in this paper we investigate the likelihood of cascades of defaults in an interbank loan market .the novel feature of this investigation is that we use empirical data to calibrate our model and compare the results with different null models .this also allows us to make a statement about the accuracy of the existing literature on cascades in financial markets .we find that stylized random network models are very likely to overestimate cascade effects after a shock to the system .overnight loans are a financial instrument to redistribute liquidity between banks in the short term .these loans are settled in two different ways .most of them are settled as over - the - counter ( otc ) agreements , about which data is not systematically collected .studies on interbank loan markets therefor often rely on data that is reconstructed from the observation of payments between banks ( see , e.g. , * ? ? ?the data that we analyze here is not from otc transactions .it comes from the italian trading platform e - mid . in principleevery bank could join this platform , and in fact roughly one third of the users are foreign banks .banks can trade loans with different maturities and different currencies . in our studywe focus on the euro - denominated ( unsecured ) overnight loans , which constitute by far the largest share of all transactions .earlier studies have shown that the italian market can be regarded as representative for the euro money market , as the relative order flow to this market was relatively stable ( see , e.g. , * ? ? ?* ) .  there is an increase of research on linkages between banks since the 2008 financial crisis .some european markets have been analyzed for example by , , and . the u.s .money market system , fed - wire , has been analyzed for example by and .the interest in these markets has risen for two reasons : the partial collapse of the interbank markets itself and the increasing need for risk assessment in the bank network in general .the contagious effects that played a big role in the events after the lehman default showed that a micro - prudential analysis of banks exposures to counterparts does not capture the systemic risks that the default of a bank can pose ( see , e.g. , * ? ? ?* ; * ? ? ?* ) .  some earlier studies on the e - mid market exist ( see , e.g. , * ?from the analysis of conditional trading volume we know that no pronounced clusters of banks exist in the italian market .the development of spreads and lending relationships have been analyzed by , who found that banks try hard to avoid large exposures since 2008 .the network structure of the market ( see also * ? ? ?* ) can at best be described as a core - periphery structure , similar to the findings by for the german market . in parallel to our study , default cascades in the e - mid market have also been analyzed by , although their focus is on the search for an optimal network structure .we can however confirm one of their findings , which is that network structure matters mainly in fragile markets .several studies have tried to analyze systemic risk in interbank loan networks , for example by simulating knock - on effects in the case of a default . from relatively simple models like , who study cascade effects in a bernoulli random graph ,research has evolved to more fine - tuned models like in , where more realistic network structures and liquidity effects are taken into account. mechanisms of loss amplification have been discussed by and . show for austria that the network structure between banks is somewhat close to a scale - free graph , and that loan and bank size distributions can be approximated by a power law . analyzes cascades in the system of interbank loans for switzerland , analyze the brazilian market , using a more detailed data set . shows how missing data on links can be approximated efficiently .the paper by is more on the theoretical side , the authors discuss cyclical interdependence and stability in a system of mutual repayments .a recent paper by builds on the latter and analyzes how likely differently distributed shocks can lead to contagion .our paper is methodologically close to the one by , but uses empirical data on the loan network .the findings of and also the findings in the aforementioned empirical studies hint at limitations of simulation studies that use homogeneously sized banks or degree . specifically , it is not possible to characterize systemically important nodes or to create realistic chains of defaults if homogeneity is assumed .cascade models certainly do not reflect all aspects of real financial markets .for the case of financial institutions , one problem is that we do not have ( and probably never will have ) data on the activity of banks for all financial products through which networks between them exist .the second problem is that we can not know how the interbank market would continue on the days and weeks after a default , since in this kind of cascade simulations the start of the cascade is defined by the ( hypothetical ) removal of one node ( bank ) from the network . in this respect ,the analysis of default cascades differs from studies on disease spreading or cascades in social networks ( see , e.g. , * ? ? ?* ; * ? ? ?* ) in which cascades with a temporal structure are analyzed .  in this studywe do not aim at a complete model of financial contagion .for this , more complex models with different layers of connections and a dynamic structure would be needed ( see * ? ? ? * ) . however , in order to build these multiplex models , certain layers have to be approximated and rules for the behavior of participants have to be calibrated .the results that we obtain from the analysis of one known layer , the overnight market , are an important input for this modeling , as our results can serve as a benchmark for calibration and help to asses the margin of error which different specifications of network models can have .for this reason , we set up an interbank network that is derived from empirical data on the italian interbank loan market ( e - mid ) .specifically , we use daily snapshots of loan contracts in this market and derive a network of netted daily exposures .we randomly choose a bank to default and analyze knock - on effects to other market participants , based on the network given by interbank loans .we generate simplified balance sheets for each bank to determine its solvency .the balance sheet items that are not given by the data on interbank lending are hypothetical , but they are generated to be consistent with their interbank behavior , i.e. , we obtain a network with a reasonable distribution of bank size , where all banks have the same a - priori ability ( relative to their size ) to absorb a shock .finally we compare the results with those from different network models ( null models ) , in which distributional properties of the network are simplified .this allows us to judge which kind of networks models should be used in cases where empirical data is not available .the remainder of the paper goes like this : in section [ sec : methods ] we explain the data set , how we derive the bank s balance sheets , and how we simulate cascade effects . in section [ sec : results ] we discuss the simulation results . in section [ sec : null ] we compare the simulation results with those of different null models from random network .section [ sec : conclusions ] concludes the paper .[ cols="^,^,^,^,^",options="header " , ]   the interbank market activity of the bank shows as lending and borrowing in the balance sheet . in our simulationthe remaining assets are exogenous .the amount of capital is always in a fixed proportion to total assets .customer deposits conclude the liabilities side of the balance sheet .[ tab : bsheet ]  hence , while the amount of interbank loans are known exactly , we have to approximate the remaining items on the banks balance sheets .the balance sheet contains external assets , interbank lending , capital , deposits , and interbank borrowing ( see table [ tab : bsheet ] ) .we approximate the total assets for each bank by assuming that it is proportional to the average of the trading volume of the last 10 days in which the bank was active in the market . only from the current day s activity , and for estimating only from interbank lending . however , the results were more noisy , and , in the latter case , lead to an indeterminacy for banks that are only borrowing . ] we assume that the value of all balance sheet items are constant fractions of the total assets . the parameter values and that we choose for the baseline scenario are the same like in .they are in line with the empirical findings of and the figures reported in the stress - test of the . in the baseline scenario , the ratio of interbank lending to total assets is set to for each bank .hence , the total assets can be written as this already determines the asset side of the balance sheet , since the external assets are then given by .the balance sheet identity implies that for each bank the ratio of capital to total assets is set to , hence .since is given by the data , this determines the liabilities side of the balance sheet :    for each day of our sample , we simulate the default of each bank that is active as a borrower .we think it is important to simulate the default of all banks , since focusing on only larger banks would bias the results towards the local properties of large banks .we assume that the initial default is triggered by an external cause , which is not in the scope of our paper . as the consequence of this default, the bank will not meet its payment obligations on the interbank loan market .hence , all banks which are net lenders to this one will suffer losses to the position .assuming that the share of a bank s exposure to the defaulting institution is , the position will be reduced to .this loss has to be compensated by other positions of the banks balance sheet .since we consider the short - run effects directly after a default , losses can only be absorbed by the capital position .summing up , this means that a bank survives a shock as long as    to check the solvency condition , we calculate the remaining amount of capital ( equity ) for each bank after the shock to the system .the simulation of defaults in this system takes place in several rounds .after the initial default of one bank , only other banks with an exposure to this bank ( net lenders ) will be effected .we have to recalculate the balance sheets of all these banks , and determine , if they could absorb the initial shock. eventually some of them default themselves . in this case, we have to calculate the 2nd round effects on all banks that are net lenders to this institution . from this point on , also banks that were not connected to the initially defaulted bank can be effected .the recalculation of balance sheets is iteratively repeated until no further bank is found to default .  on the level of the individual bank we have to consider the shock that it might receive from a loan that is not repaid .assume that bank is default , because it has received a shock .it passes a residual shock to all its lenders .the residual shock is given by the difference between shock and its capital , .let us assume that represents the amount that bank lent to bank . represents the amount that borrowed from the set of banks . in case that banks s residual shock is larger than its total borrowing , bank will receive a shock equal to the loaned amount bank will repay a fraction of the loan :      left : the distributions for in- and out - degree differ slightly .for both years the distribution is wider for the out - degree ( lending ) than for the in - degree ( borrowing ) .the distributions have a tail and remind of a power - law with cutoff .right : the ( estimated ) capital ( in million euro ) of banks is also very unevenly distributed .we show the cdf for the empirical distribution of capital for 2006 and 2011 together with a fitted normal distribution . for 2006we also show a fitted exponential and generalized pareto distribution .( for this illustration we plot the capital for all banks and all trading days . )[ fig : degree ]  put differently , this means that we assume some recovery for defaults in the successive rounds of bank failures .defaulted banks will use their capital to repay interbank liabilities as good as they can , but they will not use any of the deposits . the exact design of this recovery rate might be debatable .we think that the assumption of some recovery is more realistic than the assumption of no recovery at all ( which is implicitly assumed in some other studies ) , because it has a tendency to inflate domino effects .for the simulation we use daily empirical loan networks . we simulate cascades by iteratively choosing each bank as the possible source of a cascade , and we repeat this process for each day in the data set . for each single runwe calculate if a cascade occurred , how many banks are affected , and how large the aggregated loss is .we can then related the magnitude of the cascade to the idiosyncratic properties of the banks , i.e. , we can analyze in how far the size of the initial shock , the capital buffer , or the network structure influence the outcome .the simulation with empirical data has some features that set it apart from simulations with random graphs .one feature is that the activity of the banks varies over time . in 2011 for example , the average number of banks that are active on one day is 70 , the minimum is 38 and the maximum is 89 ( 2006 average : 128 , min : 77 , max : 144 ) . the reason are mainly the bank holidays , which are not synchronized between countries . the average degree is more stable , for 2011 its fluctuations mostly stay in the band between 4 and 5 .the degree distribution is somewhat close to a power law , the number of nodes is of course too small for a precise classification .figure [ fig : degree ] shows that the maximum of the out - degree ( borrowing ) is slightly larger than the in - degree ( lending ) .this is caused by the fact that some larger banks serve as a kind of hub for the provision of liquidity , which leads to a slight asymmetry in the market .another feature of the data is that we can have more than one component .this means that parts of the network can be separated from the giant component .this can happen , when for example one small bank is borrowing from only one other bank , which itself has no other connections .this feature arises naturally from the strong heterogeneity in the size of banks .we checked that the large majority of banks are in one large component . in 2006 ( 2011 )the average number of components was 1.7 ( 2.3 ) .       in these scatter plots of simulation results ,the shock size is measured by the amount of failed loans of the bank that initially defaults .the left panel shows the resulting cascade measured by the share of banks that default as a consequence of this shock . in the right panelthe cascade is measured by the share of interbank loans that fail as a consequence of the initial default .[ fig : cascadesize ]  whether the initial default of a bank leads to significant losses to the entire system depends heavily on the size of the initial shock . for 2006we analyze 25,914 hypothetical initial defaults , in 13,206 cases ( 50.1 % ) this default triggers the successive default of at least one other bank . in 2011 the number of transactions and also banks is much lower , thus we investigate only 13,614 cases of hypothetical initial default , 8,385 of which lead to at least one knock - on default of another bank ( 61.6 % ) . the first impression from these numbersis that the interbank market has become more fragile .one reason is that banks share risks with fewer counterparts in 2011 ( see also * ? ? ?however , it should be noted that in the simulations the ratio of interbank market activity to capital is kept constant . even though the interbank network might be backed by more capital after the financial crisis, we can still conclude that the network is structurally more prone to cascade effects in 2011 than it was in 2006 .figure [ fig : cascadesize ] shows that the cascade size in our simulation depends on the size of the shock from the initial defaulter . in network science oftena threshold is defined after which knock - on effects are regarded as a cascade , e.g. , when more than 5% of the nodes are affected .if such a measure is used in 2006 ( 2011 ) we have 904 ( 1782 ) cases , 3.5% ( 13.1% ) , in which a cascade happens .the pure number of affected nodes is not necessarily the best measure to quantify the cascade , because the banks which they represent can be of very different size .hence , the right panel shows the cascade size measured by the fraction of total lending losses .the difference is that now cases in which high initial shocks lead to the default of only few but probably big counterparts are more accurately displayed , the same holds for small initial defaults that only trigger further small banks .when we apply the same 5% threshold to this measure , we find that the difference between the years appears smaller , since in 2006 4.2% of initial defaults result in a cascade compared to 8.5% in 2011 .   a reduction in the ratio of capital to total assets has a non - linear effect on the average cascade size , because at low levels of capital the chance of domino effects increases .increasing the ratio of interbank assets to total assets leads to an almost linear increase of the average cascade size .[ fig : parameters ]  while the linkages in the interbank market for our simulation are derived from empirical data , and can thus be regarded as representative for the interbank assets , the remainder of the bank s characteristics are only approximated .the absolute numbers of certain balance sheet items are in fact not important for the results that we obtain . the parameters and , which describe the share of interbank assets and the ratio of capital to total assets , are however very important . figure [ fig : parameters ] shows their influence on the average cascade size .the important difference between them is that the cascade size increases only slowly once passes some upper bound , while lowering leads to a critical point after that defaults sky - rock .although these results are qualitatively similar to those of for the bernoulli random graph , heterogeneity of banks and network structure lead to more realistic results .different from we observe dependency from for a wider range of values , and a continuous increase in the cascade size for ( as a result of the network structure and the larger number of banks ) . as discussed in section 2.1 ,our choice for the parameter values for the baseline scenario are an approximation based on empirical findings . however , the qualitative results presented in the remainder of section 3 do not critically depend on the exact choice of these values .the heat maps show the average cascade when a bank with a specific degree defaults in 2006 .the left panel shows that the cascade size increases with the in - degree ( borrowing ) of the defaulted bank . the right panel shows the relationship of in - degree and the shock size , i.e. the amount of loans that are not re - payed by the defaulted bank .we observe that large shocks do not necessarily lead to the largest cascades ; medium sized shocks among well connected banks can affect even more banks . [fig : initial_degree ]     the average cascade size increases with the core number of the initially defaulted bank .( the vertical bar shows the standard error . )[ fig : initial_bc ]  in the following we analyze the influence of individual node characteristics on the cascade size and on the chance of survival of banks after a shock .we start with an analysis of the initially defaulting bank .we have already seen that the cascade size depends on the size of the initial shock . however , besides the pure size of the shock the connectivity of a bank and the network structure play a role ( see also * ? ? ?the left panel of figure [ fig : initial_degree ] shows the average cascade size dependent on the in- and out degree of the bank that we choose to default .interestingly , we observe only few banks that have large in- and out - degree . the cascade size increases with the in - degree ( borrowing of the defaulted bank ) , as can be expected .more interesting is to disentangle the influence of in - degree and shock size .the right panel shows that even medium sized shocks can cause large cascades , if banks are well connected .it has to be noted that some of the larger shocks in our data set stem from large international banks , which connections to other global players outside the italian e - mid platform can not be accounted for in this simulation .however , the analysis clearly shows that for any given shock size greater connectivity leads to larger cascades , i.e. , a larger number of defaulted banks .  left panel : the conditional probability of a bank to default as part of a chain reaction decreases with the in - degree ( borrowing ) . for the out - degree ( lending )we observe a hump - shaped relationship .middle panel : banks with an intermediate closeness are the ones most prone to default in our simulation .right panel : also for the core number we observe a hump - shaped relationship with the probability to default .[ fig : vulnerable ]  a related measure for centrality is the core number of a node .a k - core is defined as the largest subgraph which nodes have at least connections ( see * ? ? ?hence , if a node has a high core number , this shows that it has not only a high degree , but that it is also connected to nodes with a similar high degree .figure [ fig : initial_bc ] shows that in fact the cascade size increases with the core number .previous studies on disease spreading models have also shown this effect of coreness on the disease outbreak size ( see , e.g. , * ? ? ? * ) .after analyzing which bank is the most `` dangerous '' when it defaults , we can now turn to the analysis of the vulnerability of banks to this default .hence , we look at measures of node centrality to determine which characteristics ( besides the shock size ) determine the probability of a bank to default at any stage of a cascade process .we start with the degree , which is reported in the left panel of figure [ fig : vulnerable ] . for both years , 2006 and 2011, the conditional probability to default is decreasing with the in - degree ( borrowing ) , because banks that borrow a lot are less vulnerable ( and mostly lend less than they borrow ) .the results for the out - degree are more interesting . in general, we would expect that the default probability decreases with degree , since shocks are wider distributed among banks .however , for 2006 we observe a hump - shaped curve , the probability to default increases for a degree of up to 7 , before it slowly decreases for banks with more links .for 2011 , where the network is smaller , we barely reach the point where the probability decreases ( the results for degree > 15 are noisy due to the small sample ) .we can infer that this sample contains a lot of banks that in terms of risk sharing are too small .some of these might in fact be part of an affiliate group of banks .the analysis of some centrality measures helps us to understand these patterns .the node closeness describes how easily a node can be reached from any other one in the network ( see , e.g. , * ? ? ?the middle panel of figure [ fig : vulnerable ] shows that nodes ( banks ) that have low closeness , and are thus difficult to reach for a shock , have a low probability to default .the probability increases for values of up to 0.26 and then decreases for nodes with a higher closeness . put differently , for the connectivity of banksthere is a threshold after which the likelihood to be hit by a shock is out - weight by the benefits from spreading exposures .we obtain similar results from an analysis of the default probability dependent on the core number .the probability to default is highest for banks with a core number of 4 - 5 , and decreases for values higher than this .our analysis emphasizes that degree , centrality , and coreness are crucially related to the banks ability to withstand shocks .models with random interconnections between banks can not reproduce all these findings .therefore they demand further investigation .this is evenmore true for the unclear influence of the heterogeneity in the size of banks . in the next section ,we compare the results from the empirical cascade simulation with the results from null models to disentangle the effect of network structure and heterogeneity in the size of banks .the box - plots show the average daily cascade size for the empirical model ( left ) compared to the null models described in this section for 2011 .the rewired and random null model lead to a slightly higher cascade size .the null models that randomize both , size distribution and links , lead to much larger cascades than the empirical model .[ fig : null ]  the interbank network has several interesting structural features that are likely to have a significant impact on its stability . among those are the heterogeneity in the bank size , the directionality and assortativity of links , and the heterogeneity in the link weights . in order to investigate the role of each of these features we need to derive a benchmark; this means that we have to derive the average cascade size in a network where these features are excluded or randomized .such a benchmark network is called a null model .null models are an established tool to separate `` natural '' network effects from influences that are caused by the actual degree sequence of a network , i.e. , behavior that is likely to be strategic and planned , or the outcome of the environment .null models have been studied in ecological networks ( see , e.g. , * ? ? ?* ) , protein networks ( see , e.g. , * ? ? ?* ) , commmunication networks ( see , e.g. , * ? ? ?* ) , face - to - face interaction networks ( see , e.g. , * ? ? ?* ) and recently in trade networks ( see * ? ? ?* ) .  hence , we compare the stability of the model based on the empirical network of the interbank market with the stability of different null models . in all caseswe preserve the number of nodes , the number of links , and the directionality of the network . specifically , we generate the following null models :  in the _ rewired _ null model we preserve the degree of each nodes , and the weight of each link ( loan size ) .the algorithm randomly selects pairs of links with similar size and rewires ( switches ) the counterparts of these links , if the rewired links do not yet exist .if the rewired links are already present in the network , the algorithm picks another pair with similar weights .the banks balance sheets are adjusted to be consistent with the slight changes in the bank s interbank market activity .this null model is similar to one proposed by .the model should show in how far the non - randomness of connectivity influences the stability of the network , given the very heterogeneous link weights .in the _ random _ ( erds - rnyi ) null model we start out with the empirical set of loan contracts and generate random links between the banks using the empirical loan contract weights .this null model destroys the topological structure of the network , but preserves the daily interbank volume .the size of the banks , their capital , and their degree are also randomized . the banks balance sheets are again adjusted to be consistent with interbank market activity .  the _ fixed weight _null model preserves the number of links for all banks , but assigns constant weights to all of them . for each day , the constant weight is calculated by dividing the total lending by the number of contracts on that day .hence , this null model preserves the degree distribution , but makes the link weights homogenous .accordingly , the size of banks ( and their capital ) is proportional to the degree .finally , we assume a _ random network with fixed weights_. in this null model the link weights are constant , as described in the previous model. moreover , we randomize the connectivity of the nodes like in the random model . this model has the least similarity with the empirical network ; it also has the largest deviation in the resulting average cascade size .figure [ fig : null ] illustrates the size of the cascade in different null models compared to the results from the empirical network model of 2011 with its baseline parametrization . for the rewired modelthe average cascade size is comparable to that from the empirical network . the random null model , which destroys the degree connectivity but keeps the weights , results in a 1.4 times higher cascade size .the fixed weight model leads to an even 1.9 times higher average cascade .the random fixed weighted model results in 2.5 times more defaults than the empirical model .the analysis of the null models reveals that simulating bank networks assuming homogeneity in the bank and loan size dramatically overestimates its fragility . in comparison ,the `` error '' from ignoring the size distributions of loans and banks seems to be much greater than assuming a wrong distribution of links and degree .  in section[ sec : sensitivity ] we have shown that the cascade size depends on the lending activity and on the average amount of capital , i.e. the choice of the parameters and . similarly we can analyze the sensitivity of the relative differences in the cascade size for the different null models . to reduce the dimensionality of the analysiswe will only look at a variation of the ratio of capital to total assets .this is without loss of generality , since the system could also be described by one single parameter , the ratio of lending to capital ( the system has one free parameter ) .left : null models with randomized network structure show a exaggerated increase in cascade size for systems with low capital compared with the empirical model .right : when the cascade size is measured by the lending loss the default of smaller banks appears less severe in the fixed weight and random fixed weight model , the defaults in the random model show relatively stronger than in the empirical model .the shaded areas show standard errors .[ fig : null_all ]  the left panel of figure [ fig : null_all ] shows the average cascade size for all five models depended on ( or , more generally speaking , for different levels of fragility ) .the deviation of the null models from the empirical model differ most when is decreased to values much lower than in the baseline parametrization . while the fixed weight null model seems to result in a constant overestimation of the cascade size , the two models with a randomized network structure show exponentially increasing deviations in the cascade size for low values of . for the fixed weight modelwe observe a greater likelihood of chains of defaults of small banks than in the empirical model . in this casethe fixed weights lead to the creation of many similar sized small banks , which all can not withstand defaults of slightly larger banks .the increasing deviation of the models with the randomized network structure stems from the micro - structure of the overnight loan market . in the short runmost of the banks are predominantly active on one side of the market , i.e. , excess liquidity or demand leads to the fact that they focus on either borrowing or lending . since a bank can only pass shock to other banksif it is active on both sides of the market , the effect of randomization of the network structure is especially large when the system is rather unstable ( low ) and chains of defaults gain importance versus direct effects of an initial default .the rewired model shows cascade sizes that are almost indistinguishable from the empirical model , which leads to the conclusion that the in- and out - degree distributions together with bank size is a sufficient description of the system .alternatively we can again look at the lending loss as a measure of the severity of the cascade .the results for the different models are presented in the right panel of figure [ fig : null_all ] .although the absolute numbers are smaller here , the qualitative results are similar . in comparison with the left panel ,the models which use fixed link weights generate less lending loss for relatively high values of than the empirical model .this is caused by the default of banks that appear smaller than in the empirical model .this effect of the fixed link weights vanishes once becomes relatively small .in these cases the homogeneity in the size distribution leads to more chains of defaults .in this paper we analyzed the likelihood of cascades in an empirical interbank loan market .the novel aspect of the paper is that we assume heterogeneity in the size of banks and loans , which are derived directly and indirectly from empirical data .our results indicate that the cascade size depends not only on the size of the initial shock , but also on the in - degree of the defaulter .moreover , the position of the initial defaulter within the network , e.g. its centrality and coreness , plays an important role for its ability to start a cascade .we found that the fragility of this market , given our model , has increased significantly from 2006 until 2011 .our analysis of the vulnerability of banks shows that for 2006 , in line with theoretical results on risk sharing , the probability to default decreases with the out - degree . for 2011however , we do not observe the same pattern , instead there is strong positive correlation between the centrality and vulnerability of banks .it seems likely that the increase in vulnerability is linked to a change in lending behavior , caused by the lost trust in many borrowers , and that this change feeds back as a negative influence on overall system stability .  to analyze specific aspects of the network structure on the cascade process , we made use of different null models .the results show that the empirical network has features , which can not be captured by randomized network models .furthermore , we show that assuming homogeneity for bank size and link weights can lead to an overestimation of the fragility of the system .we believe that this study is an attempt to model more realistic financial networks .further research should focus on the influence of temporal aspects of the interbank network on cascades of defaults .another research direction is to model multiplex networks that incorporate different financial products and markets .the authors thank petter holme , martin rosvall , and thomas lux for helpful discussions .fk thanks the swedish research council .mr thanks the leibniz association for partial funding of this project .isella , l. , stehl , j. , barrat , a. , cattuto , c. , pinton , j. f. , van den broeck , w. , 2011 . what s in a crowd ?analysis of face - to - face behavioral networks .journal of theoretical biology , 271(1 ) , 166 - 180 .karsai , m. , kivel , m. , pan , r. k. , kaski , k. , kertsz , j. , barabsi , a. l. , saramki , j. , 2011 .small but slow world : how network topology and burstiness slow down spreading .physical review e , 83(2 ) , 025102 .

Abstract: we analyze cascades of defaults in an interbank loan market . the novel feature of this study is that the network structure and the size distribution of banks are derived from empirical data . we find that the ability of a defaulted institution to start a cascade depends on an interplay of shock size and connectivity . further results indicate that the ability to limit default risk by spreading the lending to many counterparts decreased with the financial crisis . to evaluate the influence of the network structure on market stability , we compare the simulated cascades from the empirical network with results from different randomized network models . the results show that the empirical network has non - random features , which can not be captured by rewired networks . the analysis also reveals that simulations assuming homogeneity for size of banks and loan contracts dramatically overestimates the fragility of the interbank market .   * keywords : * interbank loan network systemic risk cascades  null models   _ the final publication is available at link.springer.com + http://link.springer.com/article/10.1007/s10614-014-9478-z + to be published in computational economics _


all introductory physics textbooks , with or without calculus , cover the addition of both resistances and capacitances in series and in parallel . the formul for adding resistances [ eq:1 ] r & = & r_1 + r_2 +  , , + [ eq:2 ] 1r & = & 1r_1 + 1r_2 +  , , and capacitances [ eq:3 ] 1c & = & 1c_1 + 1c_2 +  , , + [ eq:4 ] c & = & c_1 + c_2 +  , , are well - known and well - studied in all the books.  in books with calculus there are often end - of - chapter problems in which students must find and using continuous versions of equations ( [ eq:1 ] ) and ( [ eq:4 ] ) .however , we have found _ none _ which includes problems that make use of continuous versions of equations ( [ eq:2 ] ) and ( [ eq:3 ] ) .students who can understand and solve the first class of problems should be able to handle the second class of problems , as well .we feel that continuous problems that make use of all four equations should be shown to the students in order to give them a global picture of how calculus is applied to physical problems .physics contains much more than mathematics .when integrating quantities in physics , _ the way we integrate them is motivated by the underlying physics_. students often forget the physical reasoning and they tend to add ( integrate ) quantities only in one way .  in this paper, we introduce an approach to solving continuous versions of equations ( [ eq:2 ] ) and ( [ eq:3 ] ) that is as straightforward and logical for the students as solving continuous versions of equations ( [ eq:1 ] ) and ( [ eq:4 ] ) .we then extend the logic to the addition of other quantities encountered in undergraduate introductory physics .this demonstrates that the method is not specific to resistors and capasitors but general and includes all quantities obeying similar addition laws .the organization of this article is as follows : section [ sec : formulae ] discusses many physical quantities taken from introductory physics that obey similar addition laws . among these quantities ,resistance and capacitance are the most well - known to students .inductance is also known but probably not mastered at the level it should be .elasticity is somewhat known , but thermal resistance , diffusion resistance , and viscous resistance are almost unkown to students . as a result , for resistance and capacitance we only remind readers of the basic formul , while for the rest quantities we expand the discussion to some length so the students will become familiar with the physical background . in section [ sec :examples ] , we present basic applications of the addition formul .this section is meant to demonstrate in a simple way how one chooses the correct addition formula ( in series or in parallel ) , given a problem . in section [ sec : problems ] we solve several additional problems that make use of the main idea . in each problem , we have chosen one representative quantity . however , the reader must realise that the same problem can be stated for any of the quantities given in section [ sec : formulae ] , not just the chosen quantity .we hope that this article will motivate teachers to explain to students the subtle points between ` straight integration ' as taught in calculus and ` physical integration ' to find a physical quantity .the basic formula to compute resistance is the formula of a uniform cylindrical resistor : where is the resistivity of the material , is the length of the conductor and the cross - sectional area . written as conductance .however , the term _ conductance _ is in uniform lingustic aggreement with the rest of the terminology . ] , this is is where is the conductivity of the material .the basic formula to compute capacitance is that of a parallel - plate capacitor filled with a uniform dielectric material : where is the dielectric constant of the dielectric , the area of the plates , and the distance between the plates .we are going to call the inverse of the capacitance the * incapacitance * of the capacitor . for a parallel - plate capacitor    calculation of inductanceis usually based on the definition l_bi , [ eq : ldef ] and therefore some discussion is necessary regarding our point of view .it is well - known that the inductance for a solenoid is given by , [ eq : lsolenoid ] where is the number of turns of the solenoid , is the cross - section and the length of the solenoid .this expression can be used as the basic formula when we compute the inductance of another geometry which involves an inductor made from a single wire by twisting it in a particular geometry and creating a number of turns .when we have some geometry in which no obvious way to count ` turns ' exists , we must use a different formula .clearly , the corresponding basic formula must come from a simple geometry giving rise to a uniform magnetic field . in figure[ fig : parallel - plates ] we see an inductor with such properties .the inductor consists of two finite - plane wires that carry opposite currents with uniform linear current density . due to the complete analogy with a parallel - plate capacitor , we call itthe * parallel - plate inductor*.    an infinite sheet with uniform linear current density creates a uniform magnetic field in space with value .the direction of the field is found using the right - hand rule .when two infinite sheets are given , the total field is the sum of two fields . given the directions of the currents of the parallel - plate inductor , the magnetic field adds to zero outside the plates and to between the plates .of course , in the case of finite plates this result is only approximate , being a good approximation when . to compute the inductance of the parallel - plate inductor ,we must compute the flux in the definition ( [ eq : ldef ] ) .we notice that the magnetic field is perpendicular to the area . in this case and .therefore notice that the only difference is that the ` solenoid ' is now made from only one turn , i.e. .therefore , in problems where the current is distributed continuously only ` one turn ' should be counted .this issue can cause to curious students considerable frustration since they can raise the following objection : in the calculation of the inductance , we concluded that since one ` turn ' was used ( incorrectly ) in the computation of flux , namely only the flux through one cross - section was used .however , there is an infinite number of cross - sections and an integration has to be done before inserting the flux into equation ( [ eq : ldef ] ) .therefore , the result is not correct . " in this calculation , we believe that it is _ not _ easy to persuade the student that she is wrong and no flux has been lost . and unfortunately , similar situations arise in all continuous problems .books fall short in providing any answers to this question they just present the solution .our method has the advantage , among others , that it answers this question in a satisfactory and unambiguous way .it shows explicitly that all flux has been counted .this is easily understood if the reader studies the problems of sections [ sec : parallel - ind ] and [ sec : coaxial ] .below , we present an independent calculation to confirm that the result found above for the inductance of the parallel - plate inductor is correct .the alternative calculation will be based on the energy stored in the inductor .the magnetic energy density for the parallel - plate inductor is the total energy stored in the magnetic field is then however , for any inductor , the energy stored in the magnetic field is also given by .when this is compared to the previous result , we conclude that l=_0dw , [ eq : lbasic ] i.e. the same result as found above .we are going to call the inverse of the inductance the * dedutance * of the inductor .therefore , the deductance of a parallel - plate inductor is    imagine a cylinder made from a uniform conducting material whose bases are kept at different temperatures .then , due to the temperature difference between the bases , heat will flow from one base to the other . the rate according to which heat is flowing , i.e. is called the * thermal current*. it is known , see for example , that i_th=_th a tl , [ eq : i1 ] where is the length of the cylinder , is its cross - section , and is a constant characteristic of the material , called the * thermal conductivity*. we define the inverse of the thermal conductivity to be the * thermal resistivity * of the material .equation ( [ eq : i1 ] ) is sometimes referred to as * fourier s law * for the flow of energy .the * thermal resistance * of the cylinder is then defined by _th ti_th .[ eq : i2 ] notice the analogy with the standard resistance : .potential difference is the reason electric current flows . here, temperature difference is the reason behind the thermal current . comparing the two formul  ( [ eq : i1 ] ) and ( [ eq : i2 ] ) we have written above, we arrive at , [ eq : i7 ] an expression almost identical to that of the electric resistance for the cylinder .we define the * thermal conductance * as this implies that for the uniform cylinder .[ eq : i8 ]   the thermal conductivity discussed in the last section is only a particular example from a more general category of processes known as * transport phenomena * .transport phenomena are irreversible processes that occur in systems that are not in statistical equilibrium . in these systems, there is a net transfer of energy , matter , or momentum .fourier s law stated in the previous section was a law for the flow of energy .we will write similar laws for the flow of matter and the flow of momentum .imagine a cylinder filled with gas such that the particle densities and of the gas at the bases are kept constant at different values .then , due to the density difference between the bases , particles will flow from one base to the other .the rate according to which particles are flowing , i.e. is called the * particle current*. it is known , see for example , that i_diff=_diff a nl , [ eq : i3 ] where is the length of the cylinder , is its cross - section , and is a constant characteristic of the material called the * diffusion coefficient*. another name , in the spirit of what we have been discussing , would be * diffusion conductivity*. the inverse of , , is named the * diffusion resistivity * of the material . equation ( [ eq : i3 ] )is sometimes referred to as * fick s law*. the * diffusive resistance * of the cylinder is then defined by _diff ni_diff .[ eq : i4 ] its inverse gives the * diffusive conductance * : imagine that the thermal agitation ( speed ) of the molecules at the two bases of the cylinder is different .then , due to the speed difference between the bases , momentum will flow from one base to the other . the rate according to which speed is flowing , i.e. is called the * momentum current*. it is known , see for example , that i=_vis a vl , [ eq : i5 ] where is the length of the cylinder , is its cross - section , and is a constant characteristic of the material called the * coefficient of viscosity*. another name , again in the spirit of what we have been discussing , would be * viscous conductivity*. the inverse of , , is the * viscous resistivity * of the material . the * viscous resistance * of the cylinderis then defined by_ vis vi .[ eq : i6 ] its inverse gives the * viscous conductivity * :   formul ( [ eq : i7 ] ) and ( [ eq : i8 ] ) we derived in the previous section are also applicable in the present cases with the appropriate index changes .although it is not the topic of our article , we mention that ultimately all transport phenomena are related via the microscopic description . in particular we can find the following expressions for the coefficients : _ th & = & 12nk_b v_ave l_free , + _ diff & = & 13 v_ave l_free , + _ vis & = & 13 nm v_ave l_free , where is the mean free path of the molecules , the average velocity of the molecules , the number of molecules per unit volume , the mass of one molecule , and boltzmann s constant .these expressions should be compared with that of the electric conductivity which is more familiar : where is the charge of a carrier .the concept of elasticity is more than a mere definition .the behavior of a rubber band or the behavior of a rod or a cable under stress is basically analogous to that of many springs connected together .let s imagine a uniform rod of length and cross - sectional area .we focus on an infinitesimal piece of length at distance from one base .if is the infinitesimal extension of this piece under the force , then hooke s law states that where is the elasticity constant for the piece .we can write the above relation as where is the elasticity per unit length and is the extension of the system per unit length , known as * linear strain*. it is known that approximately  where is the cross - section and is a constant characteristic of the material known as * young s modulus*. combining the last expressions we conclude that if is constant then where is the length of the system. the stiffness would be respectively : we have thus obtained basic formul similar to those of resistors and capacitors that allow the computation of and in any geometry .most probably these formul are well - known to engineers , but they are not well - known among physicists .however , once written down , they look familiar and natural .let be some resistors , be some capacitors , be some inductors , be some springs , be some transport conductors ( either thermal , or diffusion , or viscous ) .we will also indicate by , , , , the equivalent resistance , capacitance , inductance , stiffness , and transport resistance , respectively , either when the objects are connected in series or in parallel . either from introductory physics , or as a straightforward exercise on the definitions , the reader can persuade himself that when the objects are connected in series r & = & r_1 + r_2 +  , + 1c & = & 1c_1 + 1c_2 +  , + l & = & l_1 + l_2 +  , + 1k & = & 1k_1 + 1k_2 +  , + & = & _ 1 + _ 2 +  , and when they are in parallel & = & 1r_1 + 1r_2 +  , + c & = & c_1 + c_2 +  , + 1l & = & 1l_1 + 1l_2 +  , + k & = & k_1 + k_2 +  , + 1 & = & 1_1 + 1_2 +  . introducing the conductance , incapacitance , deductance , elasticity , and thermal conductance , we can rewrite them in a form that is always additive : where , & & , p  r , d , l , , , + & & , p  g , c , k , k , .  in the remaining section , we will demonstrate the application of the addition formul by examining specific examples from resistors and capacitors .along with the solution , several comments are made to help the reader understand some of the implicit assumptions and other details in the solution . in the section that follows ,more problems are discussed for the readers who wish to master the technique .  1 .* problem * [ cylindrical resistor ] + the cylindrical resistor shown in figure [ fig:1 ] is made such that the resistivity is a function of the distance from the axis .what is the total resistance of the resistor ? + .a potential difference is applied between the bases of the cylinder and therefore electric current is running parallel to the axis of the cylinder.,width=377 ] + * solution * + we divide the cylindrical resistor into infinitesimal resistors in the form of cylindrical shells of thickness .one of these shells is seen in red in figure [ fig:1 ] .when the current is flowing along the axis of the cylinder , the infinitesimal resistors are _ not _ connected in series .instead , all of the infinitesimal cylindrical shells of thickness are connected at the same end points and , therefore , have the same applied potential . in other words ,the shells are connected in parallel and it is the conductance that is important .specifically for the infinitesimal shell therefore g = _ cylinder dg 2 _ 0^a ( r ) r dr .for example , if , then g = 2_0 a^2 , where .the resistance is therefore r_02 a^2 .* problem * [ truncated - cone resistor ] + a resistor is made from a truncated cone of material with uniform resistivity .what is the total resistance of the resistor when the potential difference is applied between the two bases of the cone ? + * solution * + this is a well - known problem found in many of the introductory physics textbooks .we can partition the cone into infitesimal cylindrical resistors of length .one representative resistor at distance from the top base is seen in figure [ fig : cone ] .the area of the resistor is and therefore its infinitesimal resistance is given by from the figure we can see that + .,width=264 ] + the infinitesimal resistors are connected in series and therefore r=_cone dr = h(c - b ) _b^c drr^2 = hbc .[ eq : rcone ] + * comment * : however , this solution , which is common in textbooks , tacitly assumes that the disks used in the partition of the truncated cone are equipotential surfaces .this is of course not true , as can be seen quite easily .if they were equipotential surfaces , then the electric field lines would be straight lines , parallel to the axis of the cone. however , this can not be the case as , close to the lateral surface of the cone , it would mean that the current goes through the lateral surface and does not remain inside the resistor .therefore , the disks are not equipotential surfaces .one way out of this subtlety is to assume that the disks are approximate equipotential surfaces as suggested in .this is the attitude we adopt in this article as our intention is not to discuss the validity of the partitions used in each problem , but to emphasize the unified description of resistances and capacitances as additive quantities .similar questions can be raised and studied in the majority of the problems mentioned in the present manuscript . a reader with serious interests in electricity is referred to the article of of romano and price where the conical resistor is studied .once that article is understood , the reader can attempt to generalize it to the rest of the problems of our article . 3 .* problem * [ cylindrical capacitor ] + in introductory physics , the capacitance of a cylindrical capacitor is found using the definition , where is the charge on the positive plate of the capacitor and the absolute value of the potential difference between the two plates. however , this problem asks to compute the capacitance using only the formul giving the capacitance of a parallel plate , plane capacitor .+ * solution * + as shown in the left side of figure [ fig : hollowcylinder ] , the capacitor is partitioned into small cylindrical capacitors for which the distance between the plates is .for such small capacitors , the formula of a parallel - plate capacitor is valid .we notice though that all infintesimal capacitors are connected in series .therefore and the total capacitance is then +  and and height . in the left picture ,we have sliced it in infinitesimal cylindrical shells , while in the right picture we have sliced it in infinitesimal annuli.,title="fig:",height=377 ]  and and height . in the left picture , we have sliced it in infinitesimal cylindrical shells , while in the right picture we have sliced it in infinitesimal annuli.,title="fig:",height=340 ] + * comment * : one might be tempted to partition the cylindrical capacitor into infinitesimal capacitors as seen in the figure to the right ( blue section ) .such capacitors look simpler than the infitesimal cylindrical shell we used above .furthermore , they are connected in parallel ( notice that each capacitor is carrying an infinitesimal charge and ) and therefore it is enough to deal with capacitance , , and not incapacitance .+ however , with a minute s reflection the reader will see that in order to use the parallel - plate capacitor formula in the infinitesimal case , the distance between the plates must be infinitesimal which indicates that the infinitesimal capacitors must be connected in series . in the proposed ( blue )slicing , the distance between the plates of the infinitesimal capacitor is finite , namely .the infinitesimal capacitor is still a cylindrical capacitor of infitesimal height and therefore its capacitance should be expressed in a form that is not known before the problem is solved . in other words , dc =  2_0(b / a ) dz [ eq : unknown ] from which c =  2_0(b / a ) _ 0^hdz =  2_0(b / a ) h .[ eq : known ] however , the expression ( [ eq : unknown ] ) is unkown until the result ( [ eq : known ] ) is found .* problem * [ truncated - cone capacitor ] + a capacitor is made of two circular disks of radii and respectively placed at a distance such that the line that joins their centers is perpendicular to the disks .find the capacitance of this arrangement ( seen in figure [ fig : cone ] ) .+ * solution * + we partition the capacitor into infinitesimal parallel - plate capacitors of distance and plate area exactly as seen in figure [ fig : cone ] .these infitesimal capacitors are connected in series and therefore the incapacitance is the relevant additive quantity : notice that the computation is identical to that of with final result : d 1_0 hbc  c= _ 0 bch .[ eq : trunccone ] when , we recover the result of the parallel - plate capacitor .in this section we pose and solve a number of problems that will help the reader become fluid in the application of the quantities in table [ table:2 ] .a few of the problems are well - known , standard ones found in all textbooks ; these problems are re - examined and solved in this section using our technique .re - derive the well - known expression for the capacitance of a spherical capacitor ( where are the radii of the spheres with ) by partitioning it into infinitesimal capacitors .we partition the capacitor into spherical shells of thickness .the infinitesimal shells are connected in series and therefore their incapacitance is the total incapacitance is thus d = _ dd 14_0 _drr^2 14_0 b - aab , from which we find the well - known formula for the capacitance :      show that the capacitance of a spherical capacitor which is filled with a dielectric having dielectric constant , where is the distance from the center and are constants , is given by for and for .  as in the previous problem, we partition the capacitor into spherical shells of thickness . the infinitesimal shells are connected in series and therefore their incapacitance is the total incapacitance is thus d = _ dd 14_0 _ dr(r ) r^2 = 14_0 c _a^b drr^n+2 . for , for , d 14_0 c _a^b drr^n+2 14_0 c r^-n-1-(n+1)|_a^b 14_0 c(n+1 ) b^n+1-a^n+1b^n+1 a^n+1 .show that the capacitance of a cylindrical capacitor which is filled with a dielectric having dielectric constant , where is the distance from the axis and , are constants , is given by   we partition the capacitor into coaxial infinitesimal cylindrical shells of radius and thickness which are connected in series ( see figure [ fig : hollowcylinder ] ) .the infinitesimal incapacitance of such a shell is therefore d12_0 hc_a^b drr^n+1 12_0 hcb^n - a^nn a^n b^n , since .therefore i.e. exactly the advertised formula .show that the capacitance of a cylindrical capacitor which is filled with a dielectric having dielectric constant , where is the distance from the base and , are constants , is given by   we partition the capacitor into coaxial infinitesimal annuli of thickness which are connected in parallel ( see figure [ fig : hollowcylinder ] ) .each infinitesimal capacacitor has the geometry of a cylindrical capacitor and therefore its infinitesimal capacitance is given by therefore c 2_0 c ( b / a ) _0^h z^n dz 2_0 c ( b / a ) h^n+1n+1 , since .\(a ) two metallic flat annuli are placed such that they form a capacitor with the shape of a hollow truncated cone as seen in figure [ fig : hollowcone ] .partition the capacitor into infinitesimal capacitors and show that the capacitance is given by show that this result reduces to that of a cylindrical capacitor for .also , show that it agrees with the result for a parallel - plate capacitor with .\(b ) now , fill the two bases with disks of radius and argue that the capacitance of the hollow truncated cone equals that of the truncated cone minus the capacitance of the parallel - plate capacitor that we have removed .this means that the capacitance of the hollow truncated cone should equal how is it possible that this result does not agree with that of part ( a ) ?\(a ) we divide the truncated cone into annuli of height . these are parallel - plate capacitors connected in series .therefore from the similar triangles see on the left side of figure [ fig : hollowcone ] , we see that r - bc - b  dzhc - b dr . [ eq : ratio ] therefore dd 1_0 hc - b drr^2-a^2h2_0a(c - b ) ( 1r - a-1r+a dr , and d&=&h2_0a(c - b)_b^c ( 1r - a-1r+a dr h2_0a(c - b)(- .when we see that and therefore the result can not be found by simple substitution . however we can use l hospital s rule and the limit exists , then . ] : dh2_0a_cb- c - b h2_0a_cb ddc h2_0a 2ab^2-a^2 .h_0 1b^2-a^2 .this is just for a plane capacitor .  the case is obtained in the same way : d&= & h2_0(c - b)_a0 -a h2_0(c - b ) _a0dda ( - + & = & h2_0(c - b ) _ a0 ( -2cc^2-a^2 - -2bb^2-a^2 h_0 cb , as found previously .\(b ) we use a parallel - plate capacitor with circular plates of radius at a distance to fill the plates of our capacitor .this capacitor has capacitance the conical capacitor we have thus created has capacitance the original capacitor and the one we added are connected in parallel since the same voltage is applied at their plates .therefore , according to the superposition principle apparently this result does not agree with that of part ( a ) .the reason is subtle but easy to find .the superposition principle states that _ if a problem in electricity can be split in two other problems , then the solution to the original problem is the sum of the solutions of the partial problems_. but is our problem the _ exact _ sum of the two partial ones ?let s assume that each plate of the truncated cone has a charge of absolute value and constant charge density equal to on the top plate and equal to on the bottom plate . splits into and on the plates of the hollow truncated cone and the cylinder , respectively , in proportion to the areas of the plates .  on the top plate of the hollow truncated conewe have and on the top plate of the cylinder .  on the bottom plate of the hollow truncated cone we have and on the top plate of the cylinder .however , and are not and ( except when ) . the only way to ensure this is to change the charge densities on the plates .but then the problem is _ not _ a simple addition of two other problems .     a capacitor with the shape of a hollow truncated cone is now formed from two ` cylindrical ' shells .find its capacitance .as in the previous problem , we partition the capacitor into infinitesimal annuli . however , in this case they behave as infinitesimal _ cylindrical _ capacitors connected in series .the plates have radii and , and the height is . therefore dc 2_0dz . using equation ( [ eq : ratio ] ) , we can find c = 2_0hc - b _ b^c dr .= 2_0hac - b _ b / a^c / a dtt , where in the last equation we have made a change of variables .  the integral is known as the * logarithmic integral * . using this definition , we can write the previous result in the form :   when , we find and therefore we should use l hospital s rule to compute the result : c = 2_0 ha _0^c / a dtt .recall now that _0^x f(t ) dt = f(x ) .therefore , c = 2_0 ha 1a(b / a ) 2_0 h(b / a ) , i.e. the result of a cylindrical capacitor with radii and .a conductor has the shape of a hollow cylinder as seen in figure [ fig : hollowcylinder ] .show that the resistance when the voltage is applied between the inner and outer surfaces is given by   we split the conductor into infinitesimal cylindrical shells ( left picture of figure [ fig : hollowcylinder ] ) which are connected in series and have infinitesimal resistance : the total resistance of the conductor is then : r = dr2h _ b^c drr 2h .\(a ) a conductor has the shape seen in figure [ fig : hollowcone ] .show that the resistance when the voltage is applied between the upper and lower bases is given by show that this result reduces to that of a solid truncated - cone for .\(b ) argue now that the resistance of the hollow truncated - conical wire is the difference between the the resistance of the truncated - conical wire and a cylindrical wire of radius .this implies that explain why this does not agree with part ( a ) .\(a ) we partition the conductor into infinitesimal cylindrical conductors of length and cross - sectional area as seen in figure [ fig : hollowcone ] .these infinitesimal conductors are connected in series .they have resistance where we used equation ( [ eq : ratio ] ) .therefore r = h(c - b)_b^c dr(r^2-a^2 ) = h2(c - b)a- .  for , .we thus must use l hospital s rule : r&=&h2(c - b)_a0 -a = h2(c - b)_a0dda - = hcb .\(b ) using the superposition principle for figure [ fig : superposition ] , we would have written down since they are connected in parallel .then this does not agree with the result of part ( a ) since the sum of two partial problems is _ not _ the problem we are studying .one can verify this by checking the current densities on the top and bottom faces .a conductor has the shape seen in figure [ fig : hollowcone ] .show that the resistance when the voltage is applied between the inner and outer surfaces is given by show that , for , this result agrees with that of problem [ sec : cyl - conductor ] .we partition the conductor into infinitesimal annuli connected in parallel .the corresponding infinitesimal conductivity is from this , the total conductivity is found to be s2 _ b^c dz 2hac - b ( c / a)-(b / a ) , which can be inverted to give to total resistance r2 ha c - b(c / a)-(b / a ) .a conductor has the shape of a truncated wedge as seen in figure [ fig : wedge ] .show that the resistance of the conductor when the voltage is applied between the left and right faces is while the resistance when the voltage is applied between the top and bottom faces is     \(a ) we partition the conductor into infinitesimal layers of thickness along the -direction .these layers are infinitesimal resistors with the shape of square cylinders connected in series .their resistance is from the figure we see that then dr= a(c - b ) dzz , and r= a(c - b)_b^c dzz = a(c - b ) .\(b ) when the voltage is applied between the top and bottom faces , the infinitesimal resistors are connected in parallel .now the current flows through area and the length it travels is : therefore s= ac - b _b^c dzz = ac - b , from which r= c - ba 1(c / b ) .   find the inductance for a section of angular span of a toroidal inductor of radii and , height , and number of turns .we partition the toroid into infinitesimal solenoids all run by the same current and thus connected in series .this is seen in figure [ fig : toroid ] .all infinitesimal inductors have the same number of turns and they have the simple geometry of a solenoid .therefore , each has an infinitesimal inductance where is the length of the inductor at distance from the center .then l = dl & = & _ 0 n^2h _ a^b drr = _ 0 n^2h . for a full circle , and a well - known result , usually found by computing flux .notice that , using our method , it is not necessary to know the value of the magnetic field in order to find the inductance .   split the parallel - plate inductor into convenient infinitesimal inductors .then make use of equation ( [ eq : ldef ] ) to again derive equation ( [ eq : lbasic ] ) .the inductor is split in parallel infinitesimal slices as seen in figure [ fig : plane - inductor2 ] .each slice is similar to a turn of a solenoid ; it is carrying an infinitesimal current .the infinitesimal slices have a deductance of where .therefore the total deductance is then and      think of a coaxial cable made of two cylindrical shells of radii and .if the cable has length , compute its inductance .( the currents are uniformly distributed along the cross - sections of the wires . )  : the standard computation found in introductory physics books ( e.g. ) uses the definition ( [ eq : ldef ] ) .the magnetic field between the two cylinders can be easily found using ampre s law : where is the distance from the center .then we take a cross - section between the two cylinders ( seen in blue in the left picture of figure [ fig : coaxial ] ) .we split this cross - section into infinitesimal strips of width that are penetrated by infinitesimal flux the total flux through the whole cross - section is then , and therefore   obviously the standard computation falls short of explaining why only one cross - section has been used and no integration over the angular coordinate has been performed .   and and height . in the right picture , we have sliced it into infinitesimal cylindrical shells , while in the middle picture we have sliced it into infinitesimal wedges.,title="fig:",height=340 ]  and and height . in the right picture , we have sliced it into infinitesimal cylindrical shells , while in the middle picture we have sliced it into infinitesimal wedges.,title="fig:",height=340 ]  and and height . in the right picture , we have sliced it into infinitesimal cylindrical shells , while in the middle picture we have sliced it into infinitesimal wedges.,title="fig:",height=377 ]  : let s partition the cable into infinitesimal inductors with the shape of the wedge as seen in the middle picture of figure [ fig : coaxial ] .the angular span of each wedge is and each plate carries current .the wedges are connected in parallel .each wedge has an infinitesimal deductance of the total detuctance is and therefore we recover the result of the previous solution .we now see why only the flux of a single cross - section should be included .  :the computation we presented in the previous paragraph was no shorter than the standard one , as it also must compute the flux for one cross - section .it only had the advantage of explaining why the flux of a single cross - section must be included . however, now we give a short calculation based on our technique , by - passing the use of flux and relying solely on the basic formula ( [ eq : lbasic ] ) .we partition the cable into infinitesimal cylindrical shells seen in the right picture of figure [ fig : coaxial ] which behave as parallel - plate inductors connected in series .the distance between the plates is , the length of each plate is , and the width of each plate is .therefore the total inductance isthen    in the coaxial inductor of the previous problem , imagine that the current is flowing from the inner cylindrical shell towards the outer shell .compute the inductance of this configuration of this configuration .this is another problem that presents the power of our approach . using the standard computations, students would have great difficulty in solving this problem , as they must first compute the magnetic field .however , our method reduces the problem to a an almost trivial calculation !  we again partition the cable into cylindrical shells of width .these are now infinitesimal inductors that are connected in parallel , each having a deductance therefore the total deductance is k = dk1_012h _ a^b drr 1_012h , which of course implies that is probably no need for additional problems .the reader has certainly uncovered the pattern .all the quantities we have used let the generic symbol stand for any of them follow a simple additive law p & = & _ i p_i , , + p & = & dp , . for elements in the shape of a uniform cylinder of length and cross - section material is characterised by the constant ( corresponding to quantity ) , given by for infinitesimal elements  in all cases of identical geometry , the results will be identical .in fact , in many instances above we could have saved some computations but we avoided doing so in order to present the big picture first .once the reader is aware of the global picture , she can easily use it to transfer a result for a quantity in some geometry to a quantity in a similar geometry .an example of this is as follows .a sauna room has the shape of the wedge seen in figure [ fig : wedge ] .the insulation of the wall at has deteriorated and , as a result , the wall is at a lower temperature with respect to the wall at .an engineer would like to know the thermal resistance of the room .assume that all other walls are at the same temprature with the wall at .it is obvious that heat will flow from the left wall to the right wall .one might want to divide the room in infinitesimal layers parrallel to the side walls and continue as usual but this is not necessary .this problem has already been solved in section [ sec : wedge ] . based on the results there ,we can write dowm immediately that here is another problem   a cylindrical cable of radius and lenght is made of a huge number of small filaments such that the cable may be considered continuous .the filaments have been arranged in such a way that young s modulus for the cable is , where is the distance from the center of the cable and some constant . what is the stiffness of the cable ?m. abramowitz , i.a .stegun , , dover .m. alonso , e.j .finn , , addison - wesley .d. haliday , r. resnick , j. walker , , 6th ed . ,john - wiley & sons . e. hecht , , brooks / cole 1996 .p. nolan , , wm . c. brown communications 1993 .romano , r.h .price , _ the conical resistor conundrum : a potential solution _ , am. j. phys .* 64 * ( 1996 ) 1150 .serway , , 4th ed . , saunders college publishing .tipler , , 3rd ed . , worth publishers .r. wolfson , j.m .pasachoff , , 3rd ed ., addison - wesleyyoung , r.a .freedman , , 9th ed ., addison - wesley 1996 .

Abstract: we propose a unified approach to addition of some physical quantities ( among which resistors and capacitors are the most well - known ) that are usually encountered in introductory physics such that the formul required to solve problems are always simply additive . this approach has the advantage of being consistent with the intuition of students . to demonstrate the effectiveness of our approach , we propose and solve several problems . we hope that this article can serve as a resource paper for problems on the subject .  * addition laws in introductory physics *  * c.j . efthimiou * and  * r.a . llewellyn * + department of physics + university of central florida + orlando , fl 32816