Optimization of the post-crisis recovery plans in scale-free networks
Mohammad Bahrami, Narges Chinichian, Ali Hosseiny, Gholamreza Jafari, Marcel Ausloos
aa r X i v : . [ phy s i c s . s o c - ph ] O c t Optimization of the post-crisis recovery plans inscale-free networks M ohammad B ahrami N arges C hinichian A li H osseiny ∗ G holamreza J afari M arcel A usloos Department of Physics, Shahid Beheshti University, G.C. Evin, Tehran 19839, Iran Center for Network Science, Central European University, H-1051, Budapest, Hungary School of Business, College of Social Sciences, Arts, and Humanities, University of Leicester, Leicester, UK Department of Statistics and Econometrics, Bucharest University of Economic Studies, Bucharest, Romania ∗ [email protected] October 25, 2019
Abstract
General Motors or a local business, which one is it better to be stimulated in post-crisis recessions, when government stimulation is meant toovercome recessions? Due to the budget constraints, it is quite relevant to ask how government can increase the chance of economic recovery.One of the key elements to answer this question is to understand metastable features of crises in economic networks and their related hysteresis.The Ising model has been suggested for studying such features. In the homogenous networks, one needs at least a minimum budget, to forcethe network to switch its local equilibria, where such a minimum is independent of the network characteristics such as the average degree. Inthe scale free networks however, when the government aims to push the network to switch to another equilibrium, one may wonder whichnodes are to be preferably stimulated in order to minimize the cost. In this paper, it is shown that stimulation of high degree nodes costs less ingeneral. It is also found that in scale free networks, the stimulation cost depends on the networks features such as its assortativity. Although weconfine our study to the Ising model in order to tackle a problem in economics, our analysis shines lights on many other problems concerningstimulations of socio-economic systems where dynamical hysteresis appears.
I. I ntroduction I n the aftermath of the 2007-2008 economic crisis, whilethe US government was going to stimulate the economy,some controversial issues had risen. For example, the de-bate about a recovery plan for the General Motors (GM) andChrysler went up to the level of the US Senate. In that debate,some experts favored helping the big companies such as GMand Chrysler while others favored small local businesses, seeStiglitz (2010) [1] and references therein.To explain the problem more rigorously, we should recallthat firms purchase products from their "neighbors" in thetrade networks. Such a trade then results in positive corre-lations between activities of both firms. Similar to the ferro-magnet systems, the positive correlation can result in a dy-namical hysteresis for the economic networks when in deeprecessions one faces a global reduction in the activities offirms. In deep recession if the partners of a firm reduce theirproduction and the firm in a different manner works withits maximum capacity, then there is a good chance that itsproducts are not sold, but depreciate resulting in a loss. So,managers have no choice but keeping steps with their part- ∗ Corresponding author ners. Such a behavior can deepen economic harshness andmay result in a long lasting depression.In Keynesian economics, governments are suggested to in-tervene in the market and purchase from the firms, stimulat-ing them to raise their activities in order to overcome reces-sion. Due to the budget constraints, it will be critical to findthe best strategy for stimulation of a heterogeneous network,at least in a simple agent based model for firms.The questions of “whether one will better stimulate the re-cessed economy by helping the big companies or the smallones” is relevant due to the heterogeneous scale-free natureof economic networks. In other words, heterogeneity raisesthe question of finding the best “strategy” to help the sys-tem. In a homogeneous network such as a regular lattice ora small-world Watts-Strogatz network [2], all nodes are con-nected to an almost similar number of neighbors, which havethe same practical roles in the structure and have a similarpriority for stimulation at crisis.The strategy question opens a much more general prob-lem of dealing with heterogeneous networks, where one can-not easily use a mean-field solution. In such cases depend-ing on what is going to optimized, one needs to choose thebest agents of the network to be stimulated. The low-degreenodes are easy to stimulate, while the high-degree nodes are1ore difficult but also more influential. So the answer to “what is the best choice? ” will not in any way be trivial.To study this problem in a detailed way, we consider havingseveral different scale-free networks with Ising spins on theirnodes.The Ising model has been a proper choice to model a widerange of phenomena such as opinion dynamics [3, 4, 5], neu-ral function simulations [6] and many other real-world sys-tems, see for example [7, 8].The Ising model can also be considered as a proper ba-sis for our study, - addressing the correlation of activitiesin the economic network of firms [9, 10, 11]. Indeed, theIsing model has been suggested as a base to model the net-work of firms because the correlation between the activitiesof firms can be encapsulated in the interaction between somespins. Since firms trade with their neighbors in the net-work, when they increase/decrease productivity, they auto-matically force their neighbors to increase/decrease produc-tivity. In the simplest model, as a first approximation, onecan imagine that managers choose the firm level of activityfrom a binary choice being either the maximum or the mini-mum capacity of production.Since the trade is the way that firms interact with and in-fluence each other, in recessions, governments can stimulatethe economic networks via purchasing products. The budgetconstraint however limits the governments choices; therefore,which sectors or which classes of corporations are more ap-propriate choices for stimulation becomes a critical question.If the activity of a network of firms is modeled by the Isingspins, then their response to the external stimulation shouldbe studied within the literature on the kinetic Ising model.The kinetic behavior of the Ising model has been widely stud-ied; for example, see [12, 13, 14, 15, 16]. Let it be recalled thatif one imposes a magnetic field to an Ising system forcing itto switch its local equilibria, it takes some time for the sys-tem to switch. Such a metastable behavior has been widelyinvestigated for the regular networks.The probability of switching between local equilibria as afunction of the magnitude of the external field has been stud-ied, see for example [17]. While the subject of such studieshave been on regular networks, we are interested in the sametype of study on scale free networks.While regular networks are homogeneous, scale free net-works have an intrinsic heterogeneity leading to specificallyinteresting features [18, 19, 20]. Moreover, this heterogeneityraises one question: should one discuss only the magnitudeof the stimulating field? In fact or is it also relevant to ask: “Where has the stimulus field to be imposed in the network?”. The answer to these questions is the subject of this paper.Notice that although we have implemented the Ising modelas a toy model of the network of firms, this study is interest-ing also from a statistical physics point of view. It may aswell shed light on a wide range of phenomena concerning metastabilities which occur in scale free networks.In this work, we stimulate an Ising system to transfer itfrom one of its minima with all nodes first in a downward di-rection to the other minimum with almost all nodes in an up-ward direction. We will compare two general strategies: onestarting our stimulation from the high degree nodes (High-Degree-Stimulation strategy or HDS strategy) vs. anotherstarting from the low degree nodes (Low-Degree-Stimulationstrategy or LDS strategy). We investigate the amount of re-sources needed for each strategy.We will observe that different strategies need differentamount of hits, resources, or budget for success and willshow that the gap between different strategies depends onsome of the network characteristics.
II. R esults
In this Ising model of firms, if firms work with maxi-mum/minimum capacity, their state can be represented byupward/downward spins. At the beginning of each session,managers of firms and corporations look at their (collaborat-ing) neighbors and subject to their level of activities decideto increase or decrease their production level in the comingsession, e.g., resulting in hiring or firing some employees.The chance for a manager’s decision upon working withmaximum/minimum capacity is stochastic, following theprobability rates P ↑ = exp − ( N ↓ − N ↑ ) JT exp − ( N ↑ − N ↓ ) JT + exp − ( N ↓ − N ↓ ) JT P ↑ = exp − ( N ↑ − N ↓ ) JT exp − ( N ↑ − N ↓ ) JT + exp − ( N ↓ − N ↑ ) JT , (1)where P ↑ / ↓ indicates the chance for working with maxi-mum/minimum capacity and N ↑ / ↓ indicates the number ofneighbors which are working with maximum/minimum ca-pacity. The value of J indicates the level of trade and thevalue of T weights how the manager is tied to her neighbors.If the value of T is small then the manager does not takerisky actions in a sense that if the majority of their neighborsincrease/decrease their production level; then they also in-crease/decrease theirs with a high probability. In the Isingmodel for reasonably small values of T we observe symme-try breaking where if a big portion of firms decrease theirproduction, then the system can stay there for a long time.In Keynesian economics government is suggested to com-pensate decline of neighbors to move economy from itsmetastable state. Thus, a government is suggested to startpurchasing from the private parties, like stimulating the sys-tem to move to its opposite local equilibrium. This action is2 m=4 N=1000 From Top Su cce ss P erce n t ag e ∆ G D P Number of Visited NodesHDS Strategy
Success ProbabilityStimulation Cost m=4 N=1000 From Bottom Su cce ss P erce n t ag e ∆ G D P Number of Visited NodesLDS Strategy
Success ProbabilityStimulation Cost
Figure 1: Different strategies. (A) HDS strategy: In this simulation,all spins are set downward. We then start stimulation of highdegree nodes. When n highest degree nodes are stimulated withan upward magnetic field, then the system is relaxed to findthe probability for a successful stimulation which changes theequilibrium of the network with the majority of nodes upwardafter relaxation. The horizontal axis shows the number of nodesquenched in each experiment and the blue curve is the successrate for such a quench state to switch to another global state.The red line shows the desired budget for such a quench, usingan economic language. For a success rate of one needsa budget of ∆ GDP where ∆ GDP is the gap for GDPbetween expansion and recession. (B) LDS strategy: In this fig-ure, the value of n i on the horizontal axis indicates a quenchstate where n lowest degree nodes are stimulated by a govern-ment. As it can be seen, for having a success rate of oneneeds a budget equal to ∆ GDP. Moreover, as it can beseen, for a stimulation with 80% success chance, a stimulationof low degree nodes costs about 20% more than a stimulation ofhigh degree nodes. similar to stimulation of the spins with an external magneticfield. The government pays for the compensation caused bythe other firms’ declining trades. As a result, the probabili-ties are modified as follows P ↑ = exp ( − ( N ↓ − N ↑ ) J − GPT ) exp ( − ( N ↑ − N ↓ ) J − GPT ) + exp ( − ( N ↓ − N ↑ ) J + GPT ) P ↓ = exp ( − ( N ↑ − N ↓ ) J + GPT ) exp ( − ( N ↑ − N ↓ ) J − GPT ) + exp ( − ( N ↓ − N ↑ ) J + GPT ) (2)where GP is a measure of the level of the purchase by gov-ernment.As it is clear from the probabilities, the government pur-chase has some effect GP is comparable with the trade be-tween firms or strictly the value of N ↑ / ↓ J . Now, to addressthe response of the network to stimulations, one should an-alyze the kinetic behavior of the system under the changein size and strategy of stimulations. To this aim, one firstgenerates a preferential attachment scale-free network (B-AModel [21]). Then, one sets all nodes to the downward di-rection indicating a situation where all firms work with min-imum capacity.In the HDS strategy, we start the action by stimulating thehigh degree nodes. We consider a value for the number ofnodes, n , and impose a magnetic field on the n high degreenodes updating them with the probabilities in Eq 2. The stim-ulation imposed on each node is proportional to its degree GP i = k i J . (3)After a number of nodes are stimulated, we let the systemrelax along sveral Monte Carlo steps in order to see if itchanges its local equilibrium in such a way that the major-ity of firms starts working with their maximum capacity. R n = n ∑ i = GP i . (4)After a number of nodes are stimulated, we let the systemrelax in some Monte Carlo steps to see if it changes its lo-cal equilibrium in the way that the majority of firms startworking with their maximum capacity.The resource needed for each strategy denoted by R n isthe cumulative magnetic field imposed on the nodes for thestimulation R n = n ∑ i = GP i . (5)We repeat this for an ensemble of 1,000 experiments for allgiven values of n and obtain the success rate for such stimu-lation and its related resource. The result of the simulationis depicted in Fig 1. In this figure, the blue curve shows thesuccess rate for each value of n . 3or any given value of n , one then measures the cumula-tive field, i.e. R n , as the resource needed for stimulation; inan economic language, it is the cost of the stimulation. D i ff ere n ce B e t w ee n H D S & L D S S t r a t e g i e s ( % ) Network Size
Size
N=1000 m=3 D i ff ere n ce B e t w ee n H D S & L D S S t r a t e g i e s ( % ) Total Clustering
Clustering
Figure 2: Evaluation of the impact of size and clustering on ourresults. (A) The difference for successful bills between HDS and LDSstrategies for various sizes: As the size of the network growsfrom 1000 nodes to 4000 nodes, no significant difference is ob-served. In other words the gap between HDS and LDS strate-gies is independent of the size. (B) The impact of clustering:The difference between HDS and LDS stimulus bill is not influ-enced by clustering in the network.
The red curve shows the cost for each stimulation in ∆ GDP units, where ∆ GDP is the gap for the gross domes-tic product ( GDP ) between expansion and recession. Thisvalue is nothing else that the total degree of the network ortwice the number of links. The point is that in Eq 2 the pa-rameters N ↑↓ J stand for the gap for the trade from each firmin expansion and recession. As a result, the gap for the GDPin expansion and recession periods, can be addressed by thetotal value of N i J , where N i is the degree of i th . This is be-cause the gap for GDP in expansion and recession can bereflected in the decline of trade between nodes as it has beenencapsulated in N i J for each node. For more arguments con- cerning the relation between the role of GDP gap and theaverage degree of nodes see [11].In our simulation we looked for stimulations which by arate of 80% success rate to change the global state of thespins. For this success rate in HDS strategy, we need to stim-ulate about 280 high degree nodes which is equal to spend-ing 0.552 ∆ GDP in economic language or R n = mN J in the Ising language where m is the average degree in thenetwork.In LDS strategy everything is similar to HDS. We first setall spins downward and stimulate them. In this strategyhowever we stimulate low degree nodes. As shown in Fig 1to obtain a 80% success rate, one needs a stimulation equalto 0.663 ∆ GDP . This means that the cost of successful stim-ulation through the HDS strategy is about 20% less than theLDS strategy.The difference in the outcomes of different strategies is sig-nificant. We however need to solidify the results. The firstanalysis to be done is the investigation of the size depen-dency. We repeat our simulations for three different networksizes and observed that our normalized results seem to beindependent of the network size, see Fig 2.Our first investigation appearing in Fig. 1 is on a Barabasi-Albert (B-A) network. A known feature of the B-A networksis that they have lower clustering coefficients than many real-world scale-free networks. To test the effect of the clusteringcoefficient, we change the clustering of our networks withthe Holme-Kim method mentioned in the Methods section.It can be see that in Fig 2, changing the clustering coefficienthas insignificant effect on the size of the gaps.Another feature of the real networks is their different as-sortativity structure. Assortativity is an important featureof the networks. It a measure for correlation of degrees. Inother words it identifies if high degree nodes are preferablyconnected to high degree nodes, low degree nodes, or haveno preferences. So, we perform another analysis to checkthe effect of assortativity on the stimulation strategies. Oursimulation shows that, despite clustering, assortativity cansignificantly influence the gap between HDS and LDS strate-gies. When assortativity is increased in networks, the costfor LDS strategy remains unchanged. This is while the HDSstrategy becomes relatively cheaper and as a result, the gapbetween two strategies grows in size. For large values ofthe assortativity, the cost for LDS stimulation becomes morethan twice the cost for HDS stimulation. Another observa-tion is that the gap is saturated for large and small values ofassortativity, see Fig 3.
III. M ethods
The Ising model is identified by the Hamiltonian H = − J ∑ h i , j i s i s j − ∑ j h j s j , (6)4 D i ff ere n ce B e t w ee n H D S & L D S S t r a t e g i e s ( % ) AssortativityAssortativity
Figure 3: The impact of assortativity on the gap between differ-ent strategies.
The gaps between the cost for HDS and LDSstrategies grow as the assortativity of the networks grows. where s i is the spin of the i th site, J is the coupling constant,the notation h i , j i means that the sum is over the nearestneighbor sites and h j indicates the stimulating external fieldapplied on the j th node.To find the stimulation cost in Fig. 1 all spins are set down-ward. Then in HDS strategy the high degree nodes are stim-ulated with a field equal to h = k i J . (7)After imposing a stimulation on a number of nodes, the sys-tem is relaxed to see if this stimulation is unsuccessful topush the network to switch to its other equilibrium wherethe majority of spins is upward. The temperature is set to T = T c . At equilibrium on this temperature 98% of spinsare downward. For the sake of simulation cost however inour analysis we set all spins downward which brings at leasttwo percent systematic error. Such possible error howeveris small with respect to the cost gap between HDS and LDSstrategies.For the study of the Ising dynamics on different networks,the Glauber weight [22] is used: W ( s i → − s i ) = exp − β ∆ E + exp − β ∆ E , (8)where ∆ E is the energy difference for the system changingthe sign of i th unit and β = k B T . The main targets of ourstudy are scale-free networks (networks with power-law de-gree distribution).To have desired meta-stable states, the systems are consid-ered below the critical temperature (in this case T = T c ).There are some analytical ways to find the critical tempera-ture of Barabasi-Albert networks [23, 24, 21]. For the other networks introduced here, we used numeri-cal methods and simulations to find the critical temperature.There are several known ways to produce scale-free net-works [21, 25, 26, 27, 28].The methods used to produce or change them are listedbelow: • Barabasi-Albert Network was reconstructed using the al-gorithm mention in [21]; we generated the ensembles ofB-A networks with total sizes of 1000, 2000 and 4000nodes. The number of edges of new coming nodesranged from 2 to 8 in different ensembles. • To change the "clustering" of our scale-free networks, weused the "triad formation" step by Holme and Kim [29]keeping fixed the number of links, to be able to comparethe results. • The assortativity [30] of the scale-free networks ischanged through the Brunet and Sokolov reshufflingprocedure [31].
IV. D iscussion
Huge studies have been devoted to the occurrence of crisesand spread of shocks in economics. It has been shown forexample that economies having more connections within theproduction networks may suffer intensive cascades of eco-nomic shocks [32, 33]. In the homogeneous networks thecentral limit theorem rules out the chance for the system-atic failure of the market due to the local fluctuations. It hasbeen however shown that unlike the homogeneous networks,in the scale free networks, local fluctuations in the networkmay blow up and be the triggers for a crisis [34]. This meansthat the structure of the economic networks can seriously in-fluence their dynamic.While huge studies have been devoted to the occurrenceand diffusion of the crisis [32, 35, 36] in complexity eco-nomics [37, 38, 39, 40, 41, 42, 43, 44], analyses of the re-sponses of the networks to the recovery plans are lacking.The Ising model as a base model to address dynamicalproperties of economic networks definitely simplifies the realworld. While in this model heterogeneity is imposed onlyon the degree distribution, studies reveal that besides thedegree distribution, size and influence of firms obey powerlaw distributions [45, 46, 41, 47]. Despite such simplifications,however, the model not only gives insights, but also leads toreasonable results.In the deep recession of 2009, some experts includingNobel prize laureates, Paul Krugman and Joseph Stiglitzwarned that only big stimulations could help the economymoving toward a fast recovery [48, 1]. There was howeverno idea about how big this stimulation should be.In an analysis, through studying the hysteresis of the eco-nomic network along an Ising model, a threshold is sug-gested for the size of successful stimulations. This is shown5hat to overcome the recession the recovery bill should be big-ger than such a threshold [11]. For the recession of 2009 themodel predicts the threshold for successful recovery plan forthe case of the United State to be 650 billions of US $. The re-covery stimulus bill imposed by Obama administration wasbigger than this threshold and successful. Despite the UnitedStates, in the European Union stimulating bill was far belowthe threshold and failed to help a fast recovery, see [11].In homogenous networks it is shown [11] that the thresh-old for successful stimulation is universal and independentof the network properties. In this paper it was shown that notonly response of the netwrok depends on its own structure,but also it depends on the strategies chosen by the govern-ment, i.e. the sectors where the stimulating bill is imposedon.Our analysis shows that in general, it is more efficient tostart stimulation from high degree nodes. The resource gapbetween HDS and LDS strategies is independent of both thenetwork size and its clustering coefficient. The gap betweenstrategies however is influenced seriously by the assortativityvalue of the networks. Networks with the highest assortativ-ity show the largest gaps. Such results indicate that in orderto obtaining some better estimate of the gap between HDSand LDS strategies, beside the degree distribution, we needto know other features of the studied network.Back to the first question raised in the paper, our analy-sis suggests that the US government should have focused onbig companies such as Chevrolet instead of small businesses.Due to the simplicity of the model, our findings might not bereliable for policy makers at this stage, nevertheles stronglysuggesting further studies, investigations, and simulationsbased on real data. Dynamical hysteresis is not restricted tothe Ising model. It exists in a wide range of systems whereagents can influence each other. Actually positive correla-tions can lead to dynamical hysteresis. So, for more realisticmodels we expect dynamical hysteresis exists and only quan-titative results are modified. An example is a model wherefirms can raise or decline production in a continouse level.Still for such model the dynamical hysteresis is observed andsurprisingly the threshold for successful stimulation is closeto the Ising model [49].One interesting discussion can occur if we consider theconsequence of our hypothesis for the long run trends. Inthe Ising model there is a rich literature with respect tothe response of the model to the external fields, see forexample [50] and references therein. While the literatureconcerning homogenous networks is rich, such studies arelacking for heterogenous networks. Findings of the papershow that such studies will deepen our understanding ofthe metastable states in socio-economic systems. R eferenceseferences