arXiv subjects

Udit Raj

Publications and source records attributed to Udit Raj.

5 recordsLinked to original sources

arXiv2026

Nearest Graph Laplacians with Prescribed Connected Components: A Convex Framework for Network Reconstruction

We study the problem of constructing the nearest graph Laplacian matrix to a given Laplacian while enforcing a prescribed connected-component structure. Let the vertex set be partitioned into nonempty disjoint blocks $C_1,\ldots,C_k$, and let $U=[u_1,\ldots,u_k]$ be the matrix of the corresponding block-indicator vectors. The constraint $MU=0$ ensures that these prescribed indicators lie in the nullspace of the optimized Laplacian $M^\star$, and hence the associated graph has at least $k$ connected components. To guarantee exactly the prescribed components, we impose additional block-connectivity constraints on the principal blocks $M_j=M[C_j,C_j]$. These constraints ensure that each prescribed block induces a connected weighted subgraph. The resulting problem is a convex semidefinite optimization problem with a strictly convex Frobenius-norm objective. We prove existence and uniqueness of the minimizer and show that the optimized Laplacian has exactly the prescribed connected components, with nullspace $\operatorname{span}\{u_1,\ldots,u_k\}$. The framework proposed in this work provides a principled tool for quantifying the minimum structural intervention required to transform a graph-based network into one having a prescribed group-separated structure. Numerical examples, including the Sampson monastery positive-affection network, illustrate the nearest faction-consistent weighted reconstruction and the minimum Laplacian perturbation required to realize the prescribed faction structure.

math.OC↗

arXiv2026

Combinatorial metaplexes and centrality indices for identifying higher-order interactions

Complex systems consist of interacting units whose interactions may be pairwise, involving two units, or higher-order, involving more than two units simultaneously. Graphs capture pairwise interactions and represent such systems as networks, whereas simplicial complexes can capture higher-order interactions (HoIs) and represent them as higher-order networks comprising simplices. In the clique complex construction, HoIs arise whenever vertices form a clique in the underlying graph. In classical graph-theoretic and simplicial-complex models, vertices are treated as structurally indistinguishable objects. However, in many real-world systems vertices possess internal structure, and their intrinsic properties influence the HoIs present in the system. To address this limitation, we introduce the combinatorial metaplex, consisting of two interacting components: an underlying simplicial complex that serves as an admissibility structure specifying boundary-compatible higher-order simplex candidates, and a concentration layer defined by a concentration map assigning a value to each vertex and extending the map to simplices so that the resulting distribution satisfies a conservation relation between vertex weights and facet weights. This concentration layer provides a deterministic threshold rule governing the inclusion of true HoIs. Using facet-mediated adjacency and weighted walks, we define one-parameter families of degree, closeness, and harmonic centralities for non-facet simplices, interpolating between those determined solely by the simplicial complex and those determined by concentration-induced coupling. The framework is illustrated through a representative example, including a comparison of HoIs obtained from the clique complex and the combinatorial metaplex, followed by an analysis of edge centralities within the combinatorial metaplex.

math.GM↗

arXiv2025

Construction of the Nearest Nonnegative Hankel Matrix for a Prescribed Eigenpair

We study the problem of determining whether a prescribed eigenpair $(\lambda,x)$ can be made an exact eigenpair of a nonnegative Hankel matrix through the smallest possible structured perturbation. The task reduces to check the feasibility of a set of linear constraints that encode both the Hankel structure and entrywise nonnegativity. When the feasibility set is nonempty, we compute the minimum-norm perturbation $\Delta H$ such that $(H+\Delta H)x=\lambda x$. When no such perturbation exists, we compute the nearest nonnegative Hankel matrix in a residual sense by minimizing $\|(H+\Delta H)x-\lambda x\|_{2}$ subject to the imposed constraints. Because closed-form formulas for the structured backward error are generally unavailable, our method provides a fully numerical and optimization-based framework for evaluating eigenpair sensitivity under nonnegativity-preserving Hankel perturbations. Numerical examples illustrate both feasible and infeasible cases.

math.NA↗

arXiv2025

Study of higher-order interactions in unweighted, undirected networks using persistent homology

Persistent homology has been studied to better understand the structural properties and topology features of weighted networks. It can reveal hidden layers of information about the higher-order structures formed by non-pairwise interactions in a network. Studying of higher-order interactions (HoIs) of a system provides a more comprehensive understanding of the complex system; moreover, it is a more precise depiction of the system as many complex systems, such as ecological systems and biological systems, etc., demonstrate HoIs. In this study, the weighted simplicial adjacency matrix has been constructed using the concept of adjacency strength of simplices in a clique complex obtained from an unweighted, undirected network. This weighted simplicial adjacency matrix is thus used to calculate the global measure, which is called generalised weighted betweenness centrality, which further helps us in calculating the persistent homology on the given simplicial complex by constructing a filtration on it. Moreover, a local measure called maximal generalised degree centrality has also been established for better understanding of the network topology of the studied simplicial complex. All the generalizations given in this work can be reduced to the graph-theoretic case. i.e., for a simplicial complex of dimension 1. Three different filtration schemes for constructing the sequence of simplicial complexes have been given with the help of both global and local measures, and by using these measures, the topology of higher-order structures of the studied network due to the interactions of their vertices has been compared. Further, the illustration of established definitions has been given using a real-life network by calculating Betti numbers up to dimension two.

math.CO↗

arXiv2022

Simplicial structures in ecological networks

An ecological network is a formal representation of a specific type of interaction in a corresponding ecosystem. Such networks have traditionally been modelled as encoding exclusively pairwise interactions among the fundamental units of ecosystems and have been represented and analysed using graph-theoretic methods. However, many real-world ecosystems may entertain non-binary, polyadic relations between their units, which cannot be captured by the pairwise interaction methods, but require higher-order interaction framework, and consequently the corresponding ecological networks cannot be modelled using graph-theoretic framework. This work gives a structural definition of ecological network suitable for modelling all orders of interactions between the fundamental units of the corresponding ecological system, including and going beyond the pairwise interaction framework. Carbon mediation between units of some select ecosystems are studied by modelling the corresponding ecological networks as simplicial complexes following the definition. The concept of graph centrality measure has been extended to simplicial centrality, and some important centrality measures of these networks at various structural levels of the complexes have been calculated. The centrality measures reveal valuable structural information including information about those vertices that are more likely to participate in higher-order interactions, as well as inform whether there is a difference in the ranks of vertices for these higher-order networks based on graph centrality and simplicial centrality measures.

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