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In this note, we give very simple constructions of unique neighbor expander graphs starting from spectral or combinatorial expander graphs of mild expansion. These constructions and their analysis are simple variants of the constructions of…

Graph filtering is the cornerstone operation in graph signal processing (GSP). Thus, understanding it is key in developing potent GSP methods. Graph filters are local and distributed linear operations, whose output depends only on the local…

Networks are inherently vulnerable to vertex failures, making the analysis of their structural robustness a fundamental problem in graph theory. In this study, we investigate the closeness and vertex residual closeness of graphs, with a…

We establish new algorithmic guarantees with matching hardness results for coloring and independent set problems in one-sided expanders and related classes of graphs. For example, given a $3$-colorable regular one-sided expander, we compute…

Effective resistances are ubiquitous in graph algorithms and network analysis. In this work, we study sublinear time algorithms to approximate the effective resistance of an adjacent pair $s$ and $t$. We consider the classical adjacency…

Community is a universal structure in various complex networks, and community detection is a fundamental task for network analysis. With the rapid growth of network scale, networks are massive, changing rapidly and could naturally be…

A number of inference problems with sensor networks involve projecting a measured signal onto a given subspace. In existing decentralized approaches, sensors communicate with their local neighbors to obtain a sequence of iterates that…

This paper deals with local certification, specifically locally checkable proofs: given a graph property, the task is to certify whether a graph satisfies the property. The verification of this certification needs to be done locally without…

In this work we introduce the graph-theoretic notion of mendability: for each locally checkable graph problem we can define its mending radius, which captures the idea of how far one needs to modify a partial solution in order to "patch a…

Local learning of sparse image models has proven to be very effective to solve inverse problems in many computer vision applications. To learn such models, the data samples are often clustered using the K-means algorithm with the Euclidean…

Finding the densest subgraph (DS) from a graph is a fundamental problem in graph databases. The DS obtained, which reveals closely related entities, has been found to be useful in various application domains such as e-commerce, social…

In the vertex connectivity augmentation problem, we are given an undirected $n$-vertex graph $G$, a set of links $L \subseteq \binom{V(G)}{2} \setminus E(G)$, and integers $\lambda$ and $k$. The task is to insert at most $k$ links from $L$…

We consider the problem of learning the qualities of a collection of items by performing noisy comparisons among them. Following the standard paradigm, we assume there is a fixed "comparison graph" and every neighboring pair of items in…

A code is locally recoverable when each symbol in one of its code words can be reconstructed as a function of $r$ other symbols. We use bundles of projective spaces over a line to construct locally recoverable codes with availability; that…

A locally-optimal structure is a combinatorial structure such as a maximal independent set that cannot be improved by certain (greedy) local moves, even though it may not be globally optimal. It is trivial to construct an independent set in…

Network reconstruction consists in retrieving the hidden interaction structure of a system from observations. Many reconstruction algorithms have been proposed, although less research has been devoted to describe their theoretical…

Random intersection graphs have received much attention recently and been used in a wide range of applications ranging from key predistribution in wireless sensor networks to modeling social networks. For these graphs, each node is equipped…

The metric dimension of a graph is the cardinality of a minimum resolving set, which is the set of vertices such that the distance representations of every vertex with respect to that set are unique. A fault-tolerant metric basis is a…

Given a subset S of vertices of an undirected graph G, the cut-improvement problem asks us to find a subset S that is similar to A but has smaller conductance. A very elegant algorithm for this problem has been given by Andersen and Lang…

Error-correcting codes that admit local decoding and correcting algorithms have been the focus of much recent research due to their numerous theoretical and practical applications. An important goal is to obtain the best possible tradeoffs…