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On the theory of dynamic graph regression problem

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Abstract

Most of real-world graphs are dynamic, i.e., they change over time by a sequence of update operations. While the regression problem has been studied for static graphs and temporal graphs, it is not investigated for general dynamic graphs. In this paper, we study the theory of regression over dynamic graphs. First, we present the notion of update-efficient matrix embedding, that defines conditions sufficient for a matrix embedding to be effectively used for dynamic graph regression (under \(l_2\) norm). Then, we show that given a \(n \times m\) update-efficient matrix embedding (e.g., the adjacency matrix) and after an update operation in the graph, the exact optimal solution of linear regression can be updated in O(nm) time for the revised graph. Moreover, we show that this also holds when the matrix embedding is the Laplacian matrix and the update operations are restricted to edge insertion/deletion. In the end, by conducting experiments over synthetic and real-world graphs, we show the high efficiency of updating the solution of graph regression.

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Notes

  1. We note that this notion of embedding is different from the notion of embedding used in graph pattern mining (Chehreghani et al. 2011, 2020; Chehreghani and Bruynooghe 2016).

  2. Note that the graph and the information used to find the solution of dynamic graph regression, i.e., the pseudoinverse of matrix \({\textbf{M}}\) and vector \({\textbf{b}},\) change over time. However, we only care about their values before and after an update operation, as we want to find their values after the update operation, based on their values before the update operation. To keep notations as simple as possible, we do not parameterize them by time, rather, we simply use the terms before and after the update operation to distinguish these two situations.

  3. Note that when inserting a new node to a bounded-degree graph, at most a constant (bounded) number of edges are drawn between the new node and existing nodes.

  4. https://snap.stanford.edu/data/wiki-Vote.html.

  5. https://snap.stanford.edu/data/feather-lastfm-social.html.

  6. https://snap.stanford.edu/data/soc-sign-bitcoin-otc.html.

  7. https://charts.coinmetrics.io.

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  1. Department of Computer Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran

    Mostafa Haghir Chehreghani

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  1. Mostafa Haghir Chehreghani
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Correspondence to Mostafa Haghir Chehreghani.

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Communicated by Leonardo de Lima.

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Haghir Chehreghani, M. On the theory of dynamic graph regression problem. Comp. Appl. Math. 41, 362 (2022). https://doi.org/10.1007/s40314-022-02071-2

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