of all agents:
discontentment av
⎛ = ⎜ ⎝
n
⎞
∑ discontentment ⎟⎠ i
n. (11)
1
The model also allows sporadic shifting of the scene. In the case of periodical boundary values, we assume that a square scene is surrounded by four sim ilar scenes with identical positions, states, and behav iors of agents on them; i.e., with the same arrangement of the population. If an agent leaves the main scene, say, through its left boundary, a similar agent comes to the scene through the right boundary. In this way, the population arrangement is preserved, and the infinity of the system is mimicked. In the opposite case, boundaries are assumed to be impenetrable for agents. THE MAIN BEHAVIOR PATTERNS IN THE AGGRESSOR–DEFENDER MODEL The large number of parameters made the model more sophisticated and flexible. Consider the system behavior at different values of main parameters: move ment strategy, enmity, and the proportion of cowards and defenders. Note that the extended and original models match in case of movement along the median line and zero enmity, and agents move identically in the Bonabeau and aggressor–defender models. With median/antimedian movement, the popula tion shows greater discontentment in earlier iterations than with normal movement. However, the former proves to be more adaptable, as it comes to a less dis contented steady state. With normal movement, dis contentment changes little, both within groups and in the entire population. The behavior patterns of sys tems with normal and median movement having equal numbers of cowards and defenders are compared in Fig. 3.
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Fig. 3. Behavior of the model with (a) movement along median; (b) movement along normal. Parameters: population size = 100, cowards/defenders ratio 1 : 1, radius = 3, enmity = 0, speed = 1. Averaged over 100 models. Xaxis, time (iterations); Yaxis, discontentment (averaged when for the entire community).
The introduction of enmity does not produce nota ble changes in populations in which all agents move normally (cf. Figs. 4, 5). However, a feature appears in populations with median movement of agents. At small numbers of cowards (
