idea of network coding over error - free networks , pioneered in , has been a subject of active current research .the general idea of linear network coding , where intermediate nodes linearly combine incoming packets , was explored in .a simple and effective algebraic formulation of the general network coding problem was introduced in .this established a direct connection between a network information flow problem and an algebraic variety over the closure of a finite field .  using the formulations of ,the multicast network coding problem , where one source transmits at the same rate to a set of sinks , has been characterized almost completely .a linear network code exists for the multicast case in a large enough finite field and can be found in polynomial time .the insufficiency of linear coding in the non - multicast case has been demonstrated in .recent work in and has shown the restrictions imposed on the field characteristic for the scalar linear solvability of a general network coding problem .see for more non - multicast examples .  in the algebraic view , the network code is seen as a variety specified by a system of polynomial equations in multiple variables taking values from a finite field . to derive the equations corresponding to a given network coding problem , edge - to - edge gains are assigned as variables . for every node , the flow on outgoing edges is written down in terms of the flows on the incoming edges using the edge - to - edge gains .the flow propagates in this manner from the sources to the sinks .the output from the sink is equated to its demand , and polynomial equations in the edge - to - edge gains are obtained .  in this work ,we propose a method to derive the equations using path gains as the variables .the gain on every source to sink path becomes a variable in the proposed formulation . in the method of ,the path gain would be a product of several edge - to - edge gain variables .the advantage of the path gain formulation is that the final equations are only linear and quadratic , as shown in the remainder of this article . because of the low degree and the inherent nature of the scalar linear network coding problem , simplification is readily possible for the system of equations .we provide an algorithm to compute the equations in the path gain formulation , and demonstrate the efficacy of the path gain approach by illustrative examples . starting with the butterfly network and other interesting small - sized network coding problems , we show that the path gain approach provides results on solvability of the problem . on a larger network ( with 87 nodes and 161 edges ) , we show how the path gain approach continues to provide solutions to some network coding problems .the path gain formulation is equivalent to the edge - to - edge gain formulation and can be derived from it .therefore , the work presented in this article is a method to simplify the equations generated by the edge - to - edge gain variable assignment .while the number of variables in the edge - to - edge formulation is of the order of the number of edges , the number of monomial terms in these variables is exponential in the number of edges .hence , the polynomial system is of size that can be exponential in the size of the network .assigning variable names to the paths ( which can be exponential in the size of the network in number ) does not necessarily make the path gain formulation more complex than the edge - to - edge gain formulation as far as solving the equations is concerned .however , in an actual implementation , the edge - to - edge gains are to be used . to complete the path gain formulation , we provide an algorithm to compute the edge - to - edge gains from the path gains .though there are several other standard methods to simplify systems of polynomial equations ( such as grbner basis methods ) , many problems in the area of solving systems of polynomial equations ( and in network coding with multiple sources and sinks ) are either np - hard or undecidable . in this light , the path gain formulation appears to be simpler than the edge - to - edge formulation in the sense that simplifications and solutions are easier in several examples ( both small and large ) .several methods and techniques to study the network coding problem have been introduced by many researchers in this area that has seen intense recent research activity . following the information - theoretic methods in ,more information - theoretic methods were used for characterizing network coding for multimessage unicasts in .the algebraic formulation in provided an elegant and powerful method to study network coding .random network coding , which is a popular choice in practical implementations , was introduced and studied using algebraic tools .the linear programming formulation has seen applications in wireless network coding and optimizing network coding with a cost criterion .the combinatorial approach , proposed and developed in and , has provided methods for studying the field sizes in network coding problems .  in the context of the prior work cited above, the path gain formulation for algebraic network coding presents the equivalence between network coding and a maximum - degree-2 system of polynomial equations for the first time .the equivalence is achieved without introducing any new monomial terms that are not present in the original system .the equations obtained from the path gain formulation are amenable to considerable simplification in several cases of interest .hence , the path gain method can provide deterministic solutions to several linear network coding problems .the method can , in some cases , provide results on solvability .the primary utility of the method is likely to be in larger examples . as an illustration , for the network ( in fig .[ isp ] ) with 87 nodes and 161 edges , we present results of solutions to certain network coding problems with multiple sources and sinks in section [ sec : bigger - example ] .the rest of this article is organized as follows .we will start with a notational description of the network coding problem in section [ sec : nwc ] , which also introduces the edge - to - edge gain algebraic formulation of .the path gain formulation is presented in section [ sec : transform ] , where we provide a graph transformation algorithm that is used to represent and compute the equations in a transformed graph . at the end of the section, we show how the equations derived from path gain variables are amenable to easy simplifications . in section [ sec : example ] , we illustrate the advantages of the path gain formulation using various example networks drawn from the literature . we also provide results for a large internet service provider ( isp ) network . in section [ sec : reverse ] , we give an algorithm ( that uses the transformed graph ) to derive the edge - to - edge gains from the path gains . finally , we provide concluding remarks in section [ sec : conclusion ] .the communication network is modeled as a directed , acyclic multigraph , , where the node set represents the terminals and switches in the network and the edge set represents the communication links .it is assumed that all communication links are error - free and have unit capacity .   for a given edge ,we denote :   for each node , we define   let us further assume the following without loss of generality :  1 .  a node is a source node iff and all source nodes produce exactly one unit of data per unit time . 2 .a node is a sink node iff and all sink nodes demand exactly one unit of data per unit time .  in cases where a node produces ( demands ) more than one data symbol, we can add virtual source ( sink ) nodes that produce ( demand ) exactly one data symbol , have exactly one output ( input ) link connecting them to and no input ( output ) links .then , the set of source and sink nodes is defined as follows : let the sink demand the -th source . for every source , a virtual incoming edge is added for notational convenience ( as in the edge - to - edge gain formulation ) .let us now assume that we use a finite alphabet .for each edge , an edge function is then defined as a mapping , where if and otherwise . for a sink , a virtual outgoing edge is added to denote the output . the edge function on this virtual edge , which is a mapping denoted ,is called the output function of the sink .the collection of all the edge functions in a given network is defined as a _ network code_. if all the edge functions are linear maps with respect to a field alphabet , then the code is a _ scalar linear code_.  let the data symbol generated at the -th source node , , be denoted by .the data symbol demanded by the -th sink node , , is .the sources and sinks implicitly define a set of connection requirements for the given network .the connection requirement is met at a sink if the output of the function equals for all inputs .given a network , the set of source nodes and the set of sink nodes , the network coding problem is to determine all the edge functions such that all the connection requirements are satisfied .if such a set of edge functions exists , then the network coding problem is _solvable_. if a set of linear edge functions ( with respect to a finite field ) exists that satisfies all the connection requirements , then the network coding problem is _ scalar - linearly solvable_.  in a scalar linear network coded flow ( over a field ) , the edge function of an edge can be written as , where .we refer to as either the edge function of or the symbol flowing through and denote it as a vector  ] . if is a path from the source virtual incoming edge to the sink virtual outgoing edge , the polynomial contains a path gain term .conversely , each term in the polynomial is the gain along a path from the source edge to the sink edge .  in the proposed formulation , the path gain of a path from a source input virtual edge to a sink output virtual edgeis assigned as a variable denoted .suppose there are paths , denoted ( ) , from to .we see that the polynomial can be written as .the proposed approach can be summarized as follows . equating the output edge function at the sinks to unit vectors , the equations inthe koetter - mdard formulation become linear in the new path gain variables .we call these conditions as no - interference conditions . however , if two paths overlap in one or more edges , there are inter - relationships between the path gain variables .these inter - relationships are called edge compatibility conditions , and they turn out to be quadratic in the path gain variables .a simple description of the edge compatibility conditions is as follows .if two source - sink paths and overlap in an edge , we see that the relationship needs to be satisfied , since both sides are equal to .note that and are source - sink paths as well .however , several of these equations can be combined to produce the necessary set of edge compatibility conditions .this is described in more detail in the section [ sec : ecc ] .   to work with the path gain formulation for a given network and connection requirements, we need to determine source - sink paths and assign path gain variables .then , the no interference conditions and the non - trivial edge compatibility conditions have to be determined . in the remainder of this section , we provide algorithms for performing these tasks . in these algorithms ,we employ a graph transformation that is very useful in both visualizing the path gain approach and solving for the edge - to - edge gains from the path gains .an important ingredient in the algorithms is an ordering of the nodes .every directed acyclic network determines a topological order or sequencing of nodes from sources or sinks or vice versa .a standard algorithm for finding such a topological ordering of the nodes is given below for completeness .[ algo : toposort ] topological sorting + _ input _ : a directed acyclic graph , .  1 .associate with each node , a value that is initialized to .2 .  pick a node such that , do * for each edge , + .* * append to the ordering , .3 .  if any node has not been added to the ordering yet , go to step 2 .else terminate .  _ output _ : , a topological ordering of nodes .notice that the sinks occur first in the topological ordering .loosely , the ordering traverses the nodes from sinks to sources .the final algorithm that takes a network coding problem as input and outputs a set of trees that collect together all source - sink paths is given below :  graph transformation + _ input _ : a directed acyclic graph , set of sources , set of sinks , connection requirements .  1 .obtain a topological ordering for the graph using algorithm 1 .let .3 .  loop through the nodes in the order defined by , do * if , * * for each edge , add a new node to with one output link connecting it to and one input link for each such that . * * delete the old node in .  _ output _ : , a transformed network .the final transformed network is a set of directed trees such that sink is the root of the -th tree .all leaf nodes in the trees are copies of one of the source nodes .there is a one - to - one correspondence between the paths from leaf nodes , which are copies of the source , to the root in and the paths from to in the original network .each node in the transformed network will have exactly one output link and the acyclic property of the graph is maintained by the transformation .the underlying undirected graph is a set of disjoint trees , because any cycle in it must imply that either the cycle is also present in the directed graph or that one of the nodes in the directed graph has more than one output link .hence , the equivalent network is made up of a set of directed trees .the transformation maintains one output link for each node in the original graph that has .so , the only nodes that will have , and hence be the roots of these trees , are the sink nodes ( which had to start with ) .hence , each sink would be the root of a directed tree in which all edges are directed towards this root .  also , the number of input links of a copied node in the transformed graph is equal to the number of input links possessed by the original node .so , the only nodes that will have , and hence be leaf nodes in these trees , are copies of the source nodes ( which had to start with ) .  finally , since all nodes are visited in the topological order from the sinks to the sources , all paths from the sinks to the sources will be part of the final network .this results in the one - to - one correspondence in the paths .an example of this transformation applied to the butterfly network ( fig .[ butterfly]a ) can be seen in fig .[ butterfly]b .# 1#2#3#4#5   ( 9245,3339)(-599,-4003 ) ( 5941,-3166)(0,0)[lb ] ( 5941,-2671)(0,0)[lb ] ( 6751,-2761)(0,0)[lb ] ( 7381,-3256)(0,0)[lb ] ( 8011,-3706)(0,0)[lb ] ( 6661,-3526)(0,0)[lb ] ( 5311,-2581)(0,0)[lb ] ( 5311,-3391)(0,0)[lb ] ( 5311,-3931)(0,0)[lb ] ( 2431,-3706)(0,0)[lb ] ( 2431,-3211)(0,0)[lb ] ( 3241,-3301)(0,0)[lb ] ( 3871,-3121)(0,0)[lb ] ( 3286,-2671)(0,0)[lb ] ( 1846,-2581)(0,0)[lb ] ( 1846,-3031)(0,0)[lb ] ( 1846,-3931)(0,0)[lb ] ( 4591,-3031)(0,0)[lb ] ( 271,-1051)(0,0)[lb ] ( 451,-1906)(0,0)[lb ] ( 226,-2491)(0,0)[lb ] ( 856,-2491)(0,0)[lb ] ( -314,-3166)(0,0)[lb ] ( 316,-3166)(0,0)[lb ] ( 631,-3166)(0,0)[lb ] ( 1396,-3166)(0,0)[lb ] ( 811,-1051)(0,0)[lb ] ( -584,-3661)(0,0)[lb ] ( 136,-3616)(0,0)[lb ] ( 901,-3616)(0,0)[lb ] ( 1216,-3616)(0,0)[lb ] ( -134,-1726)(0,0)[lb ] ( 991,-1726)(0,0)[lb ] ( 5851,-1546)(0,0)[lb ] ( 5851,-1051)(0,0)[lb ] ( 6661,-1141)(0,0)[lb ] ( 7291,-1636)(0,0)[lb ] ( 7921,-1501)(0,0)[lb ] ( 6571,-1906)(0,0)[lb ] ( 5311,-871)(0,0)[lb ] ( 5311,-1771)(0,0)[lb ] ( 5311,-2176)(0,0)[lb ] ( 2386,-1996)(0,0)[lb ] ( 2386,-1501)(0,0)[lb ] ( 3196,-1591)(0,0)[lb ] ( 3826,-1411)(0,0)[lb ] ( 4411,-916)(0,0)[lb ] ( 3241,-961)(0,0)[lb ] ( 1801,-871)(0,0)[lb ] ( 1801,-1321)(0,0)[lb ] ( 1801,-2221)(0,0)[lb ] ( 8326,-1276)(0,0)[lb ] ( 4906,-1186)(0,0)[lb ] ( 4771,-3526)(0,0)[lb ] ( 8326,-3526)(0,0)[lb ]  to apply the graph transformation , the topological ordering of the nodes is chosen to be .nodes 7 , 8 , 9 and 10 are sink nodes , and occur first in the ordering .nodes 5 and 6 will be replicated 2 times , since they both have 2 output links .this will result in the replication of the edges , , and .node 4 will now have 4 output links and will have to be replicated as many times along with edge . similarly , node 3 will also be replicated 4 times along with edges and .finally , the source nodes 1 and 2 will be replicated 6 times each since they both now have 6 output links .   since there is a one - to - one correspondence between the leaf source nodes in the transformed network and the source - sink paths in the original network , path gain variables are assigned at the leaf source nodes .the assignment is illustrated in fig .[ butterfly]b for the butterfly network .source nodes 1 and 2 are assigned the variable names and , respectively .the subscripts are chosen tree by tree in the transformed network .in the tree with root as node 7 , the two copies of source node 1 are assigned variables and , while the single copy of source node 2 is assigned the variable . in the tree with root node 8 ,the variables are , for the two copies of node 1 , and for the single copy of node 2 .we continue in this manner to name the scaling variables at the source leaf nodes of the other two trees to get variables , , , and , , , .  once path gain variables are assigned ( from some field ) at the leaf nodes ,all edge functions are computed in the transformed network assuming that intermediate nodes perform addition only .the output function at the root ( sink ) is the sum of all incoming edge functions .for instance , in the tree with root as node 7 in fig .[ butterfly]b , the edge functions are as follows : for , ; for , ; for and , ; for , ; for , the edge function is .the output function at sink node 7 is .similarly , the edge functions can be computed for the other trees .note that the intermediate nodes perform addition as the entire path gain has been assigned as a variable at the leaf .because of the equivalence between paths from sources to a sink in the original network and leaf nodes in the tree , we see that the output function calculated in the transformed network is identical to the output function in the koetter - mdard formulation as given in section [ sec : deriv - from - koett ] .therefore , the no interference conditions are obtained by equating the output function of the root in the transformed network to its demand .  in fig .[ butterfly]b , the output function at the root nodes 7 , 8 , 9 and 10 are , , and , respectively . forthe symbol at node 7 to be equal to the required , we have and .other equations are derived similarly .hence , in the butterfly network of fig . [ butterfly ] , we get the following linear equations :    for completion , we state the general form of the no - interference conditions below . in general , each path gain variable in the transformed network is associated with exactly one source symbol and one sink ( or tree ) .let us denote the source - sink path gains by where denotes the source , denotes the sink ( or the tree ) , and is an index among all copies of the source node in the tree rooted at .then , the general form of the `` no interference '' conditions can be written as follows :    as explained before , the path gain variables of overlapping paths are related by quadratic edge - compatibility conditions . if multiple copies of an edge are present in the transformed network , then the edge is part of multiple source - sink paths in the original network .therefore , edge compatibility conditions are indicated by the presence of multiple edges in the transformed network .the edge functions in the transformed network can be used to write down the edge compatibility conditions .we first show this for the butterfly network example and later provide the general form .  in our illustrative example of fig .[ butterfly]b , the edge is copied four times .since there are ways of choosing two copies among the four , there will be six edge compatibility conditions for .the symbols on the copies of on the trees with root nodes 7 , 8 , 9 and 10 are , , and , respectively .hence , in fractional form , we need ( roots 7 and 8) , ( roots 7 and 9 ) , ( roots 7 and 10 ) , ( 8 and 9 ) , ( 8 and 10 ) and ( 9 and 10 ) .  in the degree-2 form , the edge compatibility conditions for the four copies of the edge are listed below : for the butterfly network example , we do not get any other edge compatibility conditions . for edges and , the equations are identical to the ones listed above . also , there are no equations for edges , , and since these edges have scaled versions of the same symbol flowing through them .we have seen that not all duplicated edges result in distinct compatibility conditions .in general , edge compatibility equations will be required for each edge in the original network that satisfies the following conditions :  1 .number of copies of in the transformed network ( or the edge will not be replicated at all ) .2 .  number of different source nodes having a path to ( since if two copies of carry and , these will be scalar multiples of each other for any value assigned to ) .3 .  ( or the equations will be same as that for ) .we now state the general form of the edge - compatibility conditions in terms of nodes of the transformed network . given a node in the original network , the general form of the condition for two copies of , denoted by and in , belonging to the -th and -th trees , respectively , can be written as follows : where denotes the set of leaf nodes in the -th tree that are copies of the source node and have a path to .  a careful study of the general form shows an edge compatibility condition needs to be introduced for every two copies , , of node and for every two sources such that ( a ) , ( b ) set of nodes in the -th tree , and ( c ) .  the linear no - interference conditions and the quadratic edge compatibility conditions on the path gains are necessary and sufficient conditions for existence of solutions to the scalar linear network coding problem .the sufficiency is proved by algorithm [ algo : reverse ] in section [ sec : reverse ] . before describing the sufficiency ,we show how the linear and quadratic equations in path gains can be simplified in a systematic manner to provide useful results .the linear equations ( no interference conditions ) possess the special property that each of them involves a mutually exclusive set of variables .using this property , we can simplify the system of equations in the following two ways :  1 .it is possible that some of the variables never occur in the non - linear equations ( edge compatibility conditions ) . from, we can see that is one such variable in the example of the butterfly network .it can be easily seen that the linear equation involving can be trivially satisfied for any value assigned to the other variables involved in the same linear equation by choosing an appropriate value of ( which does not have any other condition on it ) .hence , along with the linear equation it occurs in can be removed from the system as trivially solvable .+ therefore , the first simplification would involve elimination of variables ( and their corresponding linear equations ) that do not occur in any non - linear equation .2 .  since each linear equation involves a mutually exclusive set of variables , we can eliminate one variable using each linear equation easily . eliminating this variable from the non - linear equations (note that this does not increase the degree of the system ) might reduce some of them to linear equations which can again be used to eliminate more variables iteratively .  in the case of the butterfly network , after the first step of simplification , we are left with 8 variables , 4 linear equations and 6 non - linear equations .  in the second step of the simplification , after the first round of elimination of variables using the linear equations ( [ butter : ni ] ) in ( [ butter : ec ] ) , we are left with 4 variables : , , and and the 6 equations as shown below . subsequently , and can also be eliminated , using the linear equations above , leaving just 2 variables and the relation :   hence , the network coding problem for the example of the butterfly network has been reduced to solving only one ( non - trivial ) equation given in .in this section , we provide a few examples to illustrate the usefulness of the path gain approach in deriving the system of polynomial equations corresponding to a network coding problem .note that several problems in this area of polynomial equations and network coding are np - hard or undecidable , and we do not expect polynomial - time algorithms and exact step - by - step solutions to result from the path gain approach .our approach is to demonstrate the effectiveness of the path gain method in several examples of varying complexity .  in all examples ,we provide the number of equations and variables obtained from the edge - to - edge gain formulation . the path gain formulation ( after simplifications )will result in better numbers in many cases .however , we point out that this is not a comparison of the two methods , since one is simplified and the other is not .as we have shown , the path gain method can be seen as a method for simplifying the edge - to - edge gain equations .other methods for simplifying generic systems of polynomial equations , such as grbner basis methods , are useful in several networks .also , grbner basis or other methods can be used after the path - gain - based simplifications . however , in many examples , we observe that the path gain formulation appears to provide results on solvability .this is mainly because the path gain approach provides low degree equations , which are amenable to easy analysis and further simplifications .   for the butterfly network , the relationship between the path gain variables ( shown in fig .[ butterfly]b ) and the edge - to - edge gains in the koetter - mdard formulation ( fig .[ fig : bflow ] ) can be written down as follows : ( path : ) , ( path : ) , ( path : ) , ( path : ) , ( path : ) , ( path : ) , ( path : ) , ( path : ) , ( path : ) , ( path : ) , ( path : ) , ( path : ) .( 0,0 )  # 1#2#3#4#5   ( 9797,2494)(19,-3095 ) ( 451,-1321)(0,0)[lb ] ( 451,-2311)(0,0)[lb ] ( 1486,-2311)(0,0)[lb ] ( 1486,-1276)(0,0)[lb ] ( 3331,-1321)(0,0)[lb ] ( 3331,-2311)(0,0)[lb ] ( 4006,-1546)(0,0)[lb ] ( 5446,-1546)(0,0)[lb ] ( 5446,-2536)(0,0)[lb ] ( 6481,-2536)(0,0)[lb ] ( 5941,-1591)(0,0)[lb ] ( 6661,-1456)(0,0)[lb ] ( 8956,-2311)(0,0)[lb ] ( 9451,-1591)(0,0)[lb ] ( 8281,-1636)(0,0)[lb ] ( 4951,-1006)(0,0)[lb ] ( 5356,-736)(0,0)[lb ] ( 7426,-1321)(0,0)[lb ] ( 7876,-736)(0,0)[lb ] ( 8326,-736)(0,0)[lb ] ( 9091,-736)(0,0)[lb ] ( 8731,-736)(0,0)[lb ] ( 9586,-736)(0,0)[lb ] ( 2971,-736)(0,0)[lb ] ( 2566,-736)(0,0)[lb ] ( 4141,-736)(0,0)[lb ] ( 4591,-736)(0,0)[lb ] ( 3826,-736)(0,0)[lb ] ( 5806,-871)(0,0)[lb ] ( 6211,-781)(0,0)[lb ] ( 6571,-736)(0,0)[lb ] ( 7156,-736)(0,0)[lb ]  the no - interference conditions are easily obtained . for edge compatibility between the paths and the path , we get the equation .other compatibility conditions can be checked similarly .the change to path gain variables results in easy simplification of the resulting equations with no increase in degree .finally , we obtain the simple equation , , which is not obvious even when the substitution is clearly specified .consider the network shown in fig .[ fig : char2]a taken from , where it has been proved to have linear coding solutions only over fields of characteristic 2 .nodes 1 , 2 and 3 are sources producing , and respectively .nodes 12 , 13 and 14 are sinks demanding , and respectively .the trees in the equivalent transformed network are shown in fig .[ fig : char2]b , c , d .the set of equations generated by the `` no interference condition '' are :   the set of equations generated by the `` edge compatibility condition '' for edges , , and respectively are :   using the linear equations to eliminate variables iteratively , we get 9 equations in 6 variables shown below .l a_2b_3=b_1 ; a_2=-a_4b_1 ; a_4b_3=-1 ; + a_2c_4=a_4 ; c_4=a_4c_2 ; b_3c_4+c_2=0 ; + b_1c_2+b_3=0 ; b_1c_4 + 1=0 ; b_3c_4=c_2  from equations and , we can derive the relation .substituting in the above system leads to the condition , which is not possible .hence , we must have , which implies that the system is not solvable in any field with an odd characteristic . also , in characteristic 2 , setting all variables to 1 in the above equations ,is seen to be a solution .this example demonstrates that , in practice , working with the equations derived through the path gain formulation can be advantageous .  for this example , the koetter - mdard formulation , as illustrated in , results in 17 equations in 22 variables .however , as shown in , it is possible to derive from these 17 equations using other simplifications .alternatively , a grbner basis method can also be used to derive .the path gain approach should be seen as a generic technique for simplification that can be used in arbitrary network coding problems , as shown in the next two examples .an interesting example of a multicast problem , presented in , is shown in fig .[ fig : multex ] .( 0,0 )  # 1#2#3#4#5   ( 6352,4411)(425,-4400 ) ( 6526,-4036)(0,0)[lb ] ( 3001,-436)(0,0)[b ] ( 4201,-436)(0,0)[b ] ( 3001,-1636)(0,0)[b ] ( 4201,-1636)(0,0)[b ] ( 5401,-1636)(0,0)[b ] ( 5401,-2836)(0,0)[b ] ( 4201,-2836)(0,0)[b ] ( 3001,-2836)(0,0)[b ] ( 3001,-4036)(0,0)[b ] ( 4201,-4036)(0,0)[b ] ( 5401,-4036)(0,0)[b ] ( 1801,-1636)(0,0)[b ] ( 1801,-2836)(0,0)[b ] ( 1801,-4036)(0,0)[b ] ( 601,-4036)(0,0)[b ] ( 3001,-136)(0,0)[b ] ( 4201,-136)(0,0)[b ] ( 601,-4336)(0,0)[b ] ( 1801,-4336)(0,0)[b ] ( 3001,-4336)(0,0)[b ] ( 4201,-4336)(0,0)[b ] ( 5401,-4336)(0,0)[b ] ( 6601,-4336)(0,0)[b ]  the sources are nodes 1 and 2 , and the sinks are nodes 11 - 16 .this problem does not have a binary solution , as shown in .  using the edge - to - edge gain formulation ,we get 24 equations in 32 variables .the path gain method initially results in 84 equations in 48 variables .after the simplifications , we obtain 54 equations in 18 variables .significantly , there are 6 quadratic equations , each of the form .next , we can show that ( either or ) provides a contradiction in the equations .hence , we have equations of the form , which can not be solved in the binary field . with some more analysis, we can find solutions over gf(4 ) .  from this example, we see that the path gain formulation provides useful simplifications in non - trivial cases . in contrast , grbner basis methods on the edge - to - edge gain equations are not immediately useful in showing linear in - solvability over gf(2 ) .note that this does not rule out any other simplification of the edge - to - edge equations to obtain the necessary result .we merely conclude that the path gain method provides a useful simplification .consider an isp network topology shown in fig .[ isp ] taken from .the network has 87 nodes and 161 edges .edges are directed from lower - numbered nodes to higher - numbered nodes i.e. in an edge , .hence , the graph is directed and acyclic .we assume all links have unit capacity , and use fields of characteristic 2 in our simplification steps .after directing the graph , the five nodes 1 , 12 , 21 , 51 and 64 were set as sources in the example problems .sink nodes and demands were chosen at random from among the sources visible from each sink .the graph is not reduced by this choice of demands , since all nodes are visible from the five chosen sources .  1 .  _5 sources ( all rate 1 ) , 10 sinks ( all rate 1 ) . _ the edge - to - edge gain formulation gives a system of 44 equations in 30 variables .the path gain formulation initially results in 44 linear equations and 3 degree-2 equations in 316 variables .after applying the simplification steps , we are left with only 3 degree-2 equations in 7 variables assuming solution exists in a characteristic 2 field .in fact , setting all the remaining 7 variables to zero results in a valid solution to the three equations ( some other scaling variables are non - zero ) .hence , a solution over gf(2 ) is possible .2 .  _ 5 sources ( one with rate 2 , others rate 1 ) , 11 sinks ( all rate 1 ) . _ the edge - to - edge gain formulation yields a system of 50 equations in 40 variables in this case . in comparison ,the path gain formulation initially resulted in 50 linear equations and 34 degree-2 equations in 330 variables . butafter applying the simplification steps , we are left with only 13 degree-2 equations in 17 variables assuming solution exists in a characteristic 2 field .again the all - zero solution is valid for the remaining 17 variables resulting in a network code over gf(2 ) .3 .  _ 5 sources ( all with rate 2 ) , 8 sinks ( all rate 1 ) . _the edge - to - edge gain formulation yields a system of 88 equations in 180 variables .the path gain formulation initially gives 88 linear and 11198 degree-2 equations in 632 variables .but on applying the simplification steps , assuming characteristic 2 , it turns out that the system is not solvable over characteristic 2 .  to run further examples , we computed the max - flows from the sources ( 1,12,21,51,64 ) to a set of 11 nodes chosen as sinks .rates below the individual max - flows were assigned to sources and sink demands .the results are summarized below ( notation : , are source and sink sets .a source node of rate is shown as source nodes .the demands of each sink are shown in brackets ) .1 .  ,  the path gain formulation yields 1188 equations in 507 variables. after simplifications , there are 476 equations , but none of them have a constant term .hence , setting the remaining variables to zero provides a binary solution .2 .  ,  .we obtain 555 variables and 1683 equations . upon simplification , we find that a solution does not exist in characteristic 2  . the path gain formulation yields578 variables and 12048 equations .after simplifications , there are 6780 equations , but only five of them have a constant term .a linear term ( one path gain ) appears in each of these five equations , but does not appear as a linear term in any other equation .hence , setting this one path gain to 1 and the remaining variables to zero provides a binary solution .as expected , solutions are not guaranteed even if all demands are within individual max flows .we see that the number of equations and variables increases steeply in some cases .however , guessing a binary solution may be feasible .  to obtain another example , we modified the graph of fig .[ isp ] ( by changing edge connections ) to get a butterfly network as a subgraph when the nodes 62 and 63 demand the sources 1 and 12 . on the modified graph ,we set ,  , where the other demands are chosen to be below the individual max flows .we obtain 1247 equations in 503 variables .after simplifications , there are no equations with a constant term .so , setting the remaining variables to zero results in a binary solution .  hence , we see that the path - based formulation of scalar linear network coding appears to yield useful results even over large networks with a few sources and sinks .this shows the extent of simplification possible in polynomial systems defined by network coding problems .it has been shown in that the complexity of grbner basis algorithms depends , among other things , on the maximum degree of the starting basis .the degrees of the intermediate polynomials computed during grbner basis calculations has been shown to grow up to if the maximum degree of the starting basis is . due to these issues ,grbner basis algorithms become practically intractable except for small problem instances . in the light of these results ,the path gain formulation that produces degree-2 equations becomes important in reducing the running complexity of grbner basis algorithms that may be used to solve the network coding problem .the simplification provided by the path gain approach is summarized in table [ comparison ] for the various examples presented so far .c|c|c & _ path gain & _ edge gains ( unsimplified ) + _ _  c example +  + butterfly + fig .[ fig : char2]a + ( * ? ? ? * fig .5 ) + ( * ? ? ?3 ) + ( * ? ? ?3 )  &  ccc var. & deg . 2 eqns +  + 4 & 6 + 8 & 15 + 9 & 5 + 27 & 45 + 12 & 30  &  ccc var .& eqns & deg .+  + 10 & 8 & 2 + 14 & 9 & 3 + 14 & 4 & 4 + 50 & 32 & 3 + 22 & 17 & 3   +   one iteration of elimination using the linear equations  as indicated , we have shown numbers for only one round of linear equation simplification .it can be seen that , apart from having a maximum degree of only 2 , the number of variables is also lesser in many cases enabling use of methods such as for solving the system .the number of variables and equations from the edge - to - edge gain formulation are of the order of the number of edges in the network . however , the number of monomial terms possible using the edge - to - edge gain variables is exponential in the number of edges . in a large network ,depending on the number of paths from sources to sinks , a large number of monomial terms occur in the system of polynomial equations . because of the large number of variables and larger number of terms, there is no obvious method to simplify the equations other than running standard routines for grbner basis .the path gain approach is beneficial in providing results on solvability in some examples and in reducing the complexity of grbner basis methods in most cases .while the path gain variables are useful for solving the system of polynomial equations , the implementation of network coding is through edge - to - edge gains .we now describe an algorithm to obtain a network code for the original network from the path gain variables in the transformed network .note that this completes the proof of the sufficiency of edge compatibility conditions .first , we will briefly describe the algorithm and then present a notational version of the same .a solution to the system of polynomial equations in the path gain formulation consists of a set of values assigned to the path gain variables at the leaf source nodes in the transformed network such that the no interference conditions as well as the edge compatibility conditions are satisfied .the algorithm to construct a network code from such a solution consists of propagating the values of these coefficients from the source nodes to the sink nodes through the transformed network .we compute two vectors for every edge of the graph .the first vector  ] is such that the edge function on is .note that such a scaling property is guaranteed for all copies of an edge by the compatibility conditions .once the vectors are computed for all , the network code in is completely known .suppose and are known for all the incoming edges for a node .the vectors and can be computed for the outgoing edges as illustrated for a sample case in fig .[ fig : reverse ] .( 0,0 )  # 1#2#3#4#5   ( 7677,3420)(586,-3196 ) ( 1201,-211)(0,0)[lb ] ( 1876,-211)(0,0)[lb ] ( 2476,-211)(0,0)[lb ] ( 1576,-661)(0,0)[lb ] ( 1951,-661)(0,0)[lb ] ( 2326,-661)(0,0)[lb ] ( 2176,-1336)(0,0)[lb ] ( 2176,-1711)(0,0)[lb ] ( 1576,-1711)(0,0)[lb ] ( 601,-2236)(0,0)[lb ] ( 1126,-2461)(0,0)[lb ] ( 2101,-2236)(0,0)[lb ] ( 2626,-2461)(0,0)[lb ] ( 5551,-2311)(0,0)[lb ] ( 3976,-2311)(0,0)[lb ] ( 4726,-811)(0,0)[lb ] ( 4426,-586)(0,0)[lb ] ( 3451,-286)(0,0)[lb ] ( 6976,-736)(0,0)[lb ] ( 5476,-736)(0,0)[lb ] ( 4201,-286)(0,0)[lb ] ( 4876,-286)(0,0)[lb ] ( 5476,-286)(0,0)[lb ] ( 6001,-286)(0,0)[lb ] ( 6601,-286)(0,0)[lb ] ( 6151,-586)(0,0)[lb ] ( 6451,-811)(0,0)[lb ] ( 7651,-586)(0,0)[lb ] ( 7951,-811)(0,0)[lb ] ( 7876,-1261)(0,0)[lb ] ( 6376,-1261)(0,0)[lb ] ( 4651,-1261)(0,0)[lb ] ( 4426,-1711)(0,0)[lb ] ( 6151,-1711)(0,0)[lb ] ( 7651,-1711)(0,0)[lb ] ( 1726,-2911)(0,0)[lb ] ( 5551,-3136)(0,0)[lb ] ( 3826 , 89)(0,0)[lb ] ( 7426,-2311)(0,0)[lb ] ( 7876,-2536)(0,0)[lb ] ( 6976,-286)(0,0)[lb ] ( 7576,-286)(0,0)[lb ] ( 8176,-286)(0,0)[lb ] ( 3751,-736)(0,0)[lb ] ( 4426,-2536)(0,0)[lb ] ( 5251,-2536)(0,0)[lb ] ( 5551 , 89)(0,0)[lb ] ( 4801,-2836)(0,0)[lb ] ( 7501,-2836)(0,0)[lb ] ( 7126 , 89)(0,0)[lb ]  in the figure , a node with and is replicated thrice into , and in .the incoming and outgoing links are assumed to be replicated as shown in the transformed network .for instance , the edge is replicated thrice as , and .we suppose that there are three source nodes , and  ] represent the edge function sent over edge in the final linear network code in .since the edge compatibility conditions are satisfied , the edge function on each copy of in will be a scalar multiple of .for , let the edge function on be .we collect the multiplying factors , into a vector  ] collecting the multiplying factors on copies of that connect to copies of . for a fixed and with , the sets and will be in one - to - one correspondence and ordered so that .so , we have .  in fig .[ fig : reverse ] , for instance , we have , and . also , and .similarily , and .the scaling vector  ] and  ] with  ] .note that all one - to - one correspondences are being preserved in the ordering of coordinates in the scaling vectors ._ flow matrices at a vertex : _ for a vertex , incoming edge and outgoing edge , a rank - one flow matrix is defined as .the matrix is of dimension , and the -th element ( letting ) is the coding coefficient of the -th source symbol flowing in the -th copy of edge in .we readily see that each row of is the coding vector in a copy of edge in .  in fig .[ fig : reverse ] , for instance , we have =\begin{bmatrix}a_1\\b_1\end{bmatrix}[\alpha_{11}\;\alpha_{12}\;\alpha_{13}] ] . in terms of path gain variables , is equal to the sum of the path gain variables for all paths starting from ( some copy of ) the -th source and using the -th copy of edge in .let .for , let ( for all ) , and let with independent of .the -th row of the flow matrix contains the flow in the edge incident on the node for .therefore , the sum is a matrix whose -th row is equal to the sum of all incoming flows into node . by flow conservation ,the outgoing flow on the single outgoing edge from node is equal to the -th row of for .so , the rows of contain the flows in the copies of the edge in , and the edge compatibility condition ensures that the matrix is a rank - one matrix .the vectors and are initialized for an outgoing link from the source node as follows .for the -th source node and ,  ] ,  ] ,  ] ,  ] , the second row of .hence , we have , the network coding coefficients at node 3 .also ,  ] .we can then compute and then arrive at :   now we can choose  ] , the first row of . then , the network coding coefficients for node 4 are . completing the last step of the iteration , we get , \mathbf{c}_{e_7}=[1\ 0] ] and for , we have , \mathbf{c}_{e_6,e_9}=[1] ] and  ] .similarly , the network coding coefficients for node 6 can also be computed so that sinks 9 , 10 receive the required symbols .in this work , we have used path gains as variables to arrive at an algebraic formulation for the scalar linear network coding problem .this provides a useful simplification of the edge - to - edge gain formulation proposed in , as illustrated by both small and large - sized examples .given a network coding problem , we have given algorithms to construct an equivalent transformed network and arrive at a system of polynomial equations ( of maximum degree 2 ) in terms of path gains . after solving for the path gains , we have provided an algorithm to compute the edge - to - edge gains , which can be used in implementing the network code .  each monomial term occuring in a general system of polynomial equationscan be assigned a new variable to obtain linear equations along with consistency conditions involving the new variables . however , in a general polynomial system , the consistency conditions are not guaranteed to be degree-2 equations without introducing additional monomial terms not present in the original system . through this work , we have shown that the polynomial system representing a scalar network coding problem reduces to only degree-2 consistency conditions .we thank the anonymous reviewers and the editor for their comments and suggestions that resulted in several significant improvements in the presentation and content of this article .                    nicolas courtois , alexander klimov , jacques patarin , and adi shamir , `` efficient algorithms for solving overdefined systems of multivariate polynomial equations '' , in _ lec .notes in comp ._ eurocrypt , 2000 , pp .392407 .abhay t. subramanian obtained his bachelor s and master s degrees in electrical engineering from the indian institute of technology ( iit ) madras , chennai , india in 2008 .he is currently a doctoral student at the department of management science and engineering in stanford university , ca , usa . his current research interests are in the areas of applied probability , stochastic processes , optimization , statistics , algorithms , learning and mathematical finance .andrew thangaraj received his bachelor s degree in electrical engineering from the indian institute of technology ( iit ) madras , chennai , india in 1998 and a phd in electrical engineering from the georgia institute of technology , atlanta , usa in 2003 .he was a post - doctoral researcher at the gtl - cnrs telecom lab at georgia tech lorraine , metz , france from august 2003 to may 2004 . from june 2004 , he is with the department of electrical engineering , iit madras , where he is currently an associate professor .his research interests are in the areas of information theory , coding and information - theoretic aspects of cryptography .

Abstract: in the algebraic view , the solution to a network coding problem is seen as a variety specified by a system of polynomial equations typically derived by using edge - to - edge gains as variables . the output from each sink is equated to its demand to obtain polynomial equations . in this work , we propose a method to derive the polynomial equations using source - to - sink path gains as the variables . in the path gain formulation , we show that linear and quadratic equations suffice ; therefore , network coding becomes equivalent to a system of polynomial equations of maximum degree 2 . we present algorithms for generating the equations in the path gains and for converting path gain solutions to edge - to - edge gain solutions . because of the low degree , simplification is readily possible for the system of equations obtained using path gains . using small - sized network coding problems , we show that the path gain approach results in simpler equations and determines solvability of the problem in certain cases . on a larger network ( with 87 nodes and 161 edges ) , we show how the path gain approach continues to provide deterministic solutions to some network coding problems .  algebraic network coding , network coding , scalar linear network coding .


in the recent years physicists have shown increasing interest in examining the statistical properties of real market financial data and they have contributed to the extraction of the most important characteristics which are referred to as `` stylized facts '' .such stylized facts include fat tailed distribution and short time correlations for the logarithmic returns , volatility clustering , gain - loss asymmetry , etc .  to deepen our understanding of financial markets building models is essential .one approach is constructing purely mathematical models ( e.g. , arch and garch processes which are well known to and widely used by economists ) .another way which is more appealing for statistical physicists is that of the so - called multi - agent models .these are based on interacting agents using different strategies corresponding to real market behavior .the simplest of these models are probably the so - called `` minority games '' ( e.g. , the `` el farol bar '' model ) .such rather abstract models are appropriate to capture main mechanisms but a detailed correspondence to economics is hard to find .an example for more complicated models based on both economical and physical approaches is that introduced by lux and marchesi . in this modela relatively large number of parameters enables to incorporate several aspects of real financial processes .it is well known that prices on financial markets ( particularly on stock markets ) tend to fluctuate in a broad range .it has for long been a major goal of economists to understand the cause of these fluctuations , rises , and drops .experts seem to be puzzled by the fact that sometimes price changes can be easily traced back to well defined external effects like news about political events , announcements of dramatic economic data etc . , while in many cases there is no apparent reason for the major fluctuations .  in statistical physicswe distinguish between statistical fluctuations and changes due to external perturbations .if time reversal symmetry ( or detailed balance ) holds for the system like at thermal equilibrium , the famous fluctuation - dissipation theorem implies that spontaneous fluctuations and the response to small perturbations decay in the same way .of course , the condition of detailed balance does not hold for financial markets .even statitionarity can be questioned and the time reversal symmetry is broken , e.g. , agents want to maximize profit and tend to switch to winning strategies .the goal of our study is to compare fluctuations and response to external effects on financial markets . in order to have a clear cut situation we investigate the the lux - marchesi model .the ability of the model to reproduce important stylized facts together with the relative robustness of this property when changing the parameters induced us to compare the fluctuations and response ( to fundamental price changes ) within this framework .we sincerely hope that our major results hold for real market data as well .let us start with summarizing the basic features of the lux - marchesi single asset multi agent model .one of the main assumptions is that there exists a fundamental price of stocks ( the value of the company and its prospective future growth ) around which the real price fluctuates . in this modelthe agents are let to choose among the three following strategies : optimists ( who buy whatever happens ) , pessimists ( who sell ) , and fundamentalist ( who sell if the market price is above the fundamental price and vice versa ) .optimists and pessimists together are called chartists according to the usual terminology .the number of all agents is , of which the number of fundamentalists is , that of optimists , for pessimists , and for chartists .the opinion index : measures to what extent optimistic strategy dominates among the chartists . the aggregate excess demand of the agents for the stocks is computed as following :    [ cols="^,^",options="header " , ]   0.5 cm  tab .2 : values of the parameters in the four different parameter sets   during our work we found that in the lux - marchesi model the autocorrelation of volatility shows exponential decay instead of a power - law time dependence in contradiction to the general view on real market data .the power - law decay is usually attributed to some kind of scale - invariance ( regarding time ) in financial markets ( fig .[ acf ] ) implying that real markets are in a critical state , which we could not find in the lux - marchesi model even though we tried varying the parameters in a broad range .nevertheless , the phenomenon of volatility clustering is described by the model resulting in large characteristic times of the autocorrelation function which show that the model captures important aspects of the market .some other problems regarding the thermodynamic limit ( tdl ) of the model were pointed out earlier .we think that these are irrelevant from our point of view since real markets ( and our simulations ) are far from the tdl and the time scale of the mentioned effect is bigger then that of studied fluctuations and response .the dynamics of the model implies that there are continuous fluctuations around the fundamental price .we examined the decay of these fluctuations using the following simple method : when the price rose to we defined this as a fluctuation and observed the average decay for many runs .hence we did not try to determine whether the price really sank after reaching assuming that ( at least at bigger fluctuations ) the probability of further rise is much smaller than that of further decline .we observed exponential decay for the price fluctuations using all parameter sets in accordance with the well known fast decay of the correlation function .detailed simulations were undertaken for parameter set iv ( fig .[ flukt1]a ) .furthermore , in case of relatively large fluctuations we observed that the opinion index and the fraction of chartists significantly differed from their average values and exponential decay was observed for both quantities .the characteristic time of the decay of the price decreased with the size of fluctuation and was in the order of magnitude of 1 time unit ( up to fluctuations of 4 ) ( fig .[ flukt1]b ) .the opinion index decayed with approximately the same characteristic time .on the other hand the fraction of chartists ( which is closely related to volatility ) decayed much slower with characteristic times in the order of magnitude of 100 time units .to price units as a function of time , ( b ) average characteristic time of decay as a function of the size of decay ( data points computed as an average of 7625 to 120529 fluctuations ) ]  an interesting question is what causes fluctuations . in our simulations we tried to find an answer by computing the average opinion index and the average fraction of chartists that caused a fluctuation as the function of the size of the fluctuation .we found that small fluctuations are caused merely by the lack of balance inside the chartist community ( fig .[ flukt2]a ) while larger fluctuations are likely to occur only if the fraction of chartists to all agents rises as well ( fig .[ flukt2]b ) resulting in a higher market volatility ) . in other wordsthe fluctuation in the agents behaviors results in high volatility , i.e. in a nervous market which occasionally leads to major deviations from the fundamental price .however , in such cases the attractive force of the fundamental price does not show up abruptly , since the re - stabilization of the equilibrium price has to be accompanied by a gradual restoration of the balance between chartists and fundamentalists .it is well known that financial markets are exposed to many external effects from the outside world .this means that the changes in prices are only partly due to the inherent market mechanisms ( fluctuations around `` equilibrium '' ) , changes can also be caused by news from the outside world ( e.g. financial reports ; bankruptcy ; death of important personalities ; outbreak of war ; terror attacks , etc . ) .the analysis of real market data from the point of view of external news is a highly non - trivial task . on the one hand , it is difficult to set an independent level of `` importance '' of news in our age of information explosion ; furthermore , the effects of different news may overlap . on the other hand ,the reaction of the market to the news is also hard to tell .the situation is much simpler in an artificial market like the lux - marchesi model where we can immediately change the fundamental value of a company or asset ( which would be in reality a consequence of the external event ) .in our simulation we changed the fundamental price from to and examined the average of many runs as a function of time that has passed since the event . in the computer program we solved the averaging by raising the fundamental price by for 100 time steps and then decreasing it to the original value , after another 100 time units we raised it again etc .this means we used a rectangular function ( -like function ) to perturb the system and recorded the average response ( fig .[ eses2]a ) .what we saw in case of is an abrupt drop in market price ( the speed of which was only limited by the minimum time step of the model ) followed by an overshoot .economists and traders have long known that a correction exists after a very fast price change(fig .[ eses1 ] ) .a sad example for this was the reaction of the european stock markets to the terror attack on new york on the 11th september .the prices dropped fast that day ( generally losses over 10 % were recorded ) but the next day there was already an upward moving trend ( a correction after the overreaction of the events ) .we used two different parameter sets ( ii and iv ) to check whether the occurrence overshoot or its shape depend on the parameters and saw that it is a rather robust effect .a detailed survey of the phenomenon was undertaken using parameter set iv .we saw that the speed of the drop in price was only limited by the model ( this means a maximum of 0.05 price drop in 0.01 time steps ) .another interesting result is that ( in case of a price drop ) the location of the price minimum in time is independent of the price drop for a wide range of ( for parameter set iv the price minimum is located at approximately time units up to price drops of 10% ) .this means that on average one can predict when the minimum of the price occurs ( if one knows the parameters of the market ) irrespective of how big the fundamental price change is .let us define the magnitude of the overshoot as the difference of the price minimum and the new equilibrium price ( which equals the fundamental price after the event ) . shows linear dependence on the fundamental price change in a wide range of -s ( up to a 10% abrupt fundamental price change , which is already huge on market scales ) ( fig . [eses2]b ) .when examining the cause of overshoots within the model , the explanation is at hand : during the sudden drop of the price the proportion of pessimists rises sharply within the chartists . on the other hand ,the fraction of chartists among all dealers does not change .this means that the overshoot is caused only by the movements inside the chartist `` community '' .when the price first reaches the new equilibrium state , the fraction of pessimists is still very high which implies the further drop of price , resulting in the overshoot .furthermore the opinion index decays to zero ( after a sharp drop immediately after the event ) exponentially .the main result of the presented simulations is that in the lux - marchesi model of financial markets fluctuations and response to external perturbations ( events ) decay in a significantly different manner .spontaneous fluctuations decay with a relatively long characteristic time while the response to external events is practically immediate and followed by an overshoot .the absence of the validity of the onsager hypothesis for this model ( and for financial markets in general ) is not at all surprising since the continuous competition for profit ( better payoffs ) works so as to undermine the detailed balance and time reversal symmetry .the violation of the time reversal symmetry has further consequences as well , like the assymetry of the time dependant cross correlations between different stocks .the investigation of the opinion index and the fraction of chartists in case of fluctuations and responses showed that the decay mechanisms are indeed different : large spontaneous deviations from the equilibrium price occur in highly volatile markets which are accompanied by an increase of the ratio of chartists ( pessimists and optimists ) as compared to fundamentalists .the reaction of the market to an external change of the fundamental price is mainly governed by the shift in the ratio of the optimists and pessimists inside the group of chartists .the consequence of the latter is a well defined and rather robust overshoot in the price .the characteristic time is short and the size of the overshoot is small indicating that the market tries to adjust to the new situation effectively , however , at the same time , the phenomenon itself shows the limitations of this efficiency .we have demonstrated that it is worth and possible to investigate the effect of external perturbations and spontaneous fluctuations separately in a model market .it would be most interesting to try to identify the origins of deviations from average behavior on real market data .clearly , several difficulties have to be faced when trying to distinguish between the two mentioned mechanisms : the motivation of the agents is hidden , changes are not necessarily such abrupt as in the model , insider information may influence the pattern , effects of different news overlap , large fluctuations cover the overall behavior , etc . nevertheless , we believe that our study gives a hint how to approach this problem .thanks are due to thomas lux for correspondence .furthermore we kindly thank lszl kullmann for his advice and jnos trk for his help concerning the computer simulations and other computer related problems .support of otka t029985 is also acknowledged .

Abstract: in this paper we compare market price fluctuations with the response to fundamental price drops within the lux - marchesi model which is able to reproduce the most important stylized facts of real market data . major differences can be observed between the decay of spontaneous fluctuations and of changes due to external perturbations reflecting the absence of detailed balance , i.e. , of the validity of the fluctuation - dissipation theorem . we found that fundamental price drops are followed by an overshoot with a rather robust characteristic time .


deep neural networks ( dnns ) have found many successful applications in recent years . however , it is well - known that if one trains such networks with the standard back - propagation algorithm from randomly initialized parameters , one typically ends up with models that have poor prediction performance .a major progress in dnns research is the invention of pretraining techniques for deep learning .the main strategy is to employ layer - wise unsupervised learning procedures to initialize the dnn model parameters .a number of such unsupervised training techniques have been proposed , such as restricted boltzmann machines ( rbms ) in , and denoising autoencoders ( daes ) in .although these methods show strong empirical performance , the reason of their success has not been fully understood .two reasons were offered in the literature to explain the advantages of unsupervised learning procedure : the regularization effect and the optimization effect .the regularization effect says that pretraining provides regularization which initialize the parameters in the basin of attraction to a `` good '' local minimum .the optimization effect says that the pretraining leads to better optimization so that the initial value is close to a local minimum with a lower objective value than that can be achieved with random initialization .based on experimental evidences , some researchers confirm that the pretraining can learn invariant representations and selective units .* our contributions : * we study why the pretraining encourages moderate - sparseness .the main reason is that the pretraining models can be interpreted as an adaptive sparse coding .this coding is approximated by a sparse encoder , which is implemented by adaptively filtering out a lot of features that are not present in the input and suppressing the responses of some features that are not significant in the input .we further conduct experiments to demonstrate that it is a sparse regularization ( the hidden units become more sparsely activated ) .in this part we review some advantages of the pretraining methods .  distributed representations and deep architectures play an important role in deep learning methods .a distributed representation ( an old idea ) can capture a very great number of possible input configurations .deep architectures can promote the re - use of features and lead to abstract more invariant features for most local changes of the inputs .however , it is hard to use the _ back - propagation _ to train dnns with two traditional activation functions ( the sigmoid function and the hyperbolic tangent ) .luckily , proposes a unsupervised pretraining method to initialize the dnns model parameters and learn good representations . the regularization effect and the optimization effectare used to explain the main advantages of the pretraining method .  to better understand what the pretraining models learn in deep architectures , find that the pretraining methods can learn invariant representations and selective units .some researchers use the linear combination of previous units and the maximizing activation to visualize the feature detectors ( or invariance manifolds or filters ) in an arbitrary layer .fig . 1 of and fig .3 of show that the first , second and third layer can learn edge detectors , object parts , and objects respectively . based the distributed and invariant representations , confirm that the pretraining methods tend to do a better job at disentangling the underlying factors of variation , such as objects or object parts .compared to dnns with sigmoid function , we confirm that the pretraining methods encourage moderate - sparseness as the detectors filter out a lot of features that are not present in the input . in general , there is an illusion that unsupervised pretraining methods tend to learn non - sparse representations because it does not meet the conventional sparse methods .the conventional methods consider the idea of introducing a form of sparsity regularization .most ways have been proposed by directly penalizing the outputs of hidden units , such as penalty , penalty and student - t penalty .but , the pretraining methods implement sparseness by filtering out a lot of irrelevant features .there is a classic pretraining models : rbms .an rbms is an energy - based generative model defined over a visible layer and a hidden layer .the visible layer is fully connected to the hidden layer via symmetric weights , while there have no connections between units of the same layer .the number of visible units and hidden units are denoted by and , respectively. additionally , visible units and hidden units receive input from bias - and respectively .the energy function is denoted by :   the probability that the network assigns to visible units is where is the partition function or normalizing constant .because there are no direct connections between hidden ( or visible ) units , it is very easy to sample from the conditional functions taking the form : where , and is a logistic sigmoid functions : .the training is to use algorithm to minimize the likelihood of the data : .in this section , we denote that the sparse regularization with more overlapping groups in low layer or less in high layer is called the moderate - sparseness .we mainly consider the multi - class problem to explain why the unsupervised pretraining encourages moderate - sparseness since dnns with the pretraining has been used to achieve state - of - the - art results on classification tasks .there are two reasons .first , we show a new viewing that the pretraining model is an adaptive sparse coding .second , because the pretraining can train the `` good '' feature detectors , we discuss that how the feature detectors can lead to moderate - sparseness . finally , we measure the moderate - sparseness .  to start off the discussion , there are two natural assumptions to -class training set ._ assumption 1 : _ every class has a balanced number of samples and there are a lot of common raw features ( pixels or mfcc features ) among samples of the same class . _assumption 2 : _ there are some similar raw features among samples of different classes since they share some common ones .pretraining model ( such as rbms ) is an adaptive sparse coding .the explanation is as follow . by the results of , the pretraining model ( rbms training is to minimize )is also approximated by minimizing a reconstruction error criterion : where  ] is the mean - field output of the visible units given the representation sampled from .the and can be regard as a decoder and an encoder , respectively .  from the second parts of and, every hidden unit can be further interpreted as a feature detector ( or invariance manifolds or filters ) because the hidden unit is active ( or non - active ) , that means , the detector should respond strongly ( or weakly ) when the corresponding feature is present ( or absent ) in the input .amazedly the pretraining can train edge feature detectors in low layer and objects ( or object parts ) in high layer . _given an input , the feature detectors naturally filter out a lot of features that are not present in the input and suppress the responses of some features that are not significant in the input ._ clearly , those detectors result in sparseness .* relationship with sparse coding * : sparse coding is to find the dictionary and the sparse representation to minimize the most popular form : where also is a hyper - parameter . obviously , the first part of rbms is similar to the first part of sparse coding as they are decoders .the sparse coding is directly to penalize the norm of the hidden representation .but in rbms the is approximated by the sparse encoders ( feature detectors ) , which filter out a lot of irrelevant features . in next subsectionwe shall discuss that how the feature detectors can lead to moderate - sparseness .   * low - layer * : based on the assumptions the pretraining models averagely distribute all edge feature detectors to the hidden units in low layer as every class has a same number of samples .assumption 1 shows that every class has a same number of edge feature detectors and there are a lot of common edge ones in the same class .clearly , the edge feature detectors find out the edge features belonged to self - class , suppress the responses of some nonsignificant edge features , and filter out a lot of edge features related to the other classes .suppose that there are hidden units ( edge feature detectors ) and a -class dataset , every class ideally has ones .given input samples of a class , thus , the hidden units belonged to the class are activated or weakly responded and the remaining units are not activated ( corresponding sparseness that is measured by ) .moreover , there are the common activation units ( corresponding group ) .simultaneously , assumption 2 shows that there are some similar edge feature detectors among different classes .different classes share some edge feature detectors corresponded to hidden units , which are also activated . the activated units results in more overlapping activation units in low layer .the activation overlapping degree is measured by .combined with regularization effect , therefore , we obtain the first result ( a1 ) that _ the unsupervised pretraining is a sparse regularization with more - overlapping groups in low layer_.  * high - layer * : in high layer the pretraining goes on to train object or object part features detectors from the edge features .similarly to the analysis in low layer , the hidden units are more sparsely activated or weakly responded in high layer . moreover , the activation overlapping degree is lower than one in low layer because the pretraining can potentially lead to do a better job at disentangling the objects or object parts .thus , we obtain the second result ( a2 ) that _ the unsupervised pretraining is a sparse regularization with less ( or no)-overlapping groups in high layer_.  in dnns without the pretraining the most hidden units of every layer are always activated and correspond to terrible feature detectors , which are the important causes of difficult classification problems . for classification tasks , it is always desirable to extract features that are most effective for preserving class separability and collaboratively representing objects .the pretraining firmly grasps the those benefits .the more activation overlapping units can capture the collaborative features in low layer and the less or no activation overlapping units can capture the separability in high layer .   for better understanding the pretraining , we tried to find sparseness , more - overlapping and no - overlapping characteristics of dnns with or without the pretraining .so , hoyer s sparseness measure and activation overlapping degree are defined as followings .the hoyer s sparseness measure ( hspm ) is based on the relationship between the norm and the norm .the hspm of a dimensional vector is defined as follows : this measure has good properties , which is in the interval  ] , is used to measure the percentage of activation overlapping units in different classes .it s value more close to means that there are few activation overlapping units and it is easier to separate the different classes .in this section , we use deep neural networks to do experiments . a standard architecture for dnns consists of multiple layers of units in a directed graph , with each layer fully connected to the next one .the nodes of the inter - layers are called hidden units .each hidden unit is passed through a standard sigmoid functions .the objective of learning is to find the optimal network parameters so that the network output matches the target closely .the output can be compared to a target vector through a squared loss function or an negative log - likelihood loss function .we employ the standard _ back - propagation _ algorithm to train the model parameters ( the connected weights ) .we denote that dsigm : dnns with standard sigmoid functions , dprbms : dnns only pretrained with rbms and dbns : deep belief networks pretrained with rbms and finely tuned .* datasets * : we present experimental results on standard benchmark datasets : mnist and birdsong the pixel intensities of all datasets are normalized to $ ] . *mnist * dataset has 60,000 training samples and 10,000 test samples with pixel greyscale images of handwritten digits 0 - 9 . * * birdsong * * 35 bird recordings and test set has 150sec 3 mics 90 recordings .there are not labels in test set .so we divide the train set to a new train set and a new test set(we randomly select 3,000 train samples with 16 mfcc features , the rest are test samples in every recording . ) . ]dataset has 70,000 training samples and 200,690 test samples with 16 mfcc features .  to speed - up training, we subdivide training sets into mini - batches , each containing 100 cases , and the model parameter is updated after each minibatch using the averages .weights are initialized with small random values sampled from a normal distribution with zero mean and standard deviation of .biases are initialized with zeros . for simplicity, we use a constant learning rate chosen from .momentum is also used to speed up learning .the momentum starts at a value of and linearly increases to over half epochs , and stays at thereafter .the regularization parameter for the weights is fixed at .-0 in -0.3 in   -0.1 in    -0.15 in  -0.2 in  -0.1 in    -0.15 in  -0.4 in  before presenting the comparison of activation overlapping units , we first show the sparseness of pretraining compared to the more traditional sigmoid activation function .the sparseness metric hspm is the averaged value over the definition of .we perform comparisons on mnist , and results after fine - tuning training for 200 epochs are reported in table [ tab : hspm552 ] .the results show that compared to dsigm the pretraining leads to models with higher sparseness , and smaller test errors .table [ tab : hspm552 ] compares the network hspm of dbns and dprbms to that of dsigm . from table [tab : hspm552 ] , we observe that the average sparseness of three layer dprbms is about ; the resulting dbns has similar sparseness . in fig[ fig : spon ] , ( * a * ) also shows that the feature of every layer rbms is more sparse as the train epoch increases . in contract ,the hspm of dsigm is on average below .when the pretraining are trained longer enough and the number of hidden unites increases , hspm of the pretraining models will become more sparse and also has an upper bound . in fig[ fig : spon ] , ( * b * ) shows that when the number of hidden units changes from 500 to 5000 , an upper bounds of rbms is 0.68 after 1000 training epochs .as the number of layers increases , hspm of the pretraining models also has an upper bound . from fig [ fig : spon ] , (* c * ) shows that upper bounds of five hidden ( 500 and 1000 ) layers rbms are and , respectively .we observe that the hspm of the third layer pretraining is lower than one of the second layer .we empirically obtain the high hspm by increasing the number of the hidden units of high layer , for example , dbns ( 500 - 500 - 2000 ) .this observation maybe can explain why the top layer should be big .-0 in -0.3 in  we perform comparisons on birdsong , results after fine - tuning training for 100 epochs are reported in table [ tab : hspm511bird ] .the results also show that the pretraining leads to models with higher sparseness , and smaller test errors . from table [tab : hspm511bird ] , we observe that the sparseness of three layer dprbms is higher than that of dsigm and database . although after tuning the sparseness is close to dsigm , the pretraining learn `` good '' initial values to initialize the dnn model parameters .this illustrates that the pretraining also is an optimization effect .the hspm in 3th layer are lower than 2nd layer .when training 2 layers networks , the resulting has similar test errors .so , there is a inspiration that the hspm can be used to guide the number of layers and the number of hidden units .we perform comparisons on the test set of mnist . for convenience ,the test set is denoted by , where represents a set of all digits .-combinations of the set is denoted by a set , where is the number of -combinations and is a subset of distinct elements of .the average aod among classes is an average of all aod among a subset .we compare the average aod of dprbms to that of dsigm .we found that the pretraining can capture the characteristics _( a1 ) _ that there are many overlapping units in low layer and _ ( a2 ) _ that there are few ( or no ) overlapping units in high layer . from fig[ fig : aodmnist ] , the results show that average aod among classes ( changes from to ) is high in low layer and is low ( or zero ) in high layer .the average aod gets closer to as the number of layer increases in dprbms .particularly , the average aod gets closer to than data - self in the third layer .this reveals that it is easier to classify .but it is very approximate to in every layer of dsigm .since the pretraining is known to perform well on mnist , this paper mainly discusses why the unsupervised pretraining encourages moderate - sparseness .our observations make us suspect that sparseness and activation overlapping degree play more important roles in deep neural networks . from table [tab : hspm552 ] , table [ tab : hspm511bird ] and fig [ fig : aodmnist ] , the pretraining can capture the sparse hidden units , the more activation overlapping units in low layer and the less ( or no ) activation overlapping units in high layers .

Abstract: it is well known that direct training of deep neural networks will generally lead to poor results . a major progress in recent years is the invention of various pretraining methods to initialize network parameters and it was shown that such methods lead to good prediction performance . however , the reason for the success of pretraining has not been fully understood , although it was argued that regularization and better optimization play certain roles . this paper provides another explanation for the effectiveness of pretraining , where we show pretraining leads to a sparseness of hidden unit activation in the resulting neural networks . the main reason is that the pretraining models can be interpreted as an adaptive sparse coding . compared to deep neural network with sigmoid function , our experimental results on mnist and birdsong further support this sparseness observation .