PHYSICAL REVIEW E 91 , 022818 (2015)
Explosive synchronization with partial degree-frequency correlation Rafael S. P in to 1 and A lberto S aa2,t 1Institute) de FI sica “Gleb Wataghin," UNICAMP, 13083-859 Campinas, SP, Brazil 2Departamento de Matematica Aplicada, UNICAMP, 13083-859 Campinas, SP Brazil (Received 14 August 2014; revised manuscript received 26 September 2014; published 27 February 2015) Networks of Kuramoto oscillators with a positive correlation between the oscillators frequencies and the degree of their corresponding vertices exhibit so-called explosive synchronization behavior, which is now under intensive investigation. Here we study and discuss explosive synchronization in a situation that has not yet been considered, namely when only a part, typically a small part, of the vertices is subjected to a degree-frequency correlation. Our results show that in order to have explosive synchronization, it suffices to have degree-frequency correlations only for the hubs, the vertices with the highest degrees. Moreover, we show that a partial degree-frequency correlation does not only promotes but also allows explosive synchronization to happen in networks for which a full degree-frequency correlation would not allow it. We perform a mean-field analysis and our conclusions were corroborated by exhaustive numerical experiments for synthetic networks and also for the undirected and unweighed version of a typical benchmark biological network, namely the neural network of the worm Caenorhabditis elegans. The latter is an explicit example where partial degree-frequency correlation leads to explosive synchronization with hysteresis, in contrast with the fully correlated case, for which no explosive synchronization is observed. DOI: 10.1103/PhysRevE.91.022818
PACS number(s): 05.45.Xt. 89.75.He, 89.75.Fb
I. INTRODUCTION Synchronization phenom ena [1,2] m anifest them selves in many and diverse areas. Som e exam ples o f current interest in clude the biology o f interacting fireflies [3], cellular processes in populations o f yeast [4], audience clapping [5], and pow er grids [6], am ong many others. Perhaps the m ost successful attem pt to understand synchronization theoretically is the K uram oto model [7]. It has been heavily em ployed in the past few decades as the paradigm to study the onset o f synchronized behavior am ong nonidentical interacting agents, since it is one o f the few m odels, together with som e generalizations [8], that captures the essential m echanism s o f synchronization and are still am enable to som e analytical approaches [9,10]. T he so-called K uram oto m odel consists o f an ensem ble o f N oscillators, w ith phases and natural frequencies given, respectively, by 0,- and placed on the vertices o f a com plex netw ork [11], T he netw ork topology is described by the usual sym m etric adjacency m atrix A y, with elem ents A y = 1 if the vertices i and j are connected by an edge and A y = 0 otherw ise. T he oscillators interact according to the equation d0S *, ~ = coj + A. ^ Ay sin(0y - 0/),
(1)
1= '
w here 7. is the coupling constant. The global state o f the oscillators (1) can be conveniently described by using the order param eter r defined as
e-* = _ L y v \ N ^ 7=1
1539-3755/2015/91 (2)/022818(10)
N
cot = ki = Y ^ A i j .
(3)
7=1 ( 2)
w hich corresponds to the centroid o f the phases if they are considered as a sw arm o f points m oving around the unit circle.
1rsoaresp @gmail.com [email protected]
For incoherent m otion, the phases are scattered on the circle hom ogeneously and r ss N ~ l/2 for large A , as a consequence o f the central lim it theorem , w hile for a synchronized state they should move in a single lum p and, consequently, r «s I. T he general picture for the Kuram oto model is that, with very few exceptions, for small coupling strength X there is no synchronization and therefore r % 0 for large N. However, as one increases continuously the coupling constant X, after passing a critical value Xc, w hose precise value depends both on the topology o f the netw ork and on the natural frequencies &>, distribution, the order param eter r starts to increase continuously. A sort o f sm ooth second-order phase transition from incoherence to synchronization takes place here. Very recently, a new behavior for the K uram oto model was discovered. In Ref. [12], it was shown that in scale-free networks, w hen there is a positive correlation betw een the natural frequencies o f the oscillators and the degree o f the vertices on w hich they lie, an abrupt first-order transition from incoherence to synchronization, called explosive synchroniza tion (ES), takes place. Typically, we also have a hysteresis behavior, and the forw ard and backw ard continuations (r versus X diagram ) do not coincide. In the sim plest case exhibiting ES, the natural frequency a>, o f a given oscillator equals its vertex degree kiy
Explosive synchronization has also been observed in many other system s, as the retarded K uram oto m odel [13], the second-order K uram oto model [14], in networks o f FitzH ughN agum o oscillators [15], and also in a netw ork o f chaotic Rosller oscillators [16], allow ing, in this case, an experim ental observation o f ES in electronic circuits. A m ean-field ap proxim ation to explosive synchronization was applied in Ref. [17]. We can also m ention that a relation betw een explosive
022818-1
©2015 American Physical Society
PHYSICAL REVIEW E 91, 022818 (2015)
RAFAEL S. PINTO AND ALBERTO SAA
percolation [18] and the generalized Kuramoto model pro posed in Ref. [19] was discussed in Ref. [20]. We stress that there are other mechanisms capable of inducing first-order phase transitions. For instance, in Ref. [21], an analytical treatment for first-order phase transitions for synchroniza tion is presented for the case of a Kuramoto model with uniform distribution of the natural frequencies. The situation corresponding to ES differs, whereby the frequencies are not randomly distributed but subjected to the restriction (3). Many works have recently been devoted to understand and to generalize the occurrence of explosive synchronization to other settings as, for instance, for weighted networks [19,22], where the coupling constant is no longer the same for all vertices, but its value varies for each pair of connected oscillators and may depend on the values of their natural fre quencies. In Ref. [23], starting from a given natural frequencies distribution, an algorithm was described to constaict a network of oscillator exhibiting ES. However, in all these cases, rather strong conditions to obtain ES are assumed. A first step to overcome this limitation was proposed in Ref. [24], where it is shown that the addition of a quenched disorder to the degree-frequency correlation not only could maintain the ES but also could induce ES in some kinds of networks without heterogeneous degree distributions. In this paper, we take another route and investigate ES in a Kuramoto model where only a few of the vertices have a degree-frequency correlation. We notice that the problem of partial correlation was briefly analyzed in Ref. [12] for the case of random correlations. They have shown that for a scalefree network with exponent y = 2.4, no ES was seen when less than around 50% of the vertices had degree-frequency correlation. By means of a mean-field analysis, corroborated by exhaustive numerical experiments, we show that, in order to have ES, it suffices that the degree-frequency correlation holds only for the hubs, the vertices with the highest degree. We have found ES, for instance, in Barabasi-Albert networks with only 10% of the vertices subjected to degree-frequency correlation. More interestingly, we show that by restricting the degree-frequency correlation to the hubs, this not only promotes ES but also allows it to happen in networks where the full degree-frequency correlation would not allow it. As we will see, this is the case, for instance, of a typical benchmark biological network in the field: the neural network of the worm Caenorhabditis elegans.
distribution. Notice that dcoG(co,k) = P(k) and
/
dk G(co,k) - P(co)H(co —k*) + ag(w),
f
(4) where £2
ft) + Xkr sin ) -
: exp
(a) - (w))2 (31)
(To^/lrt
2°o
we have (see the appendix for the calculation details) p.v.
1 [tF _ ( (co) — Q °° dco ,, --------£ (w) ' = — ,/-e rfi f t ) — £2 ./ 0o >/2oo x exp
( 22)
(7 i(L r))2 + ( / 2(L r))2 ’
p.v.
42
X2 =
(21 )
for r 0. The calculation details for I\ and I2 are presented in the appendix. T he corresponding m ean-field approxim ation [25] for the critical coupling Xc in arises from the lim it r —> 0 + o f equation (21),
lim I\(Xr)
^
T he other cases we will consider here are those ones with vanishing (co). For these cases, we have typically £2 < k*. We can evaluate easily the second integral in (23), for instance, in the case o f a hom ogeneous g(co) w ith null average and com pact support, i.e., for
(20)
r-> 0+
(26)
In this case, w e have for a BA netw ork with f2 < k*
eQ—kkr
e
k* — £2
For this case, both integrals in (23) vanish, leading to the follow ing critical coupling Xc for a BA netw ork w ith £2 = {co) :
ft) — 1 2 s
// £2— , kkr
/:
log
co — i2 — Xkr s in 0
'•Q+kkr
+
£2
kP (k)dk
is a norm alization constant. From (14) and (18), we have for the stationary regim e 1 rr*oo oo r ee1*0 'YV = — i/ d k k — £2
r
— ds, J —00 s So
1 kn~2(k - £2)
ds
(A 16)
1 £2
1 kn~3(k — £2)
v
X
(A22)
1 k"~2
(A23)
kl~n
(A24)
for n > 3. This recurrence can be easily solved. Taking into account that A3 = £2 1 log
k*
Q
^
i
(A25)
we have, finally, A„ = £22—n log
k* — £2
hAk
(A26)
valid for any integer n > 2, from which (33) follows straight forwardly. The evaluation of (A22) for noninteger values of y can be done by exploiting, for instance, the series representation of (k - £2)~‘ for k > £2. We have OO
- S o J ----- s0g(x),
k 2~y dk £2
A„ - £2 | A„_,
and that g'W
(A21)
we have
(A 15)
oo e-jr(s+s0)2 /
^■ _ _ ds ) =■ —Tte S°erfi(s0) So
• • « ;
(A 14)
which one can calculate by using the trick of differentiating under the integral sign. Notice that I — g( 1) with g(x) = p.v.
J
and (32) follows for the standard Gaussian distribution (31). Finally, we have the evaluation of the integral
(A13)
with p given by (24). Notice that, as expected, the limit (A 14) vanishes for symmetrical g around £2, i.e., for g(£2 + &>) = g(& ~ a>)The evaluation of the principal value (32) for the standard Gaussian distribution involves the evaluation of the principal value of the integral I
p.v.
which appears in the first term of (A4) for a power-law degree distribution P{k) oc k ~y, with y > 2. Let us consider first the case of integer y = n > 2. Since
Since the integration interval in k is bounded, one can take the limit r -» 0+ directly. By using essentially the same approximation (A ll), we have lim /f(k r) = p
(A20)
is the standard imaginary error function. The integration constant A can be determined from the requirement that g(x) — 0 for so = 0, leading to A = 0. Taking x = 1 one gets
(A12)
— where p.v. stands to the Cauchy principal value for the integral. Notice that a finite limit for this integral will typically require that £2 k,. In order to evaluate I^(Xr) for r —>• 0+, let us restore the original variable u>in (A 10) I?(Xr)
dt
/ >
(A ll)
first in (A9), valid for r 0+ and k ^ £2. Assuming P(k) regular at k = £2, we have that (A9) can be approximated in the limit r -» 0+ by lim I?(Xr) o+
(A19)
where erfi(x)
1
/ l* -n | l Xkr
g(x) = e 5f,x[A - jrerfi(s0V7)].
(A10)
Since we are interested mainly in the limit r -»• 0+ for both integrals, let us consider the approximation 1 Xr
(A18)
(A17) 022818-9
(
AY = k l ~ y y y ^ t + y -2 1=0
(A27)
PHYSICAL REVIEW E 91, 022818 (2015)
RAFAEL S. PINTO AND ALBERTO SAA valid for k* > £2. For 0 < k„ < £2 w e obtain analogously
N otice that the principal value o f the integral / d c o j ^ for
(A28)
the case g(a>) a co~Y em ployed in Sec. Ill can be evaluated analogously.
[1] S. H. Strogatz, SYNC: The Emerging Science o f Spontaneous Order, 1st ed. (Hyperion, New York, 2003). [2] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge Univer sity Press, Cambridge, 2003). [3] J. Buck, Q. Rev. Biol. 63, 265 (1988). [4] S. De Monte, F. Ovidio, S. Danp, and P. G. Sorensen, Proc. Natl. Acad. Sci. U.S.A. 104, 18377 (2007). [5] Z. Neda, E. Ravasz, Y. Brechet, T. Vicsek, and A. L. Barabasi, Nature 403, 849 (2000). [6] A. E. Motter, S. A. Myers, M. Anghel, and T. Nishikawa, Nat. Phys. 9, 191 (2013). [7] Y. Kuramoto, in Proceedings o f the International Symposium on Mathematical Problems in Theoretical Physics, University o f Kyoto, Japan, Vol. 39, Lecture Notes in Physics, edited by H. Araki (Springer-Verlag, Heidelberg, Germany, 1975), p. 420. [8] J. A. Acebron, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, Rev. Mod. Phys. 77, 137 (2005). [9] S. H. Strogatz, Physica D 143, 1 (2000). [10] E. Ott and T. M. Antonsen, Chaos 18, 037113 (2008) [11] A. Arenas, A. DIaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, Phys. Rep. 469, 93 (2008). [12] J. Gomez-Gardenes, S. Gomez, A. Arenas, and Y. Moreno, Phys. Rev. Lett. 106, 128701 (2011). [13] Thomas Kaue Dal’Maso Peron and F. A. Rodrigues, Phys. Rev. E 86, 016102 (2012). [14] P. Ji, T. K. D. M. Peron, P. J. Menck, F. A. Rodrigues, and J. Kurths, Phys. Rev. Lett. 110, 218701 (2013). [15] H. Chen, G. He, F. Huang, C. Shen, and Z. Hou, Chaos 23, 033124(2013).
[16] 1. Leyva, R. Sevilla-Escoboza, J. M. Buldii, I. Sendina-Nadal, J. Gomez-Gardenes, A. Arenas, Y. Moreno, S. Gomez, R. Jaimes-Reategui, and S. Boccaletti, Phys. Rev. Lett. 108,168702 ( 2012 ). [17] Thomas Kaue DaPMaso Peron and F. A. Rodrigues, Phys. Rev. E 86, 056108 (2012). [! 8] D. Achlioptas, R. M. Souza, and J. Spencer, Science 323, 1453 (2009). [19] X. Zhang, X. Hu, J. Kurths, and Z. Liu, Phys. Rev. E 88, 010802(R) (2013). [20] X. Zhang, Y. Zou, S. Boccaletti, and Z. Liu, Sci. Rep. 4, 5200 (2014). [21] D. Pazo, Phys. Rev. E 72, 046211 (2005). [22] I. Leyva, I. Sendina-Nadal, J. A. Almendral, A. Navas, S. Olmi, and S. Boccaletti, Phys. Rev. E 88, 042808 (2013). [23] I. Leyva, A. Navas, I. Sendina-Nadal, J. A. Almendral, J. M. Buldu, M. Zanin, D. Papo, and S. Boccaletti, Sci. Rep. 3, 1281 (2013). [24] P. S. Skardal and A. Arenas, Phys. Rev. E 89, 062811 (2014). [25] T. Ichinomiya, Phys. Rev. E 70, 026116 (2004). [26] E. Jones, E. Oliphant, P. Peterson P el a!., SciPy: Open Source Scientific Tools fo r Python, 2001, http://www.scipy.org/ [online; accessed 2014-08-10]. [27] J. Gomez-Gardenes and Y. Moreno, Phys. Rev. E 73, 056124 (2006). [28] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art o f Scientific Computing (Cambridge University Press, Cambridge, 2007). [29] D. J. Watts and S. H. Strogatz, Nature 393, 440 (1998). [30] J. Gomez-Gardenes, Y. Moreno, and A. Arenas, Phys. Rev. Lett. 98, 034101 (2007).
(I)
1
Ay = - k l - y J2 1 - Y + 3 ' 1=0
022818-10
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Explosive synchronization with partial degree-frequency correlation Rafael S. P in to 1 and A lberto S aa2,t 1Institute) de FI sica “Gleb Wataghin," UNICAMP, 13083-859 Campinas, SP, Brazil 2Departamento de Matematica Aplicada, UNICAMP, 13083-859 Campinas, SP Brazil (Received 14 August 2014; revised manuscript received 26 September 2014; published 27 February 2015) Networks of Kuramoto oscillators with a positive correlation between the oscillators frequencies and the degree of their corresponding vertices exhibit so-called explosive synchronization behavior, which is now under intensive investigation. Here we study and discuss explosive synchronization in a situation that has not yet been considered, namely when only a part, typically a small part, of the vertices is subjected to a degree-frequency correlation. Our results show that in order to have explosive synchronization, it suffices to have degree-frequency correlations only for the hubs, the vertices with the highest degrees. Moreover, we show that a partial degree-frequency correlation does not only promotes but also allows explosive synchronization to happen in networks for which a full degree-frequency correlation would not allow it. We perform a mean-field analysis and our conclusions were corroborated by exhaustive numerical experiments for synthetic networks and also for the undirected and unweighed version of a typical benchmark biological network, namely the neural network of the worm Caenorhabditis elegans. The latter is an explicit example where partial degree-frequency correlation leads to explosive synchronization with hysteresis, in contrast with the fully correlated case, for which no explosive synchronization is observed. DOI: 10.1103/PhysRevE.91.022818
PACS number(s): 05.45.Xt. 89.75.He, 89.75.Fb
I. INTRODUCTION Synchronization phenom ena [1,2] m anifest them selves in many and diverse areas. Som e exam ples o f current interest in clude the biology o f interacting fireflies [3], cellular processes in populations o f yeast [4], audience clapping [5], and pow er grids [6], am ong many others. Perhaps the m ost successful attem pt to understand synchronization theoretically is the K uram oto model [7]. It has been heavily em ployed in the past few decades as the paradigm to study the onset o f synchronized behavior am ong nonidentical interacting agents, since it is one o f the few m odels, together with som e generalizations [8], that captures the essential m echanism s o f synchronization and are still am enable to som e analytical approaches [9,10]. T he so-called K uram oto m odel consists o f an ensem ble o f N oscillators, w ith phases and natural frequencies given, respectively, by 0,- and placed on the vertices o f a com plex netw ork [11], T he netw ork topology is described by the usual sym m etric adjacency m atrix A y, with elem ents A y = 1 if the vertices i and j are connected by an edge and A y = 0 otherw ise. T he oscillators interact according to the equation d0S *, ~ = coj + A. ^ Ay sin(0y - 0/),
(1)
1= '
w here 7. is the coupling constant. The global state o f the oscillators (1) can be conveniently described by using the order param eter r defined as
e-* = _ L y v \ N ^ 7=1
1539-3755/2015/91 (2)/022818(10)
N
cot = ki = Y ^ A i j .
(3)
7=1 ( 2)
w hich corresponds to the centroid o f the phases if they are considered as a sw arm o f points m oving around the unit circle.
1rsoaresp @gmail.com [email protected]
For incoherent m otion, the phases are scattered on the circle hom ogeneously and r ss N ~ l/2 for large A , as a consequence o f the central lim it theorem , w hile for a synchronized state they should move in a single lum p and, consequently, r «s I. T he general picture for the Kuram oto model is that, with very few exceptions, for small coupling strength X there is no synchronization and therefore r % 0 for large N. However, as one increases continuously the coupling constant X, after passing a critical value Xc, w hose precise value depends both on the topology o f the netw ork and on the natural frequencies &>, distribution, the order param eter r starts to increase continuously. A sort o f sm ooth second-order phase transition from incoherence to synchronization takes place here. Very recently, a new behavior for the K uram oto model was discovered. In Ref. [12], it was shown that in scale-free networks, w hen there is a positive correlation betw een the natural frequencies o f the oscillators and the degree o f the vertices on w hich they lie, an abrupt first-order transition from incoherence to synchronization, called explosive synchroniza tion (ES), takes place. Typically, we also have a hysteresis behavior, and the forw ard and backw ard continuations (r versus X diagram ) do not coincide. In the sim plest case exhibiting ES, the natural frequency a>, o f a given oscillator equals its vertex degree kiy
Explosive synchronization has also been observed in many other system s, as the retarded K uram oto m odel [13], the second-order K uram oto model [14], in networks o f FitzH ughN agum o oscillators [15], and also in a netw ork o f chaotic Rosller oscillators [16], allow ing, in this case, an experim ental observation o f ES in electronic circuits. A m ean-field ap proxim ation to explosive synchronization was applied in Ref. [17]. We can also m ention that a relation betw een explosive
022818-1
©2015 American Physical Society
PHYSICAL REVIEW E 91, 022818 (2015)
RAFAEL S. PINTO AND ALBERTO SAA
percolation [18] and the generalized Kuramoto model pro posed in Ref. [19] was discussed in Ref. [20]. We stress that there are other mechanisms capable of inducing first-order phase transitions. For instance, in Ref. [21], an analytical treatment for first-order phase transitions for synchroniza tion is presented for the case of a Kuramoto model with uniform distribution of the natural frequencies. The situation corresponding to ES differs, whereby the frequencies are not randomly distributed but subjected to the restriction (3). Many works have recently been devoted to understand and to generalize the occurrence of explosive synchronization to other settings as, for instance, for weighted networks [19,22], where the coupling constant is no longer the same for all vertices, but its value varies for each pair of connected oscillators and may depend on the values of their natural fre quencies. In Ref. [23], starting from a given natural frequencies distribution, an algorithm was described to constaict a network of oscillator exhibiting ES. However, in all these cases, rather strong conditions to obtain ES are assumed. A first step to overcome this limitation was proposed in Ref. [24], where it is shown that the addition of a quenched disorder to the degree-frequency correlation not only could maintain the ES but also could induce ES in some kinds of networks without heterogeneous degree distributions. In this paper, we take another route and investigate ES in a Kuramoto model where only a few of the vertices have a degree-frequency correlation. We notice that the problem of partial correlation was briefly analyzed in Ref. [12] for the case of random correlations. They have shown that for a scalefree network with exponent y = 2.4, no ES was seen when less than around 50% of the vertices had degree-frequency correlation. By means of a mean-field analysis, corroborated by exhaustive numerical experiments, we show that, in order to have ES, it suffices that the degree-frequency correlation holds only for the hubs, the vertices with the highest degree. We have found ES, for instance, in Barabasi-Albert networks with only 10% of the vertices subjected to degree-frequency correlation. More interestingly, we show that by restricting the degree-frequency correlation to the hubs, this not only promotes ES but also allows it to happen in networks where the full degree-frequency correlation would not allow it. As we will see, this is the case, for instance, of a typical benchmark biological network in the field: the neural network of the worm Caenorhabditis elegans.
distribution. Notice that dcoG(co,k) = P(k) and
/
dk G(co,k) - P(co)H(co —k*) + ag(w),
f
(4) where £2
ft) + Xkr sin ) -
: exp
(a) - (w))2 (31)
(To^/lrt
2°o
we have (see the appendix for the calculation details) p.v.
1 [tF _ ( (co) — Q °° dco ,, --------£ (w) ' = — ,/-e rfi f t ) — £2 ./ 0o >/2oo x exp
( 22)
(7 i(L r))2 + ( / 2(L r))2 ’
p.v.
42
X2 =
(21 )
for r 0. The calculation details for I\ and I2 are presented in the appendix. T he corresponding m ean-field approxim ation [25] for the critical coupling Xc in arises from the lim it r —> 0 + o f equation (21),
lim I\(Xr)
^
T he other cases we will consider here are those ones with vanishing (co). For these cases, we have typically £2 < k*. We can evaluate easily the second integral in (23), for instance, in the case o f a hom ogeneous g(co) w ith null average and com pact support, i.e., for
(20)
r-> 0+
(26)
In this case, w e have for a BA netw ork with f2 < k*
eQ—kkr
e
k* — £2
For this case, both integrals in (23) vanish, leading to the follow ing critical coupling Xc for a BA netw ork w ith £2 = {co) :
ft) — 1 2 s
// £2— , kkr
/:
log
co — i2 — Xkr s in 0
'•Q+kkr
+
£2
kP (k)dk
is a norm alization constant. From (14) and (18), we have for the stationary regim e 1 rr*oo oo r ee1*0 'YV = — i/ d k k — £2
r
— ds, J —00 s So
1 kn~2(k - £2)
ds
(A 16)
1 £2
1 kn~3(k — £2)
v
X
(A22)
1 k"~2
(A23)
kl~n
(A24)
for n > 3. This recurrence can be easily solved. Taking into account that A3 = £2 1 log
k*
Q
^
i
(A25)
we have, finally, A„ = £22—n log
k* — £2
hAk
(A26)
valid for any integer n > 2, from which (33) follows straight forwardly. The evaluation of (A22) for noninteger values of y can be done by exploiting, for instance, the series representation of (k - £2)~‘ for k > £2. We have OO
- S o J ----- s0g(x),
k 2~y dk £2
A„ - £2 | A„_,
and that g'W
(A21)
we have
(A 15)
oo e-jr(s+s0)2 /
^■ _ _ ds ) =■ —Tte S°erfi(s0) So
• • « ;
(A 14)
which one can calculate by using the trick of differentiating under the integral sign. Notice that I — g( 1) with g(x) = p.v.
J
and (32) follows for the standard Gaussian distribution (31). Finally, we have the evaluation of the integral
(A13)
with p given by (24). Notice that, as expected, the limit (A 14) vanishes for symmetrical g around £2, i.e., for g(£2 + &>) = g(& ~ a>)The evaluation of the principal value (32) for the standard Gaussian distribution involves the evaluation of the principal value of the integral I
p.v.
which appears in the first term of (A4) for a power-law degree distribution P{k) oc k ~y, with y > 2. Let us consider first the case of integer y = n > 2. Since
Since the integration interval in k is bounded, one can take the limit r -» 0+ directly. By using essentially the same approximation (A ll), we have lim /f(k r) = p
(A20)
is the standard imaginary error function. The integration constant A can be determined from the requirement that g(x) — 0 for so = 0, leading to A = 0. Taking x = 1 one gets
(A12)
— where p.v. stands to the Cauchy principal value for the integral. Notice that a finite limit for this integral will typically require that £2 k,. In order to evaluate I^(Xr) for r —>• 0+, let us restore the original variable u>in (A 10) I?(Xr)
dt
/ >
(A ll)
first in (A9), valid for r 0+ and k ^ £2. Assuming P(k) regular at k = £2, we have that (A9) can be approximated in the limit r -» 0+ by lim I?(Xr) o+
(A19)
where erfi(x)
1
/ l* -n | l Xkr
g(x) = e 5f,x[A - jrerfi(s0V7)].
(A10)
Since we are interested mainly in the limit r -»• 0+ for both integrals, let us consider the approximation 1 Xr
(A18)
(A17) 022818-9
(
AY = k l ~ y y y ^ t + y -2 1=0
(A27)
PHYSICAL REVIEW E 91, 022818 (2015)
RAFAEL S. PINTO AND ALBERTO SAA valid for k* > £2. For 0 < k„ < £2 w e obtain analogously
N otice that the principal value o f the integral / d c o j ^ for
(A28)
the case g(a>) a co~Y em ployed in Sec. Ill can be evaluated analogously.
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(I)
1
Ay = - k l - y J2 1 - Y + 3 ' 1=0
022818-10
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