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1. Computation of the Sample Frechet Mean for Sets of Large Graphs with Applications to Regression

   Ferguson, Daniel; Meyer, Francois G. (June 2022, SIAM International Conference on Data Mining (SDM 2022))

   Banerjee, Arindam; Zhou, Zhi-Hua (Ed.)

   To characterize the location (mean, median) of a set of graphs, one needs a notion of centrality that is adapted to metric spaces, since graph sets are not Euclidean spaces. A standard approach is to consider the Fr ́echet mean. In this work, we equip a set of graph with the pseudometric defined by the l2 norm between the eigenvalues of their respective adjacency matrix. Unlike the edit distance, this pseudometric reveals structural changes at multiple scales, and is well adapted to studying various statistical problems for graph-valued data. We describe an algorithm to compute an approximation to the sample Fr ́echet mean of a set of undirected unweighted graphs with a fixed size using this pseudometric. 

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2. The Fréchet Mean of Inhomogeneous Random Graphs

   https://doi.org/10.1007/978-3-030-93409-5_18  

   Meyer, Francois G. (January 2022, Complex Networks & Their Applications X)

   Benito, Rosa Maria; Cherifi, Chantal; Cherifi, Hocine; Moro, Esteban; Rocha, Luis M. (Ed.)

   To characterize the “average” of a set of graphs, one can compute the sample Fr ́echet mean. We prove the following result: if we use the Hamming distance to compute distances between graphs, then the Fr ́echet mean of an ensemble of inhomogeneous random graphs is obtained by thresholding the expected adjacency matrix: an edge exists between the vertices i and j in the Fr ́echet mean graph if and only if the corresponding entry of the expected adjacency matrix is greater than 1/2. We prove that the result also holds for the sample Fr ́echet mean when the expected adjacency matrix is replaced with the sample mean adjacency matrix. This novel theoretical result has some significant practical consequences; for instance, the Fr ́echet mean of an ensemble of sparse inhomogeneous random graphs is the empty graph. 

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3. Metric and Path-Connectedness Properties of the Frechet Distance for Paths and Graphs

   Erin Chambers, Brittan Fasy (August 2023, Proceedings of the Canadian Conference on Computational Geometry)

   The Frechet distance is often used to measure distances between paths, with applications in areas ranging from map matching to GPS trajectory analysis to hand- writing recognition. More recently, the Frechet distance has been generalized to a distance between two copies of the same graph embedded or immersed in a metric space; this more general setting opens up a wide range of more complex applications in graph analysis. In this paper, we initiate a study of some of the fundamental topological properties of spaces of paths and of graphs mapped to R^n under the Frechet distance, in an effort to lay the theoretical groundwork for understanding how these distances can be used in practice. In particular, we prove whether or not these spaces, and the metric balls therein, are path-connected. 

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4. On the edit distance function of the random graph

   https://doi.org/10.1017/S0963548321000353  

   Martin, Ryan R.; Riasanovsky, Alex W. (March 2022, Combinatorics, Probability and Computing)

   Abstract Given a hereditary property of graphs $$\mathcal{H}$$ and a $$p\in [0,1]$$ , the edit distance function $$\textrm{ed}_{\mathcal{H}}(p)$$ is asymptotically the maximum proportion of edge additions plus edge deletions applied to a graph of edge density p sufficient to ensure that the resulting graph satisfies $$\mathcal{H}$$ . The edit distance function is directly related to other well-studied quantities such as the speed function for $$\mathcal{H}$$ and the $$\mathcal{H}$$ -chromatic number of a random graph. Let $$\mathcal{H}$$ be the property of forbidding an Erdős–Rényi random graph $$F\sim \mathbb{G}(n_0,p_0)$$ , and let $$\varphi$$ represent the golden ratio. In this paper, we show that if $$p_0\in [1-1/\varphi,1/\varphi]$$ , then a.a.s. as $$n_0\to\infty$$ , \begin{align*} {\textrm{ed}}_{\mathcal{H}}(p) = (1+o(1))\,\frac{2\log n_0}{n_0} \cdot\min\left\{ \frac{p}{-\log(1-p_0)}, \frac{1-p}{-\log p_0} \right\}. \end{align*} Moreover, this holds for $$p\in [1/3,2/3]$$ for any $$p_0\in (0,1)$$ . A primary tool in the proof is the categorization of p -core coloured regularity graphs in the range $$p\in[1-1/\varphi,1/\varphi]$$ . Such coloured regularity graphs must have the property that the non-grey edges form vertex-disjoint cliques. 

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5. Factorization and pseudofactorization of weighted graphs

   Kristin Sheridan, Joseph Berleant (July 2023, Discrete applied mathematics)

   For unweighted graphs, finding isometric embeddings of a graph G is closely related to decompositions of G into Cartesian products of smaller graphs. When G is isomorphic to a Cartesian graph product, we call the factors of this product a factorization of G. When G is isomorphic to an isometric subgraph of a Cartesian graph product, we call those factors a pseudofactorization of G. Prior work has shown that an unweighted graph’s pseudofactorization can be used to generate a canonical isometric embedding into a product of the smallest possible pseudofactors. However, for arbitrary weighted graphs, which represent a richer variety of metric spaces, methods for finding isometric embeddings or determining their existence remain elusive, and indeed pseudofactorization and factorization have not previously been extended to this context. In this work, we address the problem of finding the factorization and pseudofactorization of a weighted graph G, where G satisfies the property that every edge constitutes a shortest path between its endpoints. We term such graphs minimal graphs, noting that every graph can be made minimal by removing edges not affecting its path metric. We generalize pseudofactorization and factorization to minimal graphs and develop new proof techniques that extend the previously proposed algorithms due to Graham and Winkler [Graham and Winkler, ’85] and Feder [Feder, ’92] for pseudofactorization and factorization of unweighted graphs. We show that any n-vertex, m-edge graph with positive integer edge weights can be factored in O(m2) time, plus the time to find all pairs shortest paths (APSP) distances in a weighted graph, resulting in an overall running time of O(m2+n2 log log n) time. We also show that a pseudofactorization for such a graph can be computed in O(mn) time, plus the time to solve APSP, resulting in an O(mn + n2 log log n) running time. 

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