**1. Manish Pandey, ***Eindhoven University of Technology (TU/e), Eindhoven, Netherlands*

**Title:** Centrality based vertex removals

**Abstract: **This poster delves into centrality measures, which are useful in determining importance of vertices in networks. Focused on the connectivity structure, our study assesses the significance of central vertices by analysing the repercussions of their removal on number of connected components and the size of the giant component within the network.

**2. Martijn Gösgens, ***Eindhoven University of Technology (TU/e), Eindhoven, Netherlands*

**Title:** The Projection Method: A Geometrical Framework for Community Detection

**Abstract:** We present the class of projection methods for community detection that generalizes many popular community detection methods. In this framework, we represent each clustering (partition) by a vector on a high-dimensional hypersphere. A community detection method is a projection method if it can be described by the following two-step approach: 1) the graph is mapped to a query vector on the hypersphere; and 2) the query vector is projected on the set of clustering vectors. This last projection step is performed by minimizing the distance between the query vector and the clustering vector, over the set of clusterings. We prove that several popular community detection methods fit this framework. We show that these different methods suffer from the same granularity problem: they have parameters that control the granularity of the resulting clustering, but choosing these to obtain clusterings of the desired granularity is nontrivial. We provide a general heuristic to address this granularity problem, which can be applied to any projection method.

**3. Bishakh Bhattacharya,** *ISI Kolkata, India*

**Title:** Fluctuations of Extreme Eigenvalues and Eigenvectors in Perturbed Wigner-Type Matrices

**Abstract:** In this talk, we will study finite rank perturbations of Wigner matrices with independent normalized entry variables satisfying specific moment conditions. We will analyze the fluctuations of the largest eigenvalue of this perturbed matrix, as well as the asymptotic alignment of the corresponding normalized eigenvector. This is an ongoing joint work with Arijit Chakrabarty and Rajat Subhra Hazra.

**4. Yogesh, ***IISER Mohali, India*

**Title:** A non-Markovian approach to preferential attachment

**Abstract:** We prove convergence results for a preferential attachment model where network growth is non-Markovian. We also present a conjecture about the asymptotic degree distribution of the tree being a power-law distribution.

**5. Nandan Malhotra,** *Leiden University, Leiden, Netherlands*

**Title:** Limiting Spectra of Inhomogeneous Random Graphs

**Abstract:** We consider the sparse inhomogeneous Erdős-Rényi random graph where connection probabilities depend on vertex weights. We characterize the limiting distribution of the eigenvalues of the adjacency matrix of this graph through moment characterization and analytic methods, and recover results for the dense regime by increasing the sparsity parameter. Based on joint work with Luca Avena and Rajat Subhra Hazra.

**6. Rounak Ray,** *Eindhoven University of Technology (TU/e), Eindhoven, Netherlands*

**Title :** Preferential attachment models: local limit & percolation

**Abstract: **Preferential attachment models are dynamic random graphs exhibiting the power law degree distributions. In this model, each new vertex joins the graph with an independently and identically distributed (i.i.d.) random number of edges and connects to existing vertices with a probability proportional to the degree of the vertex at that moment. Although there are several models in the literature depending on slight variations in the attachment rules, we prove all of these models share a common locally tree-like structure, i.e. finite neighbourhood of a uniformly chosen vertex behaves like a tree for all these variants of the preferential attachment models. Building on this particular property, we investigate the critical percolation threshold of these graphs. Using the large set expanders with bounded average degrees property of preferential attachment models, we compute the critical percolation threshold on the local limit explicitly. For calculating the explicit critical percolation threshold, we prove and use the Kesten-Stigum theorem and spine decomposition techniques for our settings.

**7. Aritra Mandal,** *ISI Bangalore, India*

**Title: **"Dynamics of random sparse graph limits"

**Abstract:** We explore random dynamics of discrete sparse graphs on n vertices for each fixed n and study the limit as n goes to infinity. We form the discrete random sparse graphs using sparse k-random graph models and impose random dynamics on the vertices of those sparse graphs using Moran model. Moran model is a population genetics model. We first impose random dynamics on the vertices. We assign types to each vertices. This types come from Moran model. Next we join two vertices based on the existence probability of the edge between those two vertices. This probability depends on the types of the vertices.By this we get a random sparse graph sequence whose underlying dynamic comes from Moran model.