\documentclass[twocolumn,showpacs,prl,floatfix,aps]{revtex4}
\usepackage{amsmath}
%\usepackage{hyperref}
\usepackage{graphicx}
\begin{document}
\title{Dynamics of Social Diversity}
\author{E.~Ben-Naim}\email{ebn@lanl.gov}
\author{S.~Redner}\email{redner@cnls.lanl.gov}
\altaffiliation{Permanent address:
Department of Physics, Boston University, Boston, Massachusetts,
02215 USA} \affiliation{Theoretical Division and Center for
Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New
Mexico 87545}
\begin{abstract}
We introduce and solve analytically a model for the development of
disparate social classes in a competitive population.  Individuals
advance their fitness by competing against those in lower classes, and
in parallel, individuals decline due to inactivity.  We find a phase
transition from a homogeneous, single-class society to a hierarchical,
multi-class society.  In the former case, the population is uniformly
poor.  In the latter case, a finite-fraction condensate that consists
of a static lower class remains.  The rest of the population consists
of an upwardly-mobile middle class, on top of which lies a tiny upper
class in the form of a thin boundary layer.
\end{abstract}
\pacs{87.23.Ge, 02.50.Ey, 05.40.-a, 89.65.Ef} \maketitle

There are many connections between social dynamics and physical
processes. For example, in urban dynamics \cite{city},
migration-driven population development has analogies with
coarsening \cite{spinodal}.  Rumor propagation and formation of
social networks are closely related to percolation
\cite{wasserman}. An appealing route for modeling social phenomena
is to identify individuals in a society as particles in a physical
system, {\it i.e.}, an agent-based description \cite{vfh,ckfl}.
This interdisciplinary approach has helped identify underlying
mechanisms for fundamental social phenomena and has led to
quantitative predictions \cite{ww,wdan,bkr,smo,ez}.

In this spirit, we seek to understand the formation of the social
hierarchies that are ubiquitously observed in animal populations
\cite{W1,A} and in human societies \cite{C}.  We introduce a
minimalist agent-based model in which competition is the underlying
mechanism for social differentiation.  Using concepts and methods from
statistical and nonlinear physics, such as scaling and asymptotic
analysis, we find a rich phenomenology for social diversity.  As a
function of the competition rate, the population undergoes a phase
transition from a homogeneous, single-class society to a hierarchical,
multi-class society.  In the latter phase, the lower class remains
destitute and static and has the character of a condensate, while the
middle class is dynamic and has a continuous upward mobility.

Our work is based on an earlier model of Bonabeau \cite{BTD}, in which
each individual is endowed with a fitness-like variable that evolves
by two opposing processes.  The first is competition: when two agents
interact, one individual becomes more fit (gains status) and the other
becomes less fit, with the initially fitter individual being more
likely to win.  Counterbalancing this competition, the winning
probability for the fitter agent decreases as the time from the last
competition increases.  This model was found to exhibit a transition
to a heterogeneous society as the relative influence of competition is
increased \cite{BTD,SS}.

In our model, we account for the interplay between advancement by competition
and decline by inactivity via a single parameter.  Each agent is endowed with
an integer fitness value $k\geq 0$ that can change due to two processes: (i)
advancement by competition and (ii) decline by inactivity.  In the
competition step, when two agents interact, their fitnesses change according
to
\begin{equation}
\label{up}
(k,j)\to (k+1,j),
\end{equation}
for $k\geq j$.  When two equally fit agents compete, both advance
\cite{note}.  Without loss of generality, the rate of this process is
set to one.  We also consider the mean-field limit where any pair of
agents is equally likely to interact.  The rationale behind this
``rich gets richer'' dynamics is obvious: fitter individuals are
better suited for, and hence benefit from, competition.  When decline
occurs, individual fitness decreases as
\begin{equation}
\label{down}
k\to k-1
\end{equation}
with a rate $r$.  This process reflects the natural tendency for social
status to decrease in the absence of interactions.  The lower limit for the
fitness is $k=0$; once an individual reaches zero fitness, there is no
further decline.  The model is characterized by a single parameter, the rate
of decline $r$.

Let $f_k(t)$ be the fraction of agents with fitness $k$ at time $t$. This
distribution obeys the nonlinear master equation
\begin{eqnarray}
\label{dis-eq}
\frac{d f_k}{dt}&=&r(f_{k+1}-f_k)+f_{k-1}F_{k-1}-f_kF_k
\end{eqnarray}
for $k> 0$, and $df_0/dt=rf_1-f_0^2$ for $k=0$. The quantity
$F_k=\sum_{j=0}^k f_j$ is the cumulative distribution. We take the
initial condition to be \hbox{$f_k(0)=\delta_{k,0}$}. In
Eq.~(\ref{dis-eq}), the first two terms account for decline, while the
last two terms account for advancement \cite{bkm}.

To understand the behavior of this system, we focus on the cumulative
distribution $F_k$, from which the individual densities are
\hbox{$f_k=F_k-F_{k-1}$}.  From the master equation (\ref{dis-eq}), the
cumulative distribution satisfies
\begin{equation}
\label{cum-eq}
\frac{dF_k}{dt}=r(F_{k+1}-F_k)+F_k(F_{k-1}-F_k),
\end{equation}
for $k\geq 0$. The boundary condition is $F_{-1}=0$ so that
$dF_0/dt=r(F_1-F_0)-F_0^2$, and the initial condition is $F_k(0)=1$.

\smallskip\noindent{\bf Homogeneous vs.\ Hierarchical Societies.}  Our
social diversity model undergoes a phase transition from a homogeneous
to a hierarchical society. This transition follows from the continuum
limit of the master equation (\ref{cum-eq}) for the cumulative
distribution
\begin{equation}
\label{cum-cont}
\frac{\partial F}{\partial t}=(r-F)\frac{\partial F}{\partial k}.
\end{equation}
For finite fitness, the cumulative distribution approaches a steady
state in the long-time limit. Then either $F=r$ or $\partial
F/\partial k=0$. Invoking the bound $F\leq 1$, we conclude that either
$F=r$ or $F=1$. Therefore, $L$, the fraction of the population with
finite fitness exhibits a phase transition
\begin{equation}
\label{mass}
L=
\begin{cases}
r&r< 1;\\
1&r\geq 1.
\end{cases}
\end{equation}
When competition is weak, the entire population has a finite fitness,
while for strong competition, only a fraction $L<1$ of the population
has a finite fitness.

\begin{figure}[t]
 \vspace*{0.cm}
\includegraphics*[width=0.41\textwidth]{fig1.eps}
\vskip -1.3in\hskip 0.9in\includegraphics*[width=0.265\textwidth]
{fig1-inset.eps} \vskip 0.25in 

\caption{The middle class. The scaled cumulative distribution
$\Phi(x)$ versus $x$ for $r=1/2$ at $t=250$ (dotted), $1000$ (dashed),
$4000$ (dot-dashed). The solid line is the theoretical prediction
(\ref{cum-sol}).  The inset shows the qualitative behavior for $r=0$
(dashed), $r\approx 1/2$ (solid), and $r\approx 1$
(dotted). \label{middle}}
\end{figure}

We shall see that the quantity $L$ is the size of the lower class,
while the complementary fraction $1-L$ is the size of the middle
class, whose fitness increases indefinitely.  Thus for $r\geq 1$, the
society is homogeneous and consists of a single lower class.  However
for $r<1$, there is a hierarchical society that contains a distinct
lower class, and a distinct a middle class.  When $r=0$, the lower
class disappears entirely.

\smallskip\noindent{\bf Middle Class Dynamics.}  The picture presented
above is confirmed by analyzing the dynamics of the middle class.
Applying dimensional analysis to the governing Eq.~(\ref{cum-eq})
suggests that the characteristic fitness of the middle class increases
linearly with time, $k\sim t$. Thus, we posit the scaling form
\begin{equation}
\label{scaling}
F_k\simeq \Phi(k/t)
\end{equation}
with the boundary condition $\Phi(\infty)=1$.  Substituting
Eq.~(\ref{scaling}) into (\ref{cum-cont}), the scaling function
satisfies \hbox{$x\,d\Phi/dx=(\Phi-r)\,d\Phi/dx$} where $x=k/t$. The
solution is either \hbox{$\Phi(x)=r+x$} or $d\Phi/dx=0$.  As a result
(Fig.~\ref{middle})
\begin{equation}
\label{cum-sol}
\Phi(x)=
\begin{cases}
r+x&x< 1-r;\\
1&x\geq 1-r.
\end{cases}
\end{equation}
Remarkably, the scaling function for the cumulative distribution is
piecewise linear and thus non-analytic. The analysis above implicitly
assumes continuity of the scaling function and indeed, the cumulative
distribution is expected to be continuous. 

\begin{figure}[t]
 \vspace*{0.cm}
\includegraphics*[width=0.41\textwidth]{fig2.eps}
\caption{The lower, middle, and upper classes. The fitness
  distribution $f_k$ versus $k$ for $r=1/2$ at $t=4000$ (solid line),
  showing the lower \hbox{($k\alt 60$)}, middle, and upper
  \hbox{($k\agt 2000$)} classes.  Also shown for reference is the
  plateau $f_k=1/t$ (dashed line).  The distribution varies as $1/k^2$
  in the lower class up to the diffusive scale $k_{\rm lower}\sim
  (2rt)^{1/2}$.  The distribution is constant in the middle class up
  to a ballistic scale $k_{\rm upper}=(1-r)t$, beyond which there is
  an upper class that has a Gaussian decay.
  \label{all}}
\end{figure}

The scaling function (\ref{cum-sol}) has a number of basic
implications.  First, the quantity $\Phi(0)=r$ is the fraction of the
population that belongs to the lower class, confirming the prediction
of Eq.~(\ref{mass}).  This behavior is reminiscent of a physical
condensate, where a finite fraction of the population occupies the
zero fitness (in scaled units) ground state. In this sense, the entire
lower class is destitute.  When competition only occurs ($r=0$), the
society consists of a continuously-improving middle class.

We can alternatively write the fitness distribution in the scaling
form $f_k\simeq t^{-1}\phi(k/t)$.  The corresponding scaling function
is \hbox{$\phi(x)=d\Phi/dx=r\delta(x)+1$} for $x\leq 1-r$ and $\phi(x)=0$
otherwise.  The middle class thus has a {\em constant\/} fitness
distribution
\begin{equation}
\label{flat}
f_k\simeq t^{-1},
\end{equation}
for $k<k_{\rm upper}=(1-r)t$.  The lot of the middle class is constantly
improving, as the fitness extends over a growing range and the
average fitness increases linearly with time.

Numerical integration of the master equation confirms these
predictions (Figs.~\ref{middle} and \ref{all}).  We used a
fourth-order Adams-Bashforth method \cite{zwillinger} with accuracy to
$10^{-10}$ in the distribution $F_k$.  Our numerical data was obtained
by integrating $F_k$ for $0\leq k<20000$.

\smallskip\noindent{\bf Lower Class Dynamics.} The fitness of the
lower class is finite; in other words, the fitness distribution is in
a steady state. This distribution can be determined by setting the
time derivative in the rate equation to zero. Writing $F_k=L(1-G_k)$,
so that the deviation $G_k$ vanishes at large $k$, Eq.~(\ref{cum-eq})
gives
\begin{equation}
\label{recursion}
r\,\frac{G_{k+1}-G_k}{G_k-G_{k-1}}=L(1-G_k).
\end{equation}

\begin{figure}[t]
 \vspace*{0.cm}
\includegraphics*[width=0.41\textwidth]{fig3.eps}
\caption{The lower class.  The cumulative distribution, normalized by
  the rate $r$, $F_k/r$ is plotted versus $k$.  Shown are simulation
  results for $r=1/4$ (circles), $r=1/2$ (squares), $r=3/4$ (diamonds)
  at time $t=10^4$.
\label{lower}}
\end{figure}

The fitness distribution is fundamentally different in the two phases.
In the homogeneous society phase ($r\geq 1$ and $L=1$), the deviation
$G_k$ decays rapidly at large fitness.  Replacing the right-hand side
of Eq.~(\ref{recursion}) by $1$ for large $k$, the solution is simply
$G_k\sim r^{-k}$.  Therefore
\begin{equation}
\label{exponential}
f_k\sim r^{-k}.
\end{equation}
The fitness distribution decays exponentially, so that the lower class
is confined to a small range of fitness values.  The characteristic
fitness $1/\ln r$ diverges as the transition is approached.  The
society is homogeneous with a single class, the lower class, that does
not evolve with time.

In the hierarchical society phase, (where \hbox{$r< 1$} and $L=r$),
the fitness distribution is universal, as the recursion relation
(\ref{recursion}) becomes $r$-independent,
\hbox{$(G_{k+1}-G_k)/(G_k-G_{k-1})=1-G_k$}. This shows that $F_k/r$ is
a universal, $r$-independent distribution (Fig.~\ref{lower}). We start
by treating $k$ as a continuous variable, because the fitness range
becomes large as $r\downarrow 1$.  We thus expand the differences in
Eq.~(\ref{recursion}) to second order.  Since $G''\ll G'$, where prime
denotes differentiation with respect to $k$, we find $G''+GG'=0$.
Integrating once and invoking $G\to 0$ as $k\to \infty$, gives
$G'+\frac{1}{2}G^2=0$.  Asymptotically, $G\simeq 2k^{-1}$, and using
$f_k=F_k-F_{k-1}$, we find
\begin{equation}
\label{powerlaw}
f_k\simeq 2r\,k^{-2}.
\end{equation}
The lower class has a power-law fitness distribution with mean
fitness that diverges logarithmically in the upper limit.  While
the lower class is still static, it is not as destitute as in the
homogeneous society phase.

The transition between the lower and middle class occurs when
$2r/k^{2}\approx 1/t$, {\it i.e.}, where the power-law distribution
(\ref{powerlaw}) matches the uniform distribution (\ref{flat}).
Consequently, the lower class is confined to a diffusive boundary
layer of thickness
\begin{equation}
\label{diffusive}
k_{\rm lower}\sim (2rt)^{1/2}.
\end{equation}
Beyond this diffusive scale, lies the middle class whose constant
density (\ref{flat}) extends over the range \hbox{$k_{\rm lower}< k<
k_{\rm upper}$}. In the hierarchical society phase, the fitness
distribution consists of the stationary component (\ref{powerlaw})
that defines the lower class and the evolving component
(\ref{scaling}) that defines the middle class. The extent of the
stationary region indefinitely grows with time. 

\begin{figure}[t]
 \vspace*{0.cm}
\includegraphics*[width=0.41\textwidth]{fig4.eps}
\caption{The upper class. Shown is the normalized tail of the fitness
  distribution: $tf_k$ versus $z^2$, with the scaling variable
  $z=(k-vt)/\sqrt{Dt}$, for $r=1/2$ at times $t=4000$ (circles) and
  $t=8000$ (squares).
\label{upper}}
\end{figure}

We thus conclude that the lower class is always static, being in a
steady-state independent of the rate of decline $r$.  In a homogeneous
society, the lower class has an exponentially decaying fitness
distribution that lies within a narrow fitness range.  In a
hierarchical society, the lower class fitness distribution decays
algebraically and its range grows diffusively with time.

\smallskip\noindent{\bf Upper Class Dynamics.}  The upper class is
defined by the subpopulation whose fitness lies beyond \hbox{$k_{\rm
upper}=(1-r)t$}.  We probe the tail of this fitness distribution by
again considering the deviation $G_k$, defined by $F_k=1-G_k$. It
obeys the Fokker-Planck equation
\begin{equation}
\frac{\partial G_k}{\partial t}+v\frac{\partial G_k}{\partial k}=
D\frac{\partial^2 G_k}{\partial k^2}
\end{equation}
with upward drift velocity $v=(1-r)$ and diffusion coefficient
$D=(1+r)/2$.  The boundary condition $G(k=vt)\propto t^{-1}$ is set by
matching the density at the top of the middle class with that at the
bottom of the upper class. Consequently, the fitness distribution,
$f=-\partial G/\partial k$, follows the scaling form
(Fig.~\ref{upper}) 
\begin{equation}
\label{f-upper}
f_k(t)\simeq t^{-1}\psi\left(\frac{k-vt}{\sqrt{Dt}}\right).
\end{equation}
The scaling function has the Gaussian tail \hbox{$\psi(z)\sim
  \exp(-z^2/2)$}, as $z\to \infty$, characteristic of a
  convection-diffusion equation.  The upper class is thus confined to
  a diffusive boundary layer that grows as $\sqrt{Dt}$.  From
  Eq.~(\ref{f-upper}), the upper class contains a fraction $\propto
  1/\sqrt{t}$ of the total population.

For completeness, we note that for the special case of $r=0$, the rate
equation $dF_k/dt=F_k(F_{k-1}-F_k)$ admits an exact solution.  We make
the transformation
\begin{equation}
F_k=\frac{P_{k-1}}{P_k},
\end{equation}
with the initial condition $P_k(0)=1$ and the boundary condition
$P_{-1}=P_0=1$.  Remarkably, this transformation reduces the
non-linear rate equations to the set of linear equations
$dP_k/dt=P_{k-1}$ for $k\geq 1$.  Solving these recursively, we obtain 
$P_k=\sum_{j=0}^{k} {t^j}/{j!}$. Therefore,
\begin{equation}
\label{F-sum}
F_k(t)=\frac{{\displaystyle 1+t+\frac{1}{2!}t^2\cdots+\frac{1}{k!}t^k}}
{{\displaystyle 1+t+\frac{1}{2!}t^2\cdots+\frac{1}{(k+1)!}t^{k+1}}}.
\end{equation}
It is possible to show that this exact solution adheres to the scaling
form (\ref{scaling}) with $\Phi(x)$ as in (\ref{cum-sol}).  Asymptotic
analysis yields the exact shape of $F_k$ in the boundary layer,
$1-F_k\simeq \sqrt{2/\pi t}\exp[-(k-t)^2/2t] /{\rm
erfc}[(k-t)/\sqrt{2t}]$.

In summary, we introduced a minimal model of social diversity in which
the two driving mechanisms are advancement by competition and decline
by inactivity.  An idealized but plausible social structure emerges:
either a homogeneous society with a single lower class, or a
hierarchical society with multiple classes. The lower class is always
static, while the middle class and the tiny upper classes are upwardly
mobile. In a hierarchical society, the lower and the upper classes are
confined to boundary layers that are much smaller than the dominant
scale that characterizes the fitness of the middle class.  It is
striking that a deceptively simple master equation exhibits such a
rich structure, with a stationary component, followed by two transient
components, as well as a non-analytic scaling function for the
asymptotic fitness distribution.

There are numerous interesting questions suggested by this work.  For
example, what is the time history of an individual?  How rigid is the
social hierarchy and how does it depend on the population size?  What
happens if each individual is also endowed with an intrinsic fitness?
Last, does non-trivial spatial organization emerge when agents move
locally in space?

We thank D. Stauffer for stimulating our interest in social
 diversity, K. Kulakowski for an informative discussion, and D. Watts
 for literature advice.  We also acknowledge financial support from
 DOE grant W-7405-ENG-36 (EB and SR) and NSF grant DMR0227670 (SR).



\begin{thebibliography}{99}

\bibitem{city} M. Marsili and Y. C. Zhang, Phys.\ Rev.\ Lett.\ {\bf 80}, 2741
  (1998);  
F. Leyvraz and S. Redner, Phys.\ Rev.\ Lett.\  {\bf  88}, 068301 (2002).

\bibitem{spinodal} I. M. Lifshitz and V. V. Slyozov, J. Phys.\ Chem.\ Solids,
{\bf 19}, 35 (1961); A. J. Bray, Adv.\ Phys.\ {\bf 43}, 357 (1994).

\bibitem{wasserman} S.~Wasserman and K.~Faust, {\it Social Network Analysis}
  (Cambridge University Press, Cambridge, 1994); M.~Girvan and
  M.~E.~J.~Newman, Proc. Natl. Acad. Sci. USA {\bf 99}, 7821
  (2002).

\bibitem{vfh} D.~Helbing, I.~Farkas, and T.~Vicsek,
             Nature {\bf 407}, 487 (2000).

\bibitem{ckfl} I.~D.~Couzin, J.~Krause, N.~R.~Franks, S.~A.~Levin,
  Nature {\bf 433}, 513 (2005).

\bibitem{ww} W.~Weidlich, {\it Sociodynamics: A Systematic Approach to
    Mathematical Modelling in the Social Sciences} (Harwood Academic
  Publishers, 2000); C.~L.~Barrett, S.~G.~Eubank, and J.~P.~Smith, Scientific
  American {\bf 292}, 54 (2005).

\bibitem{wdan} G.~Weisbuch, G.~Deffuant, F.~Amblard, and J.~P.~Nadal,
  Complexity {\bf 7}, 55 (2002).

\bibitem{bkr} E.~Ben-Naim, P.~L.~Krapivsky, and S.~Redner, Physica D {\bf
    183}, 190 (2003).

\bibitem{smo} D.~Stauffer and H.~Meyer-Ortmanns, Int.\ J. Mod.\ Phys.\ B {\bf
    15}, 241 (2004).

\bibitem{ez} V.~M.~Eguiluz and M.~G.~Zimmermann, 
             Phys. Rev. Lett. {\bf 85}, 5659 (2000).

\bibitem{W1} E. O. Wilson, {\it The Insect Societies}, (Harvard University
   Press, Cambridge, MA, 1971).

\bibitem{A} W. C. Allee, Biol.\ Symp.\ {\bf 8}, 139 (1942);
A. M. Guhl, Anim.\ Behav.\ {\bf 16}, 219 (1968).

\bibitem{C} I. D. Chase, Amer.\ Sociol.\ Rev.\ {\bf 45}, 905 (1980);
R. V. Gould, Amer.\ J. Sociology {\bf 107}, 1143 (2002).

\bibitem{BTD} E. Bonabeau, G. Theraulaz, and J.-L. Deneubourg, Physica A {\bf
    217}, 373 (1995).

\bibitem{SS} A. O. Sousa and D. Stauffer, Intl.\ J. Mod.\ Phys.\ C {\bf 5},
  1063 (2000); K. Malarz, D. Stauffer, and K. Kulakowski, preprint.

\bibitem{note} The behavior when only one agents advances is
essentially the same.

\bibitem{bkm} 
E.~Ben-Naim, P.~L.~Krapivsky, and S.~N.~Majumdar,
  Phys.\ Rev.\ E {\bf 64}, R035101 (2000).

\bibitem{zwillinger}
    D.~Zwillinger, {\it Handbook of Differential Equations}
    (Academic Press, London, 1989).

\end{thebibliography}

\end{document}
