Analyzing the Phase Transition Dynamics in Social Contagion
by an Agent-Based Inverse Simulation
Kazumoto Takayanagi1 and Setsuya Kurahashi1
1

University of Tsukuba

Abstract: This paper attempts to clarify the critical factors that determine the phase transition dynamics
occurring in social contagion by an agent-based inverse simulation equipped with genetic algorithm.
Previous studies have already shown that structural features such as network topology or social norm, affectsignificantly the macroscopic dynamics of complex social phenomena. We still need, however, aquantitative analysis of the decisive factors which could yield some particular phase transition in a society,e.g. the emergence of an epidemic. Thus we explore, by utilizing genetic algorithm, the vast parameter
space of an agent-based model that represents social contagion, so that we might identify the specific valuesof parameters resulting in phase transitions. With the results, we will seek to describe their implications for
either preventing or enhancing the contagion.

1. Introduction
The inquiry into the effects of network topology on thesocially spreading phenomena has always been a topic
attracting a large number of researchers since the lastcentury. Several previous papers have demonstrated thatsocial contagion such as the spread of obesity as well as
happiness is greatly influenced by the structure of networkin which individuals are interacting with others. Theseprevious studies, however, appear to assume a network
being its static status when they analyze the network
effects on the spreading.
Recently, some prior studies have suggested
investigating networks based on their dynamic,
evolutionary or adaptive conditions in order to realize thegenuine dynamics in a society. Especially, Sayama et al.
has proposed adaptive networks in which the co-evolutionof ‘dynamics on networks’ and ‘dynamics of networks’plays a key role. Such networks whose states andtopologies coevolve sequentially still need modelingrealistically and analyzing systematically.
Thus, we attempt first to model such an adaptive
network in the case of obesity spreading in order to really
represent the complex social phenomenon. Secondly, we
try to explore the model’s parameter space by means ofinverse simulation, so that we might identify the decisive______________________________________________
1University of Tsukuba
112-0012 3-29-1 Otsuka, Bunkyo, Tokyok.takayanagi1882@gmail.com

factors which could yield some particular phase transition
within the society.

2. Overview of the model
We will describe the outline of our model based on theODD protocol which is prevalent among researchers
exploiting agent-based modeling, except the ‘details’ part
we are just in the middle of elaborating.

2.1

PurposeThe present model aims at identifying the key factorsthat determine the dynamics of phase transition occurringin the social contagion. We begin by modeling the obesity
epidemic as a complex social system where agents andnetworks co-evolve through growth, adaptation andinteraction, so that we might represent realistically thedynamics of spreading phenomenon in a society. We will
ultimately rely on the inverse simulation equipped withgenetic algorithm in order to clarify the critical societal
conditions which lead to particular phase transitionss, e.g.
epidemic or outbreak of obesity like infectious diseases.

2.2 Entities and scales
In this model, agents correspond to individuals who
consume an amount of food at each time step. The quantityof food consumption is determined by the dual systemconstituted of ‘affective system’ and ‘deliberative system’
which are operating competitively within an individual
The agents are situated in a network which grows
gradually as new agents are added to the network andlinked with existing nodes, i.e. agents, at every time step.

The way the network develops is essentially regulated by
the “preferential attachment” algorithm: the embedded
nodes’ probability of obtaining a new link from added
nodes is basically proportional to their current degree in
the network. In the present model, however, thisprobability is fashioned differently based on two distinct
considerations which we will describe later in detail.
Each time step corresponds to a week in the model. The
simulation starts with a network constituted of 15 agents.
The overall process ends up by counting 500 agents. Whenit takes 160 time steps for a simulation to finish, it means
the simulation continues for approximately 3 years in themodel. As regards the space, no particular, concrete spacestructure is assumed except network topology the designof which we are very involved with.

2.3 Process overview and scheduling
Since agents interact with each other in the network,
their decision as to the amount of food consumption
should be affected by their conformity to the social norm
recognized through networked interactions. Thus, agentsvary their balancing on dual system adaptively, according
to the difference between their current weight status andperceived norm weight. Agents’ adaptive behavior
successively affects their weight values.
The oscillations of each agent’s weight are reflected in
the node’s ‘suitability’, measured by the comparison
between its current state and the objectively ideal one. As
the suitability is parameterized in the growing network
algorithm, its fluctuations affect the development ofnetwork topology, which in turn influences agents’interactions.
This co-evolution between agents’ adaptive behavior
and varying network structure might yield several trends
in the agents’ average weight. This model attempts,
ultimately, to identify the conditions that could generate
the phase transition in the trend of agents’ weight bymeans of the inverse simulation using genetic algorithm.

2.4 Design concepts
2.4.1
Basic principles
Our model presumes two fundamental theories, one ofwhich is concerning the agent’s decision process and theother is regarding the algorithm for generating networks.
An agent determines the amount of food to consume
based on the balancing of internal dual system which iscomposed of the affective system and the deliberativesystem. The former system has a present-bias favoring thegratification obtained from over-eating. The latter one is,contrarily, oriented for the decision that takes into
consideration the long-term consequences, e.g. health infuture. Depending on the balancing of these two systems,
an agent decides how much food to consume.
As regards modeling the network, we assume a wellknown algorithm for generating networks, i.e. Barabási–Albert model, characterized by ‘growth’ and ‘preferentialattachment’. Then, we have differently fashioned thealgorithm for preferential attachment based on two distinct
considerations. First, the degree of each node is multiplied
by a parameter η indicating agents’ “suitability” which ismeasured by the deviation from the ideal weight status: asan agent’s weight approaches the ideal weight, the valueof her suitability increases. Second, an amount δ is added
to each agent’s degree in order to realize the equalization
among agents through weakening the harshness ofpreferential attachment.
2.4.2
Emergence
The co-evolution of these two dynamics could
generate different kinds of weight trends within the society.
On the one hand, an upward trend might emerge when thenetwork effects enhance the obesogenic mechanism insocieties. On the other hand, a downward tendency could
come out under the network influences restraining theobesity. Thus, we might be able to observe a phasetransition where the weight trend changes suddenly, whichis regarded as an emergent phenomenon in itself. Ourmodel tries to demonstrate how these trends emerge by thesimulations and, ultimately, to identify the conditions
determining the occurrence of phase transition.
2.4.3
Adaptation
In our model, the way the network grows adapts itself
to the weight statuses of nodes linked in the network. In a‘basic’ scale-free network constituted of 𝑗 nodes, theprobability for an existing node 𝑖, whose degree is 𝑘𝑖 , toreceive an edge from newly added nodes is as follows :𝑘𝑖
𝑝𝑖 = 𝑛(1)∑𝑗=1 𝑘𝑗
As noted above, we modify this basic form by adding thesuitability η and equalizing factor δ.
𝜂 𝑖 𝑘𝑖 + 𝛿
𝑝𝑖 = 𝑛(2)∑𝑗=1(𝜂𝑗 𝑘𝑗 + 𝛿 )𝜂 𝑖 = 𝜉 ∙ 𝜙𝑖(3),where 𝜙𝑖 represents the ratio of 𝑖′𝑠 current weight tothe objectively ideal weight, and 𝜉 indicates thesignificance given to the suitability. With this algorithm,
the network develops while adapting to the oscillations ofnodes’ weight, which are in turn under the influence of
networked interactions.

2.4.4
Objectives
In order to clarify the requirements for different weight
trends to emerge, the present model defines temporarily
the objective functions for genetic algorithm as follows:
trend𝑢𝑝∶

trend𝑑𝑜𝑤𝑛 ∶trend𝑙𝑒𝑣𝑒𝑙∶𝑎𝑣𝑔𝑤𝑒𝑖𝑔ℎ𝑡 𝑎𝑡 𝑡𝑘 −𝑎𝑣𝑔𝑤𝑒𝑖𝑔ℎ𝑡 𝑎𝑡 𝑡0
𝑎𝑣𝑔𝑤𝑒𝑖𝑔ℎ𝑡 𝑎𝑡 𝑡0
𝑎𝑣𝑔𝑤𝑒𝑖𝑔ℎ𝑡 𝑎𝑡 𝑡𝑘 −𝑎𝑣𝑔𝑤𝑒𝑖𝑔ℎ𝑡 𝑎𝑡 𝑡0
𝑎𝑣𝑔𝑤𝑒𝑖𝑔ℎ𝑡 𝑎𝑡 𝑡0
𝑎𝑣𝑔𝑤𝑒𝑖𝑔ℎ𝑡 𝑎𝑡 𝑡𝑘 −𝑎𝑣𝑔𝑤𝑒𝑖𝑔ℎ𝑡 𝑎𝑡 𝑡0
𝑎𝑣𝑔𝑤𝑒𝑖𝑔ℎ𝑡 𝑎𝑡 𝑡0= 0.2= −0.1
= 0,where 𝑎𝑣𝑔𝑤𝑒𝑖𝑔ℎ𝑡 𝑎𝑡 𝑡𝑘 means the average weight ofagents at time step k, while 𝑎𝑣𝑔𝑤𝑒𝑖𝑔ℎ𝑡 𝑎𝑡 𝑡0 indicatesthe one at the beginning of the simulation. We are
exploring the parameter space with these distinct objectivefunctions by running the multiple simulations.
Once we have identified the parameter values for these
conditions separately, we will be able to shed light on themechanism leading to the phase transition by graphicallymapping out those parameter values.

3. Results of parameter sweep
We have run a series of simulations with our model inorder to verify the basic model behavior. By varying thevalues of a few typical parameters, we were able toconfirm some characteristics as to the dynamics of model.
3.1 Preferential attachment
Varying the parameters, ξ concerning the node’s
suitability and δ regarding the equalization of nodes, weobserved the agents’ average weight at the beginning aswell as at the end of simulations. We investigated, then,different kinds of preferential attachment with respect totheir effects on the agents’ weight by calculating the ratioof change in the average weight.
Fig.1 shows the plotting of the relations between ξ, δ,and the ratio of change in the average weight. As the blue
plane that visualizes the results of linear regression
analysis suggests, the effect of δ on the ratio of change inweight is statisticallysignificant, while ξ
within the simulated
range does not seem toinfluence the weight
decidedly. While δ
increases, the nodes’
equalization multiplies
the nodes
havingFig. 1 ξ(x), δ(y), ratio(z).
relatively many edges

through transforming
the degree distribution,
as Fig.2 illustrates. So,
we might be able toattribute the increased
average weight to theemergence of agentslinked with many‘friends’
inthenetwork.
Fig.2 degree distribution
ξ as well as δ
meaningfully boost theclustering coefficient of the network, as Fig.3 displays.
However, the clustering coefficient in itself is proved, byregression analysis, to have no influence on the ratio ofchange in average weight. Therefore, the effects of δ on
the weight may not be reducible to its effects on theclustering coefficient.
3.2 Adaptive behavior
By varying the λ indicating the threshold for beginningof agents’ adaptive behavior, and the μ representing thebreadth of change caused by the adaptation, we examinedthe effects of agents’ adaptation on the ratio of change in
the average weight. In Fig.4, the plotting of λ, μ, and theratio of change in average weight is demonstrated. The
blue plane resulting from linear regression analysis
suggests that increasing μ, i.e. the breadth of an agent’sadaptive alteration, significantly restrains the ratio ofincrement in the average weight. As for the threshold λ, itdoes not alter the agents’ average weight meaningfully.
We could infer from these results that the frequency ofadaptive behavior is less important than its range ofchange, at least in the case of agents linked with others insuch a growing network as simulated.
3.3 Direction of future research
We have obtained some implication as to the effects ofa few parameters on the agents’ weight by running
multiple simulations on our model. Especially, it ismeaningful that in a growing network, a kind of degree
distribution emerges
under some parametersettings, and it appearstoenhancetheincreasing trend inindividuals’ weight.
However, it is quite
evident that more
extensiveandFig.3 ξ(x), δ(y), clustering coef(z)thorough exploration

Fig. 44.

λ(x), μ(y), ratio(z).

of parameter space (i.e.
inversesimulation
withgeneticalgorithm) will beneeded in order toidentify accurately theparameters and theirrange that decidedly
influence the societal
phenomena.

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