queries over graph databases can be classified broadly into whole graph at - a - time , and node at - a - time processing , and framed as a subgraph isomorph computation problem ( e.g. , ) under a set of label mapping constraints , generally known as graph matching .techniques such as graphql , quicksi and earlier research such as vflib and ullmann fall in the former category while tale , and sapper are representative of the latter .the advantage of the node at - a - time graph processing approach is its inherent ability to prune search space based on target node matching conditions .node indices are the most common pruning aid used in most of these processing methods although indices on paths , frequent structures , node distances , etc .have also been used .the key difference is in the ways indices are exploited for the construction of the target database graphs from their parts ( i.e. , the edges ) .the effectiveness of the node at - a - time methods , however , largely depends on the query type such as subgraph isomorphism , approximate matching , path queries and so on , as well as on the index structure used . in other words ,universal indexing methods are not always suitable for all queries , and therefore specialized indices are often constructed to process a query ( e.g. , graphgrep , tale , and sapper ) and never maintained .thus , it is not apparent if the index structure is switched , how an algorithm will perform leaving an open question if generic index structures can be leveraged in a way similar to relational query processing with popular indices such as b+ trees , extendible hashing and inverted files .  in order to decouple the index selection from the query expressions , and to subsequently use indices as a strategic instrument to compute alternative query plans , we focus on a representation method for graphs that is independent of the underlying access structures .our goal is to propose the hub " as the unit of graph representation that tells us all we need to know about a node or vertex of a graph .intuitively , each node in a graph as hub covers " all the edges involving its neighbors and itself .for example , the hub in figure [ fig : example](a ) covers the edges and as a unit structure ( shown as the purple edges ) .the concept of hub we have in mind can be thought of as a convenient extension of ullmann s adjacency matrix and feature structure indexing in that we localize the adjacency matrix at the node level and consider only a single feature , edges among the neighbors .consequently only structures that are part of a hub are stars ( neighbors with no shared edge with other neighbors ) and triangles ( neighbors sharing edge with other neighbors ) . in figure[ fig : example](a ) , vertex has two triangles and , and a star ( in this case just an edge ) . whereas vertices and in figure [ fig : example](b ) have a star ( with two edges and with as their center ) , and three triangles ( , , and ) respectively .our goal is to use these atomic structural cues to match shapes for the purpose of graph matching .for example , to match the query graph in figure [ fig :example](a ) with the data graph in figure [ fig : example](b ) , we look for individual node structures that are identically connected and depending on the matching requirement , have identical labels . the next step is to piece together these individual matches to see if the composed structure is the target graph .the cost of this search usually is dominated by the cost of piecing together the components and testing if the process is yielding the target graph .  in this example , we can contemplate several different types of graph matching that can be conceived as the variants of subgraph isomorphism though in the literature , only the structural isomorphs and match isomorphs defined below are prevalent .we therefore will consider only these two types of matching in the remainder of this paper .  *_ structural subgraph isomorph _ , where only the node ids ( not the labels ) are mapped from query graphs to data graphs using and injective function .* _ label subgraph isomorph _, on the other hand , requires an injective mapping of both node ids and node labels from query graph to data graph .* _ full subgraph isomorph_ extends label subgraph isomorphic matching to include edge labels in the mapping . *_ match subgraph isomorph _ uses an equality function on the definition of full subgraph isomorph to achieve exact matching of node and edge labels while maps node ids using an injective function .  among the above four modes of matching , structural subgraph isomorphism is the least restrictive or selective , and so most computationally expensive .while the match subgraph isomorphic matching ( in the literature it is known as labeled graph matching ) is the most restrcitive / selective , full and and label subgraph isomorphism are increasingly less so .the idea here is that by combining different selection and mapping constraints , called _ matching mode _ , we can capture most popular graph matching concepts and go beyond current definitions .traditional deep equality operator in object - oriented databases can be used to test if two graphs ( or a subgraph ) are equal ( or contained in the other graph ) by requiring that node ids and labels be identical . finally , by requiring that the two graphs have equal number of nodes, we can also achieve graph isomorphism for each case above .the vertex labeled graphs in figure [ fig : example ] show two query graphs and , and a data graph respectively . in these graphs ,each node has a unique node i d such as , and , and a label such as , and ( shown in uppercase with unique color codes ) .if all the labels are empty , the graph is considered unlabeled .if we matched query graph with the data graph for structural subgraph isomorphism , we will compute the green , brown and red matches , among others , as we are only required to map node ids .however , if we were to compute match subgraph isomorphs , only solution we can compute then is the green subgraph .if label subgraph isomorphs were being sought instead , the solution will include the brown subgraph where we map pink ( c ) to pink , green ( d ) to blue ( a ) , mustard ( b ) to mustard , and blue to green , along with the obvious green subgraph .let us illustrate this naive matching process using figures [ fig : example ] and [ fig : cand ] .  since there are six nodes in the graph , a total of six hubs are possible .similarly , the graph can be represented by a total of eleven hubs .let us first match with in match subgraph isomorphism mode . under this constraint , for hub ( shown clock - wise rotated by in purple in figure [ fig : cand](a ) ), we can only find one hub , hub shown in figure [ fig : cand](b ) , in which is a supergraph of hub and can produce a structure identical to ( shown as red edges ) on proper mapping of the node ids .these structures are called candidate graphlets . however ,if we choose to match in structural or label subgraph isomorphism mode , we can find more hubs as capable of generating candidates structures .for example , in label subgraph isomorphism mode , hubs and can also generate candidates ( possible candidate structures are shown in red , purple and green ) as shown in figures [ fig : cand](c ) and [ fig : cand](d ) .  to complete the matching , we can continue to match all the other hubs in in a similar way , and by applying the substitutions generated for each set of previous matches to the candidate hubs to eventually compute the green match as shown in figure [ fig : example](b ) .these graphlets can be joined or pieced together in both bottom - up fashion using a process similar to natural join , or in top - down manner using a depth - first search . in this algorithm the structure mapping function that generates the mapping , is the substitution list from previous steps ,  ] and unmapped subsets ( shown in uppercase ) . since an instantiated dot nodemay only fetch a single hub at the most , this ordering is likely to result in a least cost plan .set .+ choose the most selective graphlet as using node selectivity .+   let us also assume that we have a hash index that given a node i d ( i.e. , ) , returns the graphlet corresponding to , i.e. , .furthermore , we also have another index similar to such that it returns a set of graphlets given a set of nodes , and an integer , i.e. , is now a set of graphlets , or basically a set of nested tuples . ] , such that for each , , and . in other words, returns all graphlets that contain the set of nodes in and have a minimum of neighbors .the graph query processor for exact matching is simple .it is composed of a depth - first search algorithm * find solutions * that maps a sequence of graphlets in a query graph presented in order to a set of graphlets of a data graph using a composable term mapping function .algorithm [ alg : find ] is assumed to have access to a data graph , one of four graph matching modes , and the indices and .let . + apply substitution to , i.e. , $ ] .+   we formally present the computational characterization of mhc in terms of its complexity class , and using the decision version of the mhc problem ( mhc - d ) show that the mhc problem is indeed difficult and that it is np - complete .mhc - d is np - complete .  to show that mhc - d is np - complete , we answer the following question : given a graph and an integer ,does there exist a subset with such that for every edge , or or there exists a vertex such that both and ?first , we argue that mhc - d is in np because for a given yes - instance and with , we can verify in polynomial time that every edge in is covered by .we now complete the proof by a reduction from the mvc problem on triangle free graphs . in triangle - free graphs , for all there does not exist a vertex such that and .thus , either , or must be in to cover the edge .consequently , the mhc and the mvc of a graph are equivalent in this class of graphs and mvc is np - complete for this class of graphs .the np - completeness of mhc speaks to the hardness of the solvability in general .the optimization strategy and query plan generation presented in section [ sec : prel ] , however , calls for the computation of all mhcs of a given query , i.e. , , which is indeed a hard problem . in this section ,our goal is to design a set of experiments using different graph types , size and graph density parameters to study how the solvability and the quality of mhc solutions depend on these parameters .the goal is to experimentally identify problem classes for which available solutions are practical and acceptable , and the classes for which new heuristic solutions are warranted .we therefore employ different algorithms to show the trade - offs between optimality and computation time over different graph types .although our experiments and analysis involve producing only one optimal solution for a graph , the insights gained can be leveraged to develop new algorithms to compute or to choose other strategies when computing is infeasible .the optimal linear programming ( lp ) and ip solutions are obtained by ilog ibm cplex 12.4 on a personal computer with an intel core 2 dual processor and 3.25 gb of ram . in all problem instances , the upper limit on the computation is set at 3,600 seconds .the batch processing of the instances is carried out through simple c++ scripts .our data set includes a total of 830 instances .we have 5 different instances for each combination of a graph type , size , and density parameter to be able to draw conclusions .we have chosen to use the benchmark database graph instances in and our own synthetically generated data set for our numerical study .this is a very large database of different graph types and sizes designed specifically to test the sophistication of ( sub)graph isomorphism algorithms . since we are using subgraph isomorphism as a basic vehicle for graph matching , the instances selected are thus representative of the class of queries we are likely to handle when we solve the mhc problem .the descriptions of the graph instances we have chosen from this collection are listed below .[ [ randomly - connected - graphs ] ] randomly connected graphs + + + + + + + + + + + + + + + + + + + + + + + + +  these graphs have no special structure and the number of vertices range from 20 to 1000 ( = 20 , 60 , 100 , 200 , 600 , 1000 ) . the parameter denotes the probability of having an edge between any pair of vertices .thus , this parameter , in a sense , specifies the sparsity of a graph . in the database , three different values of ( 0.01 , 0.05 , and 0.10 )are considered .our data set includes a set of graphs of different sizes for each value of .[ [ bounded - valence - graphs ] ] bounded valence graphs + + + + + + + + + + + + + + + + + + + + + +  the vertices of the graphs in this class have the same degree ( _ fixed valence _ ) .the sizes of the instances are similar to those of the problem class ( a ) .we use three different values of valence 3 , 6 and 9 to obtain graphs of different size and valence .[ [ irregular - bounded - valence - graphs ] ] irregular bounded valence graphs + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  these graphs are generated by introducing irregularities in the problem class ( b ) .irregularity comes from randomly deleting edges and adding them elsewhere in the graph . with this modification ,the average degree is again bounded but some of the vertices may have higher degrees .the sizes of the instances are similar to those of problem classes ( a ) and ( b ) .[ [ regular - meshes - with-2d-3d - and-4d ] ] regular meshes with 2d , 3d , and 4d + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  in graphs with 2d , 3d , and 4d meshes , each vertex has connections with 4 , 6 , and 8 neighbors , and the numbers of vertices range from 16 to 1024 , 27 to 1000 , and 16 to 1296 , respectively .similar to the problem classes ( a ) ,( b ) , and ( c ) , we have a set of graphs for each combination of size and dimension .  [[ irregular - meshes ] ] irregular meshes + + + + + + + + + + + + + + + +  as in class ( c ) , irregular meshes are generated by introducing small irregularities to the regular meshes .irregularity comes from the addition of a certain number of edges to the graph . the number of edges added to the graph is scale - free graphs + + + + + + + + + + + + + + + + +  this problem class includes the graphs that follow a power - law distribution of the form where is the probability that a randomly selected vertex has exactly edges , is the normalization constant , and is a fixed parameter .we employed the scale - free graph generator of c++ boost graph library . the generator ( power law out degree algorithm )takes three inputs .these are the number or vertices , and . increasing the value of increases the average degree of vertices . on the other hand , increasing the value of decreases the probability of observing vertices with high degrees .the sizes of the instances range from 20 to 1000 ; to be precise .we considered two values for and three values of .graphs in social networks , protein - protein interaction networks , and computer networks are examples of this class .we choose three solution methods to compute mhc ( i ) an exact method to solve the problem to optimality , ( ii ) adapt two approximation algorithms from the vertex cover literature capable of computing feasible solutions fast , and ( iii ) a mathematical programming - based heuristic originally proposed for solving the scp .the ip formulation - is solved by an off - the - shelf solver to optimality .since the mhc problem is shown to be np - hard , this approach may have practical value only for small - to - medium - scale graphs .however , it sets a definitive benchmark for comparing the performances of various heuristics .we implemented two different approximation algorithms .first algorithm selects the vertex with the highest degree at each iteration .the aim is to cover as many edges as possible .next , all covered edges as well as the vertices in the cover are removed from the graph .the algorithm ends when there is no uncovered edge in the graph .the algorithm is called the _ -approximation algorithm _ ( gr1 ) for the mvc problem . here, is the maximum degree in the graph , and is evaluated by the second algorithm ( gr2 ) , the _2-approximation algorithm _ , is an adaptation of originally proposed for computing a near - optimal solution for the mvc problem .unlike the previous algorithm , it selects an edge arbitrarily , then both vertices incident to that edge are added to the cover .yelbay et al . propose a heuristic ( mbh ) that uses the dual information obtained from the lp relaxation of the ip model of scp .they show the efficacy of the heuristic on a large set of scp instances . in their work ,the dual information is used to identify the most promising columns and then form a restricted problem with those columns .then , an integer feasible solution is found by one of the two approaches . in the first approach ( mbh ) , the exact ip optimal solution is obtained by solving _ the restricted problem_. in the second approach , a metaraps local search heuristic ( lslp ) is applied over those promising columns .we use both of these approaches .we focus on analyzing and understanding the mhc solution methods in section [ subsec : methods ] on the instances in section [ subsec : probclass ] in three different axes : ( i ) optimal solvability of mhc , ( ii ) quality of the solutions , and ( iii ) computational cost of optimal solution .these analyses are aimed at understanding which problem classes are inherently more difficult relative to others so that depending on the application and query , a suitable algorithm can be selected to compute mhc .we also discuss the factors that increase the complexity of the problems .figure [ fig : opt ] shows how the optimal solution time of cplex , an exact method , varies depending on the problem size , class , and structure .the -axis and the -axis represent the number of vertices and the average computation time , respectively .the right - most data point on a line shows the size of the largest instance that can be solved to optimality in a group . in general , its performance is good for small to medium scale graphs .however , in our study , 39 out of 90 , 73 out of 285 and 39 out of 180 instances in problem classes ( a ) , ( e ) and ( f ) , respectively , could not be solved optimally using cplex within the time limit .this observation opens the door for heuristics to find acceptable but possibly suboptimal solutions .[ [ random - graphs ] ] random graphs + + + + + + + + + + + + +  from figure [ fig : randopt ] we conclude that for randomly connected graphs with more than 200 nodes , optimal solution is not achievable within the bounded time .it also suggests that the density of graphs is a factor that affects the solvability .the solver does increasingly better as the density goes down ( up to 0.01 ) for the same number of vertices .its sensitivity with respect to the size and density is apparent in the plots for equal to 0.05 and 0.10 , i.e. , a 16 fold increase in solution time .[ [ bounded - valance - graphs ] ] bounded valance graphs + + + + + + + + + + + + + + + + + + + + + +  compared to random graphs , figure [ fig : boundopt ] shows an improved performance on bounded valence graphs solving all instances under 0.3 seconds .the reason for the performance difference may be due to the considerably higher number of edges in a randomly connected graph ( which forces the number constraints in the ip model to go higher ) than that of a bounded valence graph . however , although we expect higher solution time for graphs with larger valence , figure [ fig : boundopt ] shows substantially higher time for valence 3 than valences 6 and 9 suggesting other factors may also be playing a role .  [ [ irregular - bounded - valance - graphs ] ] irregular bounded valance graphs + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  although the degree distribution is neither constant nor fully randomly distributed , cplex performs similarly to bounded valence graphs .as figure [ fig : irboundopt ] shows , all solutions are computed in less than .25 seconds , and that the computation time increases with the increase in valence .[ [ regular - mesh - graphs ] ] regular mesh graphs + + + + + + + + + + + + + + + + + + +  figure [ fig : meshopt ] shows that the size of meshes ( 2d , 3d or 4d ) usually does not have any influence on the performance barring the abrupt behavior of the 4d mesh graph . in general , the solution time appears to linearly increase with the increase in graph size , though the increase in time is extremely small .[ [ irregular - mesh - graphs ] ] irregular mesh graphs + + + + + + + + + + + + + + + + + + + + +  unlike the irregular bounded valence graphs , mesh graphs are more susceptible to irregularity and the computation time substantially increases with the degree of irregularity .figures [ fig : irmeshopt2 ] through [ fig : irmeshopt4 ] show that the sizes of the problems that can be solved to optimality decrease and the computation times increase with increasing degree of irregularity .this result is quite reasonable and expected because increasing irregularity increases the number of edges , and thus the computation time as well .this is also because randomly adding edges to a mesh graph makes it structurally more similar to random graphs , which , as discussed earlier , is inherently hard to solve .[ [ scale - free - graphs-1 ] ] scale - free graphs + + + + + + + + + + + + + + + + +  we consider the effect of the two parameters and on the solvability of the problems .on one hand , increasing makes the degree distribution sharper , i.e , we observe smaller number of vertices with high degrees . on the other hand , increasing the value of increases the degrees of non - hub nodes .figures [ fig : scalefree1 ] and [ fig : scalefree2 ] represent the optimal solution times of scale - free instances .it is clear that the difficulty of the problem is closely related to parameters and .the figures show that computation times decrease significantly with increasing values of . when , the instances with more than 100 vertices can not be solved to optimality within the time limit .when , however , all of the instances can be solved optimally in less than 0.06 seconds .these figures also show that the computation time increases with increasing values of .this means that increasing degrees of non - hub nodes makes the problem more difficult .to study the quality of solutions generated by other solution methods with respect to the optimal solutions computed using cplex , we refer to figures [ fig : perfrand ] through [ fig : perfirmesh ] .these plots are called performance profiles of algorithms that depict the fraction of problems for which the algorithm is within a factor of the best solution .thus , they compare the performance of an algorithm on an instance with the best performance observed by any other algorithm on the same instance .the x - axis represents the _ performance ratio _ given by where is the number of hub nodes in the hub cover when the instance is solved by algorithm and is the set of all benchmark algorithms .the y - axis shows the percentage of the instances that gives a solution that is less than or equal to times the best solution .recall that cplex can not solve all of the instances in problem classes ( a ) , ( e ) and ( f ) to optimality .however , the solver is able to find feasible solutions for some of those unsolved instances ( 11 out of 39 , 44 out of 73 , and 24 out of 39 in ( a ) , ( e ) , and ( f ) respectively ) .figure [ fig : perfpro ] includes all instances except those for which cplex can not find either feasible or optimal solutions within the time limit .we first analyze how much we sacrifice from the optimality by employing mathematical programming - based heuristics mbh and lslp . recall that mbh and lslp solve the same restricted problem . while mbh tries to solve the problem to optimality , the latter visits alternate solutions in the feasible region .figure [ fig : perfrand ] shows that 12% of the instances in class ( a ) where both mbh and lslp find better feasible solutions than that of cplex .note that , this can happen if and only if cplex returns a feasible solution rather than an optimal solution within the time limit . for other problem classes ,the performances of the cplex and mbh are quite similar .moreover , these figures show that lslp is outperformed by mbh and cplex on almost 40% and 30% of instances in problem classes ( b ) and ( c ) respectively . for the remaining problem classes ,the performance of lslp is also comparable to mbh and cplex .  the greedy algorithms return feasible but sub - optimal solutions quickly . except the scale - free networks ,the performances of the greedy algorithms do not change with respect to problem classes .gr2 is known as 2-approximation algorithm for mvc .figures [ fig : perfirbound ] and [ fig : perfscale ] show that there are some instances for which performance ratios of gr2 are higher than 2 .obviously , the approximation ratio of gr1 for mhc problem is higher than 2 . intuitively , the performance of gr1 is supposed to be better when the degree distribution of the vertices is not uniform .since the average degrees of the vertices are identical or are quite similar for the instances in problem classes ( a)-(e ) , the performance of gr1 does not vary for these problem classes .however , figure [ fig : perfscale ] shows that gr1 finds the optimal or the best solution in 30% of the scale - free instances .this means that the performance of the gr1 is better for the graphs that follow the power - law distribution .   the previous two analyses focused on the optimal solvability and the quality of the solutions .in this section we turn our attention to the cost of computing a feasible or optimal mhc solutions in terms of time .figures [ fig : timerand ] through [ fig : timescale ] summarize the distribution of the average computation times of the algorithms over the problem classes . in these plots , the instances for which feasible solutions were not found by any of the algorithm within a time limit are excluded .each bar in the figure represents the percentage of the instances that are solved within the time interval stated in the legend , e.g. , the blue bar for 0.0 to 0.05 seconds . since lslp is a local search algorithm , we show both the total computation time and the first time when the best solution is found .the results clearly show that the solution times of greedy algorithms ( gr1 , gr2 ) are much shorter than that of the other algorithms .mbh can solve the restricted problem to optimality in a reasonable amount of time for a great majority of the instances .we have already discussed earlier that the performance of the mbh is good in terms of its solution quality .however , the main drawback for mbh is its inability to solve the restricted problem to optimality . in such cases, lslp may serve as an alternative to mbh as it is comparable to mbh in terms of both solution quality and time , and because lslp is a local search algorithm , it is also guaranteed to produce a feasible solution .however , the performance of lslp is dependent upon prudent selection of algorithmic parameters , e.g. , the total number of iterations , the number of improvement iterations ( see for details ) .there is a trade - off between solution time and the solution quality .decreasing the total number of iterations may result in a decrease in the total solution time .however , it may increase the optimality gap .constraint solvers such as cplex usually do not offer all optimal solutions .such solutions also do not exploit database meta - data in computing the most desired solution .for example , for the graph in figure [ fig : example](c ) , there is no guarantee that the solver will produce the desirable solution when the data graph is known to be .therefore , we are required to compute all possible mhcs of , i.e. , , so that we are able to identify this least cost query plan . unfortunately , it is not guaranteed that solvers can even always compute a solution , let alone the whole family of solutions of .the discussion in section [ subsec : eval ] also suggests that although the cost of computing mhc is significantly low for small graphs , it still remains high for many graph types when the query graph is large .therefore , even though some of the existing solvers may be useful for applications involving small scale - free graphs such as protein - protein interaction networks , they may not be a great candidate for big data applications in social networks and world wide web .though it may be challenging , we believe the low mhc solution time for many graph types offers hope that designing algorithms for is feasible for most practical applications , but remains as an interesting problem .it is also important to recognize that while developing the least cost mhc using meta - data may be feasible for a single data graph , devising such algorithm for a large set of data graphs may not be feasible .it is thus worth investigating if a general but a single optimal query plan ( without computing ) , for which we have a solution , can be dynamically adjusted for best performance over a set of graphs .finally , it remains an open question if a suboptimal mhc produced by a greedy algorithm can be improved enough to defeat or match the overall processing performance using an optimal mhc solution , i.e. , total cost of mhc , plan generation , plan selection and execution .in this paper we have formally introduced the idea of graphlets as a basic unit for graph representation in a way similar to rdf triple store , and the concept of minimum hub cover of graphlets as a basic ingredient toward graph query optimization .we have demonstrated on intuitive grounds that such an approach can leverage generic access structures such as hash and set indices for query optimization .though computationally hard , we have also demonstrated that query processing and optimization using mhc and subgraph isomorphism is computationally feasible and intellectually intriguing .in particular , we have shown that for many application domains of current interest such as social networks , and protein - protein interaction networks , existing constraint solvers are capable of delivering optimal solutions for mhc , and therefore can be used to develop optimization strategies .it is our thesis that covering based graph processing we have presented opens up new research directions and holds enormous promise .the logical next step is to develop a query processor by integrating the algorithms in sections [ sec : cost ] and [ sec : qp - alg ] , with new algorithms outlined in section [ sec : fut ] .these are some of the tasks we plan to continue as our future research .this study was supported partially by tubitak 2214 ph.d .research scholarship program .m. mongiov , r. d. natale , r. giugno , a. pulvirenti , a. ferro , and r. sharan , `` sigma : a set - cover - based inexact graph matching algorithm , '' _ j. bioin . and comp . bio ._ , vol . 8 , no . 2 , pp .199218 , 2010 .f. fouss , a. pirotte , j .-m . renders , and m. saerens , `` random - walk computation of similarities between nodes of a graph with application to collaborative recommendation , '' _ tkde _ , vol .19 , no . 3 , pp .355369 , 2007 .f. gomes , c. meneses , p. pardalos , and g. viana , `` experimental analysis of approximation algorithms for the veretx cover and set covering problems , '' _ computers and operations research _ , vol .35203534 , 2006 .m. haouari and j. s. chaouachi , `` a probabilistic greedy search algorithm for combinatorial optimization with application to the set covering problem , '' _ journal of the operational research society _ ,792799 , 2002 .

Abstract: as techniques for graph query processing mature , the need for optimization is increasingly becoming an imperative . indices are one of the key ingredients toward efficient query processing strategies via cost - based optimization . due to the apparent absence of a common representation model , it is difficult to make a focused effort toward developing access structures , metrics to evaluate query costs , and choose alternatives . in this context , recent interests in covering - based graph matching appears to be a promising direction of research . in this paper , our goal is to formally introduce a new graph representation model , called _ minimum hub cover _ , and demonstrate that this representation offers interesting strategic advantages , facilitates construction of candidate graphs from graph fragments , and helps leverage indices in novel ways for query optimization . however , similar to other covering problems , minimum hub cover is np - hard , and thus is a natural candidate for optimization . we claim that computing the minimum hub cover leads to substantial cost reduction for graph query processing . we present a computational characterization of minimum hub cover based on integer programming to substantiate our claim and investigate its computational cost on various graph types .


we propose a model for a distributed information network in which each node at a given time can obtain information from a limited number of other nodes to which it is connected . from the information residing in these nodes ,each one of them can perform specific tasks ( calculation , classification , etc . ) involving all the information contained in the network nodes . in a system where information is distributed at different sites , the access to this information can be an expensive procedure if the number of sites is high .moreover , if a set of nodes want to do at any given time a calculation of a magnitude involving information contained in all other ones , the direct access of all to all nodes requires a number of requests of order , which is very high if is large .here we pose the problem of a set of elements ( nodes ) , each characterized by the value of a magnitude , in general different for each node , so that each and every one of the elements wish to calculate the average value of ( or a function of ) in the set , by accessing at one time to information contained in other elements of the set ( ) with which it is connected .this is in line with systems using epidemic protocols or gossip ones , such as the newscast protocol . in these systemsthe goal is not to enable both point - to - point communication between nodes , but rapid and efficient dissemination of information .the system intends to perform a specific collective task ( for example , calculating the average value of a variable in the set of nodes , setting the position of each node in a ranking according to the set value of the variable , etc . ) so that , eventually , all nodes have access to the result obtained from the set . to do this we consider a network in which each element is connected to other ones ( neighbors ) , from which it extracts information .the neighborhood of each element can be changed along the process of calculation , so that each node can exchange a neighbor randomly with another node .we discuss issues such as scalability and convergence , considering different sizes and data sets .we also compare the results obtained for the dynamically changing network with those obtained from the fixed static network . to do this we compare network configurations from different times of evolution with those corresponding to the initial fixed network ( which may take the form of a conventional cellular automaton ) .in a gossip framework there is an exchange of information among system elements in this exchange of information , which is a dynamic process , one receives information from another ( also can be a reciprocal exchange ) . in turn, the receiver of information can give information to other peers .overall the transmission process basically comprises three aspects : peer selection , data exchanged or transmitted between peers and data processing . in our modelwe consider a network arranged in a two - dimensional lattice ( we do this initially to compare with a conventional cellular automaton with moore s neighborhood ) with sites forming a square lattice . to each nodeis randomly assigned the value of a variable , in general different for each node .the objective is that each node `` knows '' the average value of this variable in the whole at the end of the process .each node has access to limited information on each step of calculation . to obtain specific results, it has been considered that each node , at each instant , can only store eight values of other nodes , and that initially each node has access to the data of the eight nearest neighbors in the lattice .this can be matched with a two - dimensional cellular automaton with moore s neighborhood . throughout the processthe neighborhood of each site is not fixed , but each node exchanges a neighbor with another randomly chosen node .this implies a double selection ( with appropriate calls to random numbers routines ) , one of them for the site of exchanging and the other for choosing the neighbor exchanged .the average value of the variable of each site and its eight neighbors ( inputs ) is assigned to the variable of the corresponding site for each system update .the updating of all sites is simultaneous , although it could also be carried out sequentially . since in the course of evolution neighborhoods randomly change , the initial regular structure of the network is irrelevant ( only holds the fact that each node has eight input and eight outputs , generally different ) .however , in order to formally create a by analogy with the above mentioned cellular automaton , we initially identify each node as a point on a square lattice ( figure [ fig : networks]a ) , but now , after the dynamic exchange of information , the system is turned into a complex directed network and connections do not generate a square lattice ( figure [ fig : networks]b ) a and b are represented in a circular layout in order to compact the graph , as an orthogonal layout ( square lattice like ) is not friendly looking due to the high number of nodes . ] .    to test the scalability we have considered various system sizes ( , , and ) .also we have considered two data distributions : ( a ) a uniform random distribution , ( b ) a distribution of values grouped into four quadrants as shown schematically in figure [ fig : grouped_values ] . for comparison , as mentioned above , three cases are considered : the network that dynamically changes , the fixed network with the configuration that is reached at a given moment of the dynamical evolution and the fixed network with the initial square lattice structure ( cellular automaton ) . in the latter two cases ,the neighborhoods do not change and each node has permanent access to its pre - established eight neighbors .we defined a parameter , , to evaluate how the computation progresses in each system update , i.e. , convergence to the desired value with a given accuracy .  we have imposed the condition that the calculation is stopped when the variation of between two successive updates is less than a preset value .we have also established a limited number of system updates . moreover , using the cytoscape software , we have analyzed some of the properties of the network [ 11 - 12 ] to establish the relationship between its topology and the degree of computation efficiency .we have taken the value of as the its limit of and have scored the number of updates required to reach a value of less than the successive powers of 10 from to .the table [ tab : tab1 ] shows the results for the evolving network for a set of data randomly distributed according to a uniform distribution . as can be seen , the number of updates is low and the scalability is very good , because for large variations in the size of the system the number of updates nearly remains the same .each node has to perform a small number of updates . since for each update of the entire system one must perform updates of the values of the nodes , then , the total number of individual operations is virtually proportional to system size , i.e. escalates as ( a direct calculation escalates as ) .figure [ fig : results1]a graphically show this behavior where the number of updates slowly increases when is diminished .it must be noticed that the number of updates remains the same for a certain independently of the network size . & & & & & + & 3 & 6 & 8 & 11 & 13 + & 3 & 6 & 9 & 11 & 14 + & 3 & 6 & 9 & 11 & 14 + & 3 & 6 & 9 & 11 & 14 +   value in an evolutionary network for a randomly distributed data set .the bar graph shows a linear increase of the number of updates with the required precision and remains constant when increasing the network size .( b ) number of updates required to reach the value in two fixed networks for a randomly distributed data set : ( i ) evolutionary network configuration when .( ii ) evolutionary network configuration when .as in the case of using evolutionary network , the number of updates needed to reach a certain precision in the calculus increases linearly with this precision . in this case , the increase in the network size makes the number of updates to remain nearly constant .( c , d ) number of updates required to reach the value applying a cellular automaton with moore neighborhood for a randomly distributed data set .( c ) varying .( d ) varying the network size .contrary to the two anterior cases of study , the use of cellular automata makes the number updates increase in an exponential tendency as the precision and the network size grow.,width=377 ]  let s now change the computation order . instead of updating the system as the network evolves , the evolution takes first place and then , for a fixed network successive updates are done .the results for this case can be seen in table [ tab : tab2 ] . herethe configurations that are considered are those reached with the evolutionary network when ( i ) and ( ii ) . in these cases ,the differences with the evolutionary network are not very significant , but for larger values of the limit of convergence is slightly faster , and for lower values of this limit convergence is slightly slower in the case ( i ) and slightly faster in the case ( ii ) .again , the number of updates slowly increases linearly when is diminished in both cases ( i ) and ( ii ) ( figure [ fig : results1]b ) .+ & & & & & + & 2 & 5 & 8 & 12 & 16 + & 2 & 5 & 9 & 13 & 17 + & 2 & 4 & 7 & 9 & 12 + & 2 & 5 & 7 & 10 & 12 +  table [ tab : tab3 ] shows the results obtained for the same random distribution of data using the fixed network forming a square lattice ( cellular automaton with moore s neighborhood ) .as can be seen , the number of updates required to achieve a given accuracy rapidly grows with size . to be concrete, the number of updates increases exponentially when is diminishes ( figure [ fig : results1]c ) and the size of the network increases ( figure [ fig : results1]d ) . & & & & & + & 2 & 5 & 8 & 12 & 16 + & 4 & 206 & 1608 & 3350 & 5255 + & 4 & 419 & 10678 & 29568 & ( * ) + & 4 & 420 & 40255 & 109068 & ( * ) +  to analyze the relationship of these results with the network structure , we have made an analysis of some of the properties of these networks .we have considered the clustering coefficient , the diameter and the characteristic path length ( cpl ) .the clustering coefficient of a node is the ratio , where is the number of links between the neighboring nodes and is the maximum number of links that would be possible among them ; the clustering coefficient of the network is the average value of clustering coefficients of all the network nodes .the network diameter is the maximum distance between two nodes .the characteristic path length gives the expected distance between any two nodes .it is defined as the average number of steps along the shortest paths for all possible pairs of network nodes .as can be seen , the shorter the cpl the better the communication along the network . by constructionthe number of links of each node in these networks is the same ( eight inputs and eight outputs ) .the clustering coefficient of the evolutionary network has a very low value and that of the cellular automaton is high .the two significant parameters are the diameter of the network and the characteristic path length .the values of these parameters for the fixed networks considered are shown in table [ tab : tab4 ] . as can be seen ,whereas in the four evolutionary network configurations considered the diameter and the characteristic path length have small values , and do not change significantly with the limit of , in the cellular automaton these values are much higher and rapidly growing with size ( figure [ fig : topol_param]a ) . for the cellular automaton convergence is very slow , the number of updates required to reach the limits of is very high and the scalability is very poor .therefore this fact suggests that the speed of convergence and scalability of the calculus in the network is strongly associated with the values of these two parameters .this can be seen as manifesting the property of `` small world '' , which appears in different types of complex networks .topologically speaking , the evolutionary networks are distributed in a more sparse way than the cellular automaton , according to the clustering coefficient values . regarding cpl and diameter values ,the evolutionary networks are much ` better ' connected than the cellular automaton , in terms of information dissemination .a network with low cpl means that the information contained in one node is more accessible to the rest of the nodes than in a network with high cpl .all this gives a method that re - distributes a network in a way where information is easy to share , and so , computationally speaking , with a low cost in collective computations .network & diameter & characteristic path length & clustering coefficient + evolutionary network when reaching & 5 & 3.5 & 0.007 + evolutionary network when reaching & 6 & 4.6 & 0.001 + evolutionary network when reaching & 8 & 5.9 & 0.036 + evolutionary network when reaching & 8 & 5.7 & 0.000 + cellular automaton & 16 & 10.7 & 0.429 + cellular automaton & 50 & 33.3 & 0.429 + cellular automaton & 160 & 106.7 & 0.429 +   value in a specific case using evolutionary networks when the data is not randomly distributed but strongly grouped . in this case the scalability is still very good as in the case with randomly distributed data .the number of updates also slowly increases in a linear way as in the rest of studied cases for the evolutionary network.,width=226 ]  next we have examined the influence of the way in which data are distributed . instead of a random distribution ,we have considered other one in which data is strongly grouped .this consists of four different values distributed in the nodes associated to each of the four quadrants shown in figure [ fig : grouped_values ] .the results for the evolutionary network are shown in table [ tab : tab5 ] . as can be seenscalability is very good as in the case of the data with random distribution , although the number of updates required is somewhat greater than in the case of random data distribution . as in the rest of the studied cases ,the number of updates slowly increases in a linear way ( figure [ fig : topol_param]b ) . & & & & + & 8 & 11 & 13 & 16 + & 8 & 11 & 13 & 16 + & 8 & 11 & 13 & 16 + & 8 & 11 & 13 & 16 +  finally , note that if during the process you add new nodes are added or deleted , the update could proceed as follows :  * if a node is deleted , its input links could be redirected to its output target nodes , so that those nodes that had access to it now are addressed to those to which the deleted node accessed .* if a node is inserted , it provides access to eight of the existing nodes ( as input links ) , and in exchange , each of these nodes will deliver to the new node a link to one of its neighbors ( as output nodes ) .this will allow to perform a system robustness analysis .if the variation in the number of nodes is not very sudden , it is expected the overall system to be not severely impacted , given the information transmission speed . even in the case of a change in many nodes, it can be expected that a good accuracy in the result of the global parameter will be reached again in a reduced number of updates , i.e. resilience is very good .in a networked system , which nodes have a limited access to information only from a few neighbors , the collective computation involving the whole data set is very efficient when the network changes dynamically , or with a fixed structure generated by the same dynamic process . in this case, a fast convergence towards the average value is achieved .also the number of computing updates shows a good scalability with the size of the network .this is associated with the diameter and the characteristic path length of the network . given the high rate with which the system converges to the desired value, the system is also is expected to be robust to changes in the number of nodes or in the values distribution assigned to the network .these results clearly contrast with what happens in a fixed regular network in which the convergence is slow and the computation required hugely grows when the size of the system increases .this work is partially financed by the project fis2008 - 04120 of the spanish ministry for science and innovation ( micinn ) .p. shannon , a. markiel , o. ozier , n. s. baliga , j. t. wang , d. ramage , n. amin , b. schwikowski , t. ideker , cytoscape : a software environment for integrated models of biomolecular interaction networks , genome res . 13( 2003 ) 24982504 .

Abstract: we analyze a distributed information network in which each node has access to the information contained in a limited set of nodes ( its neighborhood ) at a given time . a collective computation is carried out in which each node calculates a value that implies all information contained in the network ( in our case , the average value of a variable that can take different values in each network node ) . the neighborhoods can change dynamically by exchanging neighbors with other nodes . the results of this collective calculation show rapid convergence and good scalability with the network size . these results are compared with those of a fixed network arranged as a square lattice , in which the number of rounds to achieve a given accuracy is very high when the size of the network increases . the results for the evolving networks are interpreted in light of the properties of complex networks and are directly relevant to the diameter and characteristic path length of the networks , which seem to express `` small world '' properties .


molecular self - assembly in the laboratory is a promising way of making useful materialsglotzer2007anisotropy , barth2005engineering , whitesides2002self , blake1999inorganic .self - assembly mediated by dnaeeman1998dna , in particular , has been used to create a wide range of nanostructureswinfree1998design , wei2012complex , fujibayashi2007toward , some of which can perform basic functions  .recent work has demonstrated the self - assembly of dna ` brick ' nanostructures , which are solid , equilibrium structures in which each component or brick is of a distinct type and has a defined positionke2012three , ke2014dna .these structures self - assemble in solution because inter - component interactions , which are mediated by dna basepairs , are stronger between components designed to be adjacent in the assembled structure than between components not designed to be adjacent in the assembled structure .a growing body of theoretical workfujibayashi2009error , schulman2004one , chen2005error , winfree2004proofreading , rothemund2004algorithmic , halverson2013dna , reinhardt2014numerical , hedges2014growth , murugan2015undesired , jacobs2015communication indicates that this principle of component - type complementarity might be used quite generally to create defined , multicomponent assemblies of e.g. colloids or other nanoscale building blocks . `information - rich ' materialszuckermann2011peptoid of this kind have considerable technological potentialfrenkel2015order , cademartiri2015programmable .however , theoretical work also suggests that some obstacles must be overcome before we can self - assemble information - rich structures of arbitrary size ( dna brick nanostructures are about 1000 components in sizeke2012three , ke2014dna ) .for one , the growth rate of a structure made of distinguishable components will likely be of order times less than that of the corresponding indistinguishable - particle structure , because only about 1 in inter - component collisions can promote growthedges2014growth .for another , the energy scale of inter - component bonds must grow on the scale of in order to render the desired structure thermodynamically stable , because bond energies must counter the entropy associated with permuting component types in solutionedges2014growth , murugan2015undesired , frenkel2015order ( see appendix a ) .this energy scale is , for macroscopic objects with , in excess of .molecular self - assembly in the presence of interaction energies large on the scale of generally results in kinetic trapping , because strong bonds can prevent the correction of mistakes that happen when components undergoing brownian motion collide randomlyjack2007fluctuation , hagan2006dynamic , new_rapaport , wilber2007reversible , zaccarelli2007colloidal , lu2008gelation , rob .this is so even if the only interactions present are those designed to stabilize the intended structure we will call these _ native _ interactions so suggesting a limit to the scale on which information - rich structures may be self - assembled .  to overcome this limitation one must either seek to make defined _ nonequilibrium _ structuresanz2007evidence , peters2009competing , scarlett2010computational , whitelam2014self , or else engineer an assembly pathway that is immune to the kinetic traps associated with strong native inter - component interactions .this paper focuses on the second of these options .we will focus on a 2d assembly scheme , a version of which has been used in experiment to create dna nanotile latticespark2006finite ( that scheme aimed to produce many copies of a small target structure ; here we will be concerned with making one copy of a large target structure ) . in this scheme, each stage of assembly involves the formation of distinct squares , stabilized by four internal bonds .each bond is in general mediated by multiple components . in the presence of native interactions only, each square can be formed only from four particular pieces .no matter the strength of the bonds between these four pieces , or the order in which bonds are made , no mis - binding can happen during construction of the square .in other words , this scheme is immune to the kinetic traps associated with strong native inter - component interactions . to enable such an assembly pathway one can either combine reactants in stagespark2006finite , or ` switch on ' inter - component interactions in stages , as illustrated in ( a ) . in the first stage of assembly , selected interactions between monomers promote the assembly of squares of size 4 .one then turns on additional interactions to promote the formation of second - stage structures , squares of size 16 , and so on , with stage structures being of size .this process can continue as long as structures can be brought into contact , so permitting the generation of an arbitrarily large information - rich array .this array could provide a template for building in the third dimension , with e.g. subsequent layers deposited one at a timeademartiri2015programmable ; see panel ( b ) .the chief drawback of this scheme it that it is , like other forms of hierarchical assemblyvillar2009self , haxton2013hierarchical , madge2015design , particularly susceptible to kinetic traps caused by ` undesigned ' or _ non - native _ interactions , by which we mean interactions that are not required to stabilize the intended structure . such interactions include those between components _ not _ designed to be adjacent in the assembled structure , or interactions between intended neighbors that promote binding incompatible with the target structure .non - native interactions might arise in experiment because of ` accidental ' complementarity between dna sequences , or because of nonspecific van der waals attractions . `non - hierarchical ' many - component self - assembly can proceed in the presence of such interactionske2012three , reinhardt2014numerical , provided that the difference in energy scales between the set of native and non - native interactions is large enoughedges2014growth , murugan2015undesired . but attractive non - native interactions are tolerated poorly by a stage - by - stage hierarchical scheme , in which the effective building block size is ever - increasing , because undesirable interactions between two clusters increase in strength in proportion to their surface area . therefore , above some size scale , kinetic trapping of a hierarchical assembly scheme in the presence of attractive non - native interactions would seem to be inevitable .    in view of these considerations ,the hierarchical scheme is likely to be inferior to a non - hierarchical scheme if non - native interactions are not intentionally suppressed .indeed , the authors of found for dna tiles that the ` square - upon - square ' hierarchical assembly scheme was inferior to a non - hierarchical scheme ( in which all component types were combined directly ) , even when used to make a relatively small array .despite this finding , we argue here that the hierarchical scheme can in principle be used to build larger information - rich structures than can the non - hierarchical scheme ( which can not tolerate strong native interactions ) , _ if _ one can suppress non - native interactions .perhaps this could be done by combining dna - mediated interactions with some form of inter - particle repulsion ( we discuss this point further in the conclusions section ) .if so , the stage - by - stage hierarchical assembly pathway offers a potential route to the assembly of large information - rich structures in thermal equilibrium .  in section [ model ]we introduce an off - lattice particle - based computer model that can be used to study stage - by - stage hierarchical assembly . in section [ results ]we show that simulations of this model confirm that stage - by - stage assembly results in error - free formation of a desired structure , even in the presence of arbitrarily strong native interactions .we show that the same scheme fares poorly in the presence of attractive non - native interactions .we also discuss the efficiency of a conventional self - assembly process in the presence of irreversible or reversible native interactions .we conclude in section [ conc ] .for the trajectory shown in , we plot as a function of time the fraction of the system s mass contained within ( natively - bonded ) clusters of size .the completion of each stage of assembly corresponds to the point at which each line reaches the value unity , at which point the interactions required to promote the next stage of assembly are automatically switched on .assembly went to completion at time . ]the idea of assembling structures in a stage - by - stage manner has been studied theoretically in the dna tile literaturedemaine2011one , patitz2010self , abel2010shape , where the possibility of extending stage - by - stage assembly to large scales was noted .the particular case of square - upon - square assembly sketched in ( a ) has been demonstrated in experiments in which dna tiles formed many distinct arrayspark2006finite . herewe study this process numerically , using off - lattice molecular simulation , in order to demonstrate that it can in principle be used to avoid kinetic traps associated with very strong native interactions .for the reasons discussed in section [ intro ] , such avoidance is a necessary feature of a process designed to make large information - rich structures in thermal equilibrium . we shall also show that this scheme is particularly vulnerable to kinetic trapping caused by non - native interactions .our model of stage - by - stage assembly comprises 1024 hard discs of radius on a smooth , two - dimensional substrate .each disc is of a distinct type , one of 1024 possibilities , labeled , with .discs are decorated with 4 sticky patches , each of opening angle , arranged in a regular way ( i.e. neighboring patch bisectors are separated by an angle ) , so as to allow self - assembly of a square lattice structure .as shown in ( a ) , patches are numbered 0 , 1 , 2 , 3 in a clockwise fashion .discs possess attractive pairwise interactions that are square - well in both anglekern2003fluid and range .if the centers of two discs and are separated by a distance that satisfies , and if the line joining disc centers cuts through one patch on each disc , then discs receive a pairwise energetic reward of , where  e = f(i , j , p;i,j,p ) _ native + ( 1-f(i , j , p;i,j,p))_non - native .  here and are the energy scales of native and non - native interactions . unless otherwise stated we took and .the function depends on the disc- and patch identities involved in the pairwise contact : is the patch number of disc involved in the pairwise contact , and is the patch number of disc involved in the pairwise contact . is chosen so that the structure shown in ( b ) is the thermodynamically stable one .this structure , the target for self - assembly , is a square lattice in which disc type is found at the intersection of the the column and row of the lattice , numbered from the bottom left .that is , the bottom left disc is of type ( 0,0 ) , and the top right disc is of type ( 31,31 ) .disc types are colored so that the target structure possesses a green - to - blue - to - red color gradient . to stabilize this structure we require to be 1 if patch 1 on disc interacts with patch 3 on disc , or if patch 0 on disc interacts with patch 2 on disc ; these are ` native ' interactions .otherwise is 0 , indicating a ` non - native ' interaction .in equation form these requirements read [ direct ] f(i , j , p;i,j,p ) = ( i+1,i ) ( j , j ) ( p,1 ) ( p,3 ) + + ( j+1,j)(i , i ) ( p,0 ) ( p,2 ) , where is 1 if , and 0 otherwise . in pictures , in general , patches are shown white when engaged in a native contact , and black when unbound or engaged in a non - native contact .  to allow assembly of the target structure in a hierarchical manner we further distinguish disc native interactions by ` stage ' . interactions up to stage the self - assembly of distinct squares of discs .thus , interactions of stage 1 can promote the formation of squares of 4 discs ( of 256 possible types ) ; stage 1 and 2 interactions allow the formation of 64 distinct square 16-mers ; stage 1 , 2 , and 3 interactions allow the formation of 16 distinct square 64-mers ; stage 1 , 2 , 3 and 4 interactions allow the formation of 4 distinct square 256-mers ; and stage 1,2,3,4 and 5 interactions allow the formation of one square of size 1024 .maximally - bonded configurations that result from interactions of all stages up to 2 , 4 and 5 are shown in ( b , c ) .    in equation formwe distinguish interactions that participate in different stages of assembly by modifying to read  [ hierarchical ] f(i , j , p;i,j,p ) = ( i+1,i ) ( j , j ) ( p,1 ) ( p,3 ) _n=1 ^ 5 _ n(i ) + ( j+1,j)(i , i ) ( p,0 ) ( p,2 ) _n=1 ^ 5 _ n(j ) ,  where if is a multiple of , and is unity otherwise .the parameter , which we control externally , is either 0 , signaling that stage interactions are ` turned off ' , or 1 , signaling that stage interactions are ` turned on ' .we shall restrict ourselves to the case in which implies for all , e.g. if stage 3 interactions are ` turned on ' ( ) , then so too are stage 1 and stage 2 interactions .we did simulations of 1024 discs , one of each type , in the ensemble .simulation boxes were square , with periodic boundaries , of size such that the disc hard - core packing fraction was 32% . to demonstrate the fact that error - free hierarchical assembly can occur in the presence of strong interactions we took , so that contacts , once made , could not be broken .disc structures therefore do not dissociate , although they possess internal flexibility on account of the finite range and angular specificity of disc - disc interactions .discs were initially randomly dispersed and oriented , subject to there being no hard - core overlaps .we used the virtual - move monte carlo algorithmwhitelam2007auk described in the appendix of to move discs according to an approximation of overdamped motion .translation and rotation of interacting clusters of discs occurs under this dynamical scheme , and the rate of cluster diffusion can be controlled to a degree by the user .we chose to reject moves of clusters of hydrodynamic radius so that , approximately , translational and rotational cluster diffusion constants scaled as and ( see the si of axton2012design ) .different choices may be appropriate for different types of surface .we chose the basic scale of disc displacement so that an isolated disc will diffuse a characteristic length equal to its own diameter in 33.3 monte carlo sweeps .  to promote stage - by - stage assembly we began simulations with only stage 1 interactions turned on , i.e. we set and for . when the first stage of assembly was complete , i.e. when the simulation box contained 256 distinct square 4-mers , we turned on stage 2 interactions , i.e. we set ( with the condition unchanged ). stage 2 interactions allow the 256 square 4-mers to assemble hierarchically into 64 distinct square 16-mers . when stage 2 of self - assembly was complete we turned on stage 3 interactions , and so on until stage 5 of self - assembly was complete .for comparison , we also carried out hierarchical simulations in which non - native patch - patch interactions ( those that return zero on the right - hand side of ) were nonzero ( ( b ) , , ) , and we did ` non - hierarchical ' simulations in which all native interactions were turned on from the start of the simulation , i.e. we set for ( , , ) .* stage - by - stage assembly works in the presence of arbitrarily strong native interactions . * in we show snapshots of simulation configurations , ordered clockwise from the top left , from a single dynamic trajectory of the stage - by - stage assembly procedure described in section [ model ] .the scheme works the result is the low - energy , 1024-particle structure with no errors even though native interactions are unbreakably strong ( non - native interactions are absent ) .assembly is a diffusion - limited process , and no mis - bound configurations exist .this scheme is also unaffected by the ` monomer starvation ' kinetic trap seen in the study of viral capsid self - assemblyndres2002model , hagan2006dynamic or in models of dna brick assembly if one aims to produce multiple copies of a target structuremadge2015design : all pieces have a prescribed set of binding partners , and so it is not possible to use up the monomer supply by making ( say ) a large number of trimers that can not be completed .one needs to wait long enough for components to find their native partners , but the result is guaranteed to be free of error .  in we show , as a function of time ( number of monte carlo sweeps ) , the fraction of the system s mass contained within natively - bonded clusters of size .the completion of each stage of assembly corresponds to the point at which the individual lines reach the value unity , at which point the interactions required to promote the next stage of assembly are automatically switched on . from trajectory to trajectory we found that that time required to complete each stage varied , the more so as assembly progressed .this variability can be seen in ( a ) , which shows for 4 independent trajectories the size of the largest ( natively - bonded ) cluster as a function of time . in panel( b ) we show the mean and standard deviation of the time required to complete each stage of assembly , averaged over 8 independent trajectories .stage completion time generally increases as assembly progresses , because larger clusters diffuse more slowly than smaller ones , even though the effective number of component types diminishes ( see appendix b ) .non - stage - by - stage self - assembly results in kinetic trapping in the presence of strong native interactions . herewe show a structure that results from self - assembly in the presence of the same set of interactions use to produce ; here , however , all stages of interaction were ` switched on ' from the start . ]* stage - by - stage assembly is vulnerable to kinetic traps associated with non - native attractions .* we have shown that the stage - by - stage assembly scheme works in the presence of strong native interactions . however , its major weakness is that it is susceptible to kinetic traps associated with non - native interactions .the red line ( b ) corresponds to simulations done in the presence of non - native interactions of energy , meaning that such an attraction exists between every patch - pair in the system that does not participate in a native interaction .native contacts are again unbreakably strong . at stage 4there is a slight slowing of assembly , relative to the case of zero non - native interaction , because of brief mutual adhesion of large clusters .  for stronger non - native interactions we found that assembly is arrested prior to stage 4 . in ( a )we show , as a function of time , the size of the largest natively - bonded cluster in the system , from independent trajectories run in the presence of attractive non - native interaction energies of , , and . in all cases ,native contacts are unbreakably strong .assembly in the presence of non - native interactions becomes very slow at some point , the earlier the stronger the interaction . in panel( b ) we show the fraction of possible native contacts that have been made at time for all trajectories ; these fractions show behavior similar to that of .  the cause of this dynamical slowdown can be seen clearly in configuration snapshots . in ( a )we show a time - ordered series of snapshots from a simulation run in the presence of attractive non - native contacts of energy .the square - upon - square hierarchical assembly process is subverted by a second organizational process , that of square 4-mers into a structure in which cluster types are arranged essentially at random ( compare the color pattern to that of the target structure ) . as a result, 4-mers combine to form natively - bonded higher - order structures only slowly . in panel ( b )we show pictures of structures assembled in the presence of attractive non - native contacts of energy , , and .these pictures show that assembly of compositionally - random structures occurs from smaller building blocks as non - native contact energy decreases .non - native interactions of energy cause a compositionally - nonspecific assembly of monomers , so preventing even stage 1 of the hierarchical assembly process from happening .* ` conventional ' self - assembly does not work in the presence of arbitrarily strong native interactions . *the hierarchical assembly procedure s chief virtue is its ability to tolerate arbitrarily strong native interactions . this ability is not common in molecular self - assembly , and is not shared by a self - assembly process involving the same set of components whosenative interactions are all turned on from the start of the simulation .a typical result of this process is shown in . even though only native interactions operate , if these form in the wrong order which , inevitably , they do then the target structure can not assemble because of steric obstructions .  however , conventional self - assembly done in the presence of reversible native interactions can result in the target structure with few errors . by running a large number of simulations with different values of the native interaction energy, we found that , for native interaction energy of 8.6 , nucleation happened after some delay but was still rapid enough to be seen in direct simulation .data from two such simulations are shown in and .some mistakes result when multiple nucleation events occur and the resulting large clusters fuse in a way that blocks certain binding sites .but assembly is largely successful , reiterating the results of refs . and  : nucleation and growth of an ` information - rich ' structure of about components can occur in the presence of reversible interactions of fixed strength .size of the largest natively - bound cluster in the system , for the two trajectories shown in. the green ( resp .cyan ) line here corresponds to the top ( resp .bottom ) panel in .the final yield of assembly is slightly lower for the case of the cyan / bottom trajectory . ]ensuring the thermodynamic stability of a structure made from distinguishable components requires the energy scale of inter - component interactions to grow in proportion to .but self - assembly done in the presence of strong interactions results in general in kinetic trapping , suggesting a limit to the size of a self - assembled structure built from distinguishable components . herewe have used simulation to show that the stage - by - stage assembly of squares demonstrated in experiment in can work even in the presence of arbitrarily strong interactions , as long as those interactions are the ` native ' ones required to stabilize the target structure .this property is not shared by most other forms of self - assembly , and is required if one wishes to make large , information - rich structures in equilibrium .note that we have assembled a single structure of size to demonstrate that error - free hierarchical assembly can occur even in the face of arbitrarily strong interactions .we stopped at stage 5 of the procedure because simulating larger systems is too time - consuming , but the hierarchical procedure can be used in principle to assemble a single `` information - rich '' structure of any size .one way to see this is to consider a visual proof by induction . shows the hierarchical assembly of a -particle structure from monomer building blocks .imagine now that we have such reactions occurring in parallel .provided that disc - disc native interactions are chosen in the appropriate manner , combining the results of all reactions would lead to the hierarchical assembly of a -particle structure : imagine a time - series like that shown in in which the smallest discs shown are not monomers but the -particle assemblies that result from the first set of reactions .and so on . size limitationsthen result from practical considerations .one such consideration is that the stage - by - stage scheme is particularly susceptible to kinetic trapping caused by attractive non - native interactions , and so these must be suppressed if the scheme is to be used to make large objects . with dna - mediated interactions alone , as noted in , it seems difficult to guarantee strong native interactions and very weak non - native ones , on account of accidental basepair complementarity .choosing complementary sequences at random , one would expect two ` non - complementary ' 4-letter sequences of length to be ` accidentally ' complementary in places on average . to suppress non - native interactions it may be necessary to equip components with repulsive interactions intermediate in energy scale between fully - complementary interactions and accidentally - complementary ones .if this can be done , then in principle the scheme presented in and simulated here can be used to make large objects . however , technical challenges exist , of which four are as follows ( more detailed discussions of several of these points can be found in the dna tile literaturedemaine2011one , patitz2010self , abel2010shape , park2006finite ) .first , one must be able to ` switch on ' interactions selectively .perhaps if native interactions correspond to a certain number of fully - complementary basepairs , then selected strands could initially be ` turned off ' or ` protected ' by partially - complementary single strands . at the required stage of the processthe system temperature could be raised slightly , so as to allow thermal dissociation of protector strands only .protector strands of subsequent stages must be successively more strongly bound , and all must be much less strongly bound than native inter - component bonds .second , one must be able to monitor the progress of assembly with sufficient precision to know when all the structures in the current stage have assembled ( or wait long enough to be sure that this has happened ) ; premature activation of the next stage s bonds could cause kinetic trapping .perhaps ` smarter ' ways to vary inter - component interactions without requiring explicit intervention by the user can be developed , by using novel feedback schemesklotsa2013controlling or other nonequilibrium controls of assemblybyun2013unconventional , han2013large , li2014crafting .third , large clusters diffuse slowly , particular when bound to a surface , and so it is likely that stirring the solution or shaking the surface will be required to allow assembly to proceed beyond a certain lengthscale ( shaking may also help break up kinetic traps caused by residual non - native interactions ) .fourth , this scheme confronts the challenges inherent to any scheme involving a large number of component types , such as difficulties of synthesis ( the number of dna labels must increase with system size , for instance ; see the si of ) and issues of long timescales .that said , shows that square - upon - square assembly reactions can be run in parallel : one could in principle combine the outcome of separate hierarchical processes as the starting point for the next stage of assembly , thereby achieving -fold speed - up relative to the case of sequential self - assembly .table [ table_summary ] compares conventional and hierarchical assembly schemes .we note finally that the scheme proposed here can not be applied ( without introducing the idea of twist - sensitive , protein - like interactionsmannige2009geometric , levy20063d ) to two - dimensional objects free to move in three - dimensional space : flip over a square and it can bind with its intended partner in a non - ideal way .a similar observation applies to three - dimensional structures .therefore , we speculate that building 3d information - rich structures may be best done by using a hierarchical scheme to generate large 2d surface - bound assemblies , and then building layer by layer on top of them .  [ cols="^,^",options="header " , ]   [ table_summary ]  this work was done at the molecular foundry at lawrence berkeley national laboratory , supported by the office of science , office of basic energy sciences , of the u.s .department of energy under contract no .de - ac0205ch11231 .imagine that our goal is to promote the self - assembly , from solution , of a solid , equilibrium structure made of components .components generally lose translational and rotational entropy upon going from solution to a solid structure , and so the energy scale of inter - component bonds in the solid structure must be large enough that a monomer s energetic gain upon solidification exceeds temperature times the entropic cost of its removal from solution .if components are distinguishable , and if the solid structure of interest conforms to one particular arrangement of components , then upon solidification one _ also _ loses an entropy per particle , because the number of accessible configurations of the solution phase is times that of its distinguishable counterpart .the inter - component bond strength must therefore grow on the scale of in order to ensure the stability of a distinguishable structure built from components .the average bond energy per particle must be at least greater than in the corresponding single - component case , where is the number of contacts made by each particle .for macroscopic , of order , this excess bond energy exceeds when ( appropriate to the case modeled here or the experiments of ) , and is about for .a similar result can be obtained by considering the microscopic dynamics of growth .say that in the indistinguishable case the rate of arrival of a component at a given point on a solid structure is , which is proportional to component concentration .say that the rate of departure of that particle from the structure is , where and is the number of bonds made by the particle . in general , the arrival rate must exceed the departure rate in order to ensure stability of the structure with respect to dissolution .now in the _ distinguishable _ case the arrival rate , for fixed total component concentration , is , because only about 1 in encounters involves the component designed to fit at that point on the structure ( now we imagine for simplicity that the number of components in solution is very large , and does not diminish upon binding ) . so upon going from indistinguishable to distinguishable the rate of arrival of material has dropped by a factor of .to guarantee structural stability , therefore , the rate of departure of material must drop by a similar factor , i.e. . hence stability requires that must increase in proportion to .we tested this expectation by building the 1024-particle target structure and simulating it ( in the same square simulations boxes that we used for dynamic simulations ) in the presence of native interactions only , and in the presence of native and nonnative interactions of equal strength . in the former case particlesmake energetic bonds only if they adopt one particular position in the structure , while in the latter case the energy of the system is invariant under particle permutation , or if particles are rotated by multiples of .we therefore expect the extra pair energy ( in units of ) required to stabilize the structure in the native - interaction case to be ( for ) . by performing simulations for different values of binding energies we determined that in the presence of native interactions the structure melted for bond energy values between 7.3 and 7.1 , while in the presence of native and nonnative interactions of equal strength the condensed structure melted for values of between 3.8 and 3.6 .although simulations of this nature do not give us a precise measure of structural stability , the difference between these two ` melting points ' is , consistent with our expectation of 3.66 .the same considerations also suggest that the growth rate of a distinguishable structure will be a factor of less than that of its indistinguishable counterpart , i.e. upon going from indistinguishable to distinguishable .let us assume that collisions between squares are uncorrelated events . during stage of assemblythese events occur with mean time , where for translation - limited events , and for rotation - limited events . to go from stage to stage of assembly theremust occur of order events .so the mean time to complete stage of assembly scales , very roughly , as if assembly is translation - limited , and if assembly is rotation - limited .the data of ( b ) lie between these extremes but appear to generally increase with , suggesting that assembly is dominated , especially at late stages , by the rotational component of diffusion .

Abstract: self - assembly in the laboratory can now yield ` information - rich ' nanostructures in which each component is of a distinct type and has a defined spatial position . ensuring the thermodynamic stability of such structures requires inter - component interaction energies to increase logarithmically with structure size , in order to counter the entropy gained upon mixing component types in solution . however , self - assembly in the presence of strong interactions results in general in kinetic trapping , so suggesting a limit to the size of an ( equilibrium ) structure that can be self - assembled from distinguishable components . here we study numerically a two - dimensional hierarchical assembly scheme already considered in experiment . we show that this scheme is immune to the kinetic traps associated with strong ` native ' interactions ( interactions designed to stabilize the intended structure ) , and so , in principle , offers a way to make large information - rich structures . in this scheme the size of an assembled structure scales exponentially with the stage of assembly , and assembly can continue as long as random motion is able to bring structures into contact . the resulting superstructure could provide a template for building in the third dimension . the chief drawback of this scheme is that it is particularly susceptible to kinetic traps that result from ` non - native ' interactions ( interactions not required to stabilize the intended structure ) ; the scale on which such a scheme can be realized therefore depends upon how effectively this latter kind of interaction can be suppressed .