enumeration is a fundamental problem in computer science , and many algorithms have been proposed for many problems , such as cycles , paths , trees and cliques .however , their application to real world problems has not been researched very much , due to the handling needed for the huge amount of and the high computational cost .however , this situation is now changing , thanks to the rapid increase in computational power , and the emergence of data centric science .for example , the enumeration of all substructures frequently appearing in a database , i.e. , frequent pattern mining , has been intensively studied .this method is adopted for capturing the properties of databases , or for discovering new interesting knowledge in databases .enumeration is necessary for such tasks because the objectives can not be expressed well in mathematical terms .the use of good models helps reduce the amount of output , and the use of efficient algorithms enables huge databases to be more easily handled .more specifically , introducing a threshold value for the frequency , which enables controlling the number of solutions .in such areas , minimal / maximal solutions are also enumerated to reduce the number of solutions .for example , enumerating all cliques is usually not practical while enumerating all maximal cliques , i.e. cliques included in no other cliques , is often practical . in real - world sparse graphs ,the number of maximal cliques is not exponential , so , even in large - scale graphs , the maximal cliques can often be enumerated in a practically short time by a stand alone pc even for graphs with millions of vertices .however , the enumeration of maximum cliques , that have the maximum number of vertices among all cliques , is often not acceptable in practice , since the purpose of enumeration is to find all locally dense structures , and finding only maximum cliques will lose relatively small dense structures , thus it does not cover whole the data .paths and cycles are two of the most fundamental graph structures .they appear in many problems in computer science and are used for solving problems , such as optimizations ( e.g. flow problems ) and information retrieval ( e.g. connectivity and movement of objects ) .paths and cycles themselves are also used to model other objects .for example , in chemistry , the size and the fusing pattern of cycles in chemical graphs , representing chemical compounds , are considered to be essential structural attributes affecting on several important properties of chemical compounds , such as spectroscopic output , physical property , chemical reactivity , and biological activity .  for a path / cycle ,an edge connecting two vertices of but not included in is called a chord . a path / cycle without a chordis called a chordless path / cycle .since a chordless cycle includes no other cycle as a vertex set , it is considered minimal .thus , chordless cycles can be used to represent cyclic structures .for instance , the size and fusing pattern of chordless cycles in chemical graphs as well as other properties of chemical structures are taken into account when selecting data for prediction of nuclear magnetic resonance ( nmr ) chemical shift values .most chemical compounds contain cycles . in chemistry ,the term ` ring ' is used instead of ` cycle ' , for example a cycle consisting of 5 vertices is called 5-membered ring . since the character of ring structures of chemical compounds is assumed to be important to study the nature of the structure - property relationships , the ring perception is one of classical questions in the context of chemical informatics , so called chemoinformatics .several kinds of ring structures , such as all rings and the smallest set of smallest ring ( sssr ) are usually included in a basic dataset of chemical information .nmr chemical shift prediction is a successful case where the information about chordless cycles is employed to improve the accuracy of the prediction .the path / cycle enumeration is supposed to be useful also for analysis of network systems such as web and social networks .  in this paper , we consider the problem of enumerating all chordless paths ( resp . , cycles ) of the given graph .while optimization problems for paths and cycles have been studied well , their enumeration problems have not .this is because there are huge numbers of paths and cycles even in small graphs .however , we can reduce the numbers so that the problem becomes tractable by introducing the concept of chordless .the first path / cycle enumeration algorithm was proposed by read and tarjan in 1975 .their algorithm takes as input a graph and enumerates all cycles , or all paths connecting given vertices and , in time for each .the total computation time is where is the number of output cycles / paths .ferreira et al . recently proposed a faster algorithm , that takes time linear in the output size , that is the sum of the lengths of the paths .the chordless version was considered by wild .an algorithm based on the principle of exclusion is proposed , but the computational efficiency was not considered deeply . in this paper , we propose algorithms for enumerating chordless cycles and chordless paths connecting two vertices and ( reported in 2003 ) .note that chordless cycles can be enumerated by chordless path enumeration .the running time of the algorithm is for each , the same as the read and tarjan algorithm .we experimentally evaluated the practical performance of the algorithms for random graphs and real - world graphs .the results showed that its practical computation time is much smaller than , meaning that the algorithms can be used for large - scale graphs with non - huge amount of solutions .the results also showed that the number of chordless cycles is drastically small compared to the number of usual cycles .a _ graph _ is a combination of a vertex set and an edge set such that each edge is a pair of vertices .a graph with vertex set and edge set is denoted by .an edge of pair and is denoted by .we say that the edge _ connects _ and , is _ incident _ to and , and and are _ adjacent _ to each other , and call and _ end vertices _ of .an edge with end vertices that are the same vertex is called a _ self - loop_. two edges having the same end vertices and are called _ multi - edges_. we deal only with graphs with neither a self - loop nor a multi - edge .this restriction does not lose the generality of the problem formulation .a path is a graph of vertices and edges composing a sequence + satisfying and .the and are called the _end vertices _ of the path .if the end vertices of are and , the path is called an - path .when holds , a path is called a cycle . herewe represent paths and cycles by vertex sequences , such as .an edge connecting two vertices of a path / cycle and not included in is called a _ chord _ of .a path / cycle such that the graph includes no chord of is called _chordless_. figure [ chordless ] shows examples . in a set system composed of the vertex sets of cycles ( resp ., - paths ) , the vertex set of a chordless cycle ( resp ., - path ) is a minimal element .  for a graph and a vertex subset of , denotes the graph obtained from by removing all vertices of and all edges incident to some vertices in . for a vertex , denotes the _ neighbor _ of , that is , the set of vertices adjacent to . for a vertex set and a vertex , and denote and , respectively . for a path and its end vertex , denotes the path obtained by removing from .[ st - exist ] there is a chordless - path if and only if there is an - path .a chordless - path is an - path , thus only if part is true .if an - path exists , a shortest path from to is a chordless - path , and thus it always exists .[ exist ] a vertex is included in a cycle if and only if is included in a chordless cycle .if is not included in any cycle , it obviously is not included in any chordless cycle .hence , we investigate the case in which is included in a cycle . if is chordless , we are done .if has a chord , the addition of the chord splits into two smaller cycles , and is always included in one of them .we then consider the cycle as .the cycle with three vertices can not have a chord , thus we always meet a chordless cycle including . for a recursive algorithm ,an iteration means the computation from the beginning of a recursive call to its end , excluding any computation done in recursive calls generated in the iteration .if an iteration recursively calls an iteration , is called a _ child _ of , and is called the _ parent _ of .our enumeration problem is formulated as follows .+ * chordless - path enumeration problem * + for a given graph and two vertices and , enumerate all chordless - paths included in .+ we first observe that chordless cycle enumeration is done with chordless - path enumeration by repeating steps ; ( 1 ) for a vertex , enumerate chordless - paths in for each vertex adjacent to , and ( 2 ) remove from the graph . here is the graph obtained from by removing the edge .this implies that we only have to consider chordless - path enumeration .[ st ] for a vertex , is a chordless - path including if and only if is a chordless - path of the graph .if is a chordless - path in , is an - path all whose chords are incident to .since has no vertex in , no vertex of other than is adjacent to .thus , has no chord incident to , and is chordless .if is a chordless - path including , no vertex is included in , since the edge would be a chord if was included .thus , is a chordless - path in .the set of chordless - paths of is partitioned into disjoint sets of chordless - paths in the graphs for each .suppose that is a chordless - path in .then , from lemma [ st ] , includes exactly one vertex among .if includes , is a chordless - path in , thus is a chordless - path in .since is not an - path in for any , the statement holds .  from the lemma, we obtain the following algorithm .the is the sequence of vertices attached to the paths in the ancestor iterations , and set to be empty at the start of the algorithm .* enum_chordless_path * ( ) + 1 . *if * edge exists in * then * output ; * return * + 2 .* for each * s.t . a - path exists in *do * + 3 .call * enum_chordless_path * ( ) + 4 .* end for *  - path enumeration in the graph on left ; bold lines correspond to recursive calls in step 2 , and dotted lines correspond to those in step 6 . ]when a recursive call is generated in an iteration of the algorithm , is generated from by removing vertices and edges .the removed vertices and edges are kept in memory so that can be reconstructed from the graph .a removed edge or vertex is not removed again in the descendants of the iteration .thus , the accumulated memory usage for these removed vertices and edges is , and the space complexity of the algorithm is .  in step 2 , all vertices such that a - path exists in be listed .if and only if the condition in step 2 holds , there is a vertex such that a - path exists in . thus , those vertices can be listed by computing the connected component including in and checking the condition in step 2 for all for all .this can be done in time .the construction of is done in time by constructing it from .therefore , the time complexity of an iteration is .let us consider the recursion tree of the algorithm which is a tree representing the recursive structure of the algorithm .the vertex of the recursion tree corresponds to an iteration , and each iteration and its parent are connected by an edge .the leaves correspond to the iterations generating no recursive calls , and the algorithm outputs a solution on each leaf . because ofthe condition given placed on vertices in step 2 , there is always at least one - path in the given graph .this implies that at least one recursive call occurs when step 2 is executed .hence , the algorithm outputs a solution at every leaf of the recursion tree .the depth of the recursion tree is since at least one vertex is removed from the graph to generate a recursive call .we can conclude from these observations that the time complexity of the algorithm is where is the number of chordless - paths in .next , we discuss the reduction of the time complexity to .we first rewrite the above algorithm as follows .we denote the vertex next to in path by .note that although we introduce several variables , the algorithms are equivalent .* enum_chordless_path2 * ( ) + 1 .* if * is adjacent to * then output * ; * return * + 2 . a chordless - path in + 3 .* call * * enum_chordless_path2 * ( ) + 4 . * for each * * do * + 5 . *if * there is a - path in * then * + 6 .* call * * enum_chordless_path2 * ( ) + 7 . *end for *  we further rewrite the algorithm as follows .we compute the chordless - path computed in step 2 of the above algorithm , before the start of the iteration , i.e. , in its parent , and give it as a parameter to the recursive call .* enum_chordless_path3 * ( ) + 1 . *if * is adjacent to * then output * ; * return * + 2 .* call * * enum_chordless_path3 * ( ) + 3 . * for each * * do * + 4 . *if * there is an - path in * then * + 5 . a chordless - path in ( found by a breadth first search ) + 6 .* call * * enum_chordless_path3 * ( ) + 7 .* end if * + 8 .* end for *  figure [ p2 ] illustrates an example of the recursive structure of this algorithm .the tail of an arrow is a parent and the head is its child .we call the child generated in step 2 _ first child _ , and the arrow pointing at the first child is drawn with a bold line .we can make a path by following the bold - arrows , and we call a maximal such path a_ straight path_. since the bottom of a straight path is a leaf , the number of straight paths is bounded by the number of chordless paths . since the head of a non - bold arrow always points an end of a straight path , the number of non - bold arrows , that correspond to the recursive calls done in step 6 , is bounded by the number of straight paths .from these observations , we infer the following points regarding time complexity .  *an iteration takes time when a chordless path is output .this computation time is per chordless path .* steps 1 and 2 take time where is the number of edges adjacent to vertices in . this time is spent checking the adjacency of and and constructing for all .this comes from that can be constructed from by adding edges adjacent to in time . *the number of executions of the for loop in step 3 is bounded by .their sum over all iterations in a straight path does not exceed the number of edges .* steps 5 and 6 take time to find a chordless - path , and to construct . since the recursive call in step 6 corresponds to a straight path , this computation time is per chordless path . *the execution time for step 4 is .we see from the above that the bottle neck in terms of time complexity is step 4 .the other parts of the algorithm takes time per chordless - path .we speed up step 4 by using the following property . includes a - path for if and only if there is a vertex such that includes a - path .  in each iteration we put mark on vertices such that there is a - path in .step 4 is then done in time by looking at the marks on the vertices in .the marks can be put in short time , by updating the marks put in the first child .the condition of step 4 is checked by finding all vertices in from which going to is possible .this also takes time , but the time is reduced by re - using the results of the computation done for the first child . in the first child , marks are put according to the reachability to in .to put marks for , we find all vertices such that any - path in includes a vertex of .this is done by using a graph search starting from the vertices of that are adjacent to a marked vertex , and visiting only unmarked vertices .the time taken is linear in the number of edges adjacent to newly marked vertices .consider the computation time with respect to step 4 , for the iterations in a straight path .in these operations , a vertex ( resp . , an edge ) gets a mark at most once , i.e. , it never gets a mark twice .thus , the total computation time for this computation is linear in the sum of the degrees of marked vertices and vertices in , and is bounded by .the computation time for step 4 is thus reduced to per chordless - path .when a recursive call for a non - first child is made , all marks are deleted .we then perform a graph search starting from to put the marks .both steps take time . since this computation is done only when generating non - first child , the total number of occurrences of this computation is bounded by the number of maximal paths , i.e. , the number of chordless paths .thus , this computation takes time for each chordless path .the algorithm is written as follows .* enum_chordless_path4 * ( ) + 1 . *if * is adjacent to * then output * ; * go to * 11 + 2 .* call * * enum_chordless_path4 * ( ) + 3 . put mark by graph search on from vertices in + 4 . * for each * * do * + 4 . *if * a vertex adjacent to is marked * then * + 5 .delete marks from all vertices in + 6 . a chordless - path in + 7 .* call * * enum_chordless_path4 * ( ) + 8 .recover the marks deleted in step 5 , by graph search starting from on + 9 .* end if * + 10 . *end for *  the chordless - paths in a given graph can be enumerated in time per chordless path , in particular , polynomial time delay .we can see the correctness in the above .the time complexity of an iteration is , and each iteration outputs an --path .moreover , the height of the recursion tree is at most , thus the time between two consecutive output paths is bounded by .this concludes the theorem .the chordless cycles in a given graph can be enumerated in time per chordless cycle , in particular , polynomial time delay ..computational time ( in seconds ) for randomly generated graphs [ cols=">,>,>,>,>,>,>,>,>,>,>",options="header " , ]   chordless cycle enumeration has already been implemented as a part of a database system of chemoinformatics , composed of structural data of chemical compounds . in this system ,the number of chordless cycles in the chemical graph of a chemical compound is considered to be an attribute of the compound . in response to a query about the chemical structure of a compound , the system searches in the database for structures partially similar to the structure of query compound , andpredict some functions of the query compound .a chemical graph is usually sparse and is globally a tree or a combination of several large cycles .small components can be attached to the large cycles .thus , the number of chordless cycles is not so huge and is tractable in most cases .the program code was implemented in the cast / cnmr system for predicting the - nmr chemical shift .the codes and a more precise description of this system are available at http://research.nii.ac.jp/hsatoh/subjects/nmr-e.html .the information obtained about chordless cycles is used to improve the accuracy for the predicted values when the ring attributes affects the nmr spectrum . the cast / cnmr system predicts chemical shifts by using a chemical structure - spectrum database , containing mainly natural organic products and their related synthetic compounds .since most of the compounds include chains of fused rings , enumerating all rings for these compounds would greatly increase the output size , with lots of data useless for nmr prediction .therefore , the chordless cycle was adopted as a relevant ring attribute for the cast / cnmr system . for accurate nmr prediction for carbon atoms , an error within 1.0 ppm ( parts per million )is generally required .use of chordless cycle information reduced error values of -4.1 to 1.6 ppm for some problematic carbon atoms to less than 1.0 ppm .we proposed an algorithm for enumerating all chordless - paths , that is applicable to chordless cycle enumeration without increasing the time complexity . by reusing the results of the subroutines , the computation time is reduced to for each chordless path .the results of the computational experiments showed that the algorithm works well for both random graphs and real - world graphs ; the computation time was in dense graphs , and almost constant for sparse graphs .the results also showed that the number of chordless cycles is small compared to the number of usual cycles .this algorithm thus paves the way to efficient use of cycle enumeration in data mining .99 t. asai , h. arimura , t. uno and s. nakano , discovering frequent substructures in large unordered trees , lnai * 2843 * ( proc .ds 2003 ) , pp .4761 ( 2003 ) .a. t. balaban , p. filip , t. s. balaban , computer program for finding all possible cycles in graphs , j. comput . chem . * 6 * , pp 316 - 329 ( 1985 ) .g. m. downs , v. j. gillet , j. d. holiday , m. f. lynch , review of ring perception algorithms for chemical graphs , j. chem .* 29 * , pp .172187 ( 1989 ). g. m. downs , ring perception , the encyclopedia of computational chemistry * 4 * , john wiley & sons , chichester , uk , 1998 .d. eppstein , finding the k smallest spanning trees , lncs * 447 * , pp .3847 ( 1990 ) .r. ferreira , r. grossi , a. marino , n. pisanti , r. rizzi and g. sacomoto , optimal listing of cycles and st - paths in undirected graphs , soda 2013 , t. hanser , p. jauffret , g. gaufmann , a new algorithm for exhaustive ring perception in a molecular graph , j. chem .sci . * 36 * , pp .11461152 ( 1996 ) . s. kapoor and h. ramesh , an algorithm for enumerating all spanning trees of a directed graph , algorithmica * 27 * , pp .120130 ( 2000 ) .a. inokuchi , t. washio and h. motoda , complete mining of frequent patterns from graphs , machine learning * 50 * , pp .321354 ( 2003 ) . k. makino and t. uno , new algorithms for enumerating all maximal cliques , lncs * 3111 * ( proc .swat 2004 ) , pp .260272 ( 2004 ) .s. parthasarathy , m. j. zaki , m. ogihara and s. dwarkadas , incremental and interactive sequence mining , cikm 1999 , pp . 251258 ( 1999 ) .r. c. read and r. e. tarjan , bounds on backtrack algorithms for listing cycles , paths , and spanning trees , networks * 5 * , pp .237252 ( 1975 ) .e. tomita , a. tanaka and h. takahashi , the worst - case time complexity for generating all maximal cliques and computational experiments , theo .* 363 * , pp .2842 ( 2006 ) .h. satoh , h. koshino , t. uno , s. koichi , s. iwata and t. nakata , effective consideration of ring structures in cast / cnmr for highly accurate nmr chemical shift prediction , tetrahedron * 61 * , pp .7431 - 7437 ( 2005 ) .h. satoh , h. koshino , j. uzawa , t. nakata , cast / cnmr : highly accurate nmr chemical shift prediction system considering stereochemistry , tetrahedron * 59 * , pp 4539 - 4547 ( 2003 ) .uci machine learning repository , http://archive.ics.uci.edu/ml/ t. uno , a fast algorithm for enumerating bipartite perfect matchings , lncs * 2223 * ( proc .isaac 2001 ) , pp .367379 ( 2001 ) .t. uno , an output linear time algorithm for enumerating chordless cycles , ipsj , sig - al 92 ( 2003 ) ( in japanese , technical report , non - refereed ) .m. wild , generating all cycles , chordless cycles , and hamiltonian cycles with the principle of exclusion , j. discrete alg .* 6 * , pp .93102 ( 2008 ) .

Abstract: a chordless cycle ( induced cycle ) of a graph is a cycle without any chord , meaning that there is no edge outside the cycle connecting two vertices of the cycle . a chordless path is defined similarly . in this paper , we consider the problems of enumerating chordless cycles / paths of a given graph and propose algorithms taking time for each chordless cycle / path . in the existing studies , the problems had not been deeply studied in the theoretical computer science area , and no output polynomial time algorithm has been proposed . our experiments showed that the computation time of our algorithms is constant per chordless cycle / path for non - dense random graphs and real - world graphs . they also show that the number of chordless cycles is much smaller than the number of cycles . we applied the algorithm to prediction of nmr ( nuclear magnetic resonance ) spectra , and increased the accuracy of the prediction .


energy harvesting has recently received a great deal of attention in wireless communication since it provides virtually perpetual energy supplies to wireless networks through scavenging energy from the environment . in particular , harvesting energy from the far - field radio - frequency ( rf ) signal transmissions is a promising solution , which opens a new avenue for the unified study of wireless energy transfer ( wet ) and wireless information transmission ( wit ) as radio signals are able to carry energy and information at the same time .there are two main paradigms of research along this direction .one line of work aims to characterize the fundamental trade - offs in simultaneous wet and wit with the same transmitted signal in the so - called simultaneous wireless information and power transfer ( swipt ) systems ( see e.g. , - and the references therein ) .another line of research focuses on designing a new type of wireless network termed wireless powered communication network ( wpcn ) in which wireless terminals communicate using the energy harvested from wet ( see e.g. , - ) .  in our previous work , we studied a typical wpcn model , in which one hybrid access - point ( h - ap ) coordinates wet / wit to / from a set of distributed users in the downlink ( dl ) and uplink ( ul ) transmissions , respectively .it has been shown in that the wpcn suffers from a so - called `` doubly near - far '' problem , which occurs when a far user from the h - ap receives less wireless energy than a near user in the dl , but needs to transmit with more power in the ul for achieving the same communication performance due to the doubled distance - dependent signal attenuation over both the dl and ul . as a result ,unfair rate allocations among the near and far users are inured when their sum - throughput is maximized . in , we proposed to assign shorter / longer time to the near / far users in their ul wit to solve the doubly near - far problem , which is shown to achieve more fair rate allocations among the users in a wpcn .  on the other hand ,user cooperation is an effective way to improve the capacity , coverage , and/or diversity performance in conventional wireless communication systems . assuming constant energy supplies at user terminals, cooperative communication has been thoroughly investigated in the literature under various protocols such as decode - and - forward and amplify - and - forward ( see e.g. , , and the references therein ) .recently , cooperative communication has been studied in energy harvesting wireless communication and swipt systems ( see e.g. - ) . however , how to exploit user cooperation in the wpcn to overcome the doubly near - far problem and further improve the network throughput and user fairness still remains unknown , which motivates this work .    in this paper , we study user cooperation in the wpcn for throughput optimization . for the purpose of exposition , we consider a two - user wpcn , as shown in fig .[ fig_systemmodel ] , where one h - ap broadcasts wireless energy to two distributed users with different distances in the dl , and the two users transmit their independent information using individually harvested energy to the h - ap in the ul through time - division - multiple - access ( tdma ) . to enable user cooperation, we propose that the near user which has a better channel than the far user for both dl wet and ul wit uses part of its allocated ul time and dl harvested energy to first help to relay the information of the far user to the h - ap and then uses the remaining time and energy to transmit its own information . under this protocol, we characterize the maximum weighted sum - rate ( wsr ) of the two users by jointly optimizing the time and power allocations in the network for both wet in the dl and wit in the ul , subject to a given total time constraint .the achievable throughput gain in the wpcn by the proposed user cooperation scheme is shown both analytically and through simulations over the baseline scheme in without user cooperation .the rest of this paper is organized as follows .section [ systemmodel ] presents the system model of the wpcn with user cooperation .section [ resourcealloc ] presents the time and power allocation problem to maximize the wsr in the wpcn , and compares the solutions and achievable throughput regions with versus without user cooperation .section [ simulationresult ] presents more simulation results under practical fading channel setups .finally , section [ conclusion ] concludes the paper .as shown in fig . [ fig_systemmodel ] , this paper considers a two - user wpcn with wet in the dl and wit in the ul .the network consists of one hybrid access point ( h - ap ) and two users ( e.g. , sensors ) denoted by and , respectively , operating over the same frequency band .the h - ap and the users are assumed to be each equipped with one antenna .furthermore , it is assumed that the h - ap has a constant energy supply ( e.g. , battery ) , whereas and need to replenish energy from the received signals broadcast by the h - ap in the dl , which is then stored and used to maintain their operations ( e.g. , sensing and data processing ) and also communicate with the h - ap in the ul .we assume without loss of generality that is nearer to the h - ap than , and hence denote the distance between the h - ap and , that between the h - ap and , and that between the and as , , and , respectively , with .we also assume that so that can more conveniently decode the information sent by than the h - ap , to motivate the proposed user cooperation to be introduced next . assuming that the channel reciprocity holds between the dl and ul , then the dl channel from the h - ap to user and the corresponding reversed ul channel are both denoted by a complex random variable with channel power gain , , which in general should take into account the distance - dependent signal attenuation and long - term shadowing as well as the short - term fading .in addition , the channel from to is denoted by a complex random variable with channel power gain .if only the distance - dependent signal attenuation is considered , we should have and due to the assumptions of and . furthermore, we consider block - based transmissions over quasi - static flat - fading channels , where , , and are assumed to remain constant during each block transmission time , denoted by , but can vary from one block to another . in each block, it is further assumed that the h - ap has the perfect knowledge of , , and , and knows perfectly .we extend the harvest - then - transmit protocol proposed in for the two - user wpcn to enable user cooperation , as shown in fig .[ fig_framestructure ] . in each block , during the first amount of time , , the h - ap broadcasts wireless energy to both and in the dl with fixed transmit power .the far user then transmits its information with average power during the subsequent amount of time in the ul , , using its harvested energy , and both the h - ap and decode the received signal from . to overcome the doubly near - far problem , during the remaining amount of time in each block, the near user first relays the far user s information and then transmits its own information to the h - ap using its harvested energy with average power over amount of time and with average power over amount of time , respectively , where .note that we have a total time constraint given by for convenience , we noramlize in the sequel without loss of generality .  during the dl phase , the transmitted complex baseband signal of the h - ap in one block of interest is denoted by an arbitrary random signal , , satisfying = p_0 ] , and the transmit power values of and for ul wit as  ] with and denoting the given non - negative rate weights for and , respectively .the wsr maximization problem is then formulated from ( [ eq_achievablerate_u1])-([eq_achievablerate_u2 ] ) as   notice that if we set and , then ( p1 ) reduces to the special case of wpcn without user cooperation studied in , i.e. , the near user only transmits its own information to the h - ap , but does not help the far user for relaying its information to the h - ap .note that ( p1 ) can be shown to be non - convex in the above form . to make this problem convex, we change the variables as and .since as given in ( [ eq_txpower_u1 ] ) , , , , and in ( [ eq_r1_10])-([eq_achievablerate_u2 ] ) can be re - expressed as functions of  ] .it then follows that and .accordingly , ( p1 ) can be equivalently reformulated as    where the time constraint in ( [ eq_sumenergyconstraint ] ) can be shown to be equivalent to the power constraint originally given in ( [ eq_txpower_u2 ] ) .it is worth noting that and denote the amount of time in the dl slot duration in which the harvested energy by is later allocated to relay s information and transmit its own information in the ul , respectively . by introducing the new variables and in and , joint time and power allocation in problem ( p1 ) is converted to time allocation only in problem ( p2 ) .note that , , , and are all monotonically increasing functions over each element of , , , and , respectively .let the optimal solution of ( p2 ) be denoted by  ] .then , it can be easily verified that must hold ( otherwise , we can always increase by increasing to improve the weighted sum - rate ) .similarly , it can also be verified that since , otherwise , we can allocate part of ( or ) to ( or ) until the equality holds , which will result in an increased without reducing .[ lemma_concavity ] , , , and are all concave functions of .please refer to appendix [ app_proof_lemma_concavity ] .  from lemma [ lemma_concavity ], it follows that the objective function of ( p2 ) is a concave function of , and so are the functions at the right - hand side of both ( [ eq_u1_rate_constraint ] ) and ( [ eq_u2_rate_constraint ] ) .furthermore , the constraints in ( [ eq_sumtimeconstraint_new ] ) and ( [ eq_sumenergyconstraint ] ) are both affine .therefore , problem ( p2 ) is a convex optimization problem , and furthermore it can be verified that ( p2 ) satisfies the slater s condition ; hence , it can be solved by the lagrange duality method , shown as follows . from ( [ eq_problem_epigraph])-([eq_u2_rate_constraint ] ) ,the lagrangian of ( p2 ) is given by  where  ] , is given by ,}\ ] ] = \left [ \left ( \frac{\lambda_3^*\tau_{21}^*}{\lambda_2^ * \ln 2 } - \frac{\tau_{21}^*}{\rho_2 } \right)^{+ } , \left ( \frac{\omega_2^*\tau_{22}^*}{\lambda_2^ * \ln 2 } - \frac{\tau_{22}^*}{\rho_2 } \right)^{+ }  \right],}\ ] ] with , and , , , and denoting the optimal dual solutions .moreover , , , and in ( [ eq_prop_opt_tau ] ) are given , respectively , by  finally , , , and in ( [ eq_prop_opt_tau ] ) are solutions of , , and , respectively , where   please refer to appendix [ app_proof_prop_opt_solution ] .\2 ) * repeat *  * initialize , , . ** repeat * * * obtain  ] . * * obtain  ] . ** until * converges to a predetermined accuracy .* compute  ] and  ] . with for each given , the optimal minimizing can then be found by updating using sub - gradient based algorithms , e.g. , the ellipsoid method , with the sub - gradient of , denoted as  ] , all of which have the equivalent form as ( [ eq_app_lemma_concavity ] ) .therefore , , , , and are all concave functions of .this completes the proof .denote the hessian of defined in ( [ eq_app_lemma_concavity ] ) as ,\,\,\,i , j \in\left\ { { 1,2 } \right\},}\ ] ] where is given by   given an arbitrary real vector ^t$ ] , it can be shown from ( [ eq_app_fungtion_g ] ) and ( [ eq_app_hessian_function_g ] ) that i.e. , is a negative semi - definite matrix . therefore , is a jointly concave function of both and  completes lemma [ lemma_concavity_general ] .since ( p2 ) is a convex optimization problem for which the strong duality holds , the karush - kuhn - tucker ( kkt ) conditions are both necessary and sufficient for the global optimality of ( p2 ) , which is shown below .        since must hold for ( p2 ) , we assume without loss of generality that ( only when in ( [ eq_app_prop_kkt_deri_e21 ] ) and ( [ eq_app_prop_kkt_deri_e22 ] ) , i.e. , no harvested energy at is used for ul wit ) .furthermore , it can be easily verified that must hold for ( p2 ) and thus we can also assume that with no loss of generality ( only when from ( [ eq_app_prop_kkt_deri_tau21 ] ) and ( [ eq_app_prop_kkt_deri_tau22 ] ) i.e. , no energy is transferred by the h - ap ) .  changing variable as and after mathematically manipulations , ( [ eq_app_prop_kkt_deri_tau0 ] )can be modified as , where , , and are given in ( [ eq_prop_a])-([eq_prop_c ] ) . since , we thus have in ( [ eq_prop_opt_tau ] ) from quadratic formula . furthermore , with and from ( [ eq_app_prop_kkt_deri_tau1 ] ) , we also have , where is given in ( [ eq_prop_function_z ] ) .it is worth noting that given in ( [ eq_prop_function_z ] ) is a monotonically increasing function of where , and so is .therefore , there exists a unique satisfying for given , from which we have given in ( [ eq_prop_opt_tau ] ) .similarly , by changing variables as and in ( [ eq_app_prop_kkt_deri_tau21 ] ) and ( [ eq_app_prop_kkt_deri_tau22 ] ) , respectively , we can obtain unique and which are solutions of and , from which and can be obtained .finally , we have and in ( [ eq_prop_opt_e ] ) from ( [ eq_app_prop_kkt_deri_e21 ] ) and ( [ eq_app_prop_kkt_deri_e22 ] ) , respectively .this completes the proof of proposition [ proposition_opt_solution ] .a. a. nasir , x. zhou , s. durrani , and r. a. kennedy , `` relaying protocols for wireless energy harvesting and information processing , '' _ ieee trans .wireless commun .12 , no . 7 , pp. 3622 - 3636 , july 2013 .

Abstract: this paper studies user cooperation in the emerging wireless powered communication network ( wpcn ) for throughput optimization . for the purpose of exposition , we consider a two - user wpcn , in which one hybrid access point ( h - ap ) broadcasts wireless energy to two distributed users in the downlink ( dl ) and the users transmit their independent information using their individually harvested energy to the h - ap in the uplink ( ul ) through time - division - multiple - access ( tdma ) . we propose user cooperation in the wpcn where the user which is nearer to the h - ap and has a better channel for dl energy harvesting as well as ul information transmission uses part of its allocated ul time and dl harvested energy to help to relay the far user s information to the h - ap , in order to achieve more balanced throughput . we maximize the weighted sum - rate ( wsr ) of the two users by jointly optimizing the time and power allocations in the network for both wireless energy transfer in the dl and wireless information transmission and relaying in the ul . simulation results show that the proposed user cooperation scheme can effectively improve the achievable throughput in the wpcn with desired user fairness .   [ section ] [ section ] [ section ] [ section ] [ section ] [ section ] [ section ] [ section ]


to use the limited communication network resources efficiently , the event - triggered control ( etc ) method has been introduced in the field of networked control system ( ncs ) , where a system samples its output and sends the sampled value to the controller based on the occurrences of a well - defined event .therefore , this control method can adjust the sampling rate adaptively according to what is happening within the controlled system and may reduce unnecessary communications .because of this advantage , fruitful results on event - triggered controlled systems have been reported in the literature ( see and reference therein ) .  in the framework of input - to - state stability ,the stabilization problem of nonlinear ncss was investigated by utilizing centralized and decentralized etc methods in and , respectively .the output decentralized etc for linear ncs was studied via impulsive system theory in . to enlarge inter - execution intervals ,the model - based etc was proposed also for a linear ncs in .in addition , decentralized / distributed etc for interconnected systems was developed in and by using the small gain theory for large - scale systems .  on the other hand , synchronization of dynamical networks , or interconnected systems , and the related problem of consensus of multi - agent systems ,have attracted a great deal of attention due to their extensive applications in various fields ( see recent papers and a monograph for details ) .motivated by the fact that connected nodes in some real - world networks share information over a digital platform , the problem of synchronization of dynamical networks and its related topic consensus of multi - agent systems have also been investigated under the circumstance that nodes or agents communicate to their neighbours only at certain discrete - time instants that are based on occurrences of a well - defined event ( to name a few ) .under such a circumstance , each node can only get limited information ( e.g. , its own state , the sampled states of its neighbours and only at certain discrete - time instants ) .then , the main issue is how to use these limited information to design an event - triggering rule ( etr ) for each node under which the entire network achieves synchronization asymptotically .moreover , a system with an event - triggered controller may have zeno behaviours , which is due to the continuous / discrete - time hybrid nature of the event - triggered controller .it is known that the zeno behaviour , in which consecutive execution times approach arbitrarily closely resulting in an accumulation point , is undesirable in practice ( ) .therefore , how to prevent the occurrence of such a behaviour is another key issue in the study of synchronization for dynamical networks with etrs .these issues stimulate the study of synchronization for networks with event - based communication . in , a distributed etc mechanism was developed to investigate asymptotic consensus of a multi - agent system , which was further extended to studying gain stability of the system with additive disturbances in .to prevent the occurrence of zeno behaviour , a decentralized etr with a time - varying threshold that is exponentially decreasing in time was introduced to achieve asymptotic consensus in .self - triggered strategies were proposed in and shown to be robust to skews of the local clocks , delays , and limited precision in the communication .all abovementioned works only focused on dynamical networks or multi - agent systems with simple node dynamics ( single - integrators or double - integrators ) , which do not appear to extend in a straightforward way to networks with more general node dynamics .few results studied networks with nodes that are generalized linear dynamical systems , and most of these results only studied bounded synchronization in order to exclude zeno behaviour theoretically ( ) . for example , in , estimators ( is the number of node s neighbours ) were introduced into each node , and were used to design a decentralized etr with a fixed threshold to achieve bounded synchronization .recently , a new distributed event - triggered control method was introduced in to study asymptotic synchronization of networks with generalized linear node dynamics . however , this new etr is inadequate to exclude zeno behaviours . in , the authors adopted the same estimators as in and modified the distributed etr proposed in by introducing the exponential term used in .the design distributed event - triggered controller synchronizes the network asymptotically with no zeno behaviours .this result was also extended to output synchronization of heterogeneous networks in .however , for networks with large node degrees , this control method needs a large number of estimators .  to overcome the abovementioned issue, we introduce a new sampling mechanism with which only two estimators for each node are needed to synchronize the network .one of the estimators is used to estimate its own state , and the other is used to estimate the average state of the node s neighbours .then , a distributed etr for each node is designed based on the two estimators .moreover , inspired by the method proposed in , we replace the exponential term used in by a dwell - time ( a concept that has been extensively used in switched systems , see for example ) , which will simplify the implementation of the designed etr in practice .we will show that the network with the designed distributed etr achieves synchronization asymptotically and excludes zeno behaviour successfully .we also extend the obtained results to networks with logarithmic quantizers .our contribution is to propose a control law that for the first time has the the four essential and desirable properties as follows .i ) . by introducing estimators into each nodes and the dwell - time into the designed distributed etr , the proposed distributed event - triggered controller can guarantee asymptotic synchronization of dynamical networks with generalized linear dynamics , and moreover , we can guarantee no zeno behaviour occurs theoretically ; whereas most of the existing results sacrifice synchronization performance and only study bounded synchronization in order to exclude zeno behaviour .ii ) . by introducing a new sampling mechanism ,we reduce the number of estimators needed for each node to two , whereas existing results need estimators .thus , the proposed control method can be implemented easily , in particular for networks with large node degrees .most importantly , it can be implemented in a decentralized way .iii ) . the proposed method can deal with networks which synchronize to a time - varying solution rather than to an equilibrium point .this also thanks to the two estimators introduced into each node .we also consider the quantization effects . because of the new sampling mechanism and the logarithmic quantizer, we can still guarantee asymptotic synchronization of the network under the proposed distributed etr .the rest of the paper is organized as follows .section [ sec_model ] introduces the network model and a new sampling mechanism under which two estimators are designed for each node in the network .section [ sec_sync ] studies asymptotic synchronization of the network by designing event - triggered controllers where both centralized and distributed etr are discussed .section [ sec_quan ] considers the quantization effects and extends the results proposed in section [ sec_sync ] to networks with logarithmic quantizers .section [ sec_exam ] gives two numerical examples to illustrate the obtained results .conclusions are drawn in section [ sec_con ] ._ notation _ : we denote by , , and the set of real numbers , non - negative real numbers , integers and non - negative integers , respectively ; by and the set of -dimensional real vector and real matrix . , and are the -dimensional identity matrix , -dimensional vector with all entries being and matrix also with all entries being , respectively .we use to represent the euclidean norm of a vector and also the induced norm of a matrix . " is the transpose of a vector or a matrix ,  " is the inverse of a nonsingular matrix , and `` '' is the kronecker product of two matrices . ( ) is the time right after ( right before ) . for a single , and .let be an undirected graph consisting of a node set and a link set .if there is a link between nodes and , then we say the two nodes and are adjacent , or that node is a neighbour of node and vice versa .let be the adjacency matrix of with , where and if node and node are adjacent , otherwise .if , then we say graph is unweighted , otherwise the graph is weighted .the laplacian matrix is defined by   consider a network of linear dynamical systems whose state equation is given as follows where is the state of node .the constant matrix describes the node dynamics . is the input matrix , and is the control input to be designed .generally , continuous communications between adjacent nodes are assumed in the literature , i.e. , .this yields the following network with continuous interconnections in this paper , we assume that connections in the network are realized via discrete communications , i.e. , each node in the network only gets access to information from its neighbours at certain discrete - time instants .we will present an event - triggered version of the network , and study how to design an etr for each node such that the network achieves synchronization asymptotically .we suppose that the topological structure of the network is fixed , undirected and connected .the graph topology is represented by the adjacency matrix or the laplacian matrix . for simplicity ,we only consider unweighted networks , i.e. , ; but the obtained results can be extended to weighted networks directly .  in order to achieve synchronization of such a network ,we introduce two estimators and into each node , and adopt the following control input where is the control gain to be designed , and and are states of the two estimators and , respectively , with state equations given by for each node , the estimator is used to estimate its own state , and the estimator is used to estimate the average state of all its neighbours .the increasing time sequences and , represent time instants that node sends information to its neighbours and time instants that it receives updated information from one or more of its neighbours , respectively .the mapping is the index set of node s neighbours from which node receives updated information at , where is the index set of the neighbours for node .the error vector represents the state deviation between the estimator and node .we assumes that  a1 .: :  there is no time delay for computation and execution , i.e. , the time   represents both the sampling  time instant and the time when node   broadcasts the updated information to its neighbours ; a2 . : :  the communication network is under an ideal circumstance , i.e. , there  are no time delays or data dropouts in communication .  with these assumptions , becomes   the time sequence is decided by an event - triggering function to be designed , i.e. , for , if the inequality in is satisfied at , then node samplesits state and the state of the estimator  , calculates , sends to its neighbours , and then , reinitialize the estimator at by using ( ) . on the other hand, node will reinitialize the estimator by at each time instant when it receives updated information from its neighbours .we assume the network is well initialized at , i.e. , and each node samples its state and sends to its neighbours .therefore , we have , and for all .figure [ two_estimators ] .gives an illustration of node and the two estimators .   and its estimators .]  the problem to be considered is with the given network topology , to determine the time sequence , when node should communicate to its neighbours ( i.e. , samples the states and sends updated information to its neighbours ) by designing an etr , such that network can achieve synchronization asymptotically .moreover , the zeno behaviour which is undesirable in practice may occur in the network due to the hybrid nature of the network with event - triggered controller .thus , it is important to exclude the occurrence of such a behaviour by designing a proper event - triggering function for each node , which is another main purpose of the paper .  even though the network topology is symmetric , the information flow in the corresponding communication network is usually _ asymmetric _ because each node has its own event - triggering function and decides by itself when it should communicate to its neighbours , i.e. , , for .it should be pointed out that in most of existing works , each node in the network samples its state and sends the sampled state to its neighbours .but in our paper , each node samples both its own state and the state of its estimator , calculates the error and then sends this sampled error to its neighbours .thanks to this new sampling mechanism , we only need two estimators for the design and control purpose .  to simplify the analysis , we will show that the network with , and is equivalent to the following system  [ nm_ob ]   defining gives which has the same dynamics as defined in .moreover , at , we have thus , we have for all .then , controller becomes the last equality follows from the definition of the laplacian matrix in . substituting into gives that the network with , and is equivalent to .moreover , let , we have . then , the etr can be reformulated as   in this paper , we will use the model and the etr for analysis purpose .but the obtained results can be implemented by using the controller with the two estimators , and the etr .  in , a similar controller with estimatorsis used to achieve synchronization , where is the number of neighbours ( or the degree ) of node . with assumptionsa1 and a2 , it can be shown that the two controllers are equivalent theoretically .however , the controller used in this paper only needs two estimators and .therefore , the advantage of controller , in particular for a large - scale network which has large node degrees , is obvious in practice .to finish this section , we give the definition of asymptotic synchronisation based on the network model .[ sync ] let and be a solution to the network with the initial condition  and . then, the network is said to achieve _synchronization asymptotically _ , if and exist for all , and further if for every initial condition   of course , when the communication network is not ideal , network model with and the two estimators , can not be simplified to any more .then , the problem becomes more complicated , deserving more attention .in order to achieve the goal of the paper , denote with .then , the network model can be rewritten by  because the laplacian matrix has the zero - row sum property , at least one of its eigenvalues equals to zero .the network is undirected and connected , so is irreducible and symmetric ; and there always exists an orthogonal matrix with and such that where , and with algebraic multiplicity one .furthermore , we can choose for , and therefore we have for all , which leads to with .let , and .defining gives where we use properties and for any , which are supported by the fact and .denote , and .then , the system can be simplified to by calculation , we have where .the last equality in follows from which leads to .therefore , if , then as , i.e. , the network achieves synchronization asymptotically according to definition [ sync ] .this result is summarized in the following lemma .[ lem2 ] if the system is asymptotically stable , i.e. , , then the network achieves synchronization asymptotically .it is shown in that a necessary and sufficient condition for asymptotic synchronization of a dynamical network whose nodes interact with their neighbours continuously with respect to time is that  a3 .: :  there exist positive definite matrices   such that    hence , for the network with event - triggered controller , it is reasonable to require that a3 holds , which is also a fundamental requirement for the design of the etr .so in the rest of the paper , we use a3 as another basic assumption and always assume that matrix solutions to lyapunov equations exist .  with the help of lemma [ lem2 ] andassumption a3 , now we can discuss how to achieve synchronization for the network by designing a proper etr . under the circumstance that all information of the network is available for the design purpose , we give a centralized etr in the following theorem .[ lem3 ] the network achieves synchronization asymptotically under the sampling time sequence determined by the centralized event - triggering function , i.e. , where and .moreover , no zeno behaviour occurs in , and the inter - execution intervals , satisfies where and , , and .first , we show that under etr , , , i.e. , no zeno behaviour will occur . then , we show the asymptotic synchronization of the network under .to prove the first part , we calculate the derivative of , which gives by the definition of , we have with the same procedure used in and , we have from which we can get an upper bound on , i.e. , similarly , from we get an upper bound on  combining with and yields considering the differential equation and based on the comparison theory ( ) , we have where is the solution to ordinary differential equation with the initial condition . therefore , the inter - execution intervals can be lower bounded by the positive time constant during which evolves from to , i.e. , .solving gives in .the positiveness of follows from with and .thus , no zeno behaviour occurs in network under the etr .now , select the following lyapunov function candidate where and , are positive definite matrix solutions to . then along the trajectory of the system , we have moreover , the etr guarantees that this together with yields therefore , the equilibrium point of the system is asymptotically stable , i.e. , .applying lemma [ lem2 ] proves the result .  in practice , a centralized etr is hard to implement .for one thing , it may be costly and time consuming to gather global information for the design purpose .for another , each node usually can only get limited information from its neighbours the sampled value from its neighbours , which can only be received at certain discrete - time instants , rather than for all .therefore , a distributed etr for each node which only relies on the information that the node can get is desirable .next , we will design such a distributed etr for each node by only using information of the node itself and its estimators , i.e. , , , and , .[ them_ndw ] assume that for all there exists a minimum inter - execution interval for each , i.e. , for all , then the network achieves synchronization asymptotically under the distributed etr where ; ; ; is the largest eigenvalue of ; and are defined the same as in theorem [ lem3 ] .  with the assumption of the existence of , no zeno behaviour will occur .therefore , we only need to pay attention to the asymptotic synchronization of the network under the etr .  to do this ,we first claim that the following inequality holds for any  where . by the definition of , we have where with , and the last equality comes from .let , then for any symmetric laplacian matrix , we have , i.e. , and can be diagonalized simultaneously .furthermore , we have and , where and , are the eigenvalues of .let , be the eigenvalues of the matrix .then with , we have and , , which leads to .therefore , combining with gives the inequality .now with , we can show the asymptotic synchronization of the network .since for all , with , inequality is equivalent to moreover , we have with , which leads to select the lyapunov function with . then , the derivative of along the system satisfies therefore , the equilibrium point of system is asymptotically stable , and thus the network achieves synchronization asymptotically .  in proposition [ them_ndw ] , we assume the existence of such a for each node in order to exclude zeno behaviour . in particular , for a network with two nodes , we can show that the network has no zeno behaviours under etr ( due to space limitation , it is omitted here ) .however , it is not clear yet if this result can be extended to networks with more than 2 nodes .if the answer is negative , then a further question is under what condition such a exists for the etr .these questions deserve more attention in the future study .to overcome the abovementioned issue , we give an alternative distributed etr by introducing a dwell - time into a modified etr , which will be addressed in theorem [ them5 ] .but first we give a useful lemma which will be used to prove theorem [ them5 ] .  [lemma_centr_dwell ] suppose an etr guarantees that for all with no zeno behaviours. then the system asymptotically synchronizes . in particular ,if we use the etr the system asymptotically synchronizes , and the inter - execution interval for each node satisfies where and .suppose an etr guarantees for all , and there is no zeno behaviour happens . since for all , this together with leads to .therefore , the asymptotic synchronization follows directly .  to show the existence of a positive inter - execution interval under etr, we calculate and have where we use to get the last inequality . since for all , which leads to .therefore the above inequality becomes solving the ordinary differential equation with and gives the lower bound on the inter - execution interval .  [ them5 ]the network achieves synchronization asymptotically under the distributed etr where ;  ; is given by . moreover, no zeno behaviour occurs in .the existence of the inter - execution interval for each node under the etr is guaranteed by automatically .therefore , the only thing left is to show the asymptotic synchronization of the network with the designed etr .  according to the proof of lemma [ lemma_centr_dwell ] , for , we have .so if we can show that also holds for , then asymptotic synchronisation of the network follows from lemma [ lemma_centr_dwell ] .  under the etr , for any , we have which follows from .now we claim that to show this , let and for any . obviously , . then , based on the etr , we have if , and if .therefore , since with , the above inequality is equivalent to inequality .  substituting into yields therefore, applying lemma [ lemma_centr_dwell ] completes the proof .  [ rem_last ] according to lemma [ lemma_centr_dwell ] , the network can also achieve asymptotic synchronization if each node samples its values periodically with period given by .compared with the periodic sampling mechanism , etr can reduce the sampling times significantly .see example 2 in section [ sec_exam ] for instance .  in ,the following control input was adopted a distributed etr was used , where and .it turns out that and in etr are insufficient for the design purpose , in particular for networks whose nodes synchronize to a time - varying trajectory .suppose the network achieves asymptotic synchronization under to a time - varying trajectory .as and converge to each other , may converge to zero as well .however , will not converge to zero ( see figure [ rem ] for an example ) . according to, this makes close to zero and may lead to zeno behaviour ( see the example in for instance where inter - execution interval decreases as time increases ) .  .]  therefore , etr may not exclude zeno behaviour effectively .this is the reason that we introduce estimators into each nodes . by doing so, will approach zero , which may exclude zeno behaviours at each node . estimators were used to design a decentralized etr with a fixed threshold in , but only bounded synchronization was achieved , i.e. , the synchronization errors is less than a pre - given constant , rather than converges to zero asymptotically . in , we use the same estimators and combine the distributed etr proposed in with the time - dependence etr used in to design a new distributed etr .it shows that this new etr can guarantee asymptotic synchronization of the network and prevent the occurrence of the zeno behaviour as well .however , this control method needs estimators for each node , it will have implementation issues , in particular for networks with a large number of nodes and large node degrees . in theorem [ them5 ] , we reduce the number of the estimators to two , and further remove the exponential term used in by using a dwell - time term in the designed etr which can be implemented easily in practice .the etr in theorem [ them5 ] can be implemented by where .in the previous section , we assume that the communication network is ideal , and in this section , we consider the quantization effects , i.e. , the sampled valued is quantized before sending to the neighbours .  in order to achieve asymptotic synchronization, we only consider the logarithmic quantizer .let , be fixed , and let and .let .then , the quantization function is given as follows ( ) figure [ quantizer ] gives an illustration of the quantization function . for a vector , we also use for simplicity which is defined point - wise as .for such a function , we have the following property     a natural way to implement quantization in each node is to quantize of each node and send to its neighbours , and then the neighbours use to reinitialize the corresponding estimator .the problem of doing this is that the error between and its estimators , , will not go to zero due to the quantization effect and the fact that is time - varying and normally does not go to the origin or a constant vector for most of the synchronization problems .thus , this method can not lead to asymptotic synchronization for the network even with the logarithmic quantizer .thanks to the new sampling mechanism proposed in the previous section , we do not have the abovementioned issue . instead of quantizing , we quantize the error between and at each sampling time , send to its neighbours , and reinitialize the estimator by .by doing so , we have goes to zero as long as goes to zero .this makes asymptotic synchronization of the network with the logarithmic quantizer possible .correspondingly , node will reinitialize the estimator as soon as it receives the updated information from its neighbours by .the estimators in and in become  figure [ nodes_qetc ] gives an illustration of node with the two estimators and quantizers .   with estimators and quantizers . ]  here , we assume the network is properly initialized at , i.e. , and each nodes sends to its neighbours . therefore , we have , and .we further assume that . then with assumptions a1 and a2 ,the network with , and can be simplified to  [ nm_obquan ]   for the centralized etr with logarithmic quantizers , we have the following result .[ cent - quan ] the network achieves synchronization asymptotically under the centralized etr where , and are defined in theorem [ lem3 ] .moreover , no zeno behaviour occurs in , and the inter - execution interval satisfies where , , are given in theorem [ lem3 ] and .  according to, we have at each , with the assumption that , we have since , and as there is no jump in , we have then , the proof of the existence of a positive inter - execution interval is the same as that in theorem [ lem3 ] except that the solution to the differential equation evolves from to rather than from to .thus , by solving , we can get the lower bound on the inter - execution interval .  from the etr, we have for . with , we have selecting the lyapunov function , we have along the trajectory of system .this completes the proof .  to discuss the distributed etr with a dwell - time and logarithmic quantizers , we need the following lemma .  [ lemma_centr_quant ] the network achieves synchronization asymptotically under the distributed etr where .moreover , no zeno behaviour occurs in , and the inter - execution interval for each node satisfies whose parameters are given in lemma [ lemma_centr_dwell ] .similar to the proof of theorem [ cent - quan ] , the lower bound on can be obtained from with and .  the asymptotic synchronization of the network can be proved by combining the proof of lemma [ lemma_centr_dwell ] with the proof of theorem [ cent - quan ] .thus it is omitted here .  with lemma [ lemma_centr_quant ] , we have the following result for the distributed etr with dwell - time to synchronize the network .[ them6 ] the network achieves synchronization asymptotically under the distributed etr where ; is given in theorem [ them5 ] ; is given by .moreover , no zeno behaviour occurs in .combining the proofs of lemma [ lemma_centr_dwell ] with lemma [ lemma_centr_quant ] gives the proof .  [ rem_last1 ] similar to remark [ rem_last ] , the network with logarithmic quantizers can also achieve asymptotic synchronization if each node samples its values periodically with period given by .again , etr can reduce the sampling times significantly compared with this periodic sampling mechanism ( see example 2 in section [ sec_exam ] for instance ) .this section gives a network consisting of 10 nodes to show the effectiveness of the proposed results .here , we suppose the network has the following parameters we adopt the two - nearest - neighbour graph to describe the network topology , i.e. , , . if and , then . if and , then .  since has two eigenvalues on the imaginary axis of the complex plane , the network can not asymptotically synchronize to its equilibrium point , but to a stable time - varying solution depending on its initial condition .we will simulate the network with the centralized etr in example 1 and the distributed etr with a dwell - time ( ddt ) as well as the distributed etr with a dwell - time and quantizers ( ddtq ) in example 2 .* example 1 * ( centralized etr ) figure [ fig_centralized ] gives the simulation results under the designed centralized etr with and , which shows that the network achieves synchronization asymptotically .the minimum inter - execution interval is for from 0s to 18s , which is larger than the theoretical one obtained by .* example 2 * ( ddt ddtq ) we select the same and as in example 1 , and select and for ddtq .figure [ fig_ddt ] and figure [ fig_ddqt ] give the simulation results of the network with ddt and ddtq , respectively , which shows the effectiveness of the proposed results ( theorem [ them5 ] and theorem [ them6 ] ) . for both of the cases ,we only show the sampling time instants of each node in the first 2 seconds for clarity .table [ table_ddt_mie ] and table [ table_ddt_sam ] gives the minimum inter - execution interval and sampling times of each node for ddt and ddtq during the simulation time from 0s to 18s , respectively .  according to remark [ rem_last ] and remark [ rem_last1 ], the network can also achieve synchronization asymptotically if each node samples its states periodically with a period for the case without quantizers and a period for the case with quantirzers .for these two periodic sampling mechanisms , each node in the network will sample times for the first case and times for the second case during the same simulation time .compared with table [ table_ddt_sam ] , the proposed distributed etrs reduce sampling times for each node significantly ..the minimum inter - execution interval ( ddt v.s .ddqt ) [ cols="^,^,^,^,^,^,^,^,^,^,^ " , ]   [ table_ddt_sam ]this paper has investigated the asymptotic synchronization problem of a dynamical network by using the distributed etc method . with the help of the introduced estimators , distributed etrs for each nodehave been explored , which only rely on the state of the node and the states of the introduced estimators .it has been shown that the network achieves synchronization asymptotically with the proposed etrs , and no zeno behaviour occurs during the whole evolution of the network .quantization effect has also been discussed .it is worth pointing out that time - delay and data packet dropout are common phenomena which definitely affects the synchronization of a network with event - based communication .thus , it appears that the synchronization of such networks with imperfect communication is an important issue to pursue further for both theoretical interest and practical consideration .

Abstract: this paper studies the synchronization problem of a dynamical network with event - based communication , where each node communicates to its neighbours only when an event - triggering condition is fulfilled . firstly , two estimators are introduced into each node , one to estimate its own state , and the other to estimate the average state of its neighbours . then , with the assistance of the two estimators , a distributed event - triggering rule with a dwell - time is designed such that the network achieves synchronization asymptotically , and meanwhile no zeno behaviour occurs . the designed event - triggering rule only depends on the information that each node can obtain , and thus can be implemented in a decentralized way . the quantization effects are also considered , and the logarithmic quantizer is used to achieve asymptotic synchronization . finally , numerical examples are given to show the effectiveness of the proposed results .  distributed event - triggered control , asymptotic synchronization , dynamical networks , logarithmic quantizer .


opinion dynamics have a long history and have been studied extensively in social sciences . probably the most well - known model for opinion dynamicsis the classical degroot model .various models have been proposed for opinion dynamics to understand how an individual s opinion evolves over time , including the friedkin - johnsen model , the hegselmann - krause model , the degroot - friedkin model , and the altafini model . however , there is few work concerning the self - confidence levels of the individuals in a social network .recently , a new model , called the degroot - friedkin model , has been proposed in .the model considers the situation when a group of individuals discusses a sequence of issues , and studies the evolution of the self - confidence levels of individuals ( i.e. , how confident an individual is for her opinions on the sequence of issues ) via the reflected appraisal mechanism proposed in . lately , a modified degroot - friedkin model has been proposed in which provides a time - efficient , distributed implementation of the original degroot - friedkin model .the model has been studied in with a fixed doubly stochastic relative interaction matrix , and in with time - varying doubly stochastic relative interaction matrices , while the analysis of the modified degroot - friedkin model with a fixed stochastic ( not doubly stochastic ) relative interaction matrix still remains open .both the degroot - friedkin model and the modified degroot - friedkin model are described in discrete times .sometimes a continuous - time model would be a natural choice especially when the opinions of individuals evolve gradually over time ; see for example .recently , a continuous - time distributed self - appraisal model has been proposed in , which shows that when the relative interaction matrix is fixed and there is no `` dominant neighbor '' in the network , the social - confidence levels of the individuals will asymptotically converge to a steady state , depending on the relative interaction matrix , under appropriate connectivity assumption .local exponential stability of the steady state was shown by checking the jacobian matrix . but no convergence rate result was obtained .analysis of the continuous - time distributed self - appraisal model for the case of general stochastic relative interaction matrix ( i.e. , without the assumption of no `` dominant neighbor '' ) also remains open .  in a realistic social network ,the interaction among the individuals may change from time to time . with this in mind, this paper aims to study the continuous - time distributed self - appraisal model with time - varying interactions which are described by a sequence of time - dependent relative interaction matrices , and specifically derive a convergence rate of the model .we first construct a simple example to show that the model may not converge for general time - varying stochastic ( not doubly stochastic ) relative interaction matrices . with this observation in mind ,we focus our attention on the case when the relative interaction matrices are doubly stochastic , and show that the self - appraisals of the individuals all converge to exponentially fast , where is the number of individuals in a network , and obtain an explicit expression of the convergence rate . although doubly stochastic relative interaction matrices may be artificial , this case has an important social meaning as it explains how a democratic state is formed in a social network .the main contribution of this paper is to provide an explicit expression of the convergence rate of the continuous - time distributed self - appraisal model with time - varying doubly stochastic relative interaction matrices .we extend the result in in three - fold .first , our result implies that the model converges for all fixed doubly stochastic relative interaction matrices , whereas the result in does not subsume this implication because not all doubly stochastic matrices satisfy the no `` dominant neighbor '' assumption .second , we show that the convergence is exponentially fast for doubly stochastic relative interaction matrices , whereas only asymptotic convergence was proved in .lastly , we show that exponential convergence holds for time - varying doubly stochastic relative interaction matrices and obtain an explicit expression of the convergence rate , which was not considered in .the remainder of this paper is organized as follows .some notations are introduced in section [ notation ] . in section [ model ] , the continuous - time self - appraisal model is introduced . in section [ motivating ] , an example is presented to motivate the assumptions .the main results of the paper are presented in section [ result ] , whose analysis and proofs are given in section [ analysis ] .some discussions are given in section [ discussion ] .the paper ends with some concluding remarks in section [ conclusion ] .   for a positive integer , let denote the set .we use to denote the simplex . for each , we use to denote the vector in whose element equals 1 and all the other elements equal 0 .let denote the identity matrix and let denote the all - one vector with appropriate dimensions .a row - stochastic matrix is a nonnegative matrix with each row sum equal 1 , and is simply called a stochastic matrix .a matrix is column - stochastic if its transpose is a row - stochastic matrix .a matrix is called doubly stochastic if it is both row - stochastic and column - stochastic . for any tworeal vectors , we write if for all and if for all . we use to denote the diagonal matrix with the entry being . for a scalar ,let denote the largest integer that is no larger than .in this section , we introduce the continuous - time distributed self - appraisal model proposed in .  consider a network consisting of individuals with the constraint that each individual can communicate only with certain other individuals called `` neighbors '' .the neighbor relationships among the individuals are described by a time - dependent , -vertex , directed graph whose vertices correspond to individuals and whose arcs depict neighbor relationships .specifically , we say that individual is an _ outgoing neighbor _ of individual at time if there is an arc from vertex to vertex in , and say that individual is an _ incoming neighbor _ of individual at time if there is an arc from vertex to vertex in .we use and to denote the sets of incoming and outgoing neighbors of individual at time , respectively .each individual has control over a real - valued quantity which represents the self - appraisal of individual .the self - appraisal takes values in the interval  ] , and can be set by individual herself .let for all pairs of and such that .then , condition implies that for all and time , and thus each matrix is a stochastic matrix whose diagonal entries all equal zero .the matrix is called the _ relative interaction matrix _ at time .  at initial time , the self - appraisals are scaled so that they sum to one , i.e. , .it will be shown that this initial condition guarantees that for all time .  system ( [ eq : sys1 ] ) with a fixed relative interaction ( i.e. , for all time ) was proposed and studied in .the system can be viewed as a continuous - time version of the modified degroot - friedkin model studied in .  to help readers to grasp the social meaning of the model and understand the motivations , we give a brief interpretation of the model below .see for detailed explanation .we begin with the following continuous - time opinion dynamics : where is a real number representing the opinion of individual on an issue of interest at time .note that system is a continuous - time consensus process with the dynamics of scaled by the nonnegative factor .thus , can be viewed as a measure of the total amount of opinions individual accepts from others at time , and can be regarded as the corresponding portion individual accepts from neighbor , which is consistent with the social meaning of , i.e. , is the self - appraisal of individual measuring how confident she is on her current opinion .we now turn to the justification of the model . from , the dynamics of is determined by two terms : and .we consider the latter first .recall that measures the amount of opinion individual accepts from neighbor in the opinion dynamics and is the self - appraisal of individual reflecting the importance of individual in the network .product can then be viewed as the measure of importance of individual to individual , and thus can be viewed as the measure of importance of individual to the others in the network .we then consider the other term . in view of condition ( [ eq : c2 ] ) , it follows that thus , can be interpreted as the measure of importance of others to individual .  from the preceding discussion ,the model ( [ eq : sys1 ] ) is designed for each individual to calculate , in a distributed manner , the difference between her level of importance to others and others level of importance to her .note that any equilibrium state of system ( [ eq : sys1 ] ) is a state when the difference equals zero for each individual .therefore , the distributed self - appraisal model ( [ eq : sys1 ] ) aims to drive all individuals differences to zero .  to proceed , let ^\top ] the fact that is positive invariant as will be proved in lemma [ lm:5 ] later implies and thus it follows that is a laplacian matrix for any .it is worth noting that is not necessarily a laplacian matrix even if is doubly stochastic .the difference between system ( [ eq : sys2 ] ) and the continuous - time consensus algorithm is that is a state - dependent laplacian matrix , thus resulting in a nonlinear system , while the laplacian matrix is not state - dependent .the derived convergence results for the consensus system are typically based on assumptions on the elements of the laplacian matrix such as \cup\{0\}, ] . for a random initial condition in ,when takes the form of ( [ eq : c ] ) , the system state does not converge as shown in fig .[ fig:1 ] .   switching between a doubly stochastic matrix and a non - doubly stochastic matrix .,width=245 ]the reason can be explained as follows .the equilibria other than , of system ( [ eq : sys1 ] ) corresponding to and are different .then , one can imagine that when switches between and , the system state will oscillate , since in each switching mode , the state tends to converge to the corresponding equilibrium .note that when , the system state will converge to as long as is irreducible and doubly stochastic .this motivates us to focus on the case of doubly stochastic matrices and impose some connectivity conditions to guarantee the convergence of system ( [ eq : sys3 ] ) .in this section , we present the main result of the paper .the following assumptions will be considered in the following discussion on system ( [ eq : sys3 ] ) .[ ass:1 ] each , , is a doubly stochastic matrix with zero diagonal elements , and there exists a constant such that for all nonzero .[ ass:2 ] there exists an integer such that the union graph is strongly connected for all nonnegative integers .[ ass:3 ] there exist two positive constants and such that for all .let the function is a measure of the maximum difference between the self - appraisals of the individuals in the network .if as goes to infinity , then the self - appraisals of the individuals all converge to a common value that is as will be shown .the main result of the paper is stated as follows .[ thm:1 ] suppose that and assumptions [ ass:1 ] , [ ass:2 ] and [ ass:3 ] hold .then ,  * is a positive invariant set of system ( [ eq : sys3 ] ) , i.e. , for any initial condition , for all . *if and has nonzero entries , , then for all and the convergence is exponentially fast with a rate given by for all , where with .  the intuition for the convergence of system ( [ eq : sys3 ] ) to a democratic state ^\top ] , .the derivative of at is given by therefore , there exists a positive constant such that for .it immediately follows from the previous discussion that for .hence , .  repeating this process, we can find an sequence of integers such that note that .thus , . from assumption [ ass:2 ] , can be chosen such that .one concludes that and . the desired conclusion follows .  in what follows, we will look at the evolution of and provide an explicit upper bound for , and thus the decrease of over some time interval can be quantitatively characterized .[ lm:6 ] suppose .assume that assumptions [ ass:1 ] , [ ass:2 ] , and [ ass:3 ] hold .if and for some integer , then the following inequality holds : where and are given in ( [ eq : alpha ] ) with replaced by .  in view of lemma [ lm:5 ], is a nonincreasing function and is a nondecreasing function .we will bound from above so that the inequality ( [ eq : lm6 ] ) can be established .we divide the analysis into three steps ._ let .for any and , ] inequality ( [ eq : lm6 - 3 ] ) implies that for  ] ._ define from assumption [ ass:2 ] , is well defined and satisfies that .  for any , it follows from lemma [ lm:4 ] that where \ ] ] satisfies that and similarly , one has that and , for .note that and .one has that   from the definition of , there exists some such that .it then follows from ( [ eq : lm6 - 7 ] ) that for , ] this implies that for , ] similar to the analysis in step 1 , one can obtain that   _ step 3 ._ continuing the analysis on time interval  ] as continuing this process , a time sequence and a sequence of sets are defined as for , such that . by assumption [ ass:2 ], satisfies , for .note that and hence for all and any  ] corresponding to one .assume that is the same as in ( [ eq : c ] ) .for a random initial condition in , the state evolution is shown in fig .[ fig:2 ] .it can be seen that the system state converges to an equilibrium point in .switching between and with a common left eigenvector to one.,width=245 ]in this paper , the continuous - time self - appraisal model proposed in with a time - varying relative interaction matrix has been studied .it has been shown that the self - appraisals of the individuals in a network will all reach exponentially fast if the time - varying relative interaction matrix is piece - wise constant and doubly stochastic .similar convergence result has been conjectured for the case when all different relative interaction matrices are row - stochastic and share the same dominant left eigenvector .an explicit expression of the convergence rate has been established .we are interested in further looking into the self - appraisal model with a general relative interaction matrix which is row - stochastic , not necessarily doubly stochastic .a. mirtabatabaei , p. jia , n. e. friedkin , and f. bullo . on the reflected appraisals dynamics of influence networks with stubborn agents . in _ proc . of the 2014 american control conference _ ,pages 39783983 , 2014 .w. xia , j. liu , k. h. johansson , and t. baar .convergence rate of the modified degroot - friedkin model with doubly stochastic relative interaction matrices . in _ proc .of the 2016 american control conference _ , pages 10541059 , 2016 .

Abstract: this paper studies a recently proposed continuous - time distributed self - appraisal model with time - varying interactions among a network of individuals which are characterized by a sequence of time - varying relative interaction matrices . the model describes the evolution of the social - confidence levels of the individuals via a reflected appraisal mechanism in real time . we first show by example that when the relative interaction matrices are stochastic ( not doubly stochastic ) , the social - confidence levels of the individuals may not converge to a steady state . we then show that when the relative interaction matrices are doubly stochastic , the individuals self - confidence levels will all converge to , which indicates a democratic state , exponentially fast under appropriate assumptions , and provide an explicit expression of the convergence rate .