national metrology institutes reproduce the unit of resistance , the ohm , by means of quantum hall effect ( qhe ) devices .present experiments ( see  for a recent review ) are based on multiterminal ( typically 8 terminals ) semiconductor devices where a two - dimensional electron gas ( 2deg ) is created . at low temperature , and under appropriate values of magnetic flux density ,the 2deg resistance is quantised : , where is the von klitzing constant and is the plateau index .the accurate measurement of has to reject any stray resistance ( in dc ) or impedance ( in ac ) which is added by the device contacts and the external wiring .even though a simple four - terminal connection can be sufficient for dc measurements on a single device , the series or parallel connection of several devices  , the realization of single - chip quantum hall arrays  and measurements in the ac regime  ask for more complex connection schemes , usually based on the _ double- _ or _triple - series _ connections introduced by delahaye  .  in single device double- and triple - series connections ,only adjacent terminals are short - circuited together , and this yields a resistance value of . however ,when non - adjacent terminals are short - circuited together , the resulting resistance value can be a multiple or a fraction of .fang ( * ? ? ?* ( a ) ) experimentally investigated several connections achieving resistances values of with the same 6-terminal device at a single plateau .dc analytical modelling of double- and triple - series connections of single and twin devices was performed in  and  ; in  the analysis was extended to the ac regime .the modelling process is based on the ricketts - kemeny model  of the quantum hall device , or on its derived ones  .the above analyses were carried out by applying kirchhoff s laws to the chosen model with constraints given by the external connections .then , the resulting equations were solved to determine a particular four - terminal resistance ( or impedance ) .such analytical calculations can be tedious and error - prone , and numerical simulations can have problems of convergence  .therefore , only a few connection schemes of known practical relevance have been analysed in full .  in this paper, we propose a general approach for modelling the electrical behaviour of multiterminal qhe devices with external connections .this method is based on the use of the so - called indefinite admittance matrix .four - terminal resistances can then be determined from matrix equations depending on two matrices : one that represents the unconnected device ( or devices ) ; another that represents the external connections and that can be easily written by inspection .consider the -terminal element of figure [ fig : device_representation ] and let the terminal voltages be measured with respect to an arbitrary reference point ( datum node ) .we make the following assumptions :  1 . is linear and time - invariant .whenever is connected to an external network , the complete circuit is uniquely solvable .the sum of the terminal currents is identically zero , i.e.  .the terminal currents are invariant under a change of the reference point , i.e. under the transformation .  by means of assumptions* a1 * and * a2 * , when is connected to an external network , we can write the relations between its terminal currents and voltages as where the admittance coefficients are complex quantities defined by the equation the set of equations  can be put in matrix form as where and are the column vectors and ( t denotes the transpose operation ) , and is the matrix .this matrix , which has been called by shekel  the _ indefinite admittance matrix _ , has the following properties  :  1 .the sum of the elements of each column is identically zero , i.e.  for any .this property is a direct consequence of assumption * a3*. 2 .the sum of the elements of each row is identically zero , i.e.  for any .this property is a direct consequence of assumption * a4*. indeed , these first two properties imply that is a singular matrix .if its terminal is grounded , can be regarded as an -port with port voltages . in this case , the non - singular _ short - circuit admittance matrix _ of the -port can be obtained from by deleting the row and column . moreover , since is non - singular , it can be used to determine the _ open - circuit impedance matrix _ of the -port  .  in the application of qhe devices to resistance and impedance metrology , oneis usually concerned with the determination of four - terminal transfer resistances and impedances . with reference to figure [ fig : impedances ] , we define the four - terminal impedance as the ratio of the open - circuit voltage to the current flowing into terminal and out of terminal , when all other terminals are open - circuited : throughout this work , impedances of the form ,i.e. with one terminal in common , are referred to as three - terminal impedances , and that of the form , i.e. with two terminals in common , are referred to as two - terminal impedances .one way of determining from makes use of property * p3 * above and is based on the fact that one can `` transform '' the -terminal into an -port by grounding one terminal , as shown in figure [ fig : impedances_grounded ] for the terminal . in this case , the terminal pairs and can be considered as input ports , respectively driven by the currents and , while the terminal pairs and can be considered as output ports with port voltages and , so that the ratio can be easily determined from the open - circuit impedance matrix obtained from the application of property * p3*. this method is described in detail in section [ sec : ideal_device_connections ] by means of examples .a more straightforward , albeit more cumbersome , method of determining from is described in  .figure [ fig : device_model_ideal ] shows the portion between terminals and of the ring - array dc equivalent circuit of an _ ideal _ -terminal qhe device , as proposed by delahaye  . the magnetic flux density points out of the page : reversing , reverses the sign of the current - controlled voltage sources .applying kirchhoff s voltage law ( kvl ) to nodes , , and yields where and it is assumed that .the above set of equations directly determines the indefinite admittance matrix of the ideal qhe device : which is an circulant matrix .as was pointed out by fang  and delahaye  , multiterminal qhe devices can be connected in different ways to obtain multiples and fractions of the quantised hall resistance . in this section ,we describe how to analyse single - device connections by means of the indefinite admittance matrix .+  ( a )  ( b )  ( c )  consider the simple example of figure [ fig : single_device_1](a ) , where a 6-terminal device is employed in a typical double - series connection with terminals 1,6 and 3,4 short - circuited .suppose that the two - terminal impedance is to be determined : this means , as shown in figure [ fig : single_device_1](b ) , that one has to inject a current between terminals 1 and 4 and to measure the voltage to obtain .now , in order to determine , proceed as follow ( see  for details ) .  1 .ground one terminal . in the above example , for simplicity , we have chosen to ground terminal 4 , as shown in figure [ fig : single_device_1](c ) .2 .  choose a set of independent port voltages ( in figure [ fig : single_device_1](c ) ) .in general , since short - circuiting terminals constraints the terminal voltages , it is .3 .  write down the relations between the terminal voltages and the port voltages : since these relations are linear , they can be written in matrix form as where , in general , is an matrix and . for the circuit of figure [ fig : single_device_1](c ) , and therefore the corresponding port current vector related to the terminal current vector by the equation so that the power flowing into the device is invariant , i.e.  .4 .  determine the admittance matrix which relates the port current vector to the port voltage vector , .since at least one terminal is grounded , by virtue of * p3 * , is a non - singular short - circuit admittance matrix whose inverse is the open - circuit impedance matrix . for the circuit of figure [ fig : single_device_1](c ) , with , one obtains and 5 .identify the input and the output ports . in the example of figure [ fig : single_device_1](c ), there are only one input port and one output port , and both coincide with port 1 .thus ,   a second , more significant , example is shown in figure [ fig : single_device_2](a ) .this is one of the connections experimentally analysed by fang and stiles ( * ? ? ?* ( a ) ) . in this case, suppose that the two - terminal impedance and the four - terminal impedance are to be determined .in particular , for , fang and stiles measured the value ( within few parts in ) .+  ( a )  ( b )  in figure [ fig : single_device_2](b ) , following the method described above , we have grounded one of the terminals ( terminal 4 , again , to have one input port only ) and we have defined as port voltages .thus and   the short - circuit admittance matrix and the open - circuit impedance matrix are respectively and then , to determine : i ) inject a current between terminals 1 and 4 , i.e. _ in _ port 1 , which is considered as an input port ; and ii ) determine the voltage across terminals 3 and 4 , i.e. _ across _ port 1 , which is also considered as an output port . hence which is in agreement with the value given in  . moreover , to determine : i ) inject a current between terminals 1 and 4 , i.e. _ in _ port 1 , which is considered as an input port ; and ii ) determine the voltage across terminals 2 and 3 , i.e. _ across _ port 2 and 3 , which are considered as output ports . hence the equivalent circuit of figure [ fig : device_model_nonideal ] , a contact resistance has been added in series to each terminal . applying kvl to nodes , , and yields in matrix form , the above set of equations can be written as or where is given by  and because of , the indefinite admittance matrix for the circuit of figure [ fig : device_model_nonideal ] is   actually , in analytical calculations , need not to be evaluated in full .in fact , since is typically very close to the identity matrix ( cf .equation ) , a taylor series expansion of up to the appropriate order in might be sufficient in most cases .in this section we analyse the effect of contact resistances on several examples of single device connections ( table [ tab : single_device_connections ] , first column ) .these examples , which were already analysed using other methods by delahaye  and by jeffery _et al . _ , have been chosen for their practical interest .for each connection , the second column of table [ tab : single_device_connections ] shows a suitable choice of port voltages and the third column shows the corresponding matrix .taking into account equations  and , for each connection we have determined the short - circuit admittance matrix and the open - circuit impedance matrix . from is then easy to determine the ideal values of the measured impedances and the relative errors due to the contact resistances .these results are shown in table [ tab : single_device_connections ] , and references are given for comparison .* 2>m25mm@*5c@ & port voltages & & & & & ref .+ & & & & & & ( * ? ? ?* eq . ( 17 ) ) + & & & & & & ( * ? ? ?* eq . ( 7 ) ) + & & & & & & ( * ? ? ?* eq . ( 13 ) ) +two _ unconnected _ multiterminal elements , with terminals and with terminals , can be considered as a single multiterminal element with terminals . since there is no connection between the two elements , unless terminals and belong to the same element .the indefinite admittance matrix of is thus a block diagonal matrix where the non - zero blocks are the indefinite admittance matrices of and , respectively . in the case of two non - ideal qhe devices ,the indefinite admittance matrix is then {0pt}{0pt}\bi{d}^{-1}_\textup{a}\bar{\bi{y}}_\textup{i } & 0 \\ \hline  0 & \rule{0pt}{15pt}\bi{d}^{-1}_\textup{b}\bar{\bi{y}}_\textup{i } \end{array}\right),\ ] ] where , for each device , and are given by .  given the matrix above , connections between two or more qhe devices can be treated following the same method described in section [ sec : ideal_device_connections ] for single - device connections . in particular , it is then easy to obtain the results determined by delahaye in  for multiple - series and multiple - parallel connections and to experiment with new configurations .we have described a method to model the electrical behaviour of one , or several , multiterminal qhe devices with complex external connections .the indefinite admittance matrix of the ideal unconnected device , a matrix representing the contact resistances and a matrix representing the external connections , all enter a simple matrix expression from which the two - terminal and four - terminal impedances of interest can be determined .  as examples of application ,the method has been applied to connection schemes in the dc regime of practical interest for primary metrology , already analysed in other papers .formally , the method is valid also in the ac regime ; however , the practical application requires to model the ac behaviour of connections , and parasitic parameters at the device level , and will be the subject of future investigations .the authors are grateful to stefano grivet - talocia , politecnico di torino , for pointing out the existence of the indefinite admittance matrix , and are indebted with jrgen schurr , ptb braunschweig ( germany ) : we performed together experimental testing of fang s configurations and had many discussions on the subject .10  b. jeckelmann and b. jeanneret .the quantum hall effect as an electrical resistance standard . in _ the quantum hall effect _ , volume 45 of _ progress in mathematical physics _ , pages 55131 .birkhuser basel , 2005 .

Abstract: the modelling of electrical connections of single , or several , multiterminal quantum hall effect ( qhe ) devices is relevant for electrical metrology : it is known , in fact , that certain particular connections allow i ) the realization of multiples or fractions of the quantised resistance , or ii ) the rejection of stray impedances , so that the configuration maintains the status of quantum standard . ricketts - kemeny and delahaye equivalent circuits are known to be accurate models of the qhe : however , the numerical or analytical solution of electrical networks including these equivalent circuits can be difficult . in this paper , we introduce a method of analysis based on the representation of a qhe device by means of the _ indefinite admittance matrix _ : external connections are then represented with another matrix , easily written by inspection . some examples , including the solution of double- and triple - series connections , are shown .


in most social and economic settings , individuals do not interact uniformly with the rest of a society .instead , they influence each other according to a structured network of interactions .this network can represent friendships in a social network , transactions among firms in a market , or communication links in a process of belief formation . in this context, an interesting question is to study how the network structure affects the outcome of the interactions of the agents . with this purpose , one can model strategic interactions in a networked society as a multi - player simultaneous - move game .in particular , we focus our attention on the broad class of games with linear best response function . this class of games have been used to model various settings such as belief formation , peer effects , and public goods .  in order to analyze the influence of the network structure on the game outcome , we use two recent results by bramoull et al . and ballester et al . relating the nash equilibria of the game with the largest and smallest eigenvalues of the adjacency matrix of interactions .for example , in , the authors show that uniqueness and stability of a nash equilibrium in games with linear best responses can be determined by the smallest eigenvalue of the network . in , it was also illustrated how the smallest eigenvalue of the adjacency matrix determines the capacity of the network to absorb perturbations on the actions of the agents .motivated by the results of bramoull et al . , we study how local structural properties of the network affect the smallest eigenvalue of the adjacency matrix of interactions , affecting game equilibria . therefore , our results build a bridge between structural properties of a network of interactions and the outcome of games with linear best responses .in particular , we use algebraic graph theory and convex optimization to derive bounds on the smallest eigenvalue of the adjacency matrix in terms of the distribution of degrees , cycles , and other important substructures .as we illustrate with numerical simulations in online social networks , these bounds can be used to estimate the effect of structural perturbations on the smallest eigenvalue .the paper is organized as follows . in the next subsection ,we review graph - theoretical terminology needed in our derivations . in section strategic section, we review the relationship between the equilibria of games with linear best responses in a network and the smallest eigenvalue of the adjacency matrix of interactions . in section [ spectral section ] , we use algebraic graph theory to derive closed - form expressions for the so - called spectral moments of a network in terms of local structural features . in section [ optimal bounds ] , we use convex optimization to derive optimal bounds on the smallest ( and largest ) eigenvalue of the interaction network from these moments .our bounds help us to understand how structural properties of a network impact the stability properties of the nash equilibria in the game .we illustrate our results with numerical simulations in real online social networks in section [ simulations ] .let denote an undirected graph with nodes , edges , and no self - loops .we denote by the set of nodes and by the set of undirected edges of . if we call nodes and _ adjacent _ ( or neighbors ) , which we denote by and define the set of neighbors of as .the number of neighbors of is called the _ degree _ of the node , denoted by .we define a _ walk _ of length from to to be an ordered sequence of nodes such that for . if , then the walk is closed . a closed walk with no repeated nodes ( with the exception of the first and last nodes )is called a _cycle_. for example , _ triangles _ , _ quadrangles _ and _ pentagons _ are cycles of length three , four , and five , respectively .graphs can be algebraically represented via matrices .the adjacency matrix of an undirected graph , denoted by  ] .then , a nash equilibrium for this game is a vector ^{n} ] .   using the best response function in ( [ best reply function ] ), one can determine the entire set of equilibria by simultaneously solving for the best response of each player . in , an algorithm that finds the full set of nash equilibria in exponential timeis proposed . for a vector ,let denote the set of active agents , i.e. , .let denote the vector of actions of the agents in .the set of active players induce a subgraph , with node - set and a set of edges connecting active agents .we denote by the subgraph of whose edges connect active agents in to inactive agents in .the adjacency matrices of and are denoted by and , respectively .then , one can show the following :  [ nash conditions]a profile with active agents is a nash equilibrium if and only if :  is the identity matrix and is the -dimensional vector of ones .thus , in order to determine the complete set of all nash equilibria , one can check the conditions in proposition [ nash conditions ] for each one of the possibilities of . for each possible , these conditions can be checked by computing , and checking whether .if the last inequality holds , then is an equilibrium outcome .note that using this approach to compute the set of equilibria runs in exponential time .however , we can analyze some properties of the nash equilibria , such as uniqueness and stability , by looking into the eigenvalues of the adjacency matrix .   in order to relate the equilibrium outcomes of the game to the network structure ,the authors in defined the following potential function: , they proved , using kuhn - tucker conditions , that the set of nash equilibria coincides with the critical points of the following optimization problem: a given network structure and a parameter .we can find several results in the literature providing sufficient conditions for the existence of a unique nash equilibrium in games with linear best response functions in terms of the eigenvalues of the network of influences .we enumerate below some sufficient conditions that are related with our work :  [ kranton condition]consider the class of games with linear best response functions described in section [ game description ] . for these games, we have the following sufficient conditions for the existence of a unique nash equilibrium :  ( _ i _ ) : :   , ( bramoull et  al . , ) .( _ ii _ ) : :   ,  ( ballester et al . , ) .we can compare the set of spectral conditions in proposition [ kranton condition ] using the following inequalities :  for any simple graph , we have that , where this inequality is strict when no component of is bipartite .[ active remark]hence , condition ( _ i _ ) in proposition [ kranton condition ] provides the best sufficient condition for the uniqueness of nash equilibrium in these games .furthermore , one can also prove that under condition ( i ) or ( ii ) , all players are active at the equilibrium point , i.e. ,  .  in section [ optimal bounds] , we shall derive upper bounds on in terms of structural properties of the network .these bounds , in combination with condition ( _ i _ ) in proposition [ kranton condition ] , will allow us to derive sufficient conditions for the existence of a nash equilibrium in terms of structural properties of the network .we present conditions for stability of a nash equilibrium in terms of .a nash equilibrium is asymptotically stable when the system of differential equations: locally asymptotically stable around .one can prove the following necessary and sufficient condition for an equilibrium to be asymptotically stable :  an equilibrium profile is asymptotically stable if and only if and for all inactive agents .  from the above lemma and remark [ active remark ] , we conclude that if , there is a unique nash equilibrium and it is asymptotically stable .these results show the close connection between the smallest and the largest eigenvalues of the adjacency matrix of interactions and the outcome of games with linear best response functions in a network . on the other hand ,the results in this section are applicable if we are able to compute the eigenvalues of . for many large - scale complex networks, the structure of the network can be very intricate - in many cases not even known exactly and an explicit eigenvalue decomposition can be very challenging to compute , if not impossible . in many practical settings , instead of having access to the complete network topology , we have access to local neighborhoods of the network structure around a set of nodes . in this context , it is important to understand the impact of local structural information on the eigenvalues of the adjacency matrix . in the rest of the paper ,we propose a novel methodology to compute optimal bounds on relevant eigenvalues of from local information regarding the network structure .our results allow us to study the role of local structural information in the outcome of games with linear best response functions .we study the relationship between a network s local structural properties and the smallest eigenvalue of its adjacency matrix. algebraic graph theory provides us with tools to relate the eigenvalues of a network with its structural properties .particularly useful is the following well - known result relating the -th spectral moment of with the number of closed walks of length in  :  [ moments from walks]let be a simple graph . the -th spectral moment of the adjacency matrix of can be written as where is the set of all closed walks of length in .the cardinality of a set . ]  from ( [ moments as walks in graph ] ) , we can easily compute the first three spectral moments of in terms of the number of nodes , edges and triangles as follows :  [ one corollary]let be a simple graph with adjacency matrix .denote by , and the number of nodes , edges and triangles in , respectively .then,  notice that the coefficients 2 ( resp .6 ) in the above expressions corresponds to the number of closed walks of length 2 ( resp .3 ) enabled by the presence of an edge ( resp .triangle ) .similar expressions can be derived for higher - order spectral moments , although a more elaborated combinatorial analysis in required .  in our case , we are interested in the following expressions , derived in pj10 , for the first five spectral moments of :  [ lemma 4th metrics]let be a simple graph .denote by , , and the total number of edges , triangles , quadrangles and pentagons in , respectively . define and , where is the number of triangles touching node .then, , \label{fourth moment subgraphs } \\m_{5}\left ( a_{\mathcal{g}}\right ) & = & \frac{1}{n}\left [ 10\pi + 10\mathcal{c}% _ { dt}-30\delta \right ] .\label{fifth moment subgraphs}\end{aligned}\ ] ]  observe how ,as we increase the order of the moments , more complicated structural features appear in the expressions . in the following example , we illustrate how to use our expressions to compute the spectral moments of an online social network from empirical structural data .    in this example, we study a subgraph of facebook obtained by exploring the online social network around a particular node , as follows . from a particular starting node, we crawl the network ( using a breadth - first search ) until we discover the set of all nodes that are within a radius from the starting node . using this set of nodes , and their interconnections ( friendships ) ,we construct a social subgraph that has nodes and edges . using this real dataset ,we compute the degrees , the number of triangles , quadrangles , and pentagons touching each node . in fig .1 , we plot the distributions of degrees and triangles , as well as a scatter plot of versus ( where each point has coordinates , in log - log scale , for all ) .we can aggregate those quantities that are relevant to compute the spectral moments to obtain the following numerical values: , using corollary [ one corollary ] and lemma [ lemma 4th metrics ] , we obtain the following values for the spectral moments :   and   in this section , we have derived expressions to compute the first five spectral moment of from network structural properties . in the next section ,we use semidefinite programming to extract information regarding eigenvalues of interest from a sequence of spectral moments .here , we introduce an approach to derive an upper bound on the smallest eigenvalues of from its sequence of spectral moments , although these bounds are not essential in our analysis . ] .since we have expressions for the spectral moments in terms of local structural properties , our bounds relate the eigenvalues of a network with these properties .there is a large literature studying the relationship between structural and spectral properties of graphs ( see cds80, , and references therein , for an extensive list of spectral results ) . for many real - world networks, there is a particular set of structural properties that play a key role in the network s functionality .for example , it is well - known that social networks contain a large number of triangles ( and other cycles ) .hence , it would be useful to have spectral bounds where these structural features are jointly represented . in this section ,we derive new upper bounds on the smallest eigenvalue of the adjacency matrix in terms of the structural properties involved in ( [ moments as averages ] ) , ( [ fourth moment subgraphs ] ) and ( [ fifth moment subgraphs ] ) .our results can be easily extended to derive lower bounds on the spectral radius of the adjacency matrix , although this bound is not of relevance in our analysis of games with linear best responses .now , we derive bounds on the smallest eigenvalue of the adjacency matrix in terms of relevant structural properties by adapting the optimization framework proposed in . we first need to introduce a probabilistic interpretation of a network eigenvalue spectrum and its spectral moments . for a simple graph , we define its spectral density as: is the dirac delta function and is the set of ( real ) eigenvalues of the ( symmetric ) adjacency matrix . consider a random variable with probability density .the moments of are equal to the spectral moments of , i.e., all .  in , lasserre proposed a technique to compute the smallest interval  ] containing the support of is equal to  ] . in our case , since we have expressions for the first five spectral moments , , we can solve the first two steps of this series of sdp s to find inner approximations \subseteq \left [ \lambda _ { n},\lambda _ { 1}\right ] ] being the smallest interval containing .observe that and are the solutions to two sdp s in one variable , since the constraint ( resp . ) indicates that the matrix ( resp . ) is positive semidefinite and this matrix has affine entries with respect to ( resp . ) .hence , they can be efficiently computed using standard optimization software ( for example , cvx ) . as we increase in proposition [ lasserre bound ] ,more moments are involved in the sdp s , and the resulting bounds become tighter , i.e. , and .  in the context of our spectral analysis , the borel measure in proposition [ lasserre bound ] corresponds to the spectral density of a graph , and the smallest interval  ] . thus , proposition [ lasserre bound ] provides an efficient numerical scheme to compute the bounds and . when we are given a sequence of five spectral moments , we can solve the sdp s in ( [ bound min eigenval ] ) and ( bound max eigenval ) analytically for . in this case , the localizing matrix is: .\label{h2c}\]]as we proved in section [ spectral section ] , the spectral moments in the localizing matrix depend on the number of nodes , edges , cycles of length 3 to 5 , the sum - of - squares of degrees , and the degree - triangle correlation .furthermore , for , the optimal values and can be analytically computed , as follows .first , note that ( resp . ) if and only if all the eigenvalues of are nonnegative ( resp .nonpositive ) . for a given sequence of five moments , the characteristic polynomial of be written as is a polynomial of degree in the variable ( with coefficients depending on the moments ) .thus , by descartes rule , all the eigenvalues of are nonpositive if and only if , for and .similarly , all the eigenvalues are nonnegative if and only if and . in fact, one can prove that the optimal value of and in ( [ bound max eigenval ] ) can be computed as the smallest and the largest roots of , which yields a third degree polynomial in the variable  .there are closed - form expressions for the roots of this polynomial ( for example , cardano s formula ) , although the resulting expressions for the roots are rather complicated .  in this subsection , we have presented a convex optimization framework to compute optimal bounds on the maximum and the minimum eigenvalues of a graph from a truncated sequence of its spectral moments .since we have expressions for spectral moments in terms of local structural properties , these bounds relate the eigenvalues of a graph with its structural properties .as we illustrated in section [ strategic section ] , there is a close connection between the largest and the smallest eigenvalues of a network and the outcome of a game with linear best response functions . in this section, we use our bounds on the support of the eigenvalue spectrum to study the role of structural properties in the existence and the stability of a nash equilibrium . for this purpose ,we analyze real data from a regional network of facebook that spans users ( nodes ) connected by friendships ( edges ) . in order to corroborate our results in different network topologies ,we extract multiple medium - size social subgraphs from the facebook graph by running a breath - first search ( bfs ) around different starting nodes .each bfs induces a social subgraph spanning all nodes 2 hops away from a starting node , as well as the edges connecting them .we use this approach to generate a set of 100 different social subgraphs centered around 100 randomly chosen nodes .  ,versus the lower bound , where each point is associated with one of the social subgraphs considered in our experiments.,scaledwidth=43.0% ]  from corollary [ one corollary ] and lemma [ lemma 4th metrics ] we can compute the first five spectral moments of a graph from the following structural properties : number of nodes ( ) , edges ( ) , triangles ( ) , quadrangles ( ) , pentagons ( ) , as well as the sum - of - squares of the degrees ( ) , and the degree - triangle correlation ( ) . for convenience ,we define as a set of relevant structural properties of . in our numerical experiment, we first measure the set of relevant properties for each social subgraph , and then compute the first five spectral moments of its adjacency matrix . from these moments , we then compute the bounds and using proposition [ lasserre bound ] .as we mentioned before , these bounds can be computed as the maximum and minimum roots of a third order polynomial , for which closed form expressions are known .we illustrate the quality of our bounds in the following figures .2 is a scatter plot where each circle has coordinates , for all . observe how _ the spectral radii _ _ of these social subgraphs are remarkably close to the theoretical lower bound _ .therefore , we can use as an estimate of for social subgraphs . in fig .3 we include a scatter plot where each circle has coordinates , for all .although is a looser bound than , we observe how there is a strong correlation between the value of and .  in these numerical experiments, we have first showed that and bound the smallest and the largest eigenvalues of the adjacency matrix , and that these bounds are tight , specially .since these bounds can be written as explicit functions of the structural properties in , we can estimate the impact of structural perturbations on the spectral radius and the smallest eigenvalue by studying and for .( details of this perturbation analysis are left for future work due to space limitations . )   versus the bound , where each point is associated with one of the social subgraphs considered in our experiments.,scaledwidth=43.0% ]in this paper , we have studied games with linear best response functions in a networked context .we have focused on analyzing the role of the network structure on the game outcome . in particular ,the existence and the stability of a unique nash equilibrium in this class of games are closely related with the smallest eigenvalue of the adjacency matrix of the network .we take this spectral result as the foundation to our work , and use algebraic graph theory and convex optimization to study how local structural properties of the network affect this eigenvalue .in particular , we have derived expressions for the first five spectral moments of the adjacency matrix in terms of local structural properties .these structural properties are : the number of nodes and edges , the number of cycles of length up to 5 , the sum - of - squares of the degrees , and the degree - triangle correlation . from this sequence of five spectral moments ,we propose a novel methodology to compute optimal bounds on the smallest and the largest eigenvalues of the adjacency matrix by solving two semidefinite programs . in our case, we are able to find analytical solutions to these optimal bounds by computing the roots of a cubic polynomial , for which closed - form expressions are available .finally , we have verified the quality of our bounds by running numerical simulations in a set of 100 online social subgraphs . for future work, we shall use the results herein presented to study the effect of structural perturbations in the relevant eigenvalues of the adjacency matrix , and in properties of the nash equilibrium .

Abstract: this paper studies -person simultaneous - move games with linear best response function , where individuals interact within a given network structure . this class of games have been used to model various settings , such as , public goods , belief formation , peer effects , and oligopoly . the purpose of this paper is to study the effect of the network structure on nash equilibrium outcomes of this class of games . bramoull et al . derived conditions for uniqueness and stability of a nash equilibrium in terms of the smallest eigenvalue of the adjacency matrix representing the network of interactions . motivated by this result , we study how local structural properties of the network of interactions affect this eigenvalue , influencing game equilibria . in particular , we use algebraic graph theory and convex optimization to derive new bounds on the smallest eigenvalue in terms of the distribution of degrees , cycles , and other relevant substructures . we illustrate our results with numerical simulations involving online social networks .


quantum state preparation is the first important step for any protocol that makes use of quantum resources .examples of such protocols are quantum state teleportation and quantum key distribution which require entangled quantum states . in order to verify the integrity of the true quantum state prepared by the source , one carries out quantum state tomography to characterize it .measurements are performed on a collection of identical copies of quantum systems ( electrons , photons , etc . )that are emitted from the source .then , the quantum state of the source is inferred from the measurement data obtained from this collection .the measurements are generically described by a set of positive operators that compose a _ probability operator measurement _such a procedure of state inference is known as quantum state estimation .when the measurement outcomes form an informationally complete set , they fully characterize the source and the measurement data obtained will contain maximal information about its state . to infer the unknown state from the data , one can search for a state estimator that maximizes the _ likelihood functional _ that yields the probability of obtaining a particular sequence of measurement detections given a quantum state the maximum - likelihood ( ml ) estimator . yet , in tomography experiments performed on complex quantum systems with many degrees of freedom , it is not possible to implement such an informationally complete set of measurement outcomes .therefore , some information about the source will be missing and its quantum state can not be unambiguously determined . for instance , if a source produces a mode of light that is described by an infinite - dimensional statistical operator , then no matter how ingeniously a measurement scheme is designed to probe incoming photons prepared by this source , an infinite amount of information about the mode of light will always remain unknown .the ml estimator obtained from these informationally incomplete data is no longer unique and there will in general be infinitely many other ml estimators that are consistent with the data . these estimators form a convex set under the likelihood plateau .  in order to choose an estimator from the convex set for statistical prediction, we can consider the maximum - entropy principle advocated by e. t. jaynes  .in doing so , one obtains a unique estimator that maximizes both the likelihood andthe von neumann entropy functional .statistically , this estimator is least - biased for the informationally incomplete data . in , we developed and applied an algorithm , based on the steepest - ascent method , to approximately look for the maximum - likelihood - maximum - entropy ( mlme ) estimator .this algorithm involves a parameter that needs to be chosen just above a minimum threshold to obtain an estimator that is as close to the actual mlme estimator as possible . in general , this threshold depends on the true state .therefore , one needs to run the algorithm a few times to estimate this threshold .this paper is organized as follows .after a brief review of this steepest - ascent algorithm in section [ sec : br ] , we introduce a numerical scheme that is based on completely different principles to directly search for the mlme estimator within the convex set in section [ sec : aasdp ] . this scheme couples two separate optimization techniques semidefinite programming ( sdp ) and a derivative - free optimization method to maximize the entropy over linear combinations of a maximal set of linearly independent ml estimators that spans the ml convex set .it will be shown that , owing to the mechanisms of sdp , one can make use of this scheme to infer the expectation values of a set of entanglement witnesses to verify the presence of entanglement in the unknown quantum state for composite systems .finally , in section [ sec : emlme ] , we establish a more robust numerical scheme that systematically generates the maximal set of linearly independent ml estimators that defines the convex set without fail . instead of sdp , this scheme utilizes a nonlinear optimization routine that finds the global maximum of a highly nonlinear functional that is used to generate this maximal set .the likelihood functional for a set of measurement data collected with a pom is given by + where refers to the number of occurrences of the outcome and .the corresponding frequencies are given by .in ml state estimation , the concave log - likelihood functional is maximized to obtain the ml estimator '' is used to denote all estimators . ] for the given set of data . if the number of linearly independent outcomes is , with being the dimension of the hilbert space , the data is informationally complete and the estimator is unique . in the case where this number is less than ,the data is informationally incomplete and there exists now a continuous set of that yield the same maximal likelihood .this set is convex since the likelihood functional is concave in .  to choose the estimator that has the highest entropy out of this convex set, we can consider the lagrange functional + where is the lagrange multiplier corresponding to the constraint of maximal entropy .in doing so , we maximize the two functionals and simultaneously .we define to be the estimator that maximizes .if , and maximizing this lagrange functional is just the procedure of ml . since the data is informationally incomplete , there exists a convex plateau structure for the log - likelihood functional and maximizing yields a convex set of estimators . for large values , the term dominates , so that the resulting estimator  takes on a very small positive value , the contribution from becomes relatively much smaller than that of , and any variation of the von neumann entropy functional is only detectable over the state space region that coincides with the likelihood plateau . in other words , maximizing is equivalent to maximizing the entropy over the plateau .therefore , .the iterative algorithm that is based on the steepest - ascent method is described in - .  in practice, there is a limit to how small can be .in particular , when is smaller than some numerical threshold , the gradient of is no longer visible . in this case, the algorithm treats as and performs ml estimation .hence , the optimal parameter is to be slightly above so that .this inevitably introduces a small bias to the estimator .determining the value of analytically is quite complicated because is actually a function of the true state and the pom , that is .since is unknown to us , one needs to estimate through repeated runs of the algorithm . in the next section, we will introduce an alternative scheme to search for the exact mlme estimator .to obtain the unique mlme estimator that has the highest entropy , a search has to be performed within the ml convex set . in this section ,we introduce a numerical procedure to estimate in two steps .the first step is to obtain a collection of boundary states of the convex set for the measurement data obtained . in the next step, the operator can be estimated using these boundary states with a standard function optimization routine .  to carry out the first step , we need to identify the boundary of the ml convex set . especially for large dimensions ,the boundary of the convex set has an extremely complicated geometry that is too difficult to be analytically determined .instead , we investigate its boundary by numerical means .we begin with the fact that the real functional , where the operator , is a linear functional of .therefore , if we try to maximize ( or minimize ) over some subspace of that has a well defined boundary , the maxima ( or minima ) of this linear functional are always on the boundary of this subspace. we can thus generate boundary states by maximizing or minimizing , for a given hermitian operator , over the ml convex set .this problem is equivalent to the following optimization task : +  this is a standard linear optimization problem , with linear and positivity constraints , that can be solved with the help of sdp .  at this point, we need to find out the minimum number of boundary states that is required to search for . to do this, we represent a generic ml estimator by a set of linearly independent hermitian operators are all linearly independent , the rank of the matrix with elements is . ]trace - orthonormal hermitian basis operators satisfying the condition . with these basis operators , we can separate into the part in the measurement subspace of dimension , and the rest of the state space that constitutes the unmeasured parameters ( the unmeasured subspace ) of dimension , where , and . the generation of the basis operators that span the measurement subspace can be done by the gram - schmidt orthonormalization procedure on the , with all conventional inner products for vectors replaced by trace inner products for operators . in other words ,the number of linearly independent pom outcomes is . to generate the rest of the basis operators that span the unmeasured subspace, we continue the gram - schmidt procedure using randomly generated positive operators instead of the .from , it can be deduced that the maximum dimension of the ml convex set is . to show this, we note that _ every _ ml estimator contains the same operator since the probabilities of are fixed and for all by definition of the trace - orthonormal basis operators .this implies that the only difference between any two ml estimators in the convex set is .as the operators and are linearly independent , it follows that any ml estimator can always be expressed as a linear combination of the unique and the linearly independent basis operators that define .this means that a set of linearly independent boundary states is enough to look for the mlme estimator . as the operator fixed by the measurement operators , the maximal number of free parameters that span the convex set is . in some cases , however , the dimension of the ml convex set is lower due to the positivity constraint imposed on the ml estimators . in the extreme case ,the convex set is restricted to a single point in state space . in these situations , we do not know its actual dimension and repeated generation of boundary states is necessary to estimate the maximum number of linearly independent boundary states .we remind the reader that with enough linearly independent states , the exact mlme estimator can be obtained up to numerical precision . in section [ sec : emlme ] , a different numerical procedure will be introduced to generate the maximal set of linearly independent ml estimators that spans the ml convex set without requiring any knowledge of the convex set .  apart from serving as a routine to numerically compute the boundary states of the convex set, sdp provides an additional useful function . for composite quantum systems , if one selects the hermitian operators to be entanglement witnesses , one can obtain information about the presence of entanglement in the unknown true state even with informationally incomplete data .these witnesses have the properties that for all separable states and for at least one entangled state .the sdp routine , described above , thus looks for the maximum ( or minimum ) value of for any chosen witness operator over the space of positive .this way , we can in fact infer a set of maximum ( or minimum ) witness expectation values over the ml convex set from this optimization procedure .the set of inferred maximum witness expectation values is particularly informative , for if the maximum value of for at least one of the randomly generated operators is negative , we can immediately conclude that the true state is entangled since this state must lie within the ml convex set that results from the incomplete data . for practical computation , we can choose the witness operators to be of the decomposable form , where is a positive operator with no product kets in its range and the symbol `` '' denotes a partial transposition with respect to the subsystem .for bipartite systems , these operators are optimal witnesses , that is no other witnesses can detect all entangled states that are detected by this witness , as well as other entangled states . for multipartite systems , these operators still serve as entanglement witnesses since , although they are no longer optimal in general . to obtain a random set of boundary states of the ml convex set, random statistical operators are generated using the relation + where is a random operator which , when expressed in the computational basis , has complex matrix elements that are distributed according to the standard normal distribution of zero mean and unit variance .the second step involves an optimization procedure to maximize the entropy using the generated set of linearly independent boundary states . for the purpose of entanglement detection ,these boundary states are obtained by maximizing the linear functionals of a set of witness operators over the ml convex set according to the recipe described above .we start by writing a generic ml estimator as a linear combination of inasmuch as + where the are normalized coefficients , such that , that are in general real such that .the task now is to look for the values of for which the function is maximum over all real normalized coefficients .the unconstrained optimization of this -dimensional function with respect to can be performed with any efficient multi - dimensional unconstrained optimization routine that is included in the standard libraries of commercialized mathematical softwares . in matlab , for instance , the function ` fminsearch ` does the job using the nelder - mead simplex method ( nms ) .when is large , it is suggested in that an adaptive version of the nelder - mead simplex method ( anms ) may be more advantageous in terms of shorter computation time .we take the resulting operator as the sdp mlme estimator .there is , however , a caveat to this optimization .since the positivity of is no longer guaranteed over the entire space of real normalized vectors , the entropy + expressed in terms of the eigenvalues of the statistical operator that is represented by its eigenbasis , can take complex values . in order to restrict the optimization to yield only positive sdp mlme estimators, one can replace the entropy function in with the conditional function + which effectively restricts the original search region to the admissible state space . to checkif the evaluated operator is positive , a highly efficient way is to determine whether or not it admits a cholesky decomposition .  for full - rank sdp mlme estimators , there is a simple way to check that is indeed the mlme estimator .we recall that the form of the mlme estimator is given by where each is a lagrange multiplier for the constraint for any ml estimator in the convex set .if is full - rank , it follows from that the operator is a linear combination of only the pom outcomes , that is resides in the measurement subspace .this is equivalent to the set of conditions for all . defining the variables , the quantity can be used to determine if is close to the actual mlme estimator up to some numerical precision .  in summary, both the mlme estimator and information about the entanglement of can be obtained with informationally incomplete data using the following algorithm , which we coin as sdp mlme : +  figure [ fig : two_qubit ] summarizes the results .+  +  in figure ( [ fig : mepart ] ) , the convergence of is consistent with the fact that , in principle , the maximal number of nine linearly independent boundary states is enough to express the mlme estimator .the entanglement detection ratio in figure [ fig : ent_det_ratio ] is computed as the ratio of the number of detected pure states to the total number of generated states for every number of boundary states used .ideally , this ratio eventually goes to one if enough witness operators are generated to detect all the random pure states since there are no positive partial - transpose entangled states in this case . for benchmarking , figure [ fig : mlme_dist ]is generated to confirm the consistency of sdp mlme with the mlme algorithm described in section [ sec : br ] .there exists an average bias in [ fig : mlme_dist ] that arises from a fixed for all the pure states .despite the usefulness of sdp in entanglement verification and boundary states generation as discussed in section [ sec : aasdp ] , the speed of the sdp routine strongly depends on the dimension of the hilbert space and the total number of linear constraints imposed by the measurement data . when the total number of pom outcomes is large , there will generally be a considerable slowdown of the sdp routine as the search accounts for a large set of linear constraints in addition to the positivity constraint .another feature of the sdp routine is that the sequence of boundary states that are generated from random hermitian operators are not guaranteed to be linearly independent of one another .this means that typically , one would need to generate a large set of ml estimators that contains the maximal number of linearly independent estimators that span the ml convex set . for convex sets of large dimensions , this approach can be time - consuming . in this section ,we propose a different search routine , in place of sdp , to directly look for linearly independent ml estimators within the ml convex set in a deterministic way . with this routine, we establish a feasible algorithm to look for the exact mlme estimator .  to begin , we recall that a given set of linearly independent ml estimators , as in , implies the existence of a full - rank positive gram matrix with elements given by + this hints a straightforward strategy to cumulatively obtain the maximal set of linearly independent ml estimators that spans the entire ml convex set : starting with a single ml estimator , the next estimator containing the same be chosen such that the smallest eigenvalue of the gram matrix for these two estimators is maximized , and so forth , with the maximization performed over positive estimators .in general , is a nonlinear functional of that has multiple local stationary points .this functional is also not differentiable and has undefined gradients at the boundary of the state space . to search for its global maximum ,an appropriate numerical method to use is a nonlinear optimization algorithm that invokes pattern searches and can cope with functionals that have ill - defined gradients .the solver for this algorithm , ` patternsearch ` , is readily available in matlab .the positivity constraint is incorporated into the optimization algorithm using the augmented lagrangian method with cholesky decomposition . another versatile feature of the proposed routine is that it can be set to terminate when the maximal number of linearly independent ml estimators is generated , such that the next ml estimator always yields a zero eigenvalue for .we can , therefore , deterministically obtain the maximal set of linearly independent ml estimators that spans the ml convex set in this manner , in contrast with the sdp routine in sdp mlme . in this sense ,the routine is operationally robust even without any prior information about the convex set . to ensure that the search is numerically stable , it is favorable to start the pattern search algorithm from a highly - mixed ml estimatorthis is obtained by performing sdp as a couple of times and defining the starting estimator as an equal mixture of the resulting ml boundary estimators and the fairly mixed ml estimator obtained from the ml algorithm starting from the maximally - mixed state in computing the ml probabilities . using this maximal set ,the mlme estimator can be directly obtained by maximizing the entropy function in over all linear combinations of the linearly independent ml estimators in the set .these lead to the following pattern search mlme algorithm ( ps mlme ) for incomplete quantum state estimation : +  the performance of ps mlme depends not only on the dimension of the unmeasured subspace , but also on the complexity of the functional . more generally , as the dimension of the hilbert space , or that of the unmeasured subspace , increases , the number of local maxima of increases .this translates to a longer computation time to locate the global maximum in the search for linearly independent ml estimators .thus , for very large dimensions , ps mlme becomes inefficient and the approximate mlme algorithm , which is based on steepest ascent , is a more practical substitute . on the other hand ,ps mlme consistently gives more accurate mlme estimators as compared to the steepest - ascent algorithm , which yields biased results for large dimensions because of the finite parameter .hence , there is a tradeoff between computation time and the accuracy of the mlme estimators .figure [ fig : cvx_sa ] compares the performances of the ps mlme and the steepest - ascent mlme algorithms for varying hilbert space dimensions .+  +  figure [ fig : accuracy ] shows the consistently more accurate results obtained with ps mlme as compared to the steepest - ascent algorithm , in exchange for its longer computation time , illustrated in figure [ fig : cputime ] , for higher dimensions .the simulations show that for the moderately large dimensions considered in figure [ fig : cvx_sa ] , much more accurate mlme estimators can be obtained using ps mlme with longer computation time that is of the same order as that with steepest ascent . to reduce the computation time of ps mlme , one can consider an approximate maximization of as long as the resulting value is sufficiently large . throughout the simulations , the duration of searching for each optimal ml estimatoris restricted to five seconds . for even larger dimensions, ps mlme can be computationally demanding even when approximate maximization is carried out , and the steepest - ascent algorithm turns out to be a more realistic option .efforts to further improve the performance of the ps mlme scheme are in the works .nevertheless , we hope that the current work can serve as a stepping stone that helps to spur interesting discussions and novel contributions in related fields on numerical optimization over convex sets of positive operators .we have introduced a scheme to look for the unique estimator that maximizes the likelihood and entropy for informationally incomplete measurement data .this involves two main procedures : a generation of linearly independent boundary maximum - likelihood estimators that spans the convex set with semidefinite programming , and an entropy maximization with these estimators using a standard function optimization routine .furthermore , for composite quantum systems , one can apply this scheme to infer the expectation values of a set of entanglement witnesses .this information allows us to verify the entanglement of any quantum state with informationally incomplete data .however , semidefinite programming does not offer a definite control over the generation of maximum - likelihood estimators .this motivated us to develop an alternative scheme that is more operationally robust than the former one to search for the maximum - likelihood - maximum - entropy estimator .this latter scheme makes use of the pattern search optimization algorithm that is suitable for maximizing a nonlinear function required to deterministically generate the maximal set of estimators that defines the convex set . with numerical simulations, we showed that the latter scheme gives much more reliable results than the mlme algorithm discussed in section [ sec : br ] at the expense of slightly longer computation time when the dimension of the reconstruction hilbert space , or the unmeasured subspace , is moderately large .this work is supported by the nus graduate school for integrative sciences and engineering and the centre for quantum technologies , which is a research centre of excellence funded by ministry of education and national research foundation of singapore , as well as the technology agency of the czech republic , project no .te01020229 ( center of digital optics ) , and the czech ministry of industry and trade , project no .fr - 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Abstract: there exists , in general , a convex set of quantum state estimators that maximize the likelihood for informationally incomplete data . we propose an estimation scheme , catered to measurement data of this kind , to search for the exact maximum - likelihood - maximum - entropy estimator using semidefinite programming and a standard multi - dimensional function optimization routine . this scheme can be used to infer the expectation values of a set of entanglement witnesses that can be used to verify the entanglement of the unknown quantum state for composite systems . next , we establish an alternative numerical scheme that is more computationally robust for the sole purpose of maximizing the likelihood and entropy .