# Method of Multilevel Adaptive Synthesis of Monitoring Object Knowledge Graphs
## 1 Introduction
Let us consider the real-world object with a multilevel structure that varies in time. The structure of such object can be described as a typed bipartite graph -\(G_{TBC}=(V,\ E,\ T_{V},\ T_{E})\), where \(V=V_{1}\cup V_{2}\cup\ldots\cup V_{m}\), \(E\) is the set of edges representing relations between the elements of different levels (vertical relations) \(m\) is the set of levels in the structure of the object, \(T_{V}\) and \(T_{E}\) are the set of possible types of vertices and edges correspondingly. Consider that the edges reflect hierarchical relations between the elements of the object \(|T_{E}|=1\). The set of possible vertex types is a non-empty finite set that is specified for each object. To describe relations between the elements at one level, attributed relation graphs are used \(G_{ARG}=(V,\ E,\ \mu,\ \eta)\), where \(V\) is the set of vertices, \(E\subseteq V\times V,\ \mu:V\to MV\) is the function of attributing the labels to typed vertices, \(\eta:E\to ME\) is the function of attributing labels to typed edges, \(MV\) is the set of typed vertices, \(ME\) is the set of typed edges. The sets of vertex types \(T_{V}\) and edge types \(T_{E}\) are non-empty finite sets. Values of vertex and edge attributes are determined by their types. In that case, the model of the multilevel object with vertical and horizontal links between the elements is as follows \(G=\big{(}V^{L},\ E^{H},\ E^{R},\ FV,\ FE,\ \zeta,\ \zeta\big{)}\), where \(V^{L}\)--is the set of vertices including vertices that belong to different levels \(V^{L}=V^{0}\cup V^{1}\cup\ldots\cup V^{l}\), \(l\) is the number of levels in the object structure, \(E^{H}=E^{0,1}\cup E^{1,2}\cup\ldots\cup E^{l-1,l}\) is the set of edges linking vertices from different levels, \(E^{p,q}=\big{\{}e_{p,qj}\big{\}}\)--is the edge linking the \(i\)th element of level \(P\) with the \(j\)th element of level \(q\), \(E^{R}=E^{R^{0}}\cup E^{R^{1}}\cup\ldots\cup E^{R^{l}}\) is the set of edges linking the elements at the levels, \(E^{R^{k}}\) are the edges reflecting the links of level \(K\), \(FV\) and \(FE\) are the sets of admissible vertical and horizontal links between the elements, \(\zeta\) is the distribution function of vertices over the levels, \(\zeta\) is the function of linking the vertices and edges. For vertices and edges of graph \(G\), their types and attributes are specified. Attributes of vertices and edges of graph \(G\) are specified as \(V^{attr}_{i}=V^{attr}_{i,1},\ V^{attr}_{i,2},\ldots,V^{attr}_{i,N}\), where \(i\) is the node identifier, \(N\) is the number of node attributes of the corresponding type, \(E^{at}_{j}=E^{atr}_{j,1},\ E^{atr}_{j,2},\ldots,E^{atr}_{j,M^{\prime}}\), where \(j\) is the edge identifier, \(M\) is the number of attributes for the edge of the corresponding type. Attributes of nodes and edges are the parameters of the complex object, the values of which characterize the state of the corresponding element of the monitoring object at a specified moment. The set of attribute values at the specified moment is read by the monitoring system and is used to analyze the state of the object in the original or transformed format. At a given time \(t_{k}\) the state of the object is described as follows \(G_{t_{k}}=(V^{L}_{t_{k}},E^{H}_{t_{k}},\ E^{R}_{t_{k}},\ FV_{t_{k}},\ FE_{t_{k}})\). For vertices \(V^{L}_{t_{k}}\) and nodes \(E^{H}_{t_{k}}\) of the graph, their attribute values are determined. The admissible sets for nodes and edges at a point of time \(t_{k}\)--\(FV_{t_{k}},\ FE_{t_{k}}\) are subsets of the sets of admissible values of \(FV\) and \(FE\). When predicting the state of the monitored object, the task is to determine the values of attributes of nodes and edges of graph \(G_{t_{k}}\) at time point \(t_{k+T}\), i.e., to construct graph \(G_{t_{k+T}}\). In order to do this, the possibility of transition \(G_{t_{k}}\to G_{t_{k+T}}\) is proved.