Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 113 https://internationalpubls.com Connected Graph with Bacterial Graphs and Network Distance T. Selvaraj1, Dr. S. Subramanian2 1Research Scholar Enroll No:16R1612458 2Professor& Head, Dept. Of Mathematics, 1,2PRIST Deemed to be University, Thanjavur-613403. Email: jobu.selvaraj@gmail.com Article History: Received: 19-04-2024 Revised: 03-06-2024 Accepted: 22-06-2024 Abstract Network theory is going to analyse the set of techniques. But in a complex system network in a network of complex it has techniques to analyse the structure in a system of interacting agents. The graph theoretic representation means the system is made to apply network. The problem may be converted into graph by two components namely nodes and edges nodes are called as entities and interactions between edges are called as edges. Keywords: Connected Graph, Network, bacterial, Distance, Components. Tracking network dynamics using graph distances Here in a graph a node is assigned to each entity (or) player and interactions are modelled by edges. Graphs have endless encapsulating structural information in data sets. It may become paradigm of indispensable here the education about every separate node marked as interactable (or) irrelevant. Intended recording each individual neuron’s activity. The neurons are modelled by brain activity from brain data. Raw data can be transferred as a group of graphs. input is taken as treatable. In the identify notes some set of aligned notes are very important, ever though distances constitutes a mile stone to any analysis of statistics simultaneously comparisons, variance (or) cluster (or) even the definition of median (or) mean. Once the distance has been selected, the crucial is the analysis graph data. The distance between two unlabelled graph are very important and it must be well studied. But our focus is different nodes are marked as identity. The relevant information must discard by permutation invariant distances. the node’s leverage function is defined by nodes of the intensity of key node locations. with the distance sensitivity but of the self-graph, properties of distanced by exhibit distinct capture various types of structure Shifts correct scale for comparing graph is needed to identify decisive parameter. If we want to do the local structural changes at an atomic level structural similarities must be studied the node communities of the level notation of the heart is the scale which is used to find out different distance similarities between graph. The distances may be local, global and meso scale. Template are used to select the data Local - structural change Global – spectral distance change There are two categories to capture in between the two meso scale is in between the local and global a fresh group of graph distances depends on spectral heat kernels and argue that should be optimum Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 114 https://internationalpubls.com and must combine both local and global the different performance is around on both synthetic, fMRI data and microbiome are controlled experiments Vertice = ~ Edges = E N=|~| count of nodes |E|= count of edges j – i neighbour nodes D = Degree of matrix Aij = { 1 if i~j and 0 otherwise D = diagram (di) = I =1,….,N Such that, A matric in symmetric AT = A L = – A+ D (Laplacian’s) Laplacian is symmetric and its true eign value decomposition L = U ^ UT U = Unitary Matrix ^ = diag(λi) is the diagonal matrix of the eign values Quantifying structural distances via local changes Distances between graph are only mutually exclusive (or) structural version. Spectral distances in local changes the graph structure at each node is different but it induces local changes. Example Different bonds in different molecules are having different properties. But in brain network these distances can be noted as (ROI) Individual region of Interest. The other is identified by graph structure change in the Laplacian eign values (or) its adjacent matrix, distance marks the graph properties how nodes are functioning with nodes (or) globally organized and interact. In brain networks change in connectivity is assured globally with spectral distance. The hamming distance The count of edge insertions and deletions necessary to transfer in between two graphs. let G̃ and G are 2 graphs on nodes N. There adjacent metrices are A and A˜ Hanning distance is defined by bounded Distance between 1 and 0 in over all graph of sizes ‘N’ Analysis of frame work in General The frame work and graph are related one. pairwire dissimilarities among the graph in the MatricH is n by n dimensional matrix H Where, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 115 https://internationalpubls.com Dimensional matrix, Hij =d(Gi ,Gj) N = number of graphs N =21 micro biome N =29 fMRI data set H can be analysed in various ways Every distance is used for the analysis of (c) the graph variability from time to time is mentioned by various distance between next to next graphs H between graphs [fig 1(A) ,E ] and their low dimensional projections. It shows how much relationship it captures between connectivity and count of years only that subject under time passed cocacine relativity. Application Fig [1] mention the result of microbiome on both bacterial using hamming distance as well as fMRI data. Result The fig (1) shows the bacterial community graph in which hamming distance is completed throughout the treatment course. In fig 1[c] the distance between 2 consecutive graphs just often antibiotic courses c Fig [1] B shows the treatment gradient Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 116 https://internationalpubls.com A - hamming distance between the bacterial graphs tops rows and brain graph located at bottom bacterial graph – A B – MDS projections of the first two principal component Colour – phaser Shape – different subject In the graph c plot the next to next distance in between bacterial graph [c] G – distance between minimum spanning tree G hamming distance F – friedman – refsky test for significance for the diff data set E – cluster map of the fMRI graph G – minimum spanning tree between brain connections induced G The Jaccard distance It is union graph with respect to volume in normalization where ||.|| * mentions the Matrix of nuclear form. The above notation denotes the sharp distance between the graphs. According to steinhus transform Produces c matric D = hamming distance C = empty graph A - distances between bacterial graphs and Kendall-correlation by Jaccard B - consecutive distances in between two bacterial group C – bacterial’s MDS projection D - Minimum Spanning Tree between bacterial graphs Edge-importance disconnected components should be penalized more. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 117 https://internationalpubls.com Initial graph G1 Edge-submodularity In a same nodes edge get charged Perturbation of G1 four edge insertions Weight awareness It denotes weight of modified Perturbation of G1: 2 edge insertions,2 deletions Focus awareness random changes in not important not even happened Jaccard and humming distance treat all edges uniformly. Comparing graph structures Spectral distances are global = eign values of either (or) adjacency matrix L = D – A L = Laplacian D =diagonal matrix A = adjacency matrix Normalized Laplacian, L̃ = I - D−1/2A D−1/2 Lp distances on the eigen values Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 118 https://internationalpubls.com ……Eqn 3.1 Applications l2 = distance by two different functions such as combinatorial Laplacian eign spectrum eqn (3.1) f (λ) = e−0.1λ top row Microbiome f (λ) = e−1.2λ bottom row fMRI p = 0.025 confirmed the MDS projections l2 = eign spectra using two function of spectral distances as in eq no 3.1 top row = Microbiome bottom row = fMRI E = brain connectomes C = first two principal axes with graphs D = low-pass filter f(λ) = e−1.2λ B = MDS projection of the bacterial A = distances between bacterial graphs Spanning tree dissimilarities G and G̃ - graphs L = D - A dST(G, G̃ ) = | log(TG) − log(TG̃ )| Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 119 https://internationalpubls.com taking network changes A = bacterial graphs based on Kendall-correlation B = bacterial graphs based on MDS projection on the first two principal axes Colour – phaser Shape – different subject C = metagraph D = fMRI data for Clustermap E = brain connectomes MDS projections on the 1st two principal components(E) T̃G = NSpanning tree of G̃ ……..Eqn (3.3) Combining (3.3) and (3.2) yields General framework For Gaussian kernel, distribution Pseudo graph, Definition of the IM distance Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 120 https://internationalpubls.com The eign values of the squares of the vibrational frequencies which is inter- preted by Laplacian matrix of the network as ωi. λi = ω2 λ0 = ω0 = 0 γ = common parameter K = normalization constant defined such that ` The spectral distance between 2 graphs A & B is defined by Jurman, EN = empty graph on N Nodes FN = Complete graph on N Nodes polynomial approach P(A) = QW QT Where, The distance between 2 graphs G1 & G2 Where, dissimilarity of the microbiome bacterial graphs which is polynomial 3 = K 0.9 = α A = Heatmap of the corresponding similarity Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 121 https://internationalpubls.com B = MDS projection C = consecutive distances between graphs The common network generally compare the dissimilarities between graphs next to next points (Gt and Gt+1) Gt , Gt+1 = Consecutive points Sij t = affinity of interaction between i with j in graph Gt C. Donnat and S. Holmes d(Gt , Gt+1) Application of fMRI study with microbiome results of heat based distance with τ =1.2 P value is below 0.05 threshold, 5-nn metagraph is obtained Fig 9(B), (C), (D), (E) are the some one of the above the Annova gives the result a P value below10−4 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 122 https://internationalpubls.com Synthetic experiments the graph topology the Erdoss–Rényi model = 81nodes probability p = 0.1 Preferential Attachment (PA) = N = 81 nodes α = 1, a Stochastic Block Model with 3 sampled equally communities on the network analysis there topology is accessed by sets of network families in different global and local to Stochastic Block Model topology - results Row Top = Smooth dynamics is compared with 0.05% edges rewired at every time step. Row Bottom = detection experiment of Change point Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 123 https://internationalpubls.com Conclusion we have come acrossed so many graphs. For different node labels nodes never sets married. Easily graph distance and graph identities are very important. We have exampled the statistics by the use of metrics on graph objects. In many sources distances are very essential to assess the variability in a dataset latent clusters (or) gradients are detected by multidimensional scaling embeddings of graphs in Euclidean space. For instance, we have not used subgraphs and motif counts the same think which was quantifying in the past already. Some times we have used graph kernels. References [1] Pellegrini Matteo, Haynor David, Johnson JM: Protein interaction networks. Expert Rev Proteomics. 2004, 1 (2): [2] Vikis HG, Guan KL: Glutathione-S-transferase-fusion based assays for studying protein-protein interactions. 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