Local Symmetry in Random Graphs

Simões, Jefferson Elbert; Figueiredo, Daniel R.; Barbosa, Valmir C.

doi:10.1109/tnse.2019.2957610

Cited by 2 publications

(5 citation statements)

References 30 publications

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“…This differs from homophily in that the immediate neighbors are not necessarily similarlyattributed. A local interpretation of this phenomenon would be that nodes of similar attributes often end up linking to the same neighbors (Simões et al 2019). As a result, HopAUC can produce an accurate assessment of node rank quality when network edges are correlated to a combination of homophily and structural equivalence; either of these dynamics can justify the application of this measure, thus extending the types of networks it can be applied on.…”

Section: Dynamics Of Local Structuresmentioning

confidence: 99%

Unsupervised evaluation of multiple node ranks by reconstructing local structures

2020

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A problem that frequently occurs when mining complex networks is selecting algorithms with which to rank the relevance of nodes to metadata groups characterized by a small number of examples. The best algorithms are often found through experiments on labeled networks or unsupervised structural community quality measures. However, new networks could exhibit characteristics different from the labeled ones, whereas structural community quality measures favor dense congregations of nodes but not metadata groups spanning a wide breadth of the network. To avoid these shortcomings, in this work we propose using unsupervised measures that assess node rank quality across multiple metadata groups through their ability to reconstruct the local structures of network nodes; these are retrieved from the network and not assumed. Three types of local structures are explored: linked nodes, nodes up to two hops away and nodes forming triangles. We compare the resulting measures alongside unsupervised structural community quality ones to the AUC and NDCG of supervised evaluation in one synthetic and four real-world labelled networks. Our experiments suggest that our proposed local structure measures are often more accurate for unsupervised pairwise comparison of ranking algorithms, especially when few example nodes are provided. Furthermore, the ability to reconstruct the extended neighborhood, which we call HopAUC, manages to select a near-best among many ranking algorithms in most networks.

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Section: Dynamics Of Local Structuresmentioning

confidence: 99%

Unsupervised evaluation of multiple node ranks by reconstructing local structures

2020

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“…In our setup, it seems intuitive that two vertices that are locally indistinguishable in a large graph would also behave indistinguishably, and therefore, can be swapped. A notion of local symmetry identifying such vertices was proposed in [15], which we adopt in this paper. We need a few definitions to make precise what we mean by two vertices being locally indistinguishable.…”

Section: Local Symmetrymentioning

confidence: 99%

“…The notion of locality we adopt in this paper hinges on these k-neighbourhoods and their induced subgraphs. If two vertices induce isomorphic subgraphs, they are indistinguishable locally and we say they are k-locally symmetric [15]. Definition 4.…”

Section: Local Symmetrymentioning

confidence: 99%

“…However, please note that, in our context, the induced subgraphs G[N k (u)] need not be trees. The following facts about local symmetry are useful for our study of lumpability [34,15]. Proposition 2.…”

Section: Local Symmetrymentioning

confidence: 99%

“…Second, there may be too few automorphisms to engender significant state space reduction [39] as many large random graphs tend to be asymmetric with high probability (see [30,26,31]). Therefore, we propose a lumping procedure based on a local notion of symmetry [15] taking into account only local (k-hop) neighbourhoods of each vertex. In our approach, we construct an equitable partition [20,Chapter 9] of V by clubbing together vertices that are locally symmetric.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Approximate lumpability for Markovian agent-based models using local symmetries

et al. 2019

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We study a Markovian agent-based model (MABM) in this paper. Each agent is endowed with a local state that changes over time as the agent interacts with its neighbours. The neighbourhood structure is given by a graph. In a recent paper [39], the authors used the automorphisms of the underlying graph to generate a lumpable partition of the joint state space ensuring Markovianness of the lumped process for binary dynamics. However, many large random graphs tend to become asymmetric rendering the automorphism-based lumping approach ineffective as a tool of model reduction. In order to mitigate this problem, we propose a lumping method based on a notion of local symmetry, which compares only local neighbourhoods of vertices. Since local symmetry only ensures approximate lumpability, we quantify the approximation error by means of Kullback-Leibler divergence rate between the original Markov chain and a lifted Markov chain. We prove the approximation error decreases monotonically. The connections to fibrations of graphs are also discussed.

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Local Symmetry in Random Graphs

Cited by 2 publications

References 30 publications

Unsupervised evaluation of multiple node ranks by reconstructing local structures

Unsupervised evaluation of multiple node ranks by reconstructing local structures

Approximate lumpability for Markovian agent-based models using local symmetries

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