For the statement of the theorem,k Nisaconstant, and we assume that for every graphGamatching extendingk-familyF(G)isspecified.

Notable instances of this architecture include, e.g., [33,37,51,105],and the spectral approaches proposed in, e.g., [14, 29, 64, 81]--all of which descend from early work in [65, 80, 102, 97]. Fork =1,the power ofthe algorithm has been completely characterized [4,63]. In general, a different mappingM()could be used, depending on the neighborhood information that we would like to aggregate. The following result relates the power of thek-WLandδ-k-WL. Proposition1(restated, Proposition 1 in the main text).

These algorithms typically recoveracausal graph uptoits Markovequivalence class anditisknowntobefundamentally impossible tolearn causal relationships solely based on observational data.

Inthis paper,weconsider thematching ofmulti-agent multi-armed bandit problem, i.e., while agents prefer arms with higher expected reward, arms also have preferences onagents.

Fortunately, one often has access to similarity relationships amongst the options that can be leveraged.

We uncover a new relation between Closeness centrality and the Condorcet principle. We define a Condorcet winner in a graph as a node that compared to any other node is closer to more nodes. In other words, if we assume that nodes vote on a closer candidate, a Condorcet winner would win a two-candidate election against any other node in a plurality vote. We show that Closeness centrality and its random-walk version, Random-Walk Closeness centrality, are the only classic centrality measures that are Condorcet consistent on trees, i.e., if a Condorcet winner exists, they rank it first. While they are not Condorcet consistent in general graphs, we show that Closeness centrality satisfies the Condorcet Comparison property that states that out of two adjacent nodes, the one preferred by more nodes has higher centrality. We show that Closeness centrality is the only regular distance-based centrality with such a property.

Graph theoretical ideas are highly utilized by computer science fields especially data mining. In this field, a data structure can be designed in the form of tree. Covering is a widely used form of data representation in data mining and covering-based rough sets provide a systematic approach to this type of representation. In this paper, we study the connectedness of graphs through covering-based rough sets and apply it to connected matroids. First, we present an approach to inducing a covering by a graph, and then study the connectedness of the graph from the viewpoint of the covering approximation operators. Second, we construct a graph from a matroid, and find the matroid and the graph have the same connectedness, which makes us to use covering-based rough sets to study connected matroids. In summary, this paper provides a new approach to studying graph theory and matroid theory.