Graphical Classification in Multi-Centrality-Index Diagrams for ... - MDPI

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Graphical Classification in Multi-Centrality-Index Diagrams for Complex Chemical Networks Yasutaka Mizui, Tetsuya Kojima, Shigeyuki Miyagi and Osamu Sakai *

ID

Electronic Systems Engineering, The University of Shiga Prefecture, 2500 Hassakacho, Hikone, Shiga 522-8533, Japan; [email protected] (Y.M.); [email protected] (T.K.); [email protected] (S.M.) * Correspondence: [email protected]; Tel.: +81-749-28-8382 Received: 22 October 2017; Accepted: 5 December 2017; Published: 9 December 2017

Abstract: Various sizes of chemical reaction network exist, from small graphs of linear networks with several inorganic species to huge complex networks composed of protein reactions or metabolic systems. Huge complex networks of organic substrates have been well studied using statistical properties such as degree distributions. However, when the size is relatively small, statistical data suffers from significant errors coming from irregular effects by species, and a macroscopic analysis is frequently unsuccessful. In this study, we demonstrate a graphical classification method for chemical networks that contain tens of species. Betweenness and closeness centrality indices of a graph can create a two-dimensional diagram with information of node distribution for a complex chemical network. This diagram successfully reveals systematic sharing of roles among species as a semi-statistical property in chemical reactions, and distinguishes it from the ones in random networks, which has no functional node distributions. This analytical approach is applicable for rapid and approximate understanding of complex chemical network systems such as plasma-enhanced reactions as well as visualization and classification of other graphs. Keywords: chemical reaction network; centrality index; statistical analysis; random graph

1. Introduction Graph theory provides for us a graphical approach to a system containing various elements with connections between them [1,2]. Chemical reaction networks are one such system, and small and simple reaction systems are visualized in linear or small graphs with sufficient understanding of reaction procedures [3–5]. On the other hand, the networks of protein and metabolic systems in a biological cell are quite complicated due to their numbers of nodes (at least, more than 1000), so that not only visualization as a graph but also statistical properties such as degree distributions are representative for characterizing their complexity [6,7]. Recently, we performed graph visualization for plasma-enhanced chemical reactions [8,9]. In low-temperature reactive plasma, high-energy electrons trigger a number of simultaneous dissociations of mother molecules, and its chemistry is more complex than other chemical systems in artificial environments for chemical plants [10–12]. After definitions of a node (for one species) and an edge (for each reaction) for a display in a graph, centrality indices of nodes derived from the graph work as representatives of chemical roles in the system, such as agents, intermediates, and products. However, except for such microscopic points of view, approaches have not been accomplished for describing macroscopic properties of graphs for chemical reaction networks that contain several tens of species. Such relatively small-sized chemical network systems create rich outputs despite limited numbers of nodes and edges in a graph. For instance, information processing in biochemical reactions revealed collective behaviors that can be interpreted using interactions among analogical spins, leading to

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similar features to electronic information processing or mechanical systems [13,14]. Another example of graphical network approaches for medium-sized chemical complexity was on numerical calculations of rate equations in plasma-enhanced chemical reactions, and the calculated results were visualized in reaction pathways in a graph to summarize complicated time evolutions of densities of species [15]. In comparison with the previous achievements based on numerical finite-difference methods [10–12,15], direct visualization of reactions based on graph theory is applied here, and we focus on graphical classification of nodes or species on statistical aspects that are missing in our previous studies [8,9]. In this study, for such systems as medium-sized chemical reaction networks, which are neither so small as several numbers of reactions nor too large with more than 1000 species, we demonstrate suitable macroscopic measures by graphical diagrams based on multi-centrality indices. Using such diagrams, semi-statistical properties with confirmation of systematic and global structures in the corresponding system can be deduced, even if the total number of nodes is limited to less than 100, as well as identifications of roles in each species, such as agents, intermediates and products. This method of classification of species in macroscopic points of view provides us with insight and understandings about predictions of global properties of chemical reactions for approximate designs of upcoming chemical reactors and rapid selection of mother chemicals for products when accurate computer aided designs are not available. In Section 2, using reactions in silane and methane plasmas, we show the diagrams with axes of betweenness and closeness centrality indices. Using such a diagram, we can understand both macroscopic and microscopic properties of graphs with tens of nodes. In Section 3, we compare these two examples with random graphs, and discuss the validities of this graphical characterization. 2. Analysis of Reaction Networks in Plasma Chemistry As examples, we use two plasma-enhanced chemical networks reported in [10,11]. We performed some analyses of methane plasma [8] and silane plasma [9] using some centrality indices, but they did not include macroscopic measures of complex chemical networks. Here, we proceed to study them using the same reaction systems to obtain their macroscopic and microscopic properties simultaneously. Table A1 in Appendix A shows active species in silane plasma with temporary indicator numbers, where we use reaction sets in [11]. There are a wide variety of species that originate initially from two species: SiH4 and Ar. The number of species or nodes in the corresponding graph is 58, and that of reactions or edges is 222. The reason why we can observe such rich diversity is based on high-energy electrons whose energy spreads up to 20 eV with electron temperature of 1–5 eV; kinetic energy with 1–20 eV induces most of reactions of decomposition, dissociation and ionization [16]. Almost all reactions are bi-molecule, and we treat all reactions as irreversible ones. To convert a chemical reaction into a graph, when we handle the following reaction, A + B → D + E,

(1)

and use the names of species in reactions as node indicators, we set directed edges from node A to node D, from node A to node E, from node B to node D and from node B to node E, as shown in Figure 1a. When one performs graph representation for an underlying system of interest, elements of the system becomes nodes and interactions between elements are displayed as edges. For chemical reactions, as interpreted in [3], species becomes nodes, and not a simple co-existence but an agent-product relation is suitable for a directed edge, leading to a form of representation in Figure 1a. In general chemical reactions for inorganic molecules, since roles of agents, intermediates and products are fairly clear with energy consumption as driving forces, regular graphs or graphs with frequent cycles are less important, and monotone graphs or trees with several nodes are sufficient in many cases [3–5]. However, in reactions enhanced by energetic electrons in plasma, cycles appear with their various sizes with increasing values of clustering coefficients [8], which makes a role of each species more complicated.

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After completion of an edge list for a graph, we calculate several centrality indices. In this study, we use betweenness centrality index C b and closeness centrality index C c . C b is defined as [2]: After completion of an edge list for a graph, we calculate several centrality indices. In this study, g jk i  i   C bcloseness , we use betweenness centrality index Cb and centrality index Cc . Cb is defined as [2]: (2) g jk j ,k j k i g jk (i ) Cb (i ) = ∑ , (2) g jk where i , j and k are species/node indicator j,k numbers and g jk is number of the shortest paths

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j 6 = k 6 =i

between nodes j and k . g jk i  indicates number of g jk passing through node i . C c is where i, j and k are species/node indicator numbers and g jk is number of the shortest paths between defined as [2]: nodes j and k. g jk (i ) indicates number of g jk passing through node i. Cc is defined as [2]: 1 C c i   1 , (3) Cc (i ) =  dij , (3) ∑j dij j

where dij is distance or edge numbers from node i to j along the shortest path. Since we where dij is distance or edge numbers from node i to j along the shortest path. Since we consider consider directed graphs, no path may exist from node i to j ; in such a case, dij is defined as directed graphs, no path may exist from node i to j; in such a case, dij is defined as number of all nodes. C ctopological number of all nodes. This fact indicates that, although representsaspects, one of the topological aspects, This fact indicates that, although Cc represents one of the it includes a measure of agents in chemical reactions. In our previous studies [8,9], we calculated simplified PageRank it includes a measure of agents in chemical reactions. In our previous studies [8,9], we calculated values [17], which represent information on roles of species, i.e., in a microscopic point of view. thisa simplified PageRank values [17], which represent information on roles of species, i.e.,In in study, we put more emphasis on analysis of topological and statistical properties of graphs, and Cb microscopic point of view. In this study, we put more emphasis on analysis of topological and and Cc are properties suitable forofthis purpose. statistical graphs, and C b and C c are suitable for this purpose. Figure 1b shows the graph of Figure 1b shows the graph ofreactions reactionswith withspecies speciesin insilane silaneplasma. plasma.Since Sincethe thenumber numberof ofnodes nodes isisnot large, we can identify an individual species. On the other hand, the number of edges is not large, we can identify an individual species. On the other hand, the number of edges ispretty pretty large, cannot trace all of For complex networks with huge canone recognize neither large,and andwe we cannot trace allthem. of them. For complex networks with size, hugeone size, can recognize of them, and they are mainly analyzed using statistical properties (i.e., Power-law tails in neither of them, and they are mainly analyzed using statistical properties (i.e., Power-law degree tails in distributions [18]) from a macroscopic point of point view, of neglecting identification of microscopic roles of degree distributions [18]) from a macroscopic view, neglecting identification of microscopic each our case, however, the number nodes isofinsufficient to obtain smooth statistical rolesspecies. of each In species. In our case, however, theofnumber nodes is insufficient to obtain smooth trends. Figure 2 shows the in-degree and out-degree distribution. Nodes scatter broadly in both sides statistical trends. Figure 2 shows the in-degree and out-degree distribution. Nodes scatter broadly in of thesides dashed linedashed that shows equal numbers the in-degrees and the out-degrees, which indicates both of the line that shows equalofnumbers of the in-degrees and the out-degrees, which both agents, with largerwith values of the out-degree, and products presentare in present balance.inWhen we When carry indicates both agents, larger values of the out-degree, andare products balance. out searches for global information in Figure 2, although it indicates that this degree distribution is not we carry out searches for global information in Figure 2, although it indicates that this degree the Poisson one, wethe cannot clarify statistical characteristic at thischaracteristic moment. distribution is not Poisson one,itswe cannot clarify its statistical at this moment.

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Figure 1. Graph for chemical reaction network in silane plasma. (a) Example of formation of nodes Figure 1. Graph for chemical reaction network in silane plasma. (a) Example of formation of nodes and and edges for one typical reaction; and (b) whole structure of graph. Nodes represent species in edges for one typical reaction; and (b) whole structure of graph. Nodes represent species in chemical chemical reactions, and edges start from agents and ends at products of each reaction. Chemical reactions, and edges start from agents and ends at products of each reaction. Chemical reactions are reactions are[11]. listed in Ref.red [11]. Closed red diamonds indicate species displayed in 3. Figures 2 and 3. listed in Ref. Closed diamonds indicate species displayed in Figures 2 and

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In-degree Figure 2. nodes) in in silane plasma displayed in 2. Distributions Distributions of ofin-degree in-degreeand andout-degree out-degreeofofspecies species(or (or nodes) silane plasma displayed Figure 1. Inset dashed lineline indicates equal cases of in-degrees andand out-degrees. in Figure 1. Inset dashed indicates equal cases of in-degrees out-degrees.

Here, we wepropose proposeaadiagram diagramwith withC Cand and to analyze a macroscopic property of a and graph c Here, Cc toCanalyze a macroscopic property of a graph to b b and to microscopic obtain microscopic roles ofsimultaneously, species simultaneously, shown 3inforFigure 3 forinthe graph in obtain roles of species as shown as in Figure the graph Figure 1. In Figure 1.both In general, and C c show centrality indices, andto a node closerhas to the centervalue, has a C b centrality general, Cb andboth Cc show indices, and a node closer the center a higher and positive is roughly expected between them. However, this diagram includes various higher value,correlation and positive correlation is roughly expected between them. However, this diagram types of information, below. For instance, nodes H and Si H are located in the includes various typesasofdescribed information, as described below. ForSiinstance, nodes and Si H Si5H 9 5 4 11 4 9 11 periphery region of Figure 1. Both of them are in the low-C area in Figure 3, but their positions in are located in the periphery region of Figure 1. Both of themb are in the low- C b area in Figure 3, but the areindifferent. Si4 Hare three directed has paths from other species and another to Si4 H 9 has 10 . theirdiagram positions the diagram different. three directed paths from other species and Si4H9 The degree, the total number of edges, is limited to 4 as shown in Figure 2, and we can recognize another to Si H10 . The degree, the total number of edges, is limited to 4 as shown in Figure 2, and the edges and4 the nodes around Si4 H9 in Figure 1. Consequently, Si4 H9 is not a significant species we can recognize edges and around in Figure 1. Consequently, Si H is not a 4H9 that affects many the reactions, and the it isnodes located in the Si low-C b area in Figure 3, while Cc of4 Si94 H9 is in significant many reactions, and it is located in the low- C b area Figure 3, while the middlespecies range. that Si5 Haffects oneindirected path to 11 also has three directed paths from other species and of is in the middle range. also has three directed paths from other species and one Cc Si Si5H another species, Si5 H12 . It is in the low-C area with very low Cc . The difference of locations between b 11 4H9 Si H and Si H is destination of out-degree Si4CHb10area is more in reactions than Si5 Hof directed path to another species, . It is inedges; the lowwithactive very low Si H 5 11 4 9 12 C c . The difference 5 12 in our model, and we can clarify such a point using this diagram. Note that both of them locations between Si4H9 and Si5H11 is destination of out-degree edges; Si4H10 is more have activethe in same statistical values of degrees, and they are in the same position in the degree distribution that is a reactions than Si5H12 in our model, and we can clarify such a point using this diagram. Note that conventional classification in theory of complex network [2]. both of them have the same statistical values of degrees, and they are in the same position in the From a macroscopic point of view, this diagram displays semi-statistical properties of the graph degree distribution that is a conventional classification in theory of complex network [2]. as a visual classification tool. The overall distribution of data points displayed here is fairly uniform From a macroscopic point of view, this diagram displays semi-statistical properties of the graph along the log(Cb ) axis, and it covers the range in 4 orders of magnitude of Cb . The Six Hy system, as a visual classification tool. The overall distribution of data points displayed here is fairly uniform which includes main species coming from the mother gas, SiH4 , is distributed throughout the entire along the logC b  axis, and it covers the range in 4 orders of magnitude of C b . The Six Hy system, range. The H system, which also originates from SiH4 , scatters in all ranges. The Ar system is in a which includesarea mainasspecies coming from thewith mother gas, SiH distributed entire less important Cb values. In the area high-value C4b,, is stable speciesthroughout such as SiHthe 4 and H2 range.and Theelectrons H system, alsooforiginates from SiH systemisisthe in amost less exist, arewhich also one the highest-value species bothininall Cbranges. and CcThe . SiHAr 3 , which 4 , scatters important precursor for Si thin film deposition using silane plasma, is also in this high-value region. area as C b values. In the area with high-value C b , stable species such as SiH4 and H2 In the and caseelectrons of the linear Cb axis the inset of Figure datainpoints are concentrated around 0, exist, are also oneshown of the in highest-value species3,both is the C c . SiH3 , which C b and and a few of them scatter in the area of high Cb , which is not a sufficient visualization of the graph. most important precursor for Si thin film deposition using silane plasma, is also in this high-value A logarithmic plot for Cb is a key manner for graphs for reaction paths and visual classifications of region. In the case of the linear C b axis shown in the inset of Figure 3, data points are concentrated chemical reactions, in particular, in plasma-enhanced chemistry. around 0, and a few of them scatter in the area of high C b , which is not a sufficient visualization of the graph. A logarithmic plot for C b is a key manner for graphs for reaction paths and visual classifications of chemical reactions, in particular, in plasma-enhanced chemistry.

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Betweenness centrality CB Figure Figure3.3.Diagram Diagramofofbetweenness-closeness betweenness-closenesscentrality centralityindices indicesof ofreaction reactionnetwork networkininsilane silaneplasma plasma displayed in Figure 1. Inset is diagram in linear scales. displayed in Figure 1. Inset is diagram in linear scales.

Anotherfact factthat thatwe wecan candeduce deducefrom fromthis thisdiagram diagramisisrough roughestimation estimationon ondifferences differencescoming coming Another fromroles rolesof ofspecies. species. Values Valuesof ofCC , whichare areshown shownalong alongthe thevertical verticalaxis, axis,become becomehigh highwhen whenthe the from c ,c which out-degree a given species is largeristhan its in-degree Figure 2 (see to confirm these out-degreeof of a given species larger than its(see in-degree Figure 2 correspondences). to confirm these For instance, Cc of SiH is larger C than that of H : the ratio, in-degree/out-degree of SiH4 , is 19/34, correspondences). For 4instance, of is larger than that of : the ratio, in-degree/out-degree H SiH 2 c 2 4 while that of H is 33/19. Electrons have also unbalanced values: the ratio is 8/34. These indicate 2 of SiH4 , is 19/34, while that of H2 is 33/19. Electrons have also unbalanced values: thefacts ratio is 8/34. that species mainly working as agents (on the left-hand side of chemical reactions, such as A and B in These facts indicate that species mainly working as agents (on the left-hand side of chemical reactions, Reaction (1), having larger out-degrees) are in the upper area of C . Products in chemical reactions c such as A and B in Reaction (1), having larger out-degrees) are in the upper area of C c . Products tend to be in the lower area. in chemical reactions tend to be in the lower area. Figure 4 shows another example of chemical reactions, CH4 system in methane plasma. In [8], Figure 4 shows another example of chemical reactions, CH system in methane plasma. In [8], we treat species except ions, but here we include all species listed4 in Ref. [10], shown in Table A2 in we treat species except ions, but here we include all species listed in Ref. [10], shown in Table A2 in Appendix A. The log(Cb ) − Cc diagram for the graph in Figure 4 is shown in Figure 5, which is quite Appendix A. The log(C b ) − C c diagram for the graph in Figure 4 is shown in Figure 5, which is quite similar to that in Figure 3; the area of log(Cb ) is quite wide in comparison with the width of the Cc quite in comparison the width the are similarAlso, to that in Figure the area of have Cc log(Cbeen b ) isso range. most of the 3; species that far wide pointed out on theirwith importance of of roles range. Also, of the species have beenpoint so farofpointed out onalso their importance of roles are located in the most higher-C In that a microscopic view, we can find similar points to the b range. located in the plasma. higher- CFor In aCH microscopic of the view, we can also findtosimilar points case of silane instance, similar locations SiH4 and SiHto3 the in 3 are in 4 and CHpoint b range. Figure 3, respectively. Our previous report [8] in which the centrality indices similar to PageRank [17] case of silane plasma. For instance, CH4 and CH3 are in the similar locations to SiH4 and SiH3 indicate of CH the result here the one in Ref. [8]. to There are a 3 , and in Figureimportance 3, respectively. Our previous report [8] is in consistent which the with centrality indices similar PageRank few points that are different from the case of silane plasma, such as the location of electrons between [17] indicate importance of CH3 , and the result here is consistent with the one in Ref. [8]. There are a Figures 3 and 5, although such a feature may be a factor coming from each specific system. few points that are different from the case of silane plasma, such as the location of electrons between Two results shown here indicate that complex reactions in plasma chemistry can be visualized Figures 3 and 5, although such a feature may be a factor coming from each specific system. in a log(Cb ) − Cc diagram, and both macroscopic and microscopic properties are derived from it. In particular, since plasma chemistry includes various levels of roles as well as wide range of contribution frequencies to reactions, the range of node distributions along the log(Cb ) axis is quite wide. Cb , given by Equation (2), includes information as reaction connection between species, while Cc indicates rather simple information about a location in a network.

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Figure 4. Graph for chemical reaction network in methane plasma. Nodes represent species in Figure 4. Graph Graphfor forchemical chemical reaction network in methane plasma. represent species in Figure 4. reaction network in methane plasma. NodesNodes represent species in chemical chemical reactions, and edges start from agents and ends at products of each reaction. Chemical chemical reactions, and edges start from agents and ends at products of each reaction. Chemical reactions, and edges start from agents and ends at products of each reaction. Chemical reactions are reactions are listed in Ref. [10]. Closed red diamonds indicate species displayed in Figure 4. reactions are listed in Ref. red [10].diamonds Closed red diamonds indicate species displayed listed in Ref. [10]. Closed indicate species displayed in Figure 4. in Figure 4. 0.010 0.010

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Figure 5. 5. Diagram Diagramof ofbetweenness-closeness betweenness-closeness centrality centrality indices indices of of reaction reaction network network in in methane methane plasma plasma Figure Figure 5. Diagram of betweenness-closeness centrality indices of reaction network in methane plasma displayed in Figure 3. Inset is diagram in linear scales. displayed in Figure 3. Inset is diagram in linear scales. displayed in Figure 3. Inset is diagram in linear scales.

3. Discussion Two results shown here indicate that complex reactions in plasma chemistry can be visualized Two results shown here indicate that complex reactions in plasma chemistry can be visualized in a log(C b ) − C c diagram, and both macroscopic and microscopic properties are derived from it. In 5 showand wide-range distributions nodes on the log(Cb ) are axis, but it might arise in a Figures diagram, both macroscopic and of microscopic properties derived from it. In logC b   C3c and particular, since plasmathat chemistry includes various levels ofweroles as well as wide range of from simple randomness also exists in random graphs. Here compare such tendencies to those particular, since plasma chemistry includes various levels of roles as well as wide range of contribution frequencies to reactions, the range of node distributions along the axis is quite log(C b ) in arbitrary random graphs. contribution frequencies to reactions, the range of node distributions along the log C b  axis is quite wide.We , given by Equation (2), includes information as reaction connection between species, while C bfixed the numbers of nodes and edges to the ones in Figure 1, and created random graphswhile with wide. C b , given by Equation (2), includes information as reaction connection between species, directed edgesrather in computation. Figure 6 shows examples, and we cannot see any common points simple information about athree location in a network. C c indicates C c indicates rather simple information about a location in a network. in allocation of roles for specific species. In all log(Cb ) − Cc diagrams shown in Figure 7, the values C Discussion and Cc for the nodes are around a certain area. In particular, the values of log(Cb ) is in a range 3. 3. bDiscussion with approximately one order of magnitude except for a few nodes. This is attributed to the fact that Figures 3 and 5 show wide-range distributions of nodes on the log (C b ) axis, but it might arise Figures 3 and 5graphs show wide-range distributions of nodes degrees of random are in the Poisson distributions [18].on the logC b  axis, but it might arise from simple randomness that also exists in random graphs. Here we compare such tendencies to from simple randomness that also exists in random graphs. Here we compare such tendencies to those in arbitrary random graphs. those in arbitrary random graphs. We fixed the numbers of nodes and edges to the ones in Figure 1, and created random graphs We fixed the numbers of nodes and edges to the ones in Figure 1, and created random graphs with directed edges in computation. Figure 6 shows three examples, and we cannot see any common with directed edges in computation. Figure 6 shows three examples, and we cannot see any common

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points in allocation of roles for specific species. In all logC b   C c diagrams shown in Figure 7, the values C b and C c for the nodes are around a certain area. In particular, the values of logC b  is in Symmetry 309 7 ofthe 11 a range 2017, with9,approximately one order of magnitude except for a few nodes. This is attributed to fact that degrees of random graphs are in the Poisson distributions [18]. To compare compare spectra spectra in in the the log scale, the the data data in in Figures Figures 3, 3, 55 and and 77 are are summarized summarized as as log(C C b) scale, To b P ininFigure histograms of Figure8.8. Values Values of of the the nodes nodes in in histograms of cumulative cumulative probabilities probabilities or or relative relative densities densities P 2 2 random graphs are localized around , and almost no changes among these three plots. This is 10 random graphs are localized around 10 , and almost no changes among these three plots. This is attributed to the fact that degrees of random graphs are in the Poisson distribution [18], and these attributed to the fact that degrees of random graphs are in the Poisson distribution [18], and these graphs have have certain certain averaged averaged profiles profiles of of parameters parameters with with some some deviations. deviations. On On the the other other hand, hand, the the graphs spectrum of silane plasma chemistry with the same numbers of the nodes and the edges is quite broad. spectrum of silane plasma chemistry with the same numbers of the nodes and the edges is quite The case ofcase methane plasma chemistry showsshows similar tendencies. This This comparison implies that broad. The of methane plasma chemistry similar tendencies. comparison implies networks of complex plasma chemistry include self-arranged systematic roles in the constituent that networks of complex plasma chemistry include self-arranged systematic roles in the constituent species. The The roles rolesofofagents agentsand andproducts productsare are partially distinguished from scattering along Cc species. partially distinguished from scattering along the the Cc axis axis the diagram. A wide range of(Clog a number of levels of intermediate C b  indicates in theindiagram. A wide range of log a number of levels of intermediate rolesroles fromfrom one b ) indicates species to another along along successive sequential reactions, where the levelsthe arelevels from are inevitable functions one species to another successive sequential reactions, where from inevitable in reactions less-frequent contributions for sub-products creations, etc. functions into reactions to less-frequent contributions for sub-products creations, etc.

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(c) Figure 6. Random graphs created randomly in computation (a–c). Figure 6. Random graphs created randomly in computation (a–c).

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Figure 7. Diagrams of betweenness-closeness betweenness-closeness centralityindices. indices. (a–c)corresponds corresponds to (a–c)ininFigure Figure Figure 7. 7. Diagrams Diagrams of Figure betweenness-closeness centrality centrality indices.(a–c) (a–c) correspondstoto(a–c) (a–c) in Figure6, 6, respectively. Inset is diagram in linear scales. respectively. Inset is is diagram in in linear scales. 6, respectively. Inset diagram linear scales.

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Figure 8. Histograms of cumulative probabilities or relative densities as function of betweenness Figure 8. 8. Histograms Histograms of of cumulative cumulative probabilities probabilities or or relative relative densities densities as as function function of of betweenness betweenness Figure centrality index. Data points are from Figures 3, 5 and 7. centrality index. index. Data Datapoints points are are from from Figures Figures 3, 3, 55 and and 7. 7. centrality

The above descriptions are based on the data set on silane and methane plasma chemistry. If The onon silane and methane plasma chemistry. If The above above descriptions descriptionsare arebased basedon onthe thedata datasetset silane and methane plasma chemistry. another sufficient set of reactions is available, one can perform similar data analysis on complex another sufficient set of reactions is available, one can perform similar data analysis on complex If another sufficient set of reactions is available, one can perform similar data analysis on complex chemical networks. As shown here, approximate estimations of roles on species and global balance chemical shown here, here, approximate approximateestimations estimationsofofroles roleson onspecies speciesand and global balance chemical networks. networks. As As shown global balance in in the reaction system are beneficial for various purposes such as rapid analysis of robustness of a in reaction system beneficial for various purposes such as analysis rapid analysis of robustness of a thethe reaction system are are beneficial for various purposes such as rapid of robustness of a reaction reaction system against impurity mixing. Another feasible and interesting contribution is selection of reaction system impurity against impurity and interesting contribution is selection of system against mixing. mixing. AnotherAnother feasible feasible and interesting contribution is selection of species species for monitoring an industrial chemical reactor; behaviors of the limited number of the species species for monitoring an industrial reactor; behaviors of thenumber limitedofnumber of the species for monitoring an industrial chemicalchemical reactor; behaviors of the limited the species detectable detectable on the edge of reaction space can predict the ongoing entire system working on by detectable on the edge of reaction space can predict the ongoing entire system working on by comparing values of indices with the detected signals. comparing values of indices with the detected signals.

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on the edge of reaction space can predict the ongoing entire system working on by comparing values of indices with the detected signals. This kind of graph analysis using diagrams composed of multi-centrality indices is applicable for analyses of other networks. In particular, networks that include less than 100 nodes are suitable for this method to investigate both macroscopic topologies and microscopic identifications of nodes. This is also applicable for larger networks to clarify roles of nodes by specifying their positions in a rough distribution of node data in a diagram. For instance, other physical and chemical processes can be analyzed by this approach when they are described in rate equations, since chemical reactions are given in one type of rate equation and successfully interpreted in this study. 4. Conclusions A diagram with axes of multi-centrality indices works to clarify both macroscopic structures of the graph and microscopic features such as roles of a species. Plasma-enhanced chemical reactions, in which tens of chemical reactions take place in parallel, are analyzed using a diagram based on Cb and Cc indices. The derived diagram indicates, from its macroscopic distribution of node values, that nodes have a wide variety of roles that are quite different from random classification. It also shows some roles of specific nodes by their locations, such as roles of products, agents, and intermediates. For plasma chemistry, this study provides us a measure for statistical properties for tens or hundreds of species in reaction systems, and for graph analysis, this analysis will become a practical and applicable method for understanding such medium-sized graphs. Acknowledgments: One of the authors (Osamu Sakai) thanks Professor Tomoyuki Murakami at Seikei University, Professor Mark J. Kushner at University of Michigan, and Professor Tsuyoshi Akiyama at The University of Shiga Prefecture for their useful comments in this study. This study was partly supported by Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology, Japan (JSPS KAKENHI Grant Number JP16K13711). Author Contributions: Yasutaka Mizui performed main analysis shown here and wrote the paper by drawing figures. Tetsuya Kojima performed numerical calculation for random graphs. Shigeyuki Miyagi prepared for calculation environments and gave advices on analysis procedures including error corrections. Osamu Sakai conceived and designed the research activities, and wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.

Appendix A The chemical reactions we analyze in this study are in two sets: silane (SiH4 ) and methane (CH4 ) plasma chemical reactions. All species treated in Figures 1–3 are shown in Table A1, and those in Figures 4 and 5 are in Table A2; they are listed in [10,11], respectively. Table A1. List of species in silane plasma chemical reactions [11]. Except charge indicators “+” and “−”, other symbols like “*” and “v” mean various electronic and mechanical excitation levels of species. M indicates arbitrary species except ions. Number

Species

Number

Species

0 1 2 3 4 5 6 7 8 9 10 11

Ar Ar * Ar ** Ar+ ArH+ e− H H* H+ H2 H2 (v(1)) H2 (v(2))

31 32 33 34 35 36 37 38 39 40 41 42

Si2 H6 ** Si2 H6 + Si2 H7 + Si3 H4 + Si3 H6 + Si3 H7 Si3 H8 Si4 H10 Si4 H2 + Si4 H6 + Si4 H8 + Si4 H9

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Table A1. Cont. Number

Species

Number

Species

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

H2 * H2 + H3 + M Si Si+ Si2 H+ Si2 H2 Si2 H2 + Si2 H3 Si2 H3 + Si2 H4 Si2 H4 + Si2 H5 Si2 H5 + Si2 H6 Si2 H6 (v(1,3)) Si2 H6 (v(2,4)) Si2 H6 *

43 44 45 46 47 48 49 50 51 52 53 54 55 56 57

Si5 H10 + Si5 H11 Si5 H12 Si5 H4 + SiH SiH+ SiH2 SiH2 − SiH2 + SiH3 SiH3 − SiH3 + SiH4 SiH4 (v(1,3)) SiH4 (v(2,4))

Table A2. List of species in methane plasma chemical reactions [10]. Except charge indicator “+”, other symbols like “*” mean various electronic and mechanical excitation levels of species. M indicates arbitrary species except ions. Number

Species

Number

Species

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

CH4 H CH3 CH2 C2 H6 * M CH C2 H5 * C C2 H4 * C2 H6 C2 H5 C2 H4 C2 H3 C2 H2 C2 H C+ CH+ H2 CH2 +

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

CH3 + CH4 + C2 H2 + C2 H3 + C2 H4 + C2 H5 + H2 + H3 + e− C4 H3 C4 H2 CH5 + C3 H4 + C3 H5 + C4 H5 + C4 H8 + C3 H6 + C3 H7 + C4 H7 + C4 H9 +

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