CONSTRUCTING TRI-TOPOLOGICAL NETWORK SPACE MODEL USING CONNECTED COMPONENT GRAPH THEORY (T3-C2G) BASED ON HOMOTOPY ALGEBRAIC INVARIANCE MODEL

The complex network contains non-deterministic topological spaces under an invariance structural approach to create failures on a continual link during communication. The non-lineardynamic topological structure leads to problematic threading links on network nodes due to a non-identical path to route the data. To resolve this problem, we propose atri-logical algebraic mathematical construction model called homotopy based tri-topological network spa- ce using connected component graph $(T^3-C^2G)$ under network nonlinear structure,The Algebraic Invariance Linear Queuing Theory (HA/I/LQT) is used to resolve the link failure route propagation to make improved communication performance. This homotopy reduction to reduce the complex nature to make continual link based on Quillen topological structure space under the covariance tri-topological structure. Further, this makes tri-logical structure resembles the sequence of triangle structured route space to make the nearest point of node adjustment from the nearest path. This balances the M/M/G-$T^3$-Max queuing theory on triangular weightage in routing schemes to specify the dynamic homotopy topological structure to make continuous routing links to reduce the complex nature of network routing.

Automorphism Group and Other Properties of Zero Component Graph over a Vector Space

In this paper, we introduce an undirected simple graph, called the zero component graph on finite-dimensional vector spaces. It is shown that two finite-dimensional vector spaces are isomorphic if and only if their zero component graphs are isomorphic, and any automorphism of a zero component graph can be uniquely decomposed into the product of a permutation automorphism and a regular automorphism. Moreover, we find the dominating number, as well as the independent number, and characterize the minimum independent dominating sets, maximum independent sets, and planarity of the graph. In the case that base fields are finite, we calculate the fixing number and metric dimension of the zero component graphs and determine vector spaces whose zero component graphs are Hamiltonian.

On twin preserving spanning subgraph

There are several approaches to lower the complexity of huge networks. One of the key notions is that of twin nodes, exhibiting the same connection pattern to the rest of the network. We extend this idea by defining a twin preserving spanning subgraph (TPS-subgraph) of a simple graph as a tool to compute certain graph related invariants which are preserved by the subgraph. We discuss how these subgraphs preserve some distance based parameters of the simple graph. We introduce a sub-skeleton graph on a vector space and examine its basic properties. The sub-skeleton graph is a TPS-subgraph of the non-zero component graph defined over a vector space. We prove that some parameters like the metric-dimension are preserved by the sub-skeleton graph.

Perancangan Media Pembelajaran Tematik Sekolah Dasar Berbasis Serious Game

Pembelajaran tematik merupakan pembelajaran terpadu yang menggunakan tema sebagai pokok pikiran atau gagasan pokok pembicaraan dengan mengaitkan beberapa mata pelajaran, sehingga dapat memberikan pengalaman bermakna kepada siswa. Transfer pembelajaran berbasis tematik ke siswa Sekolah Dasar tidak mengalami kesulitan bagi guru jika dilakukan secara langsung di sekolah. Akan tetapi, pandemi covid-19 pada saat ini menyebabkan digunakannya media pembelajaran online sehingga guru-guru mengalami kesulitan. Kesulitan penyampaian materi tersebut juga dirasakan orang tua yang juga berusaha memberikan pembelajaran berbasis tematik di rumah berdasarkan buku yang diperoleh dari sekolah. Penelitian ini merancang framework pembelajaran berbasis serious game berdasarkan buku tematik Sekolah Dasar dengan tujuan dapat mempermudah guru dalam melakukan pembelajaran jarak jauh serta mempermudah orang tua dalam menyampaikan materi yang terdapat di dalam buku tematik tingkat Sekolah Dasar. Hasil penelitian menunjukkan bahwa komponen utama dalam perancangan serious game untuk memenuhi komponen pedagogi dan fun adalah game component, graph design, database design dan fitur design, di mana konsep immersifitas merupakan hal pokok sebagai upaya yang dapat membuat pemain masuk ke dalam suasana permainan dan mendapatkan pengalaman lebih dari permainan tersebut.

Mapping graph state orbits under local complementation

Graph states, and the entanglement they posses, are central to modern quantum computing and communications architectures. Local complementation – the graph operation that links all local-Clifford equivalent graph states – allows us to classify all stabiliser states by their entanglement. Here, we study the structure of the orbits generated by local complementation, mapping them up to 9 qubits and revealing a rich hidden structure. We provide programs to compute these orbits, along with our data for each of the 587 orbits up to 9 qubits and a means to visualise them. We find direct links between the connectivity of certain orbits with the entanglement properties of their component graph states. Furthermore, we observe the correlations between graph-theoretical orbit properties, such as diameter and colourability, with Schmidt measure and preparation complexity and suggest potential applications. It is well known that graph theory and quantum entanglement have strong interplay – our exploration deepens this relationship, providing new tools with which to probe the nature of entanglement.

Exact k-Component Graph Learning for Image Clustering

The performance of graph-based clustering methods highly depends on the quality of the data affinity graph as a good affinity graph can approximate well the pairwise similarity between data samples. To a large extent, existing graph-based clustering methods construct the affinity graph based on a fixed distance metric, which is often not an accurate representation of the underlying data structure. Also, they require postprocessing on the affinity graph to obtain clustering results. Thus, the results are sensitive to the particular graph construction methods. To address these two drawbacks, we propose a k-component graph clustering (k-GC) approach to learn an intrinsic affinity graph and to obtain clustering results simultaneously. Specifically, k-GC learns the data affinity graph by assigning the adaptive and optimal neighbors for each data point based on the local distances. Efficient iterative updating algorithms are derived for k-GC, along with proofs of convergence. Experiments on several benchmark datasets have demonstrated the effectiveness of k-GC.

Multi-Component Graph Convolutional Collaborative Filtering

The interactions of users and items in recommender system could be naturally modeled as a user-item bipartite graph. In recent years, we have witnessed an emerging research effort in exploring user-item graph for collaborative filtering methods. Nevertheless, the formation of user-item interactions typically arises from highly complex latent purchasing motivations, such as high cost performance or eye-catching appearance, which are indistinguishably represented by the edges. The existing approaches still remain the differences between various purchasing motivations unexplored, rendering the inability to capture fine-grained user preference. Therefore, in this paper we propose a novel Multi-Component graph convolutional Collaborative Filtering (MCCF) approach to distinguish the latent purchasing motivations underneath the observed explicit user-item interactions. Specifically, there are two elaborately designed modules, decomposer and combiner, inside MCCF. The former first decomposes the edges in user-item graph to identify the latent components that may cause the purchasing relationship; the latter then recombines these latent components automatically to obtain unified embeddings for prediction. Furthermore, the sparse regularizer and weighted random sample strategy are utilized to alleviate the overfitting problem and accelerate the optimization. Empirical results on three real datasets and a synthetic dataset not only show the significant performance gains of MCCF, but also well demonstrate the necessity of considering multiple components.