Page 127 - Tallinn University

F s t
1
W v i h s
1
H v i S d
W s n p W
F W s p v F
a m
w i
s
n s
= =0; =0; =1 ,
!" !"
group Y or Z, when swapping to X, the probabilities and are equal (this relation holds !" !"
These correspond to any one language being the sole survivor. The fourth and fifth points,
of the recipient linguistic group, then regardless of whether an agent previously bel!onged to
p n
e ticipant g e
m

and , correspond to a situation where more than one language have active speakers. The
true for any transition, respectively): = = .
HARIDUSTEADUSE!D group Y or Z, when swappiTnagllintonaXÜl,iktohoeli üplriõopbilasbteil2it0i1e5s/20 16. õappnedaa sta PAaRreIMeAqDuTaElA(DthUiSsTÖreÖlDat/iAorntikhlioteldkosgumik
!" !"
true for any transition, respectively): = = . (2)
coordinates for these points are as follows:
In Eq. (2), a is the volatility paramete r of the system and is the prestige (in the ex
= = .
! (2)
case, indicated by subscript, of linguistic group X). Volatility is one of the main para
In Eq. (2), a is the volatility parameter of the system and is the prestige (in the example
!!! !!! ! !
!
of interest for!the model, describi!ng the rate at which languages are swapped and ho
case, in dic=ate d b=y subscript, of; li ng=uistic group X;). Vo=la0tili,ty is one of the m(7a)in parameters
In Eq. (2), a is the volatility parameter of the system and is the prestige (in the example
the society is to language shift. It defines the! shape of the probability function and !! !+ !! !+ !!
!!! !!! !!!
of interest for the m! odel, !describing th!e rate !at which languages are swapped and how inert
!!!
=1;
case, indicated by subscript, of linguistic group X). Volatility is one of the main parameters values between 0 and infinity.
the society is to language shift. It defines the shape of the probability function and holds of interest for the model, describing the rate at which languages are swapped and how inert
!
values beVtwoleaetnili0tyasnedpianrfaitneistyt.he syst!em into two d!istinct states. When a ˂ 1, the system is cons
1− − !!! !!! !!
+1+
valueVsobleattwilieteyns0epaanrdatiensfitnhietys.yste m into two distin ct states. When a ˂ 1, the system is considered
!
the society is to language shift. It defines the shape of the probability function and holds
!!
to be of high volatility, characterised by a frequent change of linguistic group by age
= (8) V!olatility separates the sy(!s!te!)m into two distinct states. When a ˂ 1!, the system is considered
!
volatility a ˃ 1, then the system is considered to be of low volatility, characterised by
1− −
to be of high volatility,! characterised by a frequent chang!e of! linguistic group by agents. If
inert agent!s, swapping language less frequently.
to be of high volatility, characterised by a frequent change of linguistic group by agents. If
volatility a ˃ 1, then the system is considered to be of low(!v!o!l)atility, characterised by more
= ; =
ine!rt agePntrse,sstiwgeaprpe!ipnrgesleantguthaegerelleastsivfer!esqtuaetunstlyo.f a given linguistic g!roup, representing the part
! (!!!)
society holding a fav!ourable view of said group, thus being fractional and summing up t
1− ! − ! !!! (!!!) !!!
!
volatility a ˃ 1, then the system is considered to be of low! volatility, characterised by more
+ 1 + 1 − ! − !
inert Pargesntitgs,esrweapprepsienngt lathnegurealgaetilveessstfaretuqsueonf talyg. iven linguistic group , representing the part of the
!
+ 1 +
!! societyholdingafavourableviewofsaidgroup ,th+u sb+ein g=fra ct.ionalandsummingupto1:
Prestige represent the relative status of a given linguistic group, representing the part of the society holdin1g.1a faFvioxuerdabploeinvtisewofotfhesamidogdr eolu+p, thu+s be=ing f. ractional and summing up to 1: (3)
For the fourth stationary point, the coordinate for Z is = 0. Mirrored variants of this
!
or the fourth stationary point, the coordinate for Z is = 0. Mirrored variants of this
volatilitsytasteaste(sa( 1), the! only s!ta!ble sol!ution wa!!s and fo!r high volatility
We found that for low volatility states (a ˃ 1), the only stable solution was and for high e found that for low volatility states (a ˃ 1), the only stable solution was and! for high
.2 Stability of the fixed points
= ==1; = 0; =01;, = 0 , (4)
!
= =0; =0; =1 , (6)
and , correspond to a situation where more than one language have active speaker
!
These correspond to any one language being the sole survivor. The fourth and fifth points,
! ! !
!!! !!!
!!!!!!!!!!! !
!
!!!! !!! !!!! 1!!! !! ! !! !
=; ! ! !+ ! ! !+ ! ! !!
etermined by the relation of its prestige to volatility.
= = ; ! = ! ; = 0 , ( 7 )
summarisable as follows: In high volatility states (a < 1 ), =all linguistic groups surviv+e1a+nd; the ummarisable as follows: In high volatility states (a < 1), all !linguistic groups!survive and the
We found the effect of prestige and volatility on the dynamics of the model to be e found the effect of prestige and volatility on the dynamics of the model to be
1− − 1 !!! !!! number of agents they hold after the model’s relaxation is de!term!ined by vo latility and
! ! ! ! ! ! 1 −! ! ! 1! − ! ! ! ! ! ! ! ! ! ! !!! !!
! = ! = (!!!) !+1+ ; ! !
prestige.Asvolatilitynears0,prestigehasastronginfluenceo!n!thenumberofa!gentsheld.
umber of agents they hold after the model’s relaxation is determined by volatility and
+1+
ith volatility increases toward 1, prestige loses infl!uence. ! ! !
restige. As volatility nears 0, prestige has a strong influence on the nu!!m!ber of agents held.
With volatility increases toward 1,=prestige loses influence.
1 !− − ! ! ! Forlowvolatilitystates,th e=stablepointsare , and ,butj+us1t+onelanguagesurv1i−ve s.− +1+ (8)
= ! !(!!!) ! ; = !! !
or low volatility states, the stable points a!re , and , but just one langu!age survives. !
(8)
!
!!!! ! !!
(!!!) Whichoneofthethreepointsischosen,is1de−te rm −in ed!!b!yprestig e(a!n!d!)volatility,1t−hr ou−gh the (!!!)
! ! ! ! 1− !− !
+1+ (!!!) 1− !− ! !!!+1+ !
points is stable – one at the corner of the phase. This is visible in Fig. (1), where we plot x(t) oints is stable – one at the corner of the phase . This is visible in ! Fig. (1), where we plot x(t) !
!!
vs y(t), thus all coordinates represent a distribution of speakers in time. All trajectories in s y(t), thus all coordinates represent a distribution of speakers in time. All trajectories in
Fig. (1) start from the center and move toward the edges. ig. (1) start from the center and move toward the edges.
!!
1− − !!! (!!!)
! ! ! 1− −
! = (!!! ) !!!!!
! ! ; = !
!
( ! ! ! ) !!! !! !!!!! !
! ! 1− − !+1+ = +1+; = !!
stationary points , ′, ′′ and . In these!zones, only one!of the first three sta!tionary tationary points , !′, !′′ an!d . In!these zones!,!!only one of the first three stationary (!!!)
!!! !!!! ! hich one of the three points is chosen, is determined by prestige and! volatility, through the
! ! ! ! ! !
Figure 1: Plot of x(t) vs y(t). Volatility is fixed at a = 1,33. Prestige is fixed to = , and = , . The Figure 1: Plot of x(t) vs y(t). Volatility is xed at a = 1,33. Prestige is xed to and . e points E4, E4’ and E4’’ are the fourth
E4’ and E4’’ are the fourth stationary points and its mirrored variants. Point E5 represents the fifth stationary poi
stationary points and its mirrored variants. Point E represents the h stationary point. Initial values for fraction of agents
valuesforfractionofagentsheldvariesforeverytrajectory.E,E andE arethefirstthreefixedpoints.
held varies for every trajectory. E1, E2 and E3 are the rst three xed points.
5
123
Some relaxation trajectories on Fig. (1) exhibit non-monotonic relaxation. Trajectori move close to the phase lines, in some cases have two of the three par s speakers in time. This means that during the process itself, a language might “h and growing, but in reality, as soon as the weaker competitor is gone, it will su
( ! ! ! ) 1 − −
(!!!)
127
appear be con