We introduce a concept of similarity between vertices of directed graphs. Let G A and G B be two directed graphs with respectively n A and n B vertices. We define a n B × n A similarity matrix S whose real entry s ij expresses how similar vertex j (in G A ) is to vertex i (in G B ) : we say that s ij is their similarity score. The similarity matrix can be obtained as the limit of the normalized even iterates of S(k +1) = BS(k)A T +B T S(k)A where A and B are adjacency matrices of the graphs and S(0) is a matrix whose entries are all equal to one. In the special case where G A = G B = G, the matrix S is square and the score s ij is the similarity score between the vertices i and j of G. We point out that Kleinberg's "hub and authority" method to identify web-pages relevant to a given query can be viewed as a special case of our definition in the case where one of the graphs has two vertices and a unique directed edge between them. In analogy to Kleinberg, we show that our similarity scores are given by the components of a dominant eigenvector of a non-negative matrix. Potential applications of our similarity concept are numerous. We illustrate an application for the automatic extraction of synonyms in a monolingual dictionary.

The virtual ant introduced by C. Langton has an interesting behavior, which has been studied in several contexts. Here we give a construction to calculate any boolean circuit with the trajectory of a single ant. This proves the P-hardness of the system and implies, through the simulation of one dimensional cellular automata and Turing machines, the universality of the ant and the undecidability of some problems associated to it. Complements and applet at

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We consider the Turing Machine as a dynamical system and we study a particular partition projection of it. In this way, we define a language (a subshift) associated to each machine. The classical definition of Turing Machines over a one-dimensional tape is generalized to allow for a tape in the form of a Cayley Graph. We study the complexity of the language of a machine in terms of realtime recognition by putting it in relation with the structure of its tape. In this way, we find a large set of realtime subshifts some of which are proved not to be deterministic in realtime. Sofic subshifts of this class correspond to machines that cannot make arbitrarily large tours. We prove that these machines always have an ultimately periodic behavior when starting with a periodic initial configuration, and this result is proved for any Cayley Graph.