Abstract
We provide a new model of attributed networks where label of each vertex is a partition of integer n into at most m integer parts and all labels are different. The metric in the space of (n, m)-partitions is introduced. First, we investigate special class of the trinomial \((m^2, m)\)-partitions as a base for synthesis of networks G(m). It turns out that algorithmic complexity (the shortest computer program that produces G(m) upon halting) of these networks grows with m as \(\log m\) only. Numerical simulations of simple graphs for trinomial \((m^2, m)\)-partition families \((m= 3, 4, \ldots , 9)\) allows to estimate topological parameters of the graphs—clustering coefficients, cliques distribution, vertex degree distribution—and to show existence of such effects as scale-free and self-similarity for evolving networks. Since the model under consideration is completely deterministic, these results put forward new mode of thought about mechanisms of similarity, preferential attachment and popularity of complex networks. In addition, we obtained some numerical results relating robust behavior of the networks to disturbances like deleting nodes or cliques.
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Notes
- 1.
There exist optimal algorithm and recurrent rules for exact computation of number of partitions in the classes P(n, m) [11].
- 2.
There is totipotency cell in living organisms, i.e. a single cell with the ability to divide and produce all of the differentiated cells in an organism.
- 3.
The closest analogy to low value for K-complexity is the problem of generating all partitions of n into at most m parts. An upper bound on K-complexity of the problem is \(\log n\) because the recurrence holds for all integers m and n but running time to generate all partitions grows exponentially with n [11].
- 4.
Perfect graph is a graph in which the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph [16].
- 5.
J-distance \(d(A,B) = 1 - (A \cap B)/(A \cup B)\) is measure dissimilarity between two sets A and B (see [14]).
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Goryashko, A., Samokhine, L., Bocharov, P. (2019). New Deterministic Model of Evolving Trinomial Networks. In: Aiello, L., Cherifi, C., Cherifi, H., Lambiotte, R., Lió, P., Rocha, L. (eds) Complex Networks and Their Applications VII. COMPLEX NETWORKS 2018. Studies in Computational Intelligence, vol 812. Springer, Cham. https://doi.org/10.1007/978-3-030-05411-3_45
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