library(igraph)
socnet <- make_graph(~ Esmeralda-+River-+Brook++Esmeralda)
plot(socnet, layout = layout_in_circle(socnet), edge.curved=0.4, vertex.size = 90)
Computational Social Science for Sustainability
Introduction
Sustainability will only be achieved if people and institutions behave sustainably. This course teaches techniques for predicting the spread of sustainable behaviors, in human social networks. Sustainability, and other pressing issues, needs a rigorous theory of how sustainable behavior spreads. This is apparent since there has been scant real progress towards limiting biodiversity loss, inequality, global warming, and other sustainability targets such as the UN Sustainable Development Goals. This suggests people, institutions, or organizations continue to behave unsustainably, and begs the question: what will it take to get sustainable behaviors to become widely practiced in different societies? Behaviors are transmitted from person to person through social learning, be it through instruction or observation. Who learns from or observes whom is structured based on who we know, who we live near, who we work with, or whose messages we read on the internet.
In this course we want to build on progress towards an empirically-motivated system for decisionmaking when promoting sustainable behavior, like Elinor Ostrom’s socio-ecological systems framework (Ostrom1990?) that identified the following design principles. These principles are written below as re-formulated by Cox, Arnold, and Tomás (2010) who evaluated their effectiveness after two more decades of use:
- Well-defined boundaries
- Congruence between appropriation and provision rules and local conditions
- Fair, flexible collective-choice arrangements
- Monitoring and monitors must be reliable and inclusive of local communities
- Graduated sanctions: penalties for non-compliance with sustainable practices should start small and increase with further non-compliance
- Access to responsive, low-cost conflict-resolution mechanisms
- Minimum recognition of rights of all stakeholders to recognize local ownership of local common-pool resources
- Successful common-pool resource management requires nested institutions so those lower in the hierarchy can achieve power parity with those higher by having greater numbers of decision-makers in those foundational decision making groups
These design principles were inferred and developed by reviewing numerous policy and economic interventions to promote sustainable use of common-pool resources in specific socio-economic systems, both before 1990 and up to 2010 with the update from Cox, Arnold, and Tomás (2010). Among other notable achievements, the design principles helped elevate indigenous and traditional sustainable ecosystem management practices (Nalau et al. 2018), and community-led adaptation more generally (McNamara et al. 2020). For example, vibrant coastal mangrove forests have long been known to support fish populations and protect against storm surges (Pearson, McNamara, and Nunn 2020). To complement these design principles, and other lessons from sustainability practitioners, we want to develop simplified mechanistic models of human sustainability behavior change to identify rigorous principles of information transmission, cooperation, and political organization.
A structure for developing models of social behavior for sustainability
In this course we learn In the coming sections we develop several different models of social behavior for this purpose. The models are always a more specific version of some more general or abstract model. A model here is a simplified representation of the world that we can use for science and decision making. One motivation for this course, in fact, is to provide a more scientific way of thinking about social responses to sustainable policy and other interventions, whether or not that thinking is aided by computational, formal, or other modeling. We start from a very abstract model with few details specified so we can identify common elements of social science models and consolidate our theoretical knowledge (Figure 1). This is practically useful for model development and analysis since we can theoretically compartmentalize the behavioral and social assumptions being made in the model. We know where to change assumptions to generate new intervention strategies and generate new scientific hypotheses about social behavior.
Abstract model workflow description.
- Create a population represented as \(N\) nodes in a social network.
- At each time step, imagine that each individual in a population “selects” or is otherwise paired with another individual, their interaction partner. Each individual may also take independent action. For example, the Legacy-Adaptive model of the diffusion of adaptations in the next section specifies that when one person doing a Legacy behavior observes another doing the Adaptive behavior, the one doing Legacy will switch to Adaptive with probability \(\alpha\). Over the duration of the course, we will consider interactions that consist of learning/adopting behaviors, deciding whether to cooperate with others for collective adaptation, and influencing opinions.
- This process is repeated over many time steps where each individual does something or interacts with another individual.
- The conditions for stopping the model, the stopping conditions, must be specified. The simplest stopping condition is to run a simulation for a certain number of time steps. Other options include stopping when the population reaches some form of equilibrium, for example, the number of people doing an adaptive behavior stays relatively constant over several time steps.
- We observe how certain outcome variables of interest, such as how many people adopt some behavior over time or how long it takes for some fraction of people to adopt an adaptive behavior.
- In full analyses of these models, we analyze these outcome variables across different hypothetical contexts, such as changing how “viral” (i.e. how likely to be adopted) adaptive behaviors are.
To use this abstract social model, proceed box by box and decide how the different parts work. To begin, decide what assumptions to make about social network structure, whether there are payoffs, how payoffs may change over time due to environmental shifts, etc. Then one must specify how individuals are paired together to interact, which in the real world could be highly random or constrained by one’s social network. Then all paired individuals interact according to some interaction rule. In the Legacy-Adaptive model introduced below, for example, individuals adopt an adaptive behavior with a certain probability if they are exposed to it.
Developing new models of social behavior, then, amounts to specifying social structure, how people get paired up and interact, and how people decide to update their behavior. The first more specific class of social models we introduce assume that there is an adaptive behavior that can be learned when it is observed, with some probability. These are diffusion of adaptation models. Next, cooperation and coordination models. We close with opinion dynamics models.
Within diffusion of adaptation models, we will start with formal models that can give us functions for diffusion that we can easily write down and visually inspect. The cost of this exactness are unrealistic assumptions, including that population sizes approach infinity and partner choice is unconstrained. We will then show why stochastic (i.e., randomized) computational models are essential to sufficiently represent the aleatoric dance of social behavior in order to understand how social outcomes may vary due to chance events, which in general is sensitive to initial conditions. We will continue on to consider more adaptive forms of social learning that blind copying of a neighbor. Then we will analyze models of the cultural evolution of cooperation and coordination, focused on the case of sustainable management of common-pool resources such as groundwater, forests, or fisheries. Finally in this course we will study opinion dynamics models where interactions take the form of persuasion, though we will abstract away nearly all features of rhetoric and focus only on how the opinions of interaction partners affect changes in each partner’s opinions or beliefs.
A primer on social networks
Social networks are at once intuitive, perhaps partly due to their prominence in pop culture, and also technically complex and theoretically deep. Social networks have nodes (aka vertices) that represent people, usually drawn as circles but any shape is ok. Relationships between people are represented by edges, drawn as lines. If the lines have arrowheads, the arrow will point in the direction of information flow, for example from a teacher to her student. If there is no arrow, or two arrows pointing different directions, this indicates a symmetric relationship where information could flow in either direction.
We need to know some basic things about social networks because all models make assumptions about them. First, a social network is a theoretical tool, a “cognitive gadget” , that represents relationships between people. Social networks are instances of graphs, an abstract mathematical structure for representing any type of entities and relationships between them. Graph nodes (aka vertices) represent the entities and edges represent the relationships between nodes. Edges are drawn as lines with or without arrows to indicate the flow of information, with information flowing to a learner, or observer more generally, from a teacher or demonstrator (Figure Figure 2). Lines with arrows are called directed edges. For a given pair of individuals, each may sometimes be teacher or learner. This case may be represented by two directed edges pointing opposite directions or a single undirected edge, i.e., a line with no arrowheads (Figure Figure 3).
Creating and measuring social networks
To start, consider the following fictional advice network at a startup, Sustainability Intervention Solutions. There are only three team members right now with different roles. Esmeralda, the CEO, advises River and Brook. Esmeralda and Brook left their previous jobs at Hooli to start the company, so Brook also advises Esmeralda. River is the newest member of the team, an expert programmer with little business or sustainability experience, so Esmeralda mentors River. Brook is knowledgable about business and sustainability, but just learning to program, so River is helping Brook develop her skills. We can use a social network to represent this social structure where arrows point from Esmeralda to River and Brook; from River to Brook; and from Brook to Esmeralda.
After a year the team have succeeded hiring clients and need another programmer to build an interface to deliver their custom relational databases on corporate sustainability decisions in response to different legislation at the national, state, and local levels. They hire Arun who has some programming expertise and experience in sustainability because he took Matt Turner’s Computational Social Science for Sustainability at Stanford. Still, he has a lot to learn and needs River’s and Esmeralda’s advice.
After a while it is clear they need a computational scientist. They hire Marco, who reports directly to River, whom he asks for help learning about sustainability. Arun asks for programming help from Marco as does Brook. Marco gets input from River and Esmeralda on programming and sustainability.
Regular networks
Regular networks are those ones for which all nodes have the same degree. A regular lattice with degree \(k\) is useful for constructing networks with the small-world property. We can use code from the socmod.R script in the root directory of the CSS4S-ProblemSets repo to create one with ten nodes, each connected to two neighbors:
source("~/workspace/CSS4S-ProblemSets/socmod.R")
Attaching package: 'assertthat'The following object is masked from 'package:tibble':
has_namelatnet <- regular_lattice(10, 2)
plot(latnet, layout=layout_in_circle(latnet), vertex.size=30)
And here is one more example, but with 20 nodes connected to six neighbors each:
source("~/workspace/CSS4S-ProblemSets/socmod.R")
latnet <- regular_lattice(20, 6)
plot(latnet, layout=layout_in_circle(latnet), vertex.size=10)
Random networks
Erdős-Rényi random networks: \(G(N,M)\) and \(G(N, p)\)
Analysis of the Medici family
In social networks, the entities are people and the relationships could be friendships, business partnerships, or, in the case of our first social network example, marriages between powerful families in Florence during the 1400s that elevated the the Medici clan developing into what was essentially one of the world’s first political parties. This follows Jackson’s introductory example in his textbook starting on p. 4 (Jackson 2008). Studying the Medici’s social network gives us our first look at the structure of social influence. During that period, the Medici family rose in wealth and political power over the lifetime of the family patriarch, Cosimo. Starting from a relatively weak position overall in the Florentine oligarchy of the time, Cosimo arranged marriages with other powerful families to subtly amplify the Medici control of politics and markets in Florence (Figure Figure 6; source: https://cran.r-project.org/web/packages/netrankr/vignettes/use_case.html). In the Florentine network, the edges are undirected, which makes sense since marriage takes two families and is therefore not directed. Note then we have no knowledge from which family came husband and wife. Note too that there could be multiple marriages between two families, but this information is not contained in this graph.
In an undirected graph, the neighborhood of an individual \(i\) (or family, organization, institution, or other social entity) can be written \(n_i\) and is a set of all the individuals who share an edge with \(i\). The set of the Medici’s network neighbors then is written mathematically as
\[ n_{\text{Medici}} = \{\text{Salviati},~\text{Acciaiuol},~\text{Barbadori},~\text{Ridolfi},~\text{Tornabuon},~\text{Albizzi}\} \] The degree of node \(i\) is the number of neighbors it has, written \(k_i\), so \(k_\text{Medici} = 6\). In the case of directed networks it is common to define in- and out-neighborhoods, referring to arrow direction, which for us correspond to teacher- and learner-neighborhoods.
library(netrankr) # Has Florentine network data we load next line.
data("florentine_m")
# Delete Pucci family (isolated)
florentine_m <- delete_vertices(florentine_m, which(degree(florentine_m) == 0))
# plot the graph
set.seed(111)
plot(florentine_m,
vertex.label.cex = 1,
vertex.size = 20,
vertex.label.color = "black",
vertex.color = NODE_COLOR,
vertex.frame.color = NODE_COLOR)
We can make a bar plot of the degrees for each family:
deglist <- degree(florentine_m)
# Create a tbl of degrees for each family then barplot using ggplot2.
tibble(Family=names(deglist), k=as.vector(deglist)) %>%
mutate(Family=fct_reorder(as_factor(Family), k)) %>%
ggplot(aes(x=k, y=Family)) + geom_bar(stat="identity", fill=NETWORK_COLOR) +
theme_classic(base_size = 18)
The degree distribution is the fraction of nodes having degree \(k\), written \(P(k)\). Since the Medici family is the only one with a degree of six, and there are fifteen families, \(P(6)=\frac{1}{15}\approx 0.066\), which we can check like so:
degdist <- degree_distribution(florentine_m)
medici_k <- 6
# Degree distribution starts at 0 and goes to maximum k; index 1, then
# correlates to k=0 and need to add 1 to the Medici index.
degdist[[medici_k + 1]][1] 0.06666667We plot the degree distribution see that the modal \(k\) is 3 and that the distribution is somewhat more concentrated in \(k\leq3\) compared to \(k > 3\), using the built-in barplot function to plot a simple histogram of \(P(k)\) using base R:
degdist_vec <- as.vector(degdist)
library(forcats)
k_vec <- as_factor(0:6)
ggplot(tibble(k = k_vec, freq = degdist_vec), aes(x=k_vec, y=degdist_vec)) +
geom_bar(stat="identity", fill=NETWORK_COLOR) + xlab("Degree, k") +
ylab("Frequency, P(k)") + theme_classic(base_size=22) 
Other resources
I found one tutorial to follow up on while preparing this, so I created this place to put this tutorial and other resources that may be useful later on.
Diffusion of adaptations
Adaptations diffuse through communication or observation from person to person when two things happen. First, a person not doing some adaptive behavior (driving an electric vehicle, using heat pumps in their home for heating and cooling, switching to no-till farming methods, etc.) encounters someone who knows or does an adaptive behavior. Second, the person not doing the adaptive behavior must successfully acquire the knowledge and desire necessary to start doing the behavior themselves. In this section we introduce four models of this process.
The first two assume that an adaptive behavior is randomly adopted by someone performing a legacy behavior with probability \(\alpha\), the adoption rate. One of these assumes that anyone doing the adaptive behavior will continue doing it forever, the Legacy-Adaptive model (Figure Figure 9). The next adds one more assumption, that people can stop doing a behavior, i.e., they can randomly _drop_the behavior, which happens for anyone doing \(A\) with probability \(\delta\) (Figure Figure 10).
The second two assume that learning is partly probabilistic and partly adaptive, where individuals can do some computation about who or what behavior may be best to learn. These are social learning models of transmission. The first of these is a conformist-biased learning model, where individuals are assumed to be more likely to learn a behavior if more people are doing it. The second is a success-biased learning model, where individuals are more likely to learn from individuals if an individual is doing relatively well compared to other neighbors. Success-biased learning is the only one of these where the benefit of doing a behavior matters, since there must be a way to acquire “payoffs”, which is a modeling term of art representing amount of benefit, usually in unspecified units.
LA and LAL compartmental “contagion” models
Contagion models like the LA and LAL models assume that when individuals encounter one another there is some probability that some individual-level attribute, in this case whether the individual \(i\) does \(A\). These models require relatively few parameters and variables, but ignore adaptive strategies people use for acquiring new information (Table 1).
Namely, these model contain two dynamic outcome variables \(L_t\) and \(A_t\) if time is discrete (\(L(T)\) and \(A(t)\) if time is continuous; see Note 1) the number of individuals performing \(L\) and the number performing \(A\). These outcomes are dynamic, so \(t\) represents time, which is discrete in our main treatment of these models, meaning that points in time can be indexed by integers, i.e., time steps are indexed by \(t_i\) where \(i \in \{1, 2, 3,\ldots\}\), e.g., \(t \in \{0, 3, 6, 9, 12, 15, \ldots\}\) could be discrete time steps for a model with \(t\) in months meant to represent some process for which we have data every three months. If time is continuous then \(t\in[0,\infty)\). Although time can continue on to infinity, we usually specify that models will stop at some time step or when some condition is met. One of the most important conditions is whether the population has fixated, meaning all agents do the same behavior. Finally, there are two parameters that determine the probability that individuals change their behavioral state (from doing \(L\) to doing \(A\) or vice-versa). In both the LA and LAL models, the probability \(A\) is adopted when an \(L\) meets an \(A\) is the adoption rate, \(\alpha\). In the LAL model, at every time step, every agent doing \(A\) might drop the behavior, i.e., stop doing the behavior, reverting to do \(L\) for the next time step. The probability one doing \(A\) reverts to \(L\) is \(\delta\), the drop rate. So, the LA and LAL models are in fact the same model where we can set \(\delta = 0\) to recover the LA model from the LAL model.
| Symbol | Description | Values |
|---|---|---|
| \(L,L_t,L(t)\) | Number of individuals doing Legacy behavior at time \(t\); like \(A\) serves as a noun for the behavior itself. | \(L_t \in \{0,\ldots,N\}\); \(L(t) \in [0, \infty)\) |
| \(A,A_t,A(t)\) | Number of individuals doing Adaptive behavior at time \(t\), but used interchangeably for the behavior itself, e.g., “an individual doing \(A\)”. | \(A_t \in \{0,\ldots,N\}\), \(A(t) \in [0, \infty)\) |
| \(t\) | Time, which could be discrete or continuous. We write \(A\) and \(L\) as a function of time, i.e., \(A_t\) if \(t\) is discrete or \(A(t)\) if continuous | \(\{0,1,\ldots\}\) or \([0,\infty)\) |
| \(\alpha\) | Adoption rate, i.e., the probability that an individual doing \(L\) adopts \(A\) if their interaction partner is doing \(A\). | \([0, 1]\) |
| \(\delta\) | Drop rate, i.e., the probability that an individual doing \(A\) stops doing \(A\) at time \(t\) | \([0, 1]\) |
| \(T\) | Fixation or stopping time. If the modeler specifies the model should stop when all individuals perform the same behavior, it is the fixation time. If the modeler specifies a certain number of steps for the model or there is some condition that causes the model to stop before fixation, it is the stopping time. | \(\{0,1,\ldots\}\) or \([0,\infty)\) |
Legacy-Adaptive (LA) model
In class we derived the recursion for the legacy-adaptive model: \[ A_{t+1} = A_{t} + \alpha A_t\left(1 - \frac{A_t}{N}\right). \] Once an individual adopts the adaptive behavior in this model, they continue doing that behavior forever (Figure 9). Below we show how to write a function to implement this recursion to calculate a time series and compare the results for two different adoption rates, \(\alpha\).
la_recursion <- function(N, A0, alpha, tmax) {
A_return <- numeric(N+1) # Create output time series vector.
# Set current A_t to be A_0
At <- A0
A_return[1] <- At
# Iterate
for (t in 1:tmax) {
# Anext is code for A_{t+1}
Anext <- At + (alpha * At * (1 - (At/N)))
A_return[t+1] <- Anext
At <- Anext
}
return (A_return)
}N <- 100
A0 <- 5
alpha_low <- 0.05
tmax <- 200
tvec <- 0:tmax
Avec_low_alpha <- la_recursion(N, A0, alpha_low, tmax)
alpha_high <- 0.8
Avec_high_alpha <- la_recursion(N, A0, alpha_high, tmax)
adopt_tbl <- tibble(timestep = rep(tvec, 2),
alpha =
as_factor(c(rep(alpha_low, length(tvec)),
rep(alpha_high, length(tvec)))),
A = c(Avec_low_alpha, Avec_high_alpha))
ggplot(adopt_tbl, aes(x=timestep, y=A, color=alpha)) +
geom_line() + theme_classic()
Discrete change in recursion, continuous (infinitesimal) change in differential equations
In the recursion LA model, the change in \(A\), \(\Delta A\) from time step \(t\) to time step \(t+1\) is \[ \Delta A = A_{t+1} - A_t = \alpha N \Pr(L,A) \tag{1}\] where \(\Pr(L,A) = \frac{L}{N}\frac{A}{N}\) is the probability an individual doing \(L\) and another doing \(A\) interact under the well-mixed assumption.
\(\cdot\quad\) The well-mixed assumption says that everyone in a population is equally likely to interact with one another. This means the probability of someone doing \(L\) encountering someone else doing \(A\) is the same for all people doing \(L\) (or vice-versa).
Plug this probability in to Equation 1 and cancel an \(N\) on top and bottom to get \[ \Delta A = \frac{\alpha}{N} L A = \alpha' L A, \tag{2}\] where \(\alpha' = \alpha / N\). Because we use \(t\) to label time steps and assumed that time steps are separated by \(\Delta t = 1\), we can multiply the right side of Equation 2 by 1 in the form of \(\Delta t\): \[ \Delta A = \alpha' LA \Delta t \tag{3}\]
\(\cdot\quad\)The general probability that a person doing behavior \(b\) interacts with a person doing \(b'\) (read “b prime”) \(\Pr(b,b')\), i.e. \(\Pr(L,A)\) for legacy-adaptive, \(\Pr(L, L)\) for legacy-legacy, etc.
Calculus, the foundation of differential equations, is the mathematics of rates of change, which was achieved through the following conceptual leap. In terms of our LA model, calculus works by assuming that \(\Delta t\) becomes “infinitesimally small”, written \(\Delta t \to 0\). Before \(\Delta t\) gets all the way to zero, we calculate \(\Delta A\) over this infinitesimally small \(\Delta t\). This infinitesimal \(\Delta t\) is written \(dt\), and the similarly infinitesimal \(\Delta A\) is written \(dA\). When \(\Delta \to 0\), then, we replace \(\Delta\) with \(d\) and Equation 3 becomes \[ dA = \alpha' L A dt \] which is more commonly written \[ \frac{dA}{dt} = \alpha' L A dt \tag{4}\] representing the flow in of people previously doing \(L\), with the coupled differential equation for \(L\) differing only in sign of the right hand side, representing the flow of people away from doing \(L\): \[ \frac{dL}{dt} = - \alpha' L A dt. \]
These equations can be solved by using the same substitution we used in the recursion to get a formula for \(A_{t+1}\) that depended only on \(A_t\), namely that \(L = N - A\), so Equation 4 becomes
\[ \frac{dA}{dt} = \alpha' A(N - A). \tag{5}\]
Calculus tells us that we can rearrange the differnetials like they were any other variable. What we want now is to arrange all \(A\) terms to one side of the equation, and all \(t\) terms to the other because this allows us to integrate the equation. We want to integrate the equation because this is the calculus procedure that converts rates of change of a variable to the accumulation of that variable. Rearranage Equation 5 to get
\[ \frac{dA}{A(N - A)} = \alpha' dt. \]
We integrate both sides of this equation, which is written
\[ \int \frac{dA}{A(N - A)} = \int \alpha' dt = \alpha't + C \tag{6}\]
where \(C\) called a constant of integration that must be determined by initial conditions for this problem, namely the value \(A(t=0)\) and we used the fact that \(\int a dx = ax + C\) where \(a\) is any constant and \(x\) is any dynamic variable. The integral on the left hand side can be integrated formally, i.e., without using a computer, but it’s significantly more complicated. After integrating the left hand side of Equation 6 it is still necessary to solve for \(A\) in terms of \(t\). See Note 1 for these details.
After all that math we obtain the logistic equation,
\[ A(t) = \frac{N}{1 + \frac{N}{A_0}e^{-\alpha N t}}, \tag{7}\]
where \(A_0 = A(t = 0)\) is the initial condition.
Britton (2003) explains in Ch. 3, p. 87, this is the same equation as single-species population dynamics in an ecosystem with a carrying capacity, \(K\). In this case, \(K = N\). See Britton Ch. 3 for even more detail on the family of compartmental susceptible-infected-recovered models that we have adapted for sustainable adaptations in the context of the field of mathematical biology.
Note 1: Integration and solution of \(A(t)\) from LA differential equations model
Perform partial fraction decomposition on \(\frac{1}{A (N - A)}\): \[ \frac{1}{A (N - A)} = \frac{1}{N} \left( \frac{1}{A} + \frac{1}{N - A} \right). \]
Integrate both sides: \[ \int \frac{1}{N} \left( \frac{1}{A} + \frac{1}{N - A} \right) dA = \int \alpha \, dt. \]
This yields: \[ \frac{1}{N} \left( \ln|A| - \ln|N - A| \right) = \alpha t + C, \] where\(C\)is the integration constant.
Simplify: \[ \ln\left(\frac{A}{N - A}\right) = \alpha N t + C', \] where \(C' = N C\).
Exponentiate to solve for\(A\): \[ \frac{A}{N - A} = e^{\alpha N t + C'}. \]
Rearranging gives: \[ A = \frac{N}{1 + e^{-C' - \alpha N t}}. \]
Finally, rewrite \(e^{-C'}\) as a constant \(K\): \[ A = \frac{N}{1 + K e^{-\alpha N t}}, \tag{8}\]
where \(K\) depends on the initial conditions.
This is the solution for \(A(t)\), and \(L(t)\) can be recovered using \(L(t) = N - A(t)\).
To find \(K\) for some initial condition, set \(t=0\), call \(A_0 = A(t=0)\), and solve for \(K\). Note that when \(t=0\), the denominator in Equation Equation 8 becomes \(1 + K\) since \(e^0 = 1\). Then we have
\[ A(0) = \frac{N}{1 + K} \]
which we can multiply by \(1 + K\) on both sides and rearrange to get
\[ K = \frac{N}{A_0}. \]
Legacy-Adaptive-Legacy (LAL) model
When considering the total change in \(A,\) written \(\Delta A,\) following a discrete time step, \(\Delta t\), from time “now” to time “next”, i.e., from \(t\) to \(t+1\), we can drop the indices since all indices are \(t\), i.e.,
\[ \Delta A = \alpha A (1 - \frac{A}{N}) - \delta A. \tag{9}\]
In this model, two new things are possible that were not possible before. First, the adaptation may completely fail to spread. If more people end up dropping the behavior than adopting it early on, there will be no one doing \(A\) that someone doing \(L\) could copy. Second, in general, everyone will not eventually adopt \(A\), at least not all at once, unlike in the LA model. In the LAL model there is an equilibrium \(A\), \(\hat A\), that the system reaches with enough time.
Will the adaptation spread?
We want to know if the adaptation will diffuse throughout the population in the LA model given just the adoption rate \(\alpha\) and drop rate \(\delta\). Formally, the adaptation diffuses if \(\Delta A > 0\). We can approximate \(\Delta A\) just after its introduction to a population that is mostly doing the legacy behavior and obtain a useful heuristic value to tell use if it will diffuse. When the adaptive behavior is only just being introduced to a relatively large population, we we say that \(A << L,N\), and so \(1 - \frac{A}{N} \approx 0\), so that Equation 9 becomes
\[ \Delta A = \alpha A - \delta A > 0. \]
Dividing out \(A\), rearranging, and dividing by \(\delta\), we find that the adaptation diffuses if
\[ R_0 = \frac{\alpha}{\delta} > 1, \]
where \(R_0\) is a rather famous measure called the basic reproduction number in epidemiology that predicts whether a new disease will become endemic or not. For us it predicts whether a sustainable adaptation will diffuse and persist in a population.
If an adaptation persists, what will the long-term adoption level be?
To know how many people will perform the adaptive behavior after the diffusion of that behavior has stabilized, we need to calculate its equilibrium value. Assuming \(R_0 > 1\), the equilibrium number of adopters, \(\hat A\), is defined as the value of \(A\) for which \(\Delta A = 0\), i.e., when \(A\) remains constant over time:
\[ \Delta A = 0 = \alpha \hat A\left(1 - \frac{\hat A}{N}\right) - \delta \hat A. \]
This can be re-arranged to find that
\[ \frac{\hat A}{N} = 1 - \frac{\delta}{\alpha}. \]
See MSB pp. 96-97 for the steps and Figure 4.9 on p. 97 for a plot of \(\Delta A\) versus \(A\) that illustrates and explains how to find \(\hat A\) graphically (MSB uses the epidemiological framing, which means \(I\) replaces \(A\).
Agent-based modeling
Together, these define an agent-based model:
- Environmental and ecological factors, including the payoff from doing Legacy and Adaptive behaviors, which could change over time.
- The agents, which are simulated people in the case of human social science.
- Social processes:
- Partner selection
- Dyadic interaction
- Non-social, individual-level behavior change (e.g. in LAL agents drop the adaptation with probability \(\delta\))
These steps were implicitly present in the recursion and differential equation versions of the LA and LAL models, as reviewed in the Introduction in these notes. In agent-based models we explicitly define code functions that do each of these steps, if the model calls for it.
To do agent-based modeling, we will use code currently located in the socmod.R file in the root directory of the ProblemSets project repository. This approach combines object-oriented programming, characterized by explicit representations of classes of entities that define attributes and capacities of the entities, with functional programming, characterized by providing functions as arguments to other functions for describing the processes that occur within and between agents and their environment.
We define an AgentBasedModel class, which is a structured way to define and store information about agents and their social networks and different model parameters specified by the params attribute of AgentBasedModel.
Let’s initialize and run a simple toy AgentBasedModel to see the different parts so they are more familiar when we use them to make the agent-based LA model in Problem Set 2. We will create agents who switch between Legacy and Adaptive behaviors randomly at each time step, initialized with half of the agents doing the adaptive behavior (and therefore half doing legacy).
With the model defined, we can now run the model. We must first define how partner selection and interaction work in functions called partner_selection and interaction. For this model there is no model_step. We then pass these functions to the run function, plus the desired number of time steps. Note that as arguments—we do not call the functions ourselves.
When we pass functions as arguments to other functions like this we are using the functional programming stye. Functions that take other functions as arguments are called higher-order functions.
run, therefore, is a higher-order function.# Source the agent-based modeling code from socmod.R in root ProblemSets
# directory. Since this is in the CSS4S/notes directory for the course website
# I need to use a different path. Use `socmod_path` for the problem set.
socmod_path <- "~/Desktop/Demo 2-3-2025/PSet2_1-3/socmod.R" # <-- comment this out for your problem set
# socmod_path <- "socmod.R" ### <-- Use this one for your Problem Set
source(socmod_path)
# Define our fake model where nothing happens except agents randomly match up, but then just adopt
# either behavior randomly.
N <- 10
# Create a list of ten initial behaviors, five of each in
# random order to be used to create
behaviors <-
sample(
c(rep("Legacy", as.integer(N/2)),
rep("Adaptive", as.integer(N/2)))
)
agents <- purrr:::map(behaviors, \(b) { Agent$new(b) })
model <- AgentBasedModel$new(agents = agents,
network = regular_lattice(N, 4),
switch_prob = 0.1)
# Define random partner selection from agent's neighbors.
partner_selection <- function(agent, model) {
partner <- model$agents[[sample(agent$neighbors, 1)]]
return (partner)
}
interaction <- function(agent1, agent2, model) {
# Not really an "interaction" for this toy model, just random behavior adoption.
agent1$prev_behavior <- agent1$curr_behavior
if (runif(1) < model$params$switch_prob) {
b_new <- sample(c("Legacy", "Adaptive"), 1)
agent1$curr_behavior <- b_new
}
}
out <- run(model, 20, partner_selection, interaction)$output
ggplot(out, aes(x=t, y=A)) + geom_line() + scale_x_continuous(breaks=0:22) + theme_classic()
I hypothesize that if switch_prob is greater, the adoption curve will be more variable over time. To test this hypothesis, let’s create another time series, but with an even higher switch_prob and compare the two. First, we’ll add a switch_prob column to our output dataframe from before:
# Rename the output from before with a switch_prob of 0.1...
out_p1 <- out
# ...and add a `switch_prob` column as a factor since we'll use it to color our graphs later.
out_p1$switch_prob <- as.factor(model$params$switch_prob)
# Now create a new set of agents initialized with the same randomized behaviors.
agents <- as.vector(purrr::map(behaviors, \(b) Agent$new(b)))
# Use them to create a new model with a higher switch_prob.
model_p9 <- AgentBasedModel$new(agents = agents, network = regular_lattice(N, 4),
switch_prob = 0.9)
# partner_selection and interaction work as before, and we still use the default model_step
out_p9 <- run(model_p9, 20, partner_selection, interaction)$output
# Add a `switch_prob` column to out_p9...
out_p9$switch_prob <- as.factor(0.9)
# ...and concatenate the two tibbles:
out_full <- rbind(out_p1, out_p9)
ggplot(out_full, aes(x=t, y=A, color=switch_prob)) + geom_line() + theme_classic()
Maybe it is more variable, but it is hard to tell from comparing just two outputs. We would need to do more processing of the outputs, perhaps coming up with a measure of variability to compare across different swtich_prob settings. Furthermore we need more trials to reliably infer whether greater switch_prob indeed results in greater variability in resulting time series.
Soon we will use the run_trials function to run several trials for several parameter settings
Cooperation and coordination
Opinion dynamics
References
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