Dissipative stochastic sandpile model on small world network : properties of non-dissipative and dissipative avalanches
Abstract
A dissipative stochastic sandpile model is constructed and studied on small world networks in one and two dimensions with different shortcut densities , where represents regular lattice and represents random network. The effect of dimension, network topology and specific dissipation mode (bulk or boundary) on the the steady state critical properties of non-dissipative and dissipative avalanches along with all avalanches are analyzed. Though the distributions of all avalanches and non-dissipative avalanches display stochastic scaling at and mean-field scaling at , the dissipative avalanches display non trivial critical properties at and in both one and two dimensions. In the small world regime (), the size distributions of different types of avalanches are found to exhibit more than one power law scaling with different scaling exponents around a crossover toppling size . Stochastic scaling is found to occur for and the mean-field scaling is found to occur for . As different scaling forms are found to coexist in a single probability distribution, a coexistence scaling theory on small world network is developed and numerically verified.
pacs:
89.75.-k,05.65.+b,64.60.aq,64.60.avI Introduction
Power-law scaling in many natural phenomena such as earthquakes Chen et al. (1991), forest fires Drossel and Schwabl (1992); *drosselPRL93, biological evolution Bak and Sneppen (1993), droplet formation Plourde et al. (1993), superconducting avalanches Field et al. (1995), etc. are found to be outcome of self organized criticality (SOC) Bak (1996); *jensen; *christensen-moloney; *pruessner which refers to the intrinsic tendency of a wide class of slowly driven systems to evolve spontaneously to a non equilibrium steady state. At the same time, self-organization on complex structures or networks are found to appear very often in nature. For example, avalanche mode of activity in the neural network of brain de Arcangelis et al. (2006); *hesseFSN14, earthquake dynamics on the network of faults in the crust of the earth Lise and Paczuski (2002), rapid rearrangement of coronal magnetic field network Hughes et al. (2003), propagation of information through a network with a malfunctioning router causing the breakdown of the Internet network Motter and Lai (2002), blackout of the electric power grid Carreras et al. (2004) and many others. On the other hand, small world network (SWN) Watts and Strogatz (1998) not only interpolates between the regular lattice and the random network but also it preserves both the properties of regular lattice and random network, namely high “clustering-coefficient”(concept of neighborhood) and “small-world-effect” (small average shortest distance between any two nodes) respectively. It is always intriguing to study the models of SOC on networks such as SWN.
Sandpile, a prototypical model to study SOC introduced by Bak, Tang, and Wiesenfeld (BTW) Bak et al. (1987); *btwPRA88. Though BTW on regular lattice gives rise to anomalous (multi) scaling DeMenech et al. (1998); *tebaldiPRL99; Lübeck (2000), it shows a mean-field scaling Christensen and Olami (1993); Bonabeau (1995); Goh et al. (2003); Bhaumik and Santra (2013) when studied on random network. A transition from non-critical to critical behaviour was reported in BTW type sandpile model on SWN in one dimension (d) Lahtinen et al. (2005) whereas a continuous crossover to mean-field behaviour was reported for the same model on SWN in two dimension (d) de Arcangelis and Herrmann (2002). However recent study of BTW model on SWN in d shows the co-existence of more than one scaling forms in the distributions of avalanche properties Bhaumik and Santra (2013). On the other hand, stochastic sandpile models (SSM) on regular lattice, which incorporates random distribution of sand grains during avalanche, exhibit a scaling behaviour with definite critical exponents that follows finite size scaling (FSS) and define a robust universality class called Manna class Manna (1991); *dharPHYA99a. More insight in avalanche size distribution statistics were obtained by classifying the avalanches into dissipative and non-dissipative avalanches. The size distribution of dissipative avalanches of BTW sandpile in d was found to follow power law scaling with a definite exponent that does not obey FSS Drossel (2000). Later, Dickman and Campelo Dickman and Campelo (2003) showed both in one and two dimensions that the dissipative and non-dissipative avalanches of SSM on regular lattice obey different FSS behaviour with certain logarithmic correction beside the power law scaling with different exponents. However, there are not many studies that report the critical behaviour of SSM and specially that of the stochastic dissipative avalanches on networks. It is then intriguing to study avalanche size distribution of dissipative and non-dissipative avalanches of a stochastic sandpile model on SWN which interpolates regular lattice and random network, and verify whether all such scaling forms would be preserved or not under bulk dissipation mode.
In this paper, a dissipative stochastic sandpile model (DSSM) is constructed on SWN and studied as a function of shortcut density in both one and two dimensions. The distribution functions of the steady state avalanche properties as well as those of dissipative and non-dissipative avalanches on regular lattice () and random network () are found to display several interesting non-trivial features which are not reported before. Moreover, in the small world regime with intermediate ( to ) Newman et al. (2010), the steady state avalanche properties exhibit coexistence of the SSM scaling and the mean-field scaling in a single distribution depending on the avalanche sizes. A coexistence scaling theory is developed and numerically verified.
Ii The Model
SWN is generated both on a d linear lattice and on a d square lattice by adding shortcuts between any two randomly chosen lattice sites which will be referred as nodes later on. The shortcut density is defined as the number of added shortcuts per existing bond ( bonds are present in a -dimensional lattice of linear size with periodic boundary conditions (PBC) and without shortcuts) and is given by . Care has been taken to avoid self-edges of any node and multi-edges between any two nodes. To study sandpile dynamics on an SWN, first an SWN is generated for a particular value of and it is then driven by adding sand grains, one at a time, to randomly chosen nodes. If the height of the sand column at the th node becomes greater than or equal to the predefined threshold value , which is equal to here, the th node topples and the height of the sand column of the th node will be reduced by . The sand grains toppled are then distributed among two of its randomly selected adjacent nodes which are connected to the toppled node either by shortcuts or by nearest neighbour bonds. During distribution of the sand grains PBC is applied. Hence, there is no open boundary in the system where dissipation of sand grains could occur. A dissipation factor is then introduced during transport of a sand grain from one node to another to avoid overloading of the system. The toppling rule of th critical node in this DSSM on SWN then can be represented as
(1) |
where is two randomly selected nodes out of adjacent nodes of the th node, is a random number uniformly distributed over . In this distribution rule, an adjacent node may receive both the sand grains. If the toppling of a node causes some of the adjacent nodes unstable, subsequent toppling follows on these unstable nodes. The process continues until there is no unstable node present in the system. These toppling activities lead to an avalanche. During an avalanche no sand grain is added to the system.
For a given SWN, is taken as , where is the average number of steps required for a random walker to reach the lattice boundary (without PBC) starting from an arbitrary lattice site. There exits a characteristic length where is the dimensionality of the lattice, below which SWN belongs to the “large world”, the regular lattice regime, and beyond which it behaves as “small world”, the random network regime Newman and Watts (1999a); *newmanPLA99; de Mendes et al. (2000). The asymptotic behaviour of with and is given by
(2) |
for a dimensional SWN. It has diffusive behaviour for and super-diffusive behaviour for . The above scaling form is numerically verified in Ref. Bhaumik and Santra (2013). The dissipation factor for a given is determined using numerically estimated values of . A few values of are listed in Table. 1 for d and d lattices.
Iii Results and discussion
Extensive computer simulations are performed to study the dynamics of DSSM on SWN in d and d. After a transient period, the system evolves to a steady state which corresponds to equal currents of sand influx and outflux resulting constant average height of the sand columns. Critical properties of DSSM on SWN are characterized studying various avalanche properties in the steady state at different values of and system size . The maximum lattice size used for d is and that for d is . Data are averaged over avalanches collected on different SWN configurations for a given and . The information of an avalanche is kept by storing the number of toppling of every node in an array which was set to zero initially. All geometrical properties of an avalanche such as avalanche size , avalanche area , etc. can be estimated in terms of as given below
(3) |
for all .
iii.1 Toppling Surface
The values of the toppling number of an avalanche at different nodes of SWN define a surface called toppling surface Ahmed and Santra (2010) which serves as an important geometrical quantity to visualize an avalanche. The toppling surfaces for typical large avalanches in the steady state, generated on a d lattice of size , are presented for and respectively in Figs. 1(a) and 1(b). Toppling surfaces generated on a d square lattice of size are presented in Fig. 1(c) for and for in Fig. 1(d). In both the dimensions, the maximum height of the surfaces on regular lattice () are much higher than that on random network (). Though the maximum height is very less in d for , all the lattice site toppled more than once whereas in d, the toppling surface on random network consists of mostly singly toppled sites, only of the sites toppled more than once. The toppling surfaces are found very different on regular lattice and random network in different dimensions.
iii.2 Moment analysis at
The critical steady state of sandpile model is mostly characterized by power-law scaling of the probability distributions of avalanche size () occurring in the steady state. For a given and , the probability to have an avalanche of size is given by where is the number of avalanches of size out of total number of avalanches generated at the steady state. The distribution of follows a power law scaling with a well defined exponent and obeys FSS DeMenech et al. (1998); *tebaldiPRL99. The FSS form of the probability distribution of in DSSM is given by
(4) |
where is a dependent scaling function and is the capacity dimension. Very often the power law scaling is found to sustain over a short range of avalanche sizes and hinders precise extraction of the exponent from the slope of the plot of against in double logarithmic scale. A more reliable estimate of the exponent can be made employing moment analysis Karmakar et al. (2005); Lübeck (2000). For a given , the th moment of is defined as
(5) |
where,
(6) |
is the moment scaling function for (for , ). Values of are estimated from the slope of the plots of versus in double logarithmic scale for equidistant values of between and . The value of can be measured from the saturated value of in large limit. The derivative is determined numerically by finite-difference method. Once is known the exponent can be estimated from Eq. (6) using the value of .
All avalanches : of all the avalanches for various values of are presented in Fig. 2 for d and d for and . Reasonable power law scaling are observed for these extreme values of in both the dimensions. The flat tail in for in d is due to large dissipative avalanches which will be discussed later separately. Employing moment analysis, values of and are estimated at all four situations. For , estimates of are found to be and for d and d respectively. Since for , in both the dimensions, the values of estimated from Eq. (6) are in d and in d. As expected, the exponents are found very close to the reported values for SSM on regular lattice in respective dimensions Dickman and Campelo (2003); Dickman et al. (2000); Huynh et al. (2011); *huynhPRE12, for instance in d , and in d , for SSM. Whereas for , the values of are found to be and in d and d respectively. In d, the avalanches on random network () consist mostly single toppled nodes, hence is expected whereas the value of in d suggests that the avalanches consist of multiple toppled nodes. In both the dimensions, the value of for is , the mean-field value as obtained in branching processes Christensen and Olami (1993); Bonabeau (1995); Goh et al. (2003). The values of the exponents are listed in Table 2. The FSS form of is verified by plotting the scaled distribution against the scaled variable in the respective lower inset of Fig. 2 using the respective values of the critical exponents obtained.
Non-dissipative and dissipative avalanches: Avalanches are now classified into non-dissipative and dissipative avalanches. During the evolution, a dissipative avalanche must dissipate at least a sand grain once whereas no sand grain be dissipated in a non-dissipative avalanche. The avalanche size distribution can be written in terms of and , the distributions of non-dissipative and dissipative avalanches, as
(7) |
with
(8) |
where and are number of non-dissipative and dissipative avalanches of size out of total avalanches. First, the analysis of non-dissipative avalanches is given and then that of dissipative avalanches is presented.
The FSS form of the distribution is assumed to be
(9) |
where is a scaling function, and are the respective exponents. for and are plotted in Fig. 3 for several values of for both d and d. Performing moment analysis, the values of are found as for and for in both d and d. It could be recalled here that the dissipation factor is chosen from the inverse of . On an average the avalanche of size must dissipate at least one sand grain (the factor is for one toppling consists two sand transfer). Since as due to diffusive behaviour of random walker on regular lattice and as for super-diffusive behaviour of random walker on random network Bhaumik and Santra (2013), the cutoff of must scales with in the same way as scales with . Knowing the values of and , the values of are estimated. The values of are found as for and for in d. Accordingly, for and for in d. The power law scaling of is found similar to that of as the values of and are found more or less same for both the distributions for and . Whereas in d, the values of the exponents are found as for (since ) and for as . On regular lattice it is the SSM result whereas on random network it is the mean-field result. The values of and for non-dissipative avalanches are listed in Table 2. Using the values of and , a reasonable data collapse is obtained for as shown in the lower insets of Fig. 3.
non-dissipative | dissipative | all | ||
---|---|---|---|---|
d | ||||
d | ||||
The size distribution of dissipative avalanches for several values of are presented in Fig. 4 for and in both the dimensions. Interestingly, the distributions are very different in nature than the corresponding . Preliminary estimate of the size distribution exponent by linear least square fit to the data points in double-logarithmic scale reveals that except for in d. Following Christensen and co-workers Farid and Christensen (2006); *christensenEPJB08, a new scaling form of is proposed as
(10) |
where is a new scaling function, and are exponents for dissipative avalanches. The moment of such a distribution is obtained as where . Noticeably, the moment exponent becomes independent of the size distribution exponent . Performing moment analysis for both d and d, the values of for dissipative avalanches are found close to that of the all avalanches as presented in Table 2. In the limit , the scaling function becomes a constant and the distribution is given by . Consequently, for . The exponent is then estimated from the slope of the plot of vs in double logarithmic scale as presented in the upper insets of Figs. 4 (a), 4(b), and 4(c). The values of are estimated as for and for in d. It is interesting to note that such a flat distribution (i.e for in d) is also reported by Amaral and Lauritsen Amaral and Lauritsen (1996) for the dissipative avalanches of d rice pile model. In contrary to the present observation, Dickman and Campelo Dickman and Campelo (2003) found a power law scaling of with exponent for SSM with boundary dissipation on d regular lattice. In d, the exponent is obtained here as for whereas Dickman and Campelo Dickman and Campelo (2003) reported for SSM on d regular lattice with boundary dissipation. Thus the scaling behaviour of dissipative avalanches of DSSM is very different from that of dissipative avalanches of SSM with boundary dissipation in both the dimensions at . Such difference in the scaling behaviour for dissipative avalanches with different modes of dissipation is probably due to different topological properties of network in the bulk and at the boundary because the degree of a node at the boundary is different from that of a node in the bulk. Moreover, it should be noted that for the model of boundary dissipation, Dickman and Campelo introduced a logarithmic correction in the distribution of dissipative avalanches and the distribution was given by
(11) |
where is an another exponent. In order to verify whether such correction to scaling is present in the present model with bulk dissipation or not, the scaling function given in Eq. (10) for is plotted in Figs. 5(a) and 5(b) for d and d respectively. For comparison, the scaling function that of the model with boundary dissipation given in Eq. (11) is also plotted in the respective plots. It can be seen that without any correction, the scaling function is reasonably constant over a wide range of in double logarithmic scale in the case of bulk dissipation whereas that requires a correction to scaling, , in the case of boundary dissipation for () as observed by Dickman and Campelo Dickman and Campelo (2003). Hence the scaling forms considered here for the model with bulk dissipation are not subject to any logarithmic correction. However, the scaling behaviour of all avalanches are found to be same for both the models as reported in ref Malcai et al. (2006). This is because the leading singularity is provided by non-dissipative avalanches. In order to verify the form of the scaling function given the Eq. (10), data collapse has been performed by plotting against for different values of . Reasonable data collapse for different are obtained as shown in the respective lower insets of Figs. 4. For in d, the FSS form of the distribution is expected to follow the usual distribution as given in Eq. (4). From the plot of vs as given in upper inset of Fig. 4(d), is found to be , again close to the value of all avalanches. The value of is estimated from Eq. (6) as little less than mean-field result as obtained for the all avalanches. However, taking and the best data collapse is obtained, given in lower inset of Fig. 4(d), which confirms the respective form of the scaling function. It should be noted here that the value of for dissipative avalanches are very close to the value of of the all avalanches for both the extreme values of in both the dimensions because the large avalanches which are responsible for cutoff of the distribution of all avalanches are mostly dissipative, and in the moment analysis the leading contribution comes from those large dissipative avalanches.
iii.3 Small world regime
Scaling properties: Since SWN preserves both the characteristic of regular lattice and random network, it is important to study the critical properties of the avalanche size distribution in the SWN regime, . The size distribution of all the avalanches are plotted in Fig. 6(a) for d and in Fig. 6(b) for d. In Fig. 6(c) and (d), are plotted for d and d respectively. For d and d, the values of used are and respectively for both the distributions. Interestingly, both the distributions and exhibit their respective scaling forms on regular lattice () and random network () in the same distribution. The straight lines with respective slopes in these plots are guide to eye. The crossover from one scaling form to other occurs at their respective crossover avalanche size for and . For , the avalanches are small, compact and mostly confined on regular lattice whereas for , they are large, sparse and mostly exposed to random network. Since and have similar scaling behaviour, display a similar crossover scaling as that of . The coexistence of more than one scaling forms in the same distribution of avalanche properties for different sandpile model are already reported in the literature Bhaumik and Santra (2013); Lahtinen et al. (2005); Hoore and S. (2013); *araghiPRE15; *moosaviPRE15. The crossover scaling is found to occur for a wide range of within SWN regime for both and . As one expects the scaling form of regular lattice as and that of random network as , the value of is found to depend on for both the distributions. The dependence of on is assumed as
(12) |
where is an exponent. The value of for all avalanches can be obtained by simple arguments. From the conditional expectation of avalanche size for a fixed avalanche area, one expects , where is the average avalanche area for the avalanches of size , is an exponent Christensen and Olami (1993), and is the crossover length scale below which SWN behaves as regular lattice Newman and Watts (1999a); *newmanPLA99; de Mendes et al. (2000). As , one obtains and has . However, a dissipative avalanche occurs only after a required number of toppling equivalently . As in the large limit corresponding to random network, one expects with . For a given the value of is estimated from the intersection point of the straight lines with required slope in the respective regions. The estimated values of is then plotted against in double logarithmic scale in the respective insets of Fig. 6. It can be seen that in all cases shows a reasonable power-law scaling with . By linear least square fit through the data points the values of for all avalanches are found to be for d and for d which are very close to the values at in both the dimensions Nakanishi and Sneppen (1997); Ben-Hur and Biham (1996); *santraPRE07. On the other hand, for dissipative avalanches it is found that for d and for d again close the inverse of respective dimensions.
Coexistence scaling: Since FSS forms of and are found to be satisfied both on regular lattice and on random network, they should also be satisfied on SWN. Instead of FSS form, the dependence of these distributions are then verified on SWN for a fixed . A generalized scaling form for for the all avalanches on SWN is proposed as
(13) |
where and are the respective scaling functions and , are the corresponding critical exponents in and regions respectively. At for a given , the limiting values of from both the regions must be same. As , then one should have . Hence, the independent scaled distribution can be obtained as
(14) |
in terms of a single scaling function Bhaumik and Santra (2013). Such scaling form is also found to exist in the dynamic scaling of roughness of fractured surfaces López and Schmittbuhl (1998); *morelPRE98. To verify the scaling forms given in Eq. (14), the scaled probabilities for all avalanches are plotted against the scaled variable in Figs. 7(a) and 7(b) for d and d respectively taking . It can be seen that a good data collapse is obtained using , for d and using , for d. The straight lines with required slopes in the respective regions are guide to eye. It confirms the validity of the proposed scaling function form given in Eq. (13). Similarly, a generalized size distribution function can be written for dissipative avalanches around its crossover size taking . The scaled probabilities for for dissipative avalanches are plotted against the scaled variable in Figs. 7(c) and 7(d) for d and d respectively. Reasonable data collapse is obtained as expected. It is then important to notice that if a dynamical model like sandpile is studied on SWN, multiple scaling forms of an event size will coexist in the distribution of the same.
Iv Summary and Conclusion
A dissipative stochastic sandpile model is developed and its critical properties are studied on SWNs both in d and d for a wide range of shortcut density . The non-dissipative avalanches display usual stochastic scaling of SSM on regular lattice () and mean-field scaling on random network () as that of all avalanches. However, the dissipative avalanches represent a number of novel scaling properties on regular lattice as well as on random network in both d and d. The scaling behaviour of these avalanches on regular lattice is found to be very different from Dickman-Campelo scaling as observed with open boundary in both the dimensions. The bulk dissipation is found to have non-trivial effect on dissipative avalanches over the boundary dissipation. No logarithmic correction to scaling is found to occur as it was required for these avalanches on regular lattice with boundary dissipation. A set of new scaling exponents are found to describe the scaling of dissipative avalanches on regular lattice and random network. On SWN, in the intermediate range of , the model exhibits coexistence of more than one scaling forms both in d and d around a crossover size . For non-dissipative and dissipative avalanches, however, the crossover size scales with with two different exponents. The small, compact avalanches of size mostly confined on regular lattice are found to obey the usual SSM scaling whereas the large, sparse avalanches of size exposed to random network are found to obey mean-field scaling. A coexistence scaling form of the avalanche size distribution function around is proposed and numerically verified. Therefore, SWN can be considered as a segregator of several scaling forms that appear in the event size distribution in a dynamical system.
Acknowledgments: This work is partially supported by DST, Government of India through project No. SR/S2/CMP-61/2008. HB thanks MHRD, Government of India for financial assistance. Availability of computational facility, “Newton HPC” under DST-FIST project Government of India, of Department of Physics, IIT Guwahati is gratefully acknowledged.
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