Using metagraph approach for complex domains description

Using metagraph approach for complex domains description
 © Valeriy M. Chernenkiy, © Yuriy E. Gapanyuk, © Georgiy I. Revunkov,
 © Yuriy T. Kaganov, © Yuriy S. Fedorenko, © Svetlana V. Minakova
 Bauman Moscow State Technical University,
 Moscow, Russia
 chernen@bmstu.ru, gapyu@bmstu.ru, revunkov@bmstu.ru,
 kaganov.y.t@bmstu.ru, fedyura11235@mail.ru, morgana_93@mail.ru
 Abstract. This paper proposes an approach for complex domains description using complex network
 models with emergence. The advantages of metagraph approach are discussed. The formal definitions of the
 metagraph data model and metagraph agent model is given. The examples of data metagraph and metagraph
 rule agent are discussed. The metagraph and hypergraph models comparison is given. It is shown that the
 hypergraph model does not fully implement the emergence principle. The metagraph and hypernetwork
 models comparison is given. It is shown that the metagraph model is more flexible than hypernetwork model.
 Two examples of complex domains description using metagraph approach are discussed: neural network
 representation and modeling the polypeptide chain synthesis. The textual representation of metagraph model
 using predicate approach is given.
 Keywords: metagraph, metavertex, metaedge, hypergraph, hypernetwork, neural network, polypeptide
 chain, lambda architecture.
1 Introduction authors [2]. According to [2]: = ⟨ , , , ⟩,
 where – metagraph; V – set of metagraph vertices;
 Currently, models based on complex networks are MV – set of metagraph metavertices; E – set of metagraph
increasingly used in various fields of science from edges, ME – set of metagraph metaedges.
mathematics and computer science to biology and A metagraph vertex is described by the set of
sociology. This is not surprising because the domains are attributes: = { }, ∈ , where – metagraph
becoming more and more complex. vertex; – attribute.
 Therefore, now it is important to offer not only a A metagraph edge is described by the set of attributes,
model that is capable of storing and processing Big Data the source and destination vertices and edge direction
but also a model that is capable of handling the flag: = ⟨ , , , { }⟩, ∈ , = | ,
complexity of data. That is why the development of a where – metagraph edge; – source vertex
universal model for complex domains description is an (metavertex) of the edge; – destination vertex
actual task. (metavertex) of the edge; eo – edge direction flag
 One of the varieties of such models is “complex (eo=true – directed edge, eo=false – undirected edge);
networks with emergence”. The emergent element means atrk – attribute.
a whole that cannot be separated into its component parts.
 The metagraph fragment: = { }, ∈ ( ∪
 As far as the authors know, currently there are two
“complex networks with emergence” models: ∪ ∪ ), where – metagraph fragment; 
hypernetworks and metagraphs. The hypernetwork – an element that belongs to union of vertices,
model is mature and it helps to understand many aspects metavertices, edges and metaedges.
of complex networks with an emergence. The metagraph metavertex: =
 But from the author's point of view, the metagraph ⟨{ }, ⟩, ∈ , where – metagraph
model is more flexible and convenient for use in metavertex belongs to set of metagraph metavertices MV;
information systems. – attribute, – metagraph fragment.
 This paper discusses the metagraph model and Thus, a metavertex in addition to the attributes
compares it with other complex graph models. includes a fragment of the metagraph. The presence of
 private attributes and connections for a metavertex is
2 Complex networks models comparison distinguishing feature of a metagraph. It makes the
 In this section, the metagraph model will be formally definition of metagraph to be holonic – a metavertex may
described and it will be compared with hypergraph and include a number of lower-level elements and in turn,
hypernetwork models. may be included in a number of higher level elements.
 From the general system theory point of view, a
2.1 Metagraph model formalization metavertex is a special case of the manifestation of the
 emergence principle, which means that the metavertex
 A metagraph is a kind of complex network model,
 with its private attributes and connections becomes a
proposed by A. Basu and R. Blanning [1] and then
 whole that cannot be separated into its component parts.
adapted for information systems description by the
 The example of metagraph is shown in figure 1.

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e7 and metavertices are added. Thus, metaedge allows
 binding the stages of nested metagraph fragment
 e8 development to the steps of the process described with
 mv1 mv2 metaedge.
 e1 e4
 vv22 2.2 Metagraph and hypergraph models comparison
 vv44
 vv11 e2
 e6 In this section, the hypergraph model will be
 e3 vv33 e5 vv55 examined and compared with metagraph model.
 According to [3]: = ⟨ , ⟩, ∈ , ℎ ∈ ,
 where – hypergraph; – set of hypergraph vertices;
 mv3 – set of non-empty subsets of called hyperedges;
 – hypergraph vertex; ℎ – hypergraph hyperedge.
 Figure 1 Example of metagraph A hypergraph may be directed or undirected. A
 hyperedge in an undirected hypergraph only includes
 This example contains three metavertices: mv1, mv2, vertices whereas, in a directed hypergraph, a hyperedge
 and mv3. Metavertex mv1 contains vertices v1, v2, v3 and defines the order of traversal of vertices. The example of
 connecting them edges e1, e2, e3. Metavertex mv2 an undirected hypergraph is shown in figure 3.
 contains vertices v4, v5 and connecting them edge e6. This example contains three hyperedges: he1, he2, and
 Edges e4, e5 are examples of edges connecting vertices he3. Hyperedge he1 contains vertices v1, v2, v4, v5.
 v2-v4 and v3-v5 respectively and they are contained in Hyperedge he2 contains vertices v2 and v3. Hyperedge he3
 different metavertices mv1 and mv2. Edge e7 is an contains vertices v4 and v5. Hyperedges he1 and he2 have
 example of an edge connecting metavertices mv1 and a common vertex v2. All vertices of hyperedge he3 are
 mv2. Edge e8 is an example of an edge connecting vertex also vertices of hyperedge he1.
 v2 and metavertex mv2. Metavertex mv3 contains Comparing metagraph and hypergraph models it
 metavertex mv2, vertices v2, v3 and edge e2 from should be noted that the metagraph model is more
 metavertex mv1 and also edges e4, e5, e8 showing the expressive then the hypergraph model. According to
 holonic nature of the metagraph structure. Figure 1 figures 1 and 3 it is possible to note some similarities
 shows that metagraph model allows describing complex between the metagraph metavertex and the hypergraph
 data structures and it is the metavertex that allows hyperedge, but the metagraph offers more details and
 implementing emergence principle in data structures. clarity because the metavertex explicitly defines
 The vertices, edges, and metavertices are used for metavertices, vertices and edges inclusion, whereas the
 data description and the metaedges are used for process hyperedge does not. The inclusion of hyperedge he3 in
 description. hyperedge he1 in fig. 3 is only graphical and informal,
 The metagraph metaedge: = because according to hypergraph definition a hyperedge
 ⟨ , , , { }, ⟩, ∈ , = | , inclusion operation is not explicitly defined.
 where – metagraph metaedge belongs to set of
 metagraph metaedges ME; – source vertex
 (metavertex) of the metaedge; – destination vertex vv11 vv22 vv44 vv55 he3 he1
 (metavertex) of the metaedge; eo – metaedge direction
 flag (eo=true – directed metaedge, eo=false – undirected
 metaedge); – attribute, – metagraph fragment. vv33
 The example of directed metaedge is shown in figure 2.
 ... ... VE
 VS Vi e7 he2
 e8
 mv1 mv1 mv2 mv1 mv2
 e1 e1 e4 e1 e4
 v22 v22 v22
 vv11 e2 vv11 e2
 vv44
 e6 vv11 e2
 vv44
 e6
 Figure 3 Example of undirected hypergraph
 e3 vv33 e3 vv33 e5 vv55 e3 vv33 e5 vv55
 Thus the metagraph is a holonic graph model whereas
 mv3
 the hypergraph is a near flat graph model that does not
 fully implement the emergence principle. Therefore,
 Figure 2 Example of directed metaedge hypergraph model doesn’t fit well for complex data
 structures description.
 The directed metaedge contains metavertices vS, …
 vi, … vE and connecting them edges. The source vertex 2.3 Metagraph and hypernetwork models
 contains a nested metagraph fragment. During the comparison
 transition to the destination vertex, the nested metagraph The amazing fact is that the hypernetwork model was
 fragment becomes more complex, as new vertices, edges, invented twice. The first time the hypernetwork model
 was invented by Professor Vladimir Popkov with
Proceedings of the XIX International Conference colleagues in 1980s. Professor V. Popkov proposes
“Data Analytics and Management in Data Intensive several kinds of hypernetwork models with complex
Domains” (DAMDID/RCDL’2017), Moscow, Russia,
October 10–13, 2017

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formalization and therefore only main ideas of hypergraph model the hypernetwork model is a full
hypernetworks will be discussed in this section. model with emergence. Consider the differences between
According to [4] given the hypergraphs ≡ the hypernetwork and metagraph models.
 0 , 1 , 2 , ⋯ . The hypergraph ≡ 0 is According to the definition of a hypernetwork it is a
called primary network. The hypergraph is called a a layered description of graphs. It is assumed that the
secondary network of order i. Also given the sequence of hypergraphs may be divided into homogeneous layers
mappings between networks of different orders: and then mapped with mappings or combined with
Ф Ф −1 Ф1 hypersimplices. Metagraph approach is more flexible. It
→ −1 → ⋯ 1 → . Then the hierarchical
 allows combining arbitrary elements that may be layered
abstract hypernetwork of order K may be defined as
 or not using metavertices.
 = ⟨ , 1 , ⋯ ; Ф1 , ⋯ Ф ⟩. The emergence in
 Comparing the hypernetwork and metagraph models
this model occurs because of the mappings Ф between
 we can make the following notes:
the layers of hypergraphs.
 • Hypernetwork model may be considered as
 The second time the hypernetwork model was
 “horizontal” or layer-oriented. The emergence
proposed by Professor Jeffrey Johnson in his monograph
 appears between adjoining levels using
[5] in 2013. The main idea of Professor J. Johnson variant
 hypersimplices. The metagraph model may be
of hypernetwork model is the idea of hypersimplex (the
 considered as “vertical” or aspect-oriented. The
term is adopted from polyhedral combinatorics).
 emergence appears between any levels using
According to [5], a hypersimplex is an ordered set of
 metavertices.
vertices with an explicit n-ary relation and hypernetwork
is a set of hypersimplices. In the hierarchical system, the • In hypernetwork model, the elements are organized
 using hypergraphs inside layers and using mappings
hypersimplex combines k elements at the N level (base)
with one element at the N+1 level (apex). Thus, or hypersimplices between layers. In metagraph
hypersimplex establishes an emergence between two model, metavertices are used for organizing elements
adjoining levels. both inside layers and between layers. Hypersimplex
 The example of hypernetwork that combines the may be considered as a special case of metavertex.
ideas of two approaches is shown in figure 4. • Metagraph model allows organizing the results of
 previous organizations. The fragments of the flat
 graph may be organized into metavertices,
 vv55 vv66 he3 WS1 metavertices may be organized in higher-level
 metavertices and so on. The metavertex organization
 is more flexible than hypersimplex organization
 hyper- Ф1
 because hypersimplex assumes base and apex usage
 simplex PS and metavertex may include general form graph.
 • Metavertex may represent a separate aspect of the
 he1 vv44 vv33 vv11 vv22 he2 organization. The same fragments of the flat graph
 may be included in different metavertices whether
 these metavertices are used for modeling different
Figure 4 Example of hypernetwork aspects.
 Thus, we can draw a conclusion that metagraph
 The primary network is formed by the vertices of model is more flexible than hypernetwork model.
hyperedges ℎ 1 and ℎ 2. The first level 1 of However, it must be emphasized that the
secondary network is formed by the vertices of hypernetwork and metagraph models are only different
hyperedge ℎ 3. Mapping Ф1 is shown with an arrow. The formal descriptions of the same processes that occur in
hypersimplex is emphasized with the dash-dotted line. the networking with the emergence.
The hypersimplex is formed by the base (vertices 3 and From the historical point of view, the hypernetwork
 4 of ) and apex (vertex 5 of 1). model was the first complex network with an emergence
 The hypernetwork model became popular for model and it helps to understand many aspects of
complex domains description. For example, Professor complex networks with an emergence.
Konstantin Anokhin [6] proposes a new fundamental
theory of the organization of higher brain functions. 3 Metagraph model processing
According to this theory, biological neural networks The metagraph model is designed for complex data
(connectomes) are organized into cognitive and process description. But it is not intended for data
hypernetworks (cognitomes). Vertices of cognitome transformation. To solve this issue, the metagraph agent
form COGs (Gognitive Groups). Each COG may be ( ) designed for data transformation is proposed.
represented as hypersimplex. The base of COG is a set of There are two kinds of metagraph agents: the metagraph
the vertices of underlying neural networks, and its apex function agent ( ) and the metagraph rule agent ( ).
is a vertex possessing a new quality at the macrolevel of Thus = | .
cognitive hypernetworks. Thus, apex combines the base The metagraph function agent serves as a function
elements and emergence appears. with input and output parameter in form of metagraph:
 It should be noted that unlike the relatively simple = ⟨ , , ⟩, (1)

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where – metagraph function agent; – input 1 in given example). Figure 5 shows both cases
parameter metagraph; – output parameter corresponding to the start metagraph fragment and to the
metagraph; – abstract syntax tree of metagraph start rule.
function agent in form of metagraph.
 mv1 mv2 mv3
 The metagraph rule agent is rule-based: ag R =
⟨MG, R, AG ST ⟩, R = {ri }, ri : MGj → OP MG , where – v31
 e31
 v32 v31
 e31
 v32 v31
 e31
 v32
 MG1
metagraph rule agent; – working metagraph, a e32 e33 e32
 v33 v34 v33
metagraph on the basis of which the rules of agent are
performed; – set of rules ; – start condition MG=MG1
 e3
(metagraph fragment for start rule check or start rule); e1
 e2
 e4
 – a metagraph fragment on the basis of which the start=true

rule is performed; – set of actions performed on antecedent consequent
 actions

metagraph.
 rule 1
 The antecedent of the rule is a condition over start=true
metagraph fragment, the consequent of the rule is a set of conditions
 ...
actions performed on metagraph. Rules can be divided
 rule N
into open and closed.
 The consequent of the open rule is not permitted to metagraph rule agent 1
change metagraph fragment occurring in rule antecedent.
In this case, the input and output metagraph fragments Figure 5 Example of metagraph rule agent
may be separated. The open rule is similar to the template
that generates the output metagraph based on the input The distinguishing feature of metagraph agent is its
metagraph. homoiconicity which means that it can be a data structure
 The consequent of the closed rule is permitted to for itself. This is due to the fact that according to
change metagraph fragment occurring in rule antecedent. definition metagraph agent may be represented as a set
The metagraph fragment changing in rule consequent of metagraph fragments and this set can be combined in
cause to trigger the antecedents of other rules bound to a single metagraph. Thus, the metagraph agent can
the same metagraph fragment. But incorrectly designed change the structure of other metagraph agents.
closed rules system can lead to an infinite loop of
metagraph rule agent. 4 The examples of complex domains
 If the agent contains only open rules it is called an description using metagraph approach
open agent. If the agent contains only closed rules it is In this section, we give two examples of complex
called a closed agent. domains description using metagraph approach.
 Thus, metagraph rule agent can generate the output
metagraph based on the input metagraph (using open 4.1 Using metagraph approach for neural network
rules) or can modify the single metagraph (using closed representation
rules). The example of metagraph rule agent is shown in
figure 5. This subsection is based on our paper [7]. We begin
 The metagraph rule agent “metagraph rule agent 1” is with simple perceptron representation using metagraph
represented as a metagraph metavertex. According to the model. According to the Rosenblatt perceptron model
definition it is bound to the working metagraph MG1 – a [8], a conventional perceptron consists of three elements:
metagraph on the basis of which the rules of the agent are S, A and R.
performed. This binding is shown with edge e4. The layer of sensors (S) is an array of input signals.
 The metagraph rule agent description contains inner The associative layer (A) is a collection of intermediate
metavertices corresponds to agent rules (rule 1 … rule elements which are triggered if a particular set of input
N). Each rule metavertex contains antecedent and signals is activated at the same time. The adder (R) is
consequent inner vertices. In given example mv2 started when a particular collection of A-elements is
metavertex bound with antecedent which is shown with activated concurrently.
edge e2 and mv3 metavertex bound with consequent According to the notation adopted in the M. Minsky
which is shown with edge e3. Antecedent conditions and and S. Papert perceptron model [8], the value of a signal
consequent actions are defined in form of attributes on an A-element can be represented as a boolean
bound to antecedent and consequent corresponding predicate φ(S), and the value of a signal in the adder layer
vertices. as a predicate ψ(A, W). According to [8], a function that
 The start condition is given in form of attribute takes either 0 or 1 is regarded as a boolean predicate.
“start=true”. If the start condition is defined as a start Depending on the particular type of perceptron, the
metagraph fragment then the edge bound start metagraph form of predicates φ(S) and ψ(A, W) can be different.
fragment to agent metavertex (edge e1 in given example) Usually, predicate φ(S) is used to check whether the total
is annotated with attribute “start=true”. If the start input signal from sensors exceeds a certain threshold or
condition is defined as a start rule than the rule not. Also predicate ψ(A, W) (where W is a weight vector)
metavertex is annotated with attribute “start=true” (rule is used to see if the weighted sum from A-elements
 exceeds a particular threshold.

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In our case, the actual form of predicates is not • – the agent responsible for the execution of the
important. What is important is that the structure of φ(S) network.
and ψ(A, W) can be represented as an abstract syntactic In figure 7 the agents are shown as metavertices by
tree. Then we can represent the perceptron structure as a dotted-line ovals.
combination of metagraph function agents. Each The network-creating agent implements the
predicate can be represented as a kind of the formula 1: rules of creating an original neural network topology.
 = ⟨S, A, ⟩, = ⟨⟨{ }, ⟩, R, ⟩. This The agent holds both the rules of creating separate
representation is shown in figure 6. neurons and rules of connecting neurons into a network.
 Ψ
 Ψ In particular, the agent generates abstract syntactic trees
 Φ
 Φ of metagraph function agents and .
 SS ASTΦ
 AST Φ
 A
 A ASTΨ
 AST Ψ
 RR The network-modification agent holds the
 W rules of modification the network topology in process of
 {-1;0;1}
 operation. It is especially important for neural networks
 with variable topology such as HyperNEAT and SOINN.
 The network-learning agent implements a
Figure 6 The perceptron representation as a particular learning algorithm. As a result of learning the
combination of metagraph function agents changed weights are written in the metagraph
 An A-element can be represented as a function agent representation of the neural network. It is possible to
 . The input parameter is the value vector S, the output implement a few learning algorithms by using different
parameter is the value vector A. The description of the sets of rules for agent .
perceptron is similar to the description of the function The network-executing agent is responsible for
agent . The input parameter is a the metagraph the start and operation of the trained neural network.
representation of a tuple holding the description of A- The agents can work separately or jointly which may
elements as agent-functions and vector W. The output be especially important in the case of variable topologies.
parameter is the amplitude of output signal R. For example when a HyperNEAT or SOINN network is
 The description of functions can contain other trained, agent can call the rules of agent to
parameters, e.g., threshold values, but we assume that change the network topology in the process of learning.
these parameters are included in the description of the In fact, each agent uses its rules to implement a
abstract syntactic tree. specific program “machine”. The use of the metagraph
 Thus, we can describe the perceptron structure as a approach allows us to implement the “multi-machine”
combination of metagraph function agents. Now we principle: a few agents having different goals implement
describe neural network operation using metagraph rule different operations on the same data structure.
agents which are shown in figure 7. Thus, we can draw a conclusion that metagraph
 approach helps to describe both the structure of separate
 neurons and the structure of neural network operation.
 agMO agML
 4.2 Using metagraph approach for modeling the
 polypeptide chain synthesis
 MC
 agMC
 ag agMR
 Molecular biology is considered to be one of the most
 difficult to study topics of biological science. It's hard to
 The metagraph believe that the complexity of functioning of the
 representation biological cell invisible to the human eye exceeds the
 complexity of functioning of a large ERP-system, which
 of a neural network
 can contain thousands of business processes. The
 difficulty of studying biological processes is also due to
Figure 7 The structure of metagraph rule agents for the fact that in studying it is impossible to abstract from
neural network operation representation the physical and chemical features that accompany these
 The metagraph representation of neural network may processes. Therefore, the development of learning
be created similarly to the previously reviewed software that helps to better understand even one
perceptron approach. Such a representation is a separate complex process is a valid task.
task that depends on neural network topology. We will review the process of synthesis of a
 In order to provide a neural network operation the polypeptide chain which is also called “translation” in
following agents are used: molecular biology. Translation is an essential part of the
• – the agent responsible for the creation of the protein biosynthesis. This process is very valid from an
 network; educational point of view because protein biosynthesis is
 considered in almost all textbooks of molecular biology.
• – the agent responsible for the modification of
 The translation process is very complicated and in
 the network;
 this section, we review it in a simplified way.
• – the agent responsible for the learning of the
 The first main actor of the translation process is
 network;
 messenger RNA or mRNA, which may be represented as

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a chain of codons. The second main actor of the input and output metagraph fragments don’t contain
translation process is ribosome consisting of the large common elements.
subunit and the small subunit. The small subunit is While processing codons of mRNA agent ag RB
responsible for reading information from mRNA and sequentially adds fragments of the polypeptide chain PK
large subunit is responsible for generating fragments of to the output metagraph MGP . Vertices PK are connected
the polypeptide chain. with undirected edges.
 According to [9] the translation process consists of The process represented in figure 8 is very high-level.
three stages. But metagraph approach allows representing related
 The first stage is initiation. At this stage, the ribosome processes with different levels of abstraction.
assembles around the target mRNA. The small subunit is For example, for each codon or peptide, we can link
attached at the start codon. metavertex with its inner representation. And we can
 The second stage is elongation. The small subunit modify this representation during translation process
reads information from the current codon. Using this using metagraph agents.
information, the large subunit generates the fragment of Thus, the metagraph approach allows us to represent
the polypeptide chain. After that ribosome moves a model of polypeptide chain synthesis which can be the
(translocates) to the next mRNA codon. basis for the learning software.
 The third stage is termination. When the stop codon
is reached, the ribosome releases the synthesized 5 The textual representation of metagraph
polypeptide chain. Under some conditions, the ribosome model
may be disassembled.
 In previous sections, the formal definition and
 In this section, we use metagraph approach for
 graphical examples of metagraph model were defined.
translation process modeling. The representation is
 But to successfully operate with metagraph model we
shown in figure 8.
 also need textual representation. As such a
 e1 eK-1 eK+1 eN representation, we use a logical predicate model that is
 meRNA СSTART ... CK ... CSTOP MGP
 close to logical programming languages e.g. Prolog.
 PSTART Logical predicates used in this section and boolean
 MGP MGP ... predicates used in subsection 4.1 should not be confused.
 PSTART PSTART PK
 The classical Prolog uses following form of
 ... predicate: ( 1 , 2 , ⋯ , ). We
 agM agM agM ... used an extended form of predicate where along with
 PK PSTOP atoms predicate can also include key-value pairs and
 nested predicates: ( , ⋯ , = ,
Figure 8 The representation of the polypeptide chain ⋯ , (⋯ ), ⋯ ). The mapping of metagraph
synthesis (translation) process based on metagraph model fragments into predicate representation is shown
approach in Table 1.
 The mRNA is shown in figure 8 as metaedge Table 1 The textual representation of metagraph model
 = ⟨ , , = , { }, ⟩,
 № Metagraph representation Textual representation
where – source metavertex (start codon); –
 mv1
destination metavertex (stop codon); eo=true – directed Metavertex(Name=mv1, v1,
 1
metaedge; – attribute, – metagraph vv11 vv22 vv33 v2, v3)
fragment, containing inner codons of mRNA ( ) linked
with edges. e1
 e1
 The codon (shown in figure 8 as an elementary 2 vv11 vv22 ↔ vv11 vv22 Edge(Name=e1, v1, v2)
vertex) may also be represented as metavertex,
containing inner vertices and edges according to the e1 (eo=false)
required level of detail. e1 Edge(Name=e1, v1, v2,
 3 vv11 vv22 ↔ vv11 vv22
 Ribosome may be represented as metagraph rule eo=false)
agent agRB = ⟨meRNA , R, ⟩, R = {ri }, ri : CK → PK , 1. Edge(Name=e1, v1, v2,
 e1 (eo=true)
where – mRNA metaedge representation used as e1 eo=true)
 4
working metagraph; – set of rules ; – start vv11 vv22 ↔ vv11 vv22 2. Edge(Name=e1, vS=v1,
codon used as start agent condition; – codon on the vE=v2, eo=true)
 mv2
basis of which the rule is performed; – the added Metavertex(Name=mv2, v1,
 e1
fragment of polypeptide chain. vv22 v2, v3,
 The antecedent of the rule approximately corresponds 5 vv11 e2 Edge (Name=e1, v1, v2),
to the small subunit of ribosome modeling. The Edge(Name=e2, v2, v3),
 e3 vv33
 Edge(Name=e3, v1, v3))
consequent of the rule approximately corresponds to the
large subunit of ribosome modeling.
 Agent ag RB is open agent generating output
metagraph MGP based on input metaedge . The

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mv2 Metavertex(Name=mv2, v1, Case 3 also shows metagraph edge which fully
 e1 v2, v3, Edge(Name=e1, complies with the formal definition of undirected edge
 vv22 vS=v1, vE=v2, eo=true), including direction flag parameter.
 6 vv11 e2 Edge(Name=e2, vS=v2,
 vE=v3, eo=true), Case 4 shows an example of directed edge. Direction
 e3 vv33 Edge(Name=e3, vS=v1, flag parameter is also used. The source and destination
 vE=v3, eo=true)) vertices may be represented as predicate atom parameters
 me1 Metaedge(Name=me1, (case 4.1) or as predicate key-value parameters (case
 mv3 vS=v2, vE=mv3,
 7 vv22 mv4
 ... ... 4.2).
 Metavertex(Name=mv4, …
 ), eo=true) Case 5 shows an example of metavertex mv1 which
 Metagraph(Name=mg0, contains three nested vertices v1, v2 and v3 joined with
 mg0 Vertex(Name=v2, …), undirected edges e1, e2, and e3. Edges are represented
 mv5 Metavertex(Name=mv3, … with separate predicates that are nested to the metavertex
 ... ),
 8 Metavertex(Name=mv5, …
 predicate. Case 6 is similar to case 5 unless edges e1, e2,
 me1
 mv4
 mv3 ), Metaedge(Name=me1, and e3 are directed.
 vv22
 ... ... vS=v2, vE=mv3, Case 7 shows an example of directed metaedge me1
 Metavertex(Name=mv4, … which joins vertex v2 and metavertex mv3 and contains
 ), eo=true))
 metavertex mv4. The metaedge is represented as a
 attribute
 9 =
 Attribute(count, 5)
 predicate with the name “Metaedge”.
 count 5
 Case 8 shows an example of metagraph fragment mg0
 v1 which contains vertex v2, metavertices mv3 and mv5 and
 attribute metaedge me1 which joins vertex v2 and metavertex mv3
 count
 =
 55 and contains metavertex mv4. The metagraph fragment
 Vertex(Name=v1,
 10 Attribute(count, 5), is represented as a predicate with the name “Metagraph”,
 attribute
 = mv2
 Attribute(reference, mv2)) the vertex as a predicate with the name “Vertex”.
 reference
 reference ... The attribute may be represented as a special case of
 metavertex containing name and value. Case 9 shows
 Agent(Name= metagraph simple numeric attribute representation. Case 10 shows
 agent 1’, an example of vertex v1 containing numeric attribute and
 WorkMetagraph=mg1, reference attribute that refers to the metavertex mv2. The
 Rules(
 Rule(Name=rule 1’, attribute is represented as a predicate with the name
 mv1 mv2 start=true, “Attribute”.
 vv11
 (k=1)
 (k=1)
 e1(flag=main)
 Sum
 Sum
 (k=3)
 (k=3)
 mg
 mg11
 Condition Case 11 shows an example of metagraph rule agent
 vv22 (WorkMetagraph=mv1, “metagraph agent 1” representation (the predicate with
 (k=2)
 (k=2)
 Vertex(Name=v1,
 start=true
 start=true
 Attribute(k, $k1)), the name “Agent” is used). As a work metagraph mg1 is
 11 used (parameter “WorkMetagraph”). The “Rules”
 antecedent
 antecedent consequent
 consequent
 Vertex(Name=v2,
 rule
 rule 11
 Attribute(k, $k2)), predicate contains rules definition (nested predicate
 ... Edge(v1, v2, Attribute(flag, “Rule” is used). As a start rule “rule 1” is used which is
 rule
 rule N main)))
 defined by “start=true” parameter. Predicate “Condition”
 N

 metagraph
 metagraph agent
 agent 11
 Action
 (WorkMetagraph=mv2, corresponds to the rule condition. Parameter
 Add(Vertex(Name=Sum, “WorkMetagraph” contains a reference to the tested
 Attribute(k, metavertex mv1. The condition tests that metavertex mv1
 Eval($k1+$k2)))))
 ), Rule(…) … )) contains vertices v1 and v2 with attribute k. Founded
 values of k attribute of vertices v1 and v2 are assigned to
 Case 1 shows the example of metavertex mv1 which the $k1 and $k2 variables. Vertices v1 and v2 should be
contains three nested disjoint vertices v1, v2, and v3. The joined with edge containing attribute “flag=main”. If
predicate corresponds to metavertex, nested vertices are condition holds and metagraph fragment is found then
isomorphic to atoms that are parameters of the predicate. actions are performed (actions are defined by predicate
As the name of the predicate, “Metavertex” is used as the “Action”). Parameter “WorkMetagraph” contains a
corresponding element of metagraph model. Key-value reference to the result metavertex mv2. The example
parameter “Name” is used to set the name of metavertex. action contains adding new elements (that is defined by
This case is the simplest, since nested vertices are predicate “Add”). The vertex “Sum” is added containing
disjoint, and metavertex in this case is isomorphic to the attribute “k=$k1+$k2”. Predicate “Eval” is used to
hypergraph hyperedge. define the calculated expression.
 Case 2 shows metagraph edge which may be Thus, we defined a predicate description of all the
represented as a special case of metavertex containing main elements of metagraph data model.
source and destination vertices. This case is also The proposed predicate model is homoiconic. Since
isomorphic to the hypergraph hyperedge. The metagraph predicate approach is used both for metagraph data
edge is represented as a predicate with the name “Edge”. model definition and for metagraph agents definition
The source and destination vertices are represented as then high-level metagraph agents may change the
predicate atom parameters. structure of low-level metagraph agents by modifying
 their predicate definition.

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The textual representation of metagraph model may Data approach, in particular with the lambda
be used for storing metagraph model elements in architecture.
relational or NoSQL databases.
 It should be noted that metagraph model is well References
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