Cascading failures in scale-free interdependent networks
Malgorzata Turalska, Keith Burghardt, Martin Rohden, Ananthram Swami, Raissa M. D'Souza
CCascading failures in scale-free interdependent networks
Malgorzata Turalska
Network Science Division, Army Research Laboratory, Adelphi, MD, USA, 20783
Keith Burghardt
Information Sciences Institute, University of Southern California, Marina del Rey, California, USA, 90292
Martin Rohden
Department of Computer Science, University of California, Davis, California, USA, 95616
Ananthram Swami
Computational and Information Science Directorate,Army Research Laboratory, Adelphi, MD, USA, 20783
Raissa M. D’Souza
Department of Computer Science, University of California, Davis, California, USA, 95616Department of Mechanical and Aerospace Engineering,University of California, Davis, California, USA, 95616 andSanta Fe Institute, Santa Fe, New Mexico, USA, 87501 (Dated: February 21, 2019)Large cascades are a common occurrence in many natural and engineered complex systems. Inthis paper we explore the propagation of cascades across networks using realistic network topologies,such as heterogeneous degree distributions, as well as intra- and interlayer degree correlations. Wefind that three properties, scale-free degree distribution, internal network assortativity, and cross-network hub-to-hub connections, are all necessary components to significantly reduce the size of largecascades in the Bak-Tang-Wiesenfeld sandpile model. We demonstrate that correlations present inthe structure of the multilayer network influence the dynamical cascading process and can preventfailures from spreading across connected layers. These findings highlight the importance of internaland cross-network topology in optimizing stability and robustness of interconnected systems.
PACS numbers: 02.50.Ey 05.65.+b 87.19.L-
I. INTRODUCTION
Occasionally, natural as well as man-made complexnetworks suffer massive cascades, which are initializedby a breakdown of a small portion of the entire sys-tem. Such cascades characterize a plethora of complexphenomena, including neural avalanches [1–4], blackoutsin power grids [5–7], secondary extinctions in ecologi-cal systems [8, 9], and systemic default of financial in-stitutions [10, 11]. Recently, research has focused onhow modular structures or interconnections between net-works affect large cascades for simple regular networktopologies[12]. More realistic topological features, suchas broad-scale degree distributions [13, 14], assortativity[15, 16], or non-random inter-connectivity between com-munities [17–20], have yet to be explored. These struc-tural features are seen, for example, in functional brainnetworks.Here we develop a systematic study to fill this gap.Our goal is to identify near-optimal architectures for pre-venting cascading failures in realistic interconnected sys-tems. We demonstrate that as well as interlayer degreecorrelations play crucial roles in affecting the occurrenceof catastrophic cascades in interdependent heterogeneousnetworks. In particular we show that vulnerability of in- dividual nodes to fail correlates with degree of assortativ-ity present in the network. This behavior illustrates theimportance of considering higher-order network proper-ties when maximizing robustness of interconnected sys-tems.We study failure cascades with the Bak-Tang-Wiesenfeld (BTW) sandpile model, which self-organizesto an apparent critical state, in which cascades sizes aredistributed as a power law [2, 12, 21–27]. This char-acteristic mimics the heavy-tailed distributions of fail-ure cascades seen in electrical blackouts [5, 6, 28], neu-ronal avalanches [3], earthquakes [29, 30] and forest fires[31, 32]. Furthermore, the BTW model captures a com-mon feature of many systems in which individual ele-ments carry a load, but have a fixed capacity [12]. Thisproperty makes the BTW model a valuable tool whenstudying how network cascades result from the individ-ual elements failing due to exceeding their capacity andshedding their load to neighboring elements.The universality of the cascade size statistics observedin numerous dynamical systems poses significant chal-lenges to the design of strategies to control the occur-rence of large catastrophic failures. Namely, becauseindependent of whether the underlying network topol-ogy is homogeneous or heterogeneous, failure sizes are a r X i v : . [ n li n . AO ] F e b FIG. 1. Probability distribution of cascade size, P ( s ) (redcircles), and cascade area, P ( a ) (blue triangles), for a neutral(left panel) and assortative (right panel) scale-free network.The dashed line corresponds to the mean-field solution to theBTW model, where P ( s ) scales as a power law with exponent= 3 /
2. Overlap of two measures demonstrates that nodestypically fail once during each cascade. Network size is N =5000, the degree distribution scale-free exponent is γ = 3 . f = 0 . characterized by heavy-tailed distributions, thus reduc-ing cascade sizes is not easily achieved through changingconnectivity alone [33]. Brummitt et al. [12], however,demonstrated that connections between networks act as acontrol mechanism regulating frequency of catastrophicfailures in coupled random regular networks. Here onenetwork can minimize the likelihood of a large cascadeby forming an intermediate amount of connections withthe other network. Changing the connectivity away fromthis point enhances the likelihood of large cascades.Our work extends upon Brummitt et al. by couplingtogether scale-free networks, rather than random regu-lar networks, to better approximate many natural andman-made systems [34]. We show that the probabilityof large cascades is significantly affected by the degreedistribution, the inter-layer degree correlations, and theintra-layer degree correlations. These results could nothave been obtained without exploring the dynamics ofthe BTW model on heterogeneous coupled networks, andmay provide insight into the evolutionary advantages ofparticular network structures seen in nature.We organize the rest of the paper as follows. We be-gin in Sec. II with a brief background on the sandpileprocess on individual complex networks. In Sec. IV weshow fundamental disparities between results of numeri-cal simulations and assumptions made by branching pro-cess approximations to the sandpile model, which moti-vates discussing the BTW model for large cascades. InSec. V we study the spread of large cascades through in-terconnected networks. Finally, we discuss our findingsin Sec. VI. II. SANDPILE PROCESS ON ISOLATEDCOMPLEX NETWORKS
The BTW sandpile model is a prototypical, idealizedmodel of cascading dynamics caused by load shedding ona network [21, 22]. Throughout this paper, the followingformulation of the dynamics is used. Consider a networkof N nodes, where each node has some capacity to holdgrains of sand, and each grain corresponds to a unit ofload. The topology of the network is fixed, while theamount of sand on individual nodes changes in time. The capacity of a node is the maximal amount of sand thatit can hold. A natural choice is for the node to topplewhen the amount of sand first equals its degree, k [27],as the node can then shed one grain of sand to eachneighbor. We therefore set the capacity of each nodeto k −
1. Hence, a ( k − k is atcapacity , meaning that it holds as much sand (load) as itcan withstand. Adding a grain to such a node brings it over capacity , and it therefore topples.The dynamics of the sandpile model consists of a se-quence of cascades on this network, defined as follows.At each discrete time step, a grain of sand is droppedon a node chosen uniformly at random. If this additiondoes not bring the initial node over capacity, then thatcascade is finished. However, if the node is over capacity,then it topples and sheds one grain to each of its neigh-bors. Any node that then exceeds its capacity topples inthe same way, shedding to its neighbors who may in-turntopple, which continues until all nodes are below or attheir capacity (i.e., equilibrium is restored). In order toprevent the system from becoming saturated with sand,a dissipation mechanism is required: whenever a grainof sand is shed from one node to another, it dissipates(is removed) with a small probability f . In this paper f = 0 .
01, unless stated otherwise. This dissipation rateis chosen so that the largest cascades topple almost theentire network.The size of a cascade is the total number of topplingevents, while the area of the cascade is the total numberof nodes that ever topple. In scale-free networks, we findthat these two quantities are essentially the same [2] (seeFig. 1), which is in contrast to the situation for regularrandom graphs [26, 27]. As our interest is in scale-freenetworks, we thus focus on the cascade size in the rest ofthe paper.The mean-field solution to the BTW model is char-acterized by a distribution of cascade sizes, P ( s ), thatexhibits a power law with exponent − / < γ ≤
3, one ob-serves cascade size distribution of exponent γ/ ( γ −
1) [2].Otherwise the mean-field value is observed [2]. Degree
FIG. 2. Chance of toppling in a cascade of size s . Top 4 pan-els: In a neutral network (red circles), nodes of various degreeare approximately equally likely to participate in cascades ofdifferent sizes. In an assortative network (blue squares) highdegree nodes topple frequently in large cascades and infre-quently in small ones, while the reverse is true for low de-gree nodes. Bottom 2 panels: Adjacency matrices for neu-tral scale-free network (left) and assortative network (right),where nodes in each figure are sorted from low degree (bottomleft corner) to high degree (top right corner). Colors indicatethe probability that, in a large cascade, a given link partic-ipates in sand redistribution. These panels further demon-strate that, in assortative networks, links connected to highdegree nodes are more likely to participate in large cascades. heterogeneity therefore creates either the -3/2 exponentor heavier-tailed cascades, and is not a priori a mecha-nism to reduce the probability of large cascades. III. INDIVIDUAL NETWORKS
Individual scale-free networks are generated using amodified version of the configuration model [36]. Ourresults in the main text are for a degree distribution P ( k ) ∼ k − γ , where γ = 3 .
00, the mean degree (cid:104) k (cid:105) = 4and the minimum degree is k min = 2. We find, however,that the main results appear to not strongly depend onthe value of γ , as seen in the Appendix (cf. Fig. 11).Next, we adopt the rewiring algorithm of [37] to obtainnetworks with positive correlations between degrees ofindividual nodes and that of their neighbors. This proce-dure leads to a modular structure where nodes of similar degree are more likely to be connected with each other.Although the assortativity of the original neutral scale-free networks is low ( a = 0 . ± . a = 0 . ± . a = 0 . ± .
05, we choose a more moderateassortativity to reduce degree-induced modularity, andto better match the assortativity of empirical networks[39].In Fig. 2 we demonstrate how the probability for nodesto topple changes with cascade size for both neutral andassortative networks. In assortative scale-free networks,low degree nodes topple more often in small cascadesthan in large ones. The opposite is true for high degreenodes. We therefore infer that high degree nodes musttopple if a large cascade is to occur. In comparison thelikelihood for nodes to topple in neutral networks is muchless dependent on degree.
IV. SAND DISTRIBUTION ASSUMPTION
In the previous section, we notice the relationship be-tween cascade size and node degree is different for neu-tral and assortative scale-free networks. In particular thealmost constant probability for nodes to topple in theneutral topology comes as a surprise, because it contra-dicts the existing assumption that the probability thata node topples is proportional to the inverse of a node’sdegree, which is commonly referred to as the 1 /k ansatz.This ansatz is often a fundamental assumption when de-termining how the dynamics approach a critical branch-ing process [2, 23, 27]. Because we observe that high-degree nodes topple with a relatively high probabilitythat strongly diverges from 1 /k , we are motivated to in-vestigate the validity of the 1 /k ansatz. In [27] the au-thors do show that the 1 /k assumption is not strictlyvalid and that higher degree nodes are more likely to beat capacity. Yet, they attribute the observed power lawdistribution of failures sizes to be due to universality. Weshown below that analyzing the out of equilibrium dis-tribution of sand on nodes for the BTW model providesthe resolution for how a critical branching process canarise from a system seemingly poised for a super-criticalbranching process. A. Equilibrium configuration of the sandpile model
We begin by considering the distribution of sand on thenetwork in equilibrium (just before an additional grain ofsand is dropped). A corollary of the 1 /k ansatz is that theprobability for a degree- k node to have i grains of sand isconstant and equal to 1 /k , such that there is no typicalamount of sand in any inactive node [2, 23, 27]. In Fig. 3, FIG. 3. A corollary of the “1 /k ansatz” is that the probabilityfor a degree- k node to have i grains of sand is constant andequal to 1 /k , as denoted by the dashed line. However, for k -regular networks, as well as neutral and assortative scale-free networks, behavior deviates strongly from this corollary.Degree −
10 nodes (left panel) and degree −
20 nodes (rightpanel) are more likely to have near-critical amounts of sand,and less likely to have low amounts of sand. Furthermore,nodes in assortative scale-free networks behave similarly tonodes in k -regular graphs of the same degree, while nodes inneutral scale-free networks show the strongest deviation fromthe 1 /k ansatz, with loads skewed strongly towards critical ca-pacity. Network size is N = 5000 and the degree distributionscale-free exponent is γ = 3 . we notice that the distribution of sand differs significantlyfrom analytic assumptions. Although past works [2, 23,27] report that the 1 /k corollary approximately holds,our observations demonstrate a more complex picture.As shown in Fig. 3, the probability that a degree- k nodehas i -sand is not 1 /k , but is strongly skewed towardslarger values of i . This observation holds for various nodedegrees, network topologies, network sizes, and variousvalues of the dissipation rate, f , as discussed in detail inthe Appendix.Furthermore, we observe a strong effect of degree corre-lations: degree − k nodes in neutral and assortative scale-free networks are characterized by different sand distribu-tions. The departure from 1 /k distribution is particularlypronounced for the neutral topology, where probabilityfor a node to be close to capacity is nearly twice what wewould expect theoretically.Additionally we notice strong similarities between i -sand distributions for k -regular and assortative networks.This property, combined with the modular nature of as-sortative network, suggests that modules of similar de-gree dynamically behave like regular graphs with thesame degree. One could interpret the BTW dynamicson assortative scale-free networks as one on a set of cou-pled regular graphs of increasing degree. We will demon-strate later that this property has significant impact onthe occurrence of catastrophic failures in interconnectedheterogeneous networks.In summary, the probability that a node topples(Fig. 2), and the probability that a node has i grainsof sand (Fig. 3) both show strong deviations from the- FIG. 4. Distribution of sand in and out of equilibrium. Leftpanel: degree −
5, middle panel: degree −
10, and right panel:degree −
20 nodes. Black lines are the equilibrium distribution,just before any sand is dropped. Red, blue, cyan and magentalines correspond to the sand distribution among nodes thatreceive grains of sand over the course of large cascades, where s > N/ /k corollary, and sand topples with probability 1 /k inagreement with the ansatz. In the figures, the networks areneutral scale-free with γ = 3 .
00 and size N = 5000. Othertopologies show similar behavior. oretical assumptions previously relied upon to explainthe critical dynamics of the BTW model. Because em-pirically we observe that the probability of a node top-pling is greater than 1 /k , it would naively imply thatthe BTW model always produces a super-critical branch-ing process. Why, then, does all past research observe apower-law tail in the distribution of cascade sizes, a sig-nature of a critical process? Figure 1, for example, showsthat the cascade size and area distribution for scale-freenetworks broadly follows a power-law distribution overseveral orders of magnitude. Larger networks produceeven stronger agreement with the theoretical power-lawdistribution. B. Dynamics out of equilibrium
To understand why the BTW model creates power-lawdistributions, we study out-of-equilibrium behavior of themodel’s dynamics. Namely, we investigate the probabil-ity of a degree- k node having i -sand as a large cascadeprogresses. We consider a cascade evolution scheme thatproceeds according to generations, in parallel with thenomenclature of the branching processes. The node top-pling as a result of the initial random deposition of agrain of sand is called a root and forms the first genera-tion of the cascade. Each successive generation is formedby the nodes that received sand from the previous gener-ation’s nodes that have toppled. Figure 4 demonstratesour results. We find that, regardless of node degree, thesand distribution on nodes in the n th generation is a bet-ter and better approximation of the analytic assumptionof 1 /k .Initial generations strongly disagree with theory. Forexample, there is a clear peak in the distribution at sec-ond generation because, in order for a cascade to be large,a high number of neighbors must topple. These initialgenerations, however, consist of very few nodes. Thesecond generation consists of at most k nodes, and thethird generation has less than k ×(cid:104) k (cid:105) nn nodes on average,where (cid:104) k (cid:105) nn is the mean degree of the nearest neighbors.Each generation g is bounded by k × ( (cid:104) k (cid:105) nn ) g − thus thebulk of the nodes in large cascades are those in the latergenerations that happen to closely approximate the 1 /k assumption.In summary, the BTW dynamics are characterizedby the equilibrium sand distribution differing from 1 /k ansatz, but, over the course of a cascade, the sand dis-tribution among nodes that receive sand evolves to thetheoretically expected distribution. There are two prop-erties of large cascades that allow the observed BTWdynamics to more closely approximate the prediction ofthe theory. First, the out-of-equilibrium statistics areonly based on nodes that receive sand, implicitly intro-ducing a biased sub-sampling of sand distribution on thenodes. This is why the equilibrium distribution disagreeswith the 1 /k assumption. Second, we observe the nodesthat toppled in earlier generations receive sand in futuregenerations, but never enough sand to topple a secondtime. For example, a node that topples during a cascadewill send sand back to the parent node that toppled it.Additionally, due to non tree-like structure of the net-work, a non-negligible fraction of nodes receive two andmore grains of sand per generation. Thus loops presentin the network affect the sand distribution on the net-work, and are responsible for skewed equilibrium i -sanddistribution. V. INTERCONNECTED SCALE-FREENETWORKS
In this section, we study how network topology affectsthe probability of large cascades with various intra- andinterlayer connectivity statistics. We focus on the excep-tionally large cascades in the BTW sandpile model due tothe disproportionate cost associated with such large ex-treme events, when compared to the cost of small events,occurring in real interconnected systems. We generatetwo scale-free networks, denoted here network (or layer)A and network (or layer) B, each with N nodes (2 × N nodes total). We connect nodes within layers either atrandom or assortatively, and connect nodes between lay-ers either at random or in a hub-to-hub fashion. Randominter-layer connectivity is created by connecting p × N random pairs of nodes together between layers. Hub-to-hub inter-layer connectivity is created with the followingalgorithm:1. Sort nodes in each network from highest-degree tolowest-degree2. Connect p × N nodes in each network together FIG. 5. The probability of a large cascade for two coupledassortative scale-free networks versus the connectivity proba-bility p . Inter-layer connectivity has a strong impact on theBTW dynamics, with the hub-to-hub coupling resulting in aconstant chance of cascading failures, while random couplingresults in an increasing occurrence of large events. Each net-work has N = 5000 and the dissipative parameter f = 0 . starting with the highest-degree node and workingdown.The parameter p , which varies between 0 and 1, dictatesthe strength of coupling between individual networks. A. Assortative scale-free networks
We first focus our attention on the effect that the pres-ence of other layers has on an individual network layer(without loss of generality, we can choose layer A). Foreach cascade, we record the number of nodes that toppledduring the process, s A and s B , separately for both layers.We then determine the probability s A > N/
2. We dif-ferentiate between the chance of large cascades occurringlocally in layer A, denoted P AA ( s A > N/ (inflicted cascades) andtraversing into layer A, denoted P BA ( s A > N/ P AA ( s A > N/
2) initially drops dramati-cally with increasing interlayer coupling and, furthermorethis probability barely changes for p > .
2. In compari-son, P BA ( s A > N/
2) increases initially but also reachesa constant value for p > .
2. The overall probabilityof large cascades stays thus constant for p > .
2. Withrandom inter-layer coupling, however, moderate to highinter-layer connectivity significantly increases the likeli-hood of large cascades that originate in both layer A or
FIG. 6. The probability of a cascade greater than N = 5000versus inter-layer connectivity p . Shown are neutral networkswith random and hub-to-hub coupling as well as assortativenetworks with random and hub-to-hub coupling. B. This latter behavior resembles one observed in ran-dom regular networks examined by Brummitt et al. [12],suggesting a similar mechanism.The decrease in the likelihood of large cascades for hub-to-hub coupled assortative networks lies in their highlymodular structure (see connectivity of nodes in Fig. 2).Hub-to-hub connectivity extends the modular structureof individual layers to the entire dual-layer system, pre-serving linkage of nodes of similar degree. Furthermore,in an individual assortative layer, the occurrence of alarge cascade is conditioned on toppling of several highdegree nodes (see Fig. 2), a rare event. It is quadrati-cally less probable that such condition will be met in adouble layer system. Coupling layers of assortative scale-free networks in a hub-to-hub fashion therefore decreasesthe likelihood of large cascades by absorbing excess loadfrom a layer.In contrast, by coupling the layers of assortative net-works randomly, there is a greater amount of connectiv-ity between degree- k modules, and the connectivity givesrise to a more homogeneous network, reminiscent of cou-pled random regular networks studied by [12]. FollowingBrummitt et al. [12], we believe that an increase in theprobability of large cascades for large p is caused by di-verted loads that return to the network. Random wiringallows for high-degree nodes in one layer to connect tolow-degree nodes in another, which allows for cascades tomore easily cross layers because low-degree nodes topplemore often. Because low-degree nodes shed their sandmore often, there is also a greater likelihood for high-degree nodes to topple, thereby triggering a large cas-cade.We can extend our results to large cascades affectingboth layers of the network as well. We show in Fig. 6 that FIG. 7. The probability of a large cascade occurring in asystem of two coupled neutral scale-free networks versus theconnectivity probability, p . This probability is virtually indis-tinguishable between hub-to-hub and random interlayer con-nectivity. As the connectivity probability p increases, cascad-ing failures reach a constant. In each network, N = 5000 andthe dissipative parameter f = 0 . for p > .
3, the regime where large cascades are sup-pressed, assortative, hub-to-hub coupled networks havethe smallest probability of large cross-network cascadeswhere s > N . Predictably, however, greater connectiv-ity increases the probability of large cascades overall. Inreal systems, however, it may be important to connectall regions together, therefore the assortative hub-to-hubtopology produces the best trade-off of inter-connectivitywithout as large a probability of large cascades.
B. Neutral scale-free networks
In this section, we focus on coupled random scale-freenetworks. This represents networks with the least intra-layer structure. As illustrated in Fig. 7, the probabilityfor large cascades that originate in network A drops sub-stantially with p , while the probability of large inflictedcascades rises. The overall probability that any cascadeoccurs in network A, P A , is reduced with introduction ofinterlayer coupling, although for p > .
3, the probabil-ity is roughly constant, similar to assortative scale-freenetworks coupled hub-to-hub. The mechanism responsi-ble for this behavior, however, differs fundamentally fromthat in the case of assortative networks.Namely, the lack of degree correlations in neutral scale-free network causes the toppling rate to be almost inde-pendent of node degree (Fig. 2), regardless of cascadesize. Thus, interlayer coupling, whether random or hub-to-hub, has approximately the same effect on the dynam-ics. With low coupling, network A benefits from sheddingload to network B, but once there is moderate coupling,the two networks act as a single random network, im-plying the probability of large cascades does not varysignificantly for p > . VI. DISCUSSION
We set out to better understand the dynamics of theBTW sandpile model, a prototypical SOC model [21, 22].In doing so, we first noticed an under-appreciated aspectof the model: the node sand distribution differs markedlyfrom the theoretical assumption. The distribution wouldseemingly imply that the dynamics are super-critical inequilibrium, but the non-equilibrium statistics demon-strate that the sand redistributes to create critical dy-namics.Although the BTW model is simplistic, it creates im-portant insights into how the spread of cascading failuresis affected by underlying network topology. We demon-strate that the robustness of interconnected systems is afunction of correlations between intra- and interlayer in-teractions. This is similar to earlier results of Reis et al .[18], based on studies of bond percolation-like process oninterconnected scale-free networks. Intriguingly, the net-work topology that we found most effective in reducinglarge cascades—assortative scale-free networks with hub-to-hub inter-connectivity—is found in functional brainnetworks [13–18]. Because neuronal avalanches appear toself-organize to a critical state [1–4], the human brain canbe modeled in a simplified manner via the BTW model.Taking these results together, we would predict that thebrain is constructed so as to prevent large cascades. Ifwe were to interpret large cascades as seizures, for ex-ample, this would make the surprising suggestion that ahealthy brain naturally reduces the likelihood of seizures,and a reduction in assortativity, or hub-to-hub inter-connectivity would make seizures more likely. This agreeswith recent findings that particular abnormal functionalbrain networks, such as those observed in schizophrenia[40, 41], increase the likelihood of seizures [42, 43]. Fur-thermore, in agreement with our model (Fig. 2), rich clubnodes (hub nodes in assortative networks) are stronglyassociated with generation seizures [44]. Overall, ourresults show that despite the BTW model’s simplicity,it can qualitatively approximate the occurrence of brainseizures. Moreover, it can provide insight into the evo-lutionary motivation of functional brain network topolo-gies.Our work suggests several avenues of future research.We find that high-degree nodes in assortative networksare likely to topple in large cascades, therefore one coulddesign protocols controlling the amount of load on thosenodes or devise quarantine scenarios in order to limit
FIG. 8. Choice of the dissipation rate f adopted in thesandpile dynamics affects the tail of the distribution of cas-cade sizes, P ( s ). the spread of catastrophic failures. Moreover, a fruit-ful avenue of research would be predicting large cascadeswhen a cascade is beginning, such as detecting whetherhub nodes shed their load. In addition, we only exploredthis behavior for the BTW model. It is an open questionto understand if these same three features of heteroge-neous degree distributions, internal network assortativ-ity and interlayer degree correlations also suppress large-scale failure for other types of cascade models, such asthreshold models. VII. ACKNOWLEDGEMENTS
We gratefully acknowledge support from the US ArmyResearch Office MURI award W911NF-13-1-0340 andCooperative Agreement W911NF-09-2-0053 (NetworkScience CTA); The Minerva Initiative award W911NF-15-1-0502; and DARPA award W911NF-17-1-0077.
VIII. APPENDIX
In this section, we discuss several details of the modelthat, in the interest of space, we leave out of the maintext.
A. Dissipation rate and system size.
After observing the sand distribution shown on Fig. 3,which differs from theoretical assumption of criticalbranching process, one might suspect that presented re-sults are side effects of one of the model’s two parameters:the dissipation rate, f , or the system size, N . However,varying those parameters does not appear to better ap-proximate the 1 /k corollary.First we consider the effect of different values of thedissipation rate f on the sandpile dynamics. Since thisconstant regulates total amount of sand on the network,higher values correspond to increased sand removal, whilelower values lead to the accumulation of load in the sys-tem. This behavior is reflected in the statistics of ob-served cascades, illustrated in Fig. 8. The former condi- FIG. 9. Probability that a degree- k node holds i grains of sandstays constant despite significant variation in the dissipationrate f of the sandpile process. Behavior of a random 10-regular network is compared with k = 10 nodes of a neutraland assortative scale-free network, respectively.FIG. 10. Probability that a degree- k node holds i grains ofsand saturates as a network size N increases. Behavior of asandpile process on a random 10-regular network with dissi-pation rate f = 0 .
01 and f = 0 .
001 is shown on the left andright panel, respectively. tion of lowering load results in decreased probability oflarge cascades, denoted by a truncation of the tail of thecascade size distribution. On the other hand excessivesand accumulation gives rise to more frequent large cas-cades, as shown by a visible peak in the P ( s ) functionfor s ∼ O ( N ).However, even though varying f significantly changes the cascade size distribution, it has little effect on thesand distribution on individual nodes (Fig. 9). As lowervalue of f leads to sand accumulation, we observe slightincrease in probability of node being at capacity, but theeffect is very subtle, especially when contrasted with theimpact that change in f has on macroscopic observables,such as P ( s ). As noted earlier, the distribution of sandon nodes of assortative scale-free network overlaps withthe distribution of sand on random regular network ofthe same degree, independent of the selected value of f . FIG. 11. Chance of large cascades occurring in a system ofrandomly connected scale-free networks, where scaling expo-nent of the individual layer degree distribution is γ = 2 . γ = 3 .
00 (compare to Fig.7and Fig.5).
Finally, we consider the impact of finite system size, N . In Fig. 10 we show, that despite increasing the systemsize by three orders of magnitude, the disparities in sanddistribution are preserved. B. Exponent of the scale-free network degreedistribution.
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