PHYSICAL REVIEW E 90, 042120 (2014)
Growing spin model in deterministic and stochastic trees Julian Sienkiewicz Faculty of Physics, Center of Excellence for Complex Systems Research, Warsaw University of Technology, Koszykowa 75, PL-00-662 Warsaw, Poland (Received 30 June 2014; revised manuscript received 9 September 2014; published 13 October 2014) We solve the growing asymmetric Ising model [J. Sienkiewicz, K. Suchecki, and J. A. Hołyst, Phys. Rev. E 89, 012105 (2014)] in the topologies of deterministic and stochastic (random) scale-free trees predicting its nonmonotonous behavior for external fields smaller than the coupling constant J . In both cases, we indicate that the crossover temperature corresponding to maximal magnetization decays approximately as (ln ln N )−1 , where N is the number of nodes in the tree. DOI: 10.1103/PhysRevE.90.042120
PACS number(s): 02.50.Ga, 75.10.Pq, 05.50.+q
I. INTRODUCTION
Although one-dimensional systems can be often used to model social dynamics [1–3], it is usually observed that the structure of the majority of online social systems such as portals or fora follows a different type of topology, namely a scale-free one that is reflected in their degree distribution [4,5]. Such nontrivial topologies have motivated several researchers to explore the behavior of one of the most fundamental approaches of statistical physics, namely the Ising model [6], which has been tested on Cayley trees [7], Barab´asi-Albert (BA) networks [8], or growing trees [9], to mention a few. However, there is no direct evidence that social processes that take place in hierarchical trees and scale-free networks can be described by this kind of dynamics. On the other hand, the results of our previous analyses [10,11] indicate that one of the most dominant phenomena seen in online portals is a strong dependence of the expressed emotion on the emotion of the last comment (i.e., the newest one). To describe this process in a setting of chronologically added comments (that form a chain), we have previously explicitly modified the Ising model Hamiltonian by taking into account only a node’s left neighbor, as well as equipping our model with a growing component (a new node is quenched after a single update) [12]. Here, we extend this concept to tree topologies (deterministic, i.e., Cayley trees and stochastic, i.e., BA trees) in order to explore the influence exerted by the intrinsic features of those systems on the behavior of the model. The paper is organized as follows: In Sec. II, we give a brief description of the basic concepts of the model introduced in [12]. Sections III and IV gather the results obtained applying dynamics to deterministic and stochastic scale-free trees, respectively. In Sec. V, we discuss the similarities and differences with other sociophysics models (e.g., opinion or epidemics spreading). Finally Sec. VI concludes the paper by exposing the differences between the topologies considered in this study and the previously modeled growing chain. II. MODEL DESCRIPTION
The basic version of the model uses the idea of a growing chain: the first node of the chain has a random spin s0 = ±1 (it can be interpreted as an emotional valence [13] of a post in online discussion), drawn with probability Pr(s0 = ±1) = 1/2. After that, another node of the chain is added to the right side 1539-3755/2014/90(4)/042120(6)
of the last one and it is initially equipped with a spin once again drawn with equal probabilities, Pr(s1 = ±1) = 1/2. In the following step, the node becomes subject to the updating procedure that is based on the Ising-like model approach. For each appearing node n we define a function En = −J sn−1 sn − hsn , where the constant J > 0 corresponds to an exchange integral in the Ising model and h is the external field. The function En can be treated as a type of emotional discomfort function felt by a user posting a message sn . After the spin is drawn, we check how flipping its sign to the opposite one (i.e., from sn = +1 to −1 or likewise) affects the change of function E as E = En − En = −(J sn−1 + h)(sn − sn ), where the term En corresponds to sn calculated when sn → sn = −sn . Then we follow the Metropolis algorithm [14], i.e., if E < 0 we accept the change, otherwise we test if the expression exp[−E(kB T )−1 ] is smaller or larger than a random value ξ ∈ [0; 1] (here kB is Boltzmann’s constant and T is temperature). If the latter occurs, we accept the change, otherwise the spin is kept as originally chosen. The procedure of adding new nodes and setting their spins is repeated until the size N of the chain is reached. It can be shown [12] that system dynamics follows a two-state Markov chain approach defined by the transition matrix P, p 1−p P= , (1) 1−q q with conditional probabilities p = Pr(+|+) and q = Pr(−|−) (which come from the above-described dynamics) given by
p = 1 − 12 e−β (h+J ) ,
∓ 12 e±β (h−J ) ,
q=
1 2
±
1 2
p=
1 2
±
1 ∓ 12 e∓β (h+J ) , 2 1 β e (h−J ) , 2
for h 0,
(2)
for h < 0,
(3)
and q =1−
where the upper signs in Eq. (2) correspond to the case h ∈ [0,J ) and the lower signs correspond to h J , while in the case of Eq. (3) the upper signs are for h ∈ (−J,0) and the lower signs are for h −J [in all cases, β˜ = 2/(kB T )] [15]. As a result, the average spin (or valence) in the nth node of the chain is given as
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sn =
p−q [1 − (p + q − 1)n ], 2−p−q
(4)
©2014 American Physical Society
JULIAN SIENKIEWICZ
PHYSICAL REVIEW E 90, 042120 (2014)
while the average spin in the whole chain can be obtained as a mean value over sn , i.e., p−q 1 1 − QN+1 s = 1+ , (5) − 1−Q N N(1 − Q) where Q = p + q − 1. The motivation is to compare the results obtained for a chain topology with those that are derived for finite Cayley and random scale-free trees.
which can be expressed explicitly for |h| < J as
III. CAYLEY TREES
The topology of a finite Cayley tree is described by two parameters: the number of children z each node gives birth to, and the depth of the tree L. The total number of vertices (except the root one) is equal to N=
l=L l=1
zl = z
zL − 1 . z−1
(6)
Figure 1 illustrates an example of a tree with z = 3 and L = 3. The first and key observation one needs to make is that in the case of the described model (Sec. II), a directed tree can be regarded as a chain of length L whose spin values are weighted by the number of nodes present at a given level (depth) l of a tree (see Fig. 2). By this reasoning, in order to obtain the formula for the average spin in the tree, st , one needs to perform the following summation: st =
l=L 1 sl zl , N l=1
FIG. 2. (Color online) Graphical representation of the weighting of spins at a given sept of the tree.
(7)
where sl is given by Eq. (4). After some algebraic calculations, one arrives at the following expression: 1 1 − (zQ)L+1 p−q t s = 1+ − , (8) 1−Q N N (1 − zQ)
h sts = tanh β
h)z]L+1 1 1 − [(1 − e−β J cosh β × 1+ − h)z] N N [1 − (1 − e−βJ cosh β
(9)
and for |h| J as
stl = sgn(h)
J − eβ |h| cosh β J − eβ|h| sinh β
J )L+1 1 − (ze−β |h| sinh β 1 − × 1+ . (10) J ) N N (1 − ze−β|h| sinh β As can be seen in Fig. 3 (for simplicity, this plot and further ones are for J = kB = 1), the above functions follow a shape that is similar to the one observed for the chain—we have a maximum in s(T ) for |h| < J and an absence of such behavior for |h| J . Having that in mind, it is interesting to examine the dependence of the crossover temperature Tc (i.e., the temperature for which sts takes the maximum) as a function of tree parameters. It can be shown that Tc ≈ 2(ln L)−1 [see Appendix A and Fig. 4(a)], which, using Eq. (6) and assuming zL 1 and z ≈ z − 1, gives Tc ≈
2 . ln ln N − ln ln z
(11)
1.0
s
t
0.8 0.6 0.4 0.2 0.0
FIG. 1. (Color online) A schematic plot of a deterministic tree with z = 3 and L = 3.
0
1
2 T
3
4
FIG. 3. (Color online) Average spin st in a tree (z = 3, L = 12) as a function of temperature for different values of the external magnetic field: h = 0.1 (squares), h = 0.5 (circles), h = 0.9 (upward triangles), h = 1 (downward triangles), and h = 2 (diamonds). Solid lines come from Eqs. (9) and (10). All data points have been averaged over M = 104 realizations.
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3.00 2.00 1.50 1.00 0.70 0.50
1.0 0.5 0.0 ln TC
TC
GROWING SPIN MODEL IN DETERMINISTIC AND . . .
0.5 1.0 1.5
0.30 0.20 0.15
(a)
1
2.0 104
100
(b)
2.5 0
106
5
10 lnln N
L
15
FIG. 4. (Color online) (a) Crossover temperature Tc vs the depth of the tree L. Symbols are numerical solutions of Eq. (9) while the solid line comes from Tc = 2(ln L)−1 . In all cases, z = 10. (b) The logarithm of the crossover temperature Tc vs the double logarithm of the number of nodes in a tree N . Symbols (squares, z = 2; circles, z = 5; triangles, z = 10) are numerical solutions of Eq. (9), while solid lines come from Eq. (11).
A comparison of the crossover temperature obtained by numerically solving Eq. (9) with the predictions of Eq. (11) is shown in Fig. 4(b). For sufficiently large values of N , all data points and analytical curves collapse, indicating a lack of dependence on the parameter z. The behavior of the crossover temperature on the number of nodes in a Cayley tree topology is much different from the results obtained in the case of a chain [12] where Tc ∼ ln−1 (Ne). However, this fact is a natural consequence of the small depth L of the tree, which substitutes in this case the effective range of dynamics influence (N in a chain as compared to ln N in a tree).
n(l) at a distance l from the root node, (ln N/2)l−1 , (l − 1)!
(12)
l=∞ 1 sl n(l), N l=1
(13)
Q−1 p−q 1−N 2 Q . 1−Q
(14)
n(l) = A
√ where A = N is the expected number of children of the root
∞ node, and it can be directly obtained from the condition n=l n(l) = N . As a consequence, the formula for the average spin in a scale-free random tree is given by ssf = which results in ssf =
IV. RANDOM SCALE-FREE TREES
Although finite Cayley tree topology (and especially the underlying branching process) is quite common in the real world, it is rather unreasonable to believe that the structure of social media systems (e.g., forum discussions) is an outcome of the procedure resulting in an exponentially growing number of nodes in consecutive tree layers. According to several previous studies, it is commonly believed that the growth of such systems could be governed by the preferential attachment process [16]. In the case of graphs (the Barab´asi-Albert model), this assumption describes a process in which the probability that the newly appearing node i will attach to a previously present node j is proportional to its degree kj . In general, the incoming node can create m new connections, thus forming a network (i.e., a graph with cycles), however if m = 1, the resulting structure is a scale-free tree whose degree distribution p(k) is described by a power law p(k) ∼ 2k −3 . The scheme for obtaining the average spin value in random scale-free trees is similar to the one presented in the preceding section. However, in this case, the underlying process of tree formation is also of stochastic origin (i.e., not only the dynamics but the topology as well follows a probabilistic rule). We use the results of Bollob´as and Riordan [17] and Szab´o et al. [18] that give the mean-field number of vertices
The above equation can be expressed explicitly for |h| < J as
J −β h)N − 12 e−β J cosh βh ssf cosh β s = tanh β h 1 − (1 − e (15) and for |h| J as
ssf l = sgn(h)
J − eβ |h| cosh β J − eβ|h| sinh β
J )N 12 (e−β |h| sinh βJ −1) . (16) × 1 − (e−β |h| sinh β A comparison of the theoretical predictions given by Eqs. (15) and (16) is shown in Fig. 5(a). It can be shown (Appendix B) using an analytical and numerical approach that the dependence of the crossover temperature Tc on tree size N is best described by Tc ≈
2 , 1 + 43 W ln4eN
(17)
where W (· · · ) is the Lambert W function. A comparison of the crossover temperature obtained by numerically solving Eq. (14) with the predictions of Eq. (17) is shown in Fig. 5(b). It is interesting to add here that for sufficiently large values of x, the function W (x) can be approximated with W (x) ≈ ln x −
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PHYSICAL REVIEW E 90, 042120 (2014)
1.0
2.0
(a)
(b)
1.5
0.6
TC
s
sf
0.8
1.0
0.4 0.2 0.0
0
1
2
3
4
1
T
105
1010
1015
1020
1025
1030
N
FIG. 5. (Color online) (a) Average spin ssf in a random scale-free tree (N = 105 ) as a function of temperature for different values of the external magnetic field: h = 0.1 (squares), h = 0.5 (circles), h = 0.9 (upward triangles), h = 1 (downward triangles), and h = 2 (diamonds). Solid lines come from Eqs. (15) and (16). All data points have been averaged over M = 105 realizations. (b) Crossover temperature Tc vs the number of nodes N . Symbols are numerical solutions of Eq. (15), the solid line comes from Eq. (17), while the dashed line comes from Eq. (B4) (see Appendix B for details).
ln ln x, which would suggest that for large values of N the crossover temperature is given by Tc ≈ (2/3 ln ln N − ln 2)−1 .
V. DISCUSSION
The choice of the tree topology has been influenced by three distinct factors: (i) it resembles the actual structure of many online discussions fora, (ii) it enables performing exact analytical calculations, and (iii) it is unambiguous with respect to system dynamics. It has to be emphasized that in contrast to several other models of emotion propagation (usually agent-based, cf. [19–23]), here a node does not correspond to a user but to a shared message. This idea defines the causality introduced into the model—one message can be a reply only to another single message, i.e., you cannot answer several messages in one turn. Therefore, it would be useless to test the behavior of the model on topologies such as BA networks with m > 1 as it would require amending the rules of dynamics. In a similar manner, our model is different from such widely used dynamics as stochastic epidemic models such as the susceptible-infected-recovered (SIR) model, whose evolution has recently been described on tree topology by the belief propagation equations [24]—also in this case, the notion of dynamics is individuals. The issue of the feasibility of analytical calculations that usually require a chain or a tree as the underlying topology can easily be seen in studies regarding the nonconsensus opinion model (NCO) [25], interacting information waves [26], attention competition [27], or the spreading of ideas (i.e., paradigm shifts) [28]. For example, even a simple comparison of those topologies in the case of two competing paradigms leads to interesting and nonintuitive conclusions: the average time T after the initial idea stops spreading in the system scales with the chain size N as N 2 ln N , while in the case of the BA trees the result reads T ∼ N α with α ≈ 2.2. Nonetheless, in all mentioned cases these are agents who are subject to an update procedure, thus the underlying network structure in fact mirrors interactions
between individuals in the system. Our approach, based on the stylized facts with respect to observed emotional homophily in online media [10], makes no assumptions about the actual structure of the interpersonal contacts among the users—instead we introduce global parameters h and T , which characterize the behavior of the participants. The key observation visible both for chain and deterministic or stochastic trees is the issue of the interplay between the initial conditions and the external field. For |h| < J and small values of T , the average magnetization of the system (opinion or valence) is determined by the initial condition: if the first spin (message) is negative, the whole system follows it (i.e., s = −1), regardless of the positive value of the external field parameter. On the other hand, if the first message is positive, then the system’s magnetization is s = 1, which gives on average s = 0 as the initial condition is random. However, if we raise the temperature, more and more spins will align according to the direction of the magnetic field, resulting in a peak of s (for higher values of T , the magnetization decays due to thermal fluctuations). This behavior is observed in all the examined topologies, with the crossover temperature being a slowly decaying function of N .
VI. CONCLUSIONS
In this paper, we extended the previously introduced model of the growing spin chain to the case of deterministic (Cayley) and random scale-free (BA) trees. We have shown that for these topologies, the analytical approach using a Markov chain concept is still valid due to the possibility of calculating the weighted spin on each level of the tree. Similarly to the chain case, the model exhibits a crossover temperature corresponding to maximal magnetization. Unlike the chain case, the crossover temperature decays very slowly [approximately as (ln ln N )−1 compared to (ln N)−1 for the chain], which is connected to the effective diameter of the considered systems.
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10
ACKNOWLEDGMENTS
Tcnum
2
8 6
TC
This work has been supported by the Polish Ministry of Science and Higher Education (Grant No. 0490/IP3/2011/71) as well as by the European Union in the framework of European Social Fund through the Warsaw University of Technology Development Programme, realized by the Center of Advanced Studies.
4 2
APPENDIX A: DERIVATION OF THE CROSSOVER TEMPERATURE FOR FINITE CAYLEY TREES
To get an analytical approximation of Tc , we use Eq. (9) h 1, which gives us the opportunity to set and assume that β h ≈ 1 and tanh β h ≈ β h: cosh β 1 1 − [(1 − e−β J )z]L+1 h 1 + s ≈ β − . (A1) N N [1 − (1 − e−βJ )z] Secondly, let us note that for zL 1 one can approximate Eq. (6) with N ≈ zL /(z − 1). Making use of this fact and assuming N 1, we get (z − 1)(1 − e−β J )L+1 h 1 + s ≈ β . (A2) 1 − (1 − e−βJ )z J 1, we arrive at Finally, also assuming that β
h[1 + (1 − e−β J )L+1 ]. s ≈ β
(A3)
It is interesting to observe here that this result does not depend on the branching factor z that confirms the behavior seen in Fig. 4(b). As the next step, we need to solve ∂s = 0, i.e., ∂T βc (L + 1)e−βc J (1 − e−βc J )L+1 = 1 + (1 − e−βc J )L .
(A4)
By taking the logarithm of both sides and assuming L 1 as well as (1 − e−βc J )L 1, we arrive at βc + Le
−βc J
= ln βc L.
(A6)
kB ln
1
2
2J
. + W − keB
(A7)
Setting J = kB = 1 leads us to the final result,
4
5
6
APPENDIX B: DERIVATION OF THE CROSSOVER TEMPERATURE FOR RANDOM SCALE-FREE TREES
h First, as in the case of Cayley trees, we assume that β h ≈ 1 and tanh β h ≈ β h in 1, which results in setting cosh β Eq. (15):
h 1 − (1 − e−βJ )N − 12 e−β J . s ≈ β (B1)
Finally, assuming that 1 − e−β J ≈ 1 and solving arrive at 1 −β cJ
1 − N−2e
∂s ∂T
1 −βc J ≈ 12 βc J e−βc J N − 2 e ln N.
= 0, we (B2)
At this point, we use the fact that for x 1 we can expand N x/2 with Taylor series as N x/2 ≈ 1 + 12 ln N x + 12 (ln N x)2 , which gives us the final equation βc J − 14 ln N e−βc J ≈ 1
(B3)
that has the solution Tc =
2J , kB 1 + W ln4eN
(B4)
where W (· · · ) is the Lambert W function. Predictions of Eq. (B4) for kB = J = 1 are shown in Fig. 5 with a dashed line, suggesting divergence with the numerical solution Tcnum of Eq. (15), which is caused by the Taylor series expansion. To overcome this issue, we propose a solution in the form Tc (a) =
eL kB
3
FIG. 6. Sum of the squares of displacement between the numerical and theoretical results of the crossover temperature for random scale-free trees vs parameter a [see Eq. (B6)].
has a solution of the form Tc ≈
0
a
(A5)
However, the above equation still fails to be solved by analytical methods. To overcome this problem, we use the approximation ln βc L = ln(2L/kB ) − ln Tc ≈ ln(eL/kB ). Then, the resulting equation βc + Le−βc J = ln(eL/kB )
0
2 , 1 + aW ln4eN
where a is chosen so that the sum 2 Tc (a) − Tcnum
(B5)
(B6)
(A8)
is minimal. Numerical minimization of the above functional gives a ≈ 4/3 (see Fig. 6).
[1] K. Sznajd-Weron and J. Sznajd, Int. J. Mod. Phys. C 11, 1157 (2000).
[2] Z. Liu, J. Luo, and Ch. Shao, Phys. Rev. E 64, 046134 (2001). [3] J. Sienkiewicz and J. A. Hołyst, Phys. Rev. E 80, 036103 (2009).
Tc ≈
2 . ln L
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PHYSICAL REVIEW E 90, 042120 (2014)
[4] A.-L. Barab´asi and R. Albert, Science 286, 509 (1999). [5] F. Wu and B. A. Huberman, Proc. Natl. Acad. Sci. (U.S.A.) 104, 17599 (2007). [6] E. Ising, Z. Phys. 31, 253 (1925). [7] T. P. Eggarter, Phys. Rev. B 9, 2989 (1974). [8] A. Aleksiejuk, J. A. Hołyst, and D. Stauffer, Physica A 310, 260 (2002). [9] T. Hasegawa and K. Nemoto, Phys. Rev. E 80, 026126 (2009). [10] A. Chmiel, J. Sienkiewicz, M. Thelwall, G. Paltoglou, K. Buckley, A. Kappas, and J. A. Hołyst, PLoS ONE 6, e22207 (2011). [11] A. Chmiel and J. A. Hołyst, Phys. Rev. E 87, 022808 (2013). [12] J. Sienkiewicz, K. Suchecki, and J. A. Hołyst, Phys. Rev. E 89, 012105 (2014). [13] L. A. Feldman, J. Personal. Soc. Psych. 69, 153 (1995). [14] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, J. Chem. Phys. 21, 1087 (1953). [15] In [12] by mistake only Eq. (2) has been given as the solution for the whole range of h. [16] R. Albert and A.-L. Barab´asi, Rev. Mod. Phys. 74, 47 (2002).
[17] B. Bollob´as and O. Riordan, Phys. Rev. E 69, 036114 (2004). [18] G. Szab´o, M. Alava, and J. Kert´esz, Phys. Rev. E 66, 026101 (2002). [19] F. Schweitzer and D. Garc´ıa, Eur. Phys. J. B 77, 533 (2010). [20] A. Chmiel, P. Sobkowicz, J. Sienkiewicz, G. Paltoglou, K. Buckley, M. Thelwall, and J. A. Hołyst, Physica A 390, 2936 (2011). [21] P. Sobkowicz, PLoS ONE 8, e80524 (2013). [22] V. Gligorijevi´c, M. Skowron, and B. Tadi´c, Physica A 392, 538 (2013). [23] J. Sienkiewicz, M. Skowron, G. Paltoglou, and J. A. Hołyst, Advs. Complex Syst. 16, 1350026 (2013). [24] F. Altarelli, A. Braunstein, L. Dall’Asta, A. Lage-Castellanos, and R. Zecchina, Phys. Rev. Lett. 112, 118701 (2014). [25] D. ben-Avraham, Phys. Rev. E 83, 050101(R) (2011). [26] A. Mirshahvalad, A. V. Esquivel, L. Lizana, and M. Rosvall, Phys. Rev. E 89, 012809 (2014). [27] U. Cetin and H. O. Bingol, Phys. Rev. E 90, 032801 (2014). [28] P. Kondratiuk, G. Siudem, and J. A. Hołyst, Phys. Rev. E 85, 066126 (2012).
042120-6
Growing spin model in deterministic and stochastic trees Julian Sienkiewicz Faculty of Physics, Center of Excellence for Complex Systems Research, Warsaw University of Technology, Koszykowa 75, PL-00-662 Warsaw, Poland (Received 30 June 2014; revised manuscript received 9 September 2014; published 13 October 2014) We solve the growing asymmetric Ising model [J. Sienkiewicz, K. Suchecki, and J. A. Hołyst, Phys. Rev. E 89, 012105 (2014)] in the topologies of deterministic and stochastic (random) scale-free trees predicting its nonmonotonous behavior for external fields smaller than the coupling constant J . In both cases, we indicate that the crossover temperature corresponding to maximal magnetization decays approximately as (ln ln N )−1 , where N is the number of nodes in the tree. DOI: 10.1103/PhysRevE.90.042120
PACS number(s): 02.50.Ga, 75.10.Pq, 05.50.+q
I. INTRODUCTION
Although one-dimensional systems can be often used to model social dynamics [1–3], it is usually observed that the structure of the majority of online social systems such as portals or fora follows a different type of topology, namely a scale-free one that is reflected in their degree distribution [4,5]. Such nontrivial topologies have motivated several researchers to explore the behavior of one of the most fundamental approaches of statistical physics, namely the Ising model [6], which has been tested on Cayley trees [7], Barab´asi-Albert (BA) networks [8], or growing trees [9], to mention a few. However, there is no direct evidence that social processes that take place in hierarchical trees and scale-free networks can be described by this kind of dynamics. On the other hand, the results of our previous analyses [10,11] indicate that one of the most dominant phenomena seen in online portals is a strong dependence of the expressed emotion on the emotion of the last comment (i.e., the newest one). To describe this process in a setting of chronologically added comments (that form a chain), we have previously explicitly modified the Ising model Hamiltonian by taking into account only a node’s left neighbor, as well as equipping our model with a growing component (a new node is quenched after a single update) [12]. Here, we extend this concept to tree topologies (deterministic, i.e., Cayley trees and stochastic, i.e., BA trees) in order to explore the influence exerted by the intrinsic features of those systems on the behavior of the model. The paper is organized as follows: In Sec. II, we give a brief description of the basic concepts of the model introduced in [12]. Sections III and IV gather the results obtained applying dynamics to deterministic and stochastic scale-free trees, respectively. In Sec. V, we discuss the similarities and differences with other sociophysics models (e.g., opinion or epidemics spreading). Finally Sec. VI concludes the paper by exposing the differences between the topologies considered in this study and the previously modeled growing chain. II. MODEL DESCRIPTION
The basic version of the model uses the idea of a growing chain: the first node of the chain has a random spin s0 = ±1 (it can be interpreted as an emotional valence [13] of a post in online discussion), drawn with probability Pr(s0 = ±1) = 1/2. After that, another node of the chain is added to the right side 1539-3755/2014/90(4)/042120(6)
of the last one and it is initially equipped with a spin once again drawn with equal probabilities, Pr(s1 = ±1) = 1/2. In the following step, the node becomes subject to the updating procedure that is based on the Ising-like model approach. For each appearing node n we define a function En = −J sn−1 sn − hsn , where the constant J > 0 corresponds to an exchange integral in the Ising model and h is the external field. The function En can be treated as a type of emotional discomfort function felt by a user posting a message sn . After the spin is drawn, we check how flipping its sign to the opposite one (i.e., from sn = +1 to −1 or likewise) affects the change of function E as E = En − En = −(J sn−1 + h)(sn − sn ), where the term En corresponds to sn calculated when sn → sn = −sn . Then we follow the Metropolis algorithm [14], i.e., if E < 0 we accept the change, otherwise we test if the expression exp[−E(kB T )−1 ] is smaller or larger than a random value ξ ∈ [0; 1] (here kB is Boltzmann’s constant and T is temperature). If the latter occurs, we accept the change, otherwise the spin is kept as originally chosen. The procedure of adding new nodes and setting their spins is repeated until the size N of the chain is reached. It can be shown [12] that system dynamics follows a two-state Markov chain approach defined by the transition matrix P, p 1−p P= , (1) 1−q q with conditional probabilities p = Pr(+|+) and q = Pr(−|−) (which come from the above-described dynamics) given by
p = 1 − 12 e−β (h+J ) ,
∓ 12 e±β (h−J ) ,
q=
1 2
±
1 2
p=
1 2
±
1 ∓ 12 e∓β (h+J ) , 2 1 β e (h−J ) , 2
for h 0,
(2)
for h < 0,
(3)
and q =1−
where the upper signs in Eq. (2) correspond to the case h ∈ [0,J ) and the lower signs correspond to h J , while in the case of Eq. (3) the upper signs are for h ∈ (−J,0) and the lower signs are for h −J [in all cases, β˜ = 2/(kB T )] [15]. As a result, the average spin (or valence) in the nth node of the chain is given as
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p−q [1 − (p + q − 1)n ], 2−p−q
(4)
©2014 American Physical Society
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PHYSICAL REVIEW E 90, 042120 (2014)
while the average spin in the whole chain can be obtained as a mean value over sn , i.e., p−q 1 1 − QN+1 s = 1+ , (5) − 1−Q N N(1 − Q) where Q = p + q − 1. The motivation is to compare the results obtained for a chain topology with those that are derived for finite Cayley and random scale-free trees.
which can be expressed explicitly for |h| < J as
III. CAYLEY TREES
The topology of a finite Cayley tree is described by two parameters: the number of children z each node gives birth to, and the depth of the tree L. The total number of vertices (except the root one) is equal to N=
l=L l=1
zl = z
zL − 1 . z−1
(6)
Figure 1 illustrates an example of a tree with z = 3 and L = 3. The first and key observation one needs to make is that in the case of the described model (Sec. II), a directed tree can be regarded as a chain of length L whose spin values are weighted by the number of nodes present at a given level (depth) l of a tree (see Fig. 2). By this reasoning, in order to obtain the formula for the average spin in the tree, st , one needs to perform the following summation: st =
l=L 1 sl zl , N l=1
FIG. 2. (Color online) Graphical representation of the weighting of spins at a given sept of the tree.
(7)
where sl is given by Eq. (4). After some algebraic calculations, one arrives at the following expression: 1 1 − (zQ)L+1 p−q t s = 1+ − , (8) 1−Q N N (1 − zQ)
h sts = tanh β
h)z]L+1 1 1 − [(1 − e−β J cosh β × 1+ − h)z] N N [1 − (1 − e−βJ cosh β
(9)
and for |h| J as
stl = sgn(h)
J − eβ |h| cosh β J − eβ|h| sinh β
J )L+1 1 − (ze−β |h| sinh β 1 − × 1+ . (10) J ) N N (1 − ze−β|h| sinh β As can be seen in Fig. 3 (for simplicity, this plot and further ones are for J = kB = 1), the above functions follow a shape that is similar to the one observed for the chain—we have a maximum in s(T ) for |h| < J and an absence of such behavior for |h| J . Having that in mind, it is interesting to examine the dependence of the crossover temperature Tc (i.e., the temperature for which sts takes the maximum) as a function of tree parameters. It can be shown that Tc ≈ 2(ln L)−1 [see Appendix A and Fig. 4(a)], which, using Eq. (6) and assuming zL 1 and z ≈ z − 1, gives Tc ≈
2 . ln ln N − ln ln z
(11)
1.0
s
t
0.8 0.6 0.4 0.2 0.0
FIG. 1. (Color online) A schematic plot of a deterministic tree with z = 3 and L = 3.
0
1
2 T
3
4
FIG. 3. (Color online) Average spin st in a tree (z = 3, L = 12) as a function of temperature for different values of the external magnetic field: h = 0.1 (squares), h = 0.5 (circles), h = 0.9 (upward triangles), h = 1 (downward triangles), and h = 2 (diamonds). Solid lines come from Eqs. (9) and (10). All data points have been averaged over M = 104 realizations.
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3.00 2.00 1.50 1.00 0.70 0.50
1.0 0.5 0.0 ln TC
TC
GROWING SPIN MODEL IN DETERMINISTIC AND . . .
0.5 1.0 1.5
0.30 0.20 0.15
(a)
1
2.0 104
100
(b)
2.5 0
106
5
10 lnln N
L
15
FIG. 4. (Color online) (a) Crossover temperature Tc vs the depth of the tree L. Symbols are numerical solutions of Eq. (9) while the solid line comes from Tc = 2(ln L)−1 . In all cases, z = 10. (b) The logarithm of the crossover temperature Tc vs the double logarithm of the number of nodes in a tree N . Symbols (squares, z = 2; circles, z = 5; triangles, z = 10) are numerical solutions of Eq. (9), while solid lines come from Eq. (11).
A comparison of the crossover temperature obtained by numerically solving Eq. (9) with the predictions of Eq. (11) is shown in Fig. 4(b). For sufficiently large values of N , all data points and analytical curves collapse, indicating a lack of dependence on the parameter z. The behavior of the crossover temperature on the number of nodes in a Cayley tree topology is much different from the results obtained in the case of a chain [12] where Tc ∼ ln−1 (Ne). However, this fact is a natural consequence of the small depth L of the tree, which substitutes in this case the effective range of dynamics influence (N in a chain as compared to ln N in a tree).
n(l) at a distance l from the root node, (ln N/2)l−1 , (l − 1)!
(12)
l=∞ 1 sl n(l), N l=1
(13)
Q−1 p−q 1−N 2 Q . 1−Q
(14)
n(l) = A
√ where A = N is the expected number of children of the root
∞ node, and it can be directly obtained from the condition n=l n(l) = N . As a consequence, the formula for the average spin in a scale-free random tree is given by ssf = which results in ssf =
IV. RANDOM SCALE-FREE TREES
Although finite Cayley tree topology (and especially the underlying branching process) is quite common in the real world, it is rather unreasonable to believe that the structure of social media systems (e.g., forum discussions) is an outcome of the procedure resulting in an exponentially growing number of nodes in consecutive tree layers. According to several previous studies, it is commonly believed that the growth of such systems could be governed by the preferential attachment process [16]. In the case of graphs (the Barab´asi-Albert model), this assumption describes a process in which the probability that the newly appearing node i will attach to a previously present node j is proportional to its degree kj . In general, the incoming node can create m new connections, thus forming a network (i.e., a graph with cycles), however if m = 1, the resulting structure is a scale-free tree whose degree distribution p(k) is described by a power law p(k) ∼ 2k −3 . The scheme for obtaining the average spin value in random scale-free trees is similar to the one presented in the preceding section. However, in this case, the underlying process of tree formation is also of stochastic origin (i.e., not only the dynamics but the topology as well follows a probabilistic rule). We use the results of Bollob´as and Riordan [17] and Szab´o et al. [18] that give the mean-field number of vertices
The above equation can be expressed explicitly for |h| < J as
J −β h)N − 12 e−β J cosh βh ssf cosh β s = tanh β h 1 − (1 − e (15) and for |h| J as
ssf l = sgn(h)
J − eβ |h| cosh β J − eβ|h| sinh β
J )N 12 (e−β |h| sinh βJ −1) . (16) × 1 − (e−β |h| sinh β A comparison of the theoretical predictions given by Eqs. (15) and (16) is shown in Fig. 5(a). It can be shown (Appendix B) using an analytical and numerical approach that the dependence of the crossover temperature Tc on tree size N is best described by Tc ≈
2 , 1 + 43 W ln4eN
(17)
where W (· · · ) is the Lambert W function. A comparison of the crossover temperature obtained by numerically solving Eq. (14) with the predictions of Eq. (17) is shown in Fig. 5(b). It is interesting to add here that for sufficiently large values of x, the function W (x) can be approximated with W (x) ≈ ln x −
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1.0
2.0
(a)
(b)
1.5
0.6
TC
s
sf
0.8
1.0
0.4 0.2 0.0
0
1
2
3
4
1
T
105
1010
1015
1020
1025
1030
N
FIG. 5. (Color online) (a) Average spin ssf in a random scale-free tree (N = 105 ) as a function of temperature for different values of the external magnetic field: h = 0.1 (squares), h = 0.5 (circles), h = 0.9 (upward triangles), h = 1 (downward triangles), and h = 2 (diamonds). Solid lines come from Eqs. (15) and (16). All data points have been averaged over M = 105 realizations. (b) Crossover temperature Tc vs the number of nodes N . Symbols are numerical solutions of Eq. (15), the solid line comes from Eq. (17), while the dashed line comes from Eq. (B4) (see Appendix B for details).
ln ln x, which would suggest that for large values of N the crossover temperature is given by Tc ≈ (2/3 ln ln N − ln 2)−1 .
V. DISCUSSION
The choice of the tree topology has been influenced by three distinct factors: (i) it resembles the actual structure of many online discussions fora, (ii) it enables performing exact analytical calculations, and (iii) it is unambiguous with respect to system dynamics. It has to be emphasized that in contrast to several other models of emotion propagation (usually agent-based, cf. [19–23]), here a node does not correspond to a user but to a shared message. This idea defines the causality introduced into the model—one message can be a reply only to another single message, i.e., you cannot answer several messages in one turn. Therefore, it would be useless to test the behavior of the model on topologies such as BA networks with m > 1 as it would require amending the rules of dynamics. In a similar manner, our model is different from such widely used dynamics as stochastic epidemic models such as the susceptible-infected-recovered (SIR) model, whose evolution has recently been described on tree topology by the belief propagation equations [24]—also in this case, the notion of dynamics is individuals. The issue of the feasibility of analytical calculations that usually require a chain or a tree as the underlying topology can easily be seen in studies regarding the nonconsensus opinion model (NCO) [25], interacting information waves [26], attention competition [27], or the spreading of ideas (i.e., paradigm shifts) [28]. For example, even a simple comparison of those topologies in the case of two competing paradigms leads to interesting and nonintuitive conclusions: the average time T after the initial idea stops spreading in the system scales with the chain size N as N 2 ln N , while in the case of the BA trees the result reads T ∼ N α with α ≈ 2.2. Nonetheless, in all mentioned cases these are agents who are subject to an update procedure, thus the underlying network structure in fact mirrors interactions
between individuals in the system. Our approach, based on the stylized facts with respect to observed emotional homophily in online media [10], makes no assumptions about the actual structure of the interpersonal contacts among the users—instead we introduce global parameters h and T , which characterize the behavior of the participants. The key observation visible both for chain and deterministic or stochastic trees is the issue of the interplay between the initial conditions and the external field. For |h| < J and small values of T , the average magnetization of the system (opinion or valence) is determined by the initial condition: if the first spin (message) is negative, the whole system follows it (i.e., s = −1), regardless of the positive value of the external field parameter. On the other hand, if the first message is positive, then the system’s magnetization is s = 1, which gives on average s = 0 as the initial condition is random. However, if we raise the temperature, more and more spins will align according to the direction of the magnetic field, resulting in a peak of s (for higher values of T , the magnetization decays due to thermal fluctuations). This behavior is observed in all the examined topologies, with the crossover temperature being a slowly decaying function of N .
VI. CONCLUSIONS
In this paper, we extended the previously introduced model of the growing spin chain to the case of deterministic (Cayley) and random scale-free (BA) trees. We have shown that for these topologies, the analytical approach using a Markov chain concept is still valid due to the possibility of calculating the weighted spin on each level of the tree. Similarly to the chain case, the model exhibits a crossover temperature corresponding to maximal magnetization. Unlike the chain case, the crossover temperature decays very slowly [approximately as (ln ln N )−1 compared to (ln N)−1 for the chain], which is connected to the effective diameter of the considered systems.
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10
ACKNOWLEDGMENTS
Tcnum
2
8 6
TC
This work has been supported by the Polish Ministry of Science and Higher Education (Grant No. 0490/IP3/2011/71) as well as by the European Union in the framework of European Social Fund through the Warsaw University of Technology Development Programme, realized by the Center of Advanced Studies.
4 2
APPENDIX A: DERIVATION OF THE CROSSOVER TEMPERATURE FOR FINITE CAYLEY TREES
To get an analytical approximation of Tc , we use Eq. (9) h 1, which gives us the opportunity to set and assume that β h ≈ 1 and tanh β h ≈ β h: cosh β 1 1 − [(1 − e−β J )z]L+1 h 1 + s ≈ β − . (A1) N N [1 − (1 − e−βJ )z] Secondly, let us note that for zL 1 one can approximate Eq. (6) with N ≈ zL /(z − 1). Making use of this fact and assuming N 1, we get (z − 1)(1 − e−β J )L+1 h 1 + s ≈ β . (A2) 1 − (1 − e−βJ )z J 1, we arrive at Finally, also assuming that β
h[1 + (1 − e−β J )L+1 ]. s ≈ β
(A3)
It is interesting to observe here that this result does not depend on the branching factor z that confirms the behavior seen in Fig. 4(b). As the next step, we need to solve ∂s = 0, i.e., ∂T βc (L + 1)e−βc J (1 − e−βc J )L+1 = 1 + (1 − e−βc J )L .
(A4)
By taking the logarithm of both sides and assuming L 1 as well as (1 − e−βc J )L 1, we arrive at βc + Le
−βc J
= ln βc L.
(A6)
kB ln
1
2
2J
. + W − keB
(A7)
Setting J = kB = 1 leads us to the final result,
4
5
6
APPENDIX B: DERIVATION OF THE CROSSOVER TEMPERATURE FOR RANDOM SCALE-FREE TREES
h First, as in the case of Cayley trees, we assume that β h ≈ 1 and tanh β h ≈ β h in 1, which results in setting cosh β Eq. (15):
h 1 − (1 − e−βJ )N − 12 e−β J . s ≈ β (B1)
Finally, assuming that 1 − e−β J ≈ 1 and solving arrive at 1 −β cJ
1 − N−2e
∂s ∂T
1 −βc J ≈ 12 βc J e−βc J N − 2 e ln N.
= 0, we (B2)
At this point, we use the fact that for x 1 we can expand N x/2 with Taylor series as N x/2 ≈ 1 + 12 ln N x + 12 (ln N x)2 , which gives us the final equation βc J − 14 ln N e−βc J ≈ 1
(B3)
that has the solution Tc =
2J , kB 1 + W ln4eN
(B4)
where W (· · · ) is the Lambert W function. Predictions of Eq. (B4) for kB = J = 1 are shown in Fig. 5 with a dashed line, suggesting divergence with the numerical solution Tcnum of Eq. (15), which is caused by the Taylor series expansion. To overcome this issue, we propose a solution in the form Tc (a) =
eL kB
3
FIG. 6. Sum of the squares of displacement between the numerical and theoretical results of the crossover temperature for random scale-free trees vs parameter a [see Eq. (B6)].
has a solution of the form Tc ≈
0
a
(A5)
However, the above equation still fails to be solved by analytical methods. To overcome this problem, we use the approximation ln βc L = ln(2L/kB ) − ln Tc ≈ ln(eL/kB ). Then, the resulting equation βc + Le−βc J = ln(eL/kB )
0
2 , 1 + aW ln4eN
where a is chosen so that the sum 2 Tc (a) − Tcnum
(B5)
(B6)
(A8)
is minimal. Numerical minimization of the above functional gives a ≈ 4/3 (see Fig. 6).
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