1 On incompressible multidimensional networks∗ Felipe S. Abrahao˜ †, Klaus Wehmuth†, Hector Zenil‡ and Artur Ziviani†, Senior Member, IEEE Abstract In order to deal with multidimensional structure representations of real-world networks, as well as with their worst-case irreducible information content analysis, the demand for new graph abstractions increases. This article presents an investigation of incompressible multidimensional networks defined by generalized graph representations. In particular, we mathematically study the lossless incompressibility of snapshot-dynamic networks and multiplex networks in comparison to the lossless incompressibility of more general forms of dynamic networks and multilayer networks, from which snapshot-dynamic networks or multiplex networks are particular cases. We show that incompressible snapshot-dynamic (or multiplex) networks carry an amount of algorithmic information that is linearly dominated by the size of the set of time instants (or layers). This contrasts with the algorithmic information carried by incompressible general dynamic (or multilayer) networks that is of the quadratic order of the size of the set of time instants (or layers). Furthermore, we prove that incompressible general multidimensional networks have edges linking vertices at non-sequential time instants (layers or, in general, elements of a node dimension). Thus, representational incompressibility implies a necessary underlying constraint in the multidimensional network topology. Index Terms Algorithmic information, Complex networks, Lossless compression, Multidimensional systems, Network topology I. INTRODUCTION Complex networks science has been showing fruitful applications to the study of biological, social, and physical systems [5,33,50]. This way, as the interest and pervasiveness of complex networks modeling and network analysis increase in graph theory and network science, proper representations of multidimensional networks into new extensions of graph-theoretical abstractions has become an important topic of investigation [30, 31, 36]. In a general sense, a multidimensional network (also known as high-order network) is any network that has additional representational structures. For example, this is the case of dynamic (i.e., time-varying) networks [19, 35, 39, 42], multilayer networks [11, 24, 30], and dynamic multilayer networks [44,45,47]. In this regard, the general scope of this paper is to study the lossless incompressibility of multidimensional networks as well as network topological properties in such generalized graph representations. Unlike traditional methods from random graphs theory, such as the probabilistic method, and from data compression algorithm analysis, which is dependent on the chosen programming language, we present a theoretical investigation of algorithmic complexity and algorithmic randomness. First, this enables worst-case compressibility and network complexity analyses that do not depend on the choice of programming language [37, 52]. Secondly, it enables us to achieve mathematical proofs of the existence of certain network topological properties in generalized graphs, which are not necessarily generated or defined by stochastic processes [14, 34]. With this purpose, algorithmic information theory [15, 28, 34] has been giving us computably universal tools for studying data compression of individual objects [6, 26, 34, 43, 51]. On the one hand, first note that the source coding theorem [20] in classical information theory ensures that, as SV (G)S = n → ∞, every recursively labeled representation of a random graph G on n vertices and edge probability p = 1~2 in the classical Erdos–R˝ enyi´ model G(n; p)—from an independent and identically distributed stochastic process—is expected to be losslessly incompressible with probability arbitrarily close to one [32,34]. In this sense, as shown in [25,33,38,52], approaches to network complexity based on classical (or statistical) information theory have presented useful tools to find, estimate, or measure underlying graph-theoretic topological properties in random graphs or in complex networks. On the other hand, for some graphs G (with n vertices) displaying maximal degree-sequence entropy [53] or exhibiting a Borel-normal distribution of presence or absence of edges [7–9], the edge set E(G) is computable and, therefore, is algorithmically compressible on a logarithmic order [15, 28, 34]. That is, even though their inner structures might seem to arXiv:1812.01170v7 [cs.IT] 29 Apr 2020 be statistically homogeneous or to be following a uniform probability distribution, these graphs G may be compressed into O (log(n)) bits and thus are very far from being incompressible objects. Moreover, it has been shown that statistical estimations are not invariant to language description in general [52]. This comes from the reliance on probability distributions that require making a choice of feature of interest relevant to the measure of interest. For example, the node degrees for the study of their distribution in a network disregards other representations of the same object (network) and its possible underlying generating mechanism. While some of these statistical approaches may be useful when a feature of interest is selected they can only capture the complexity of the representation and not the object. This is in contrast to universal measures of randomness such as ∗In [4], a preliminary version of part of this paper is presented as an extended abstract. †National Laboratory for Scientific Computing (LNCC) – 25651-075 – Petropolis, RJ – Brazil. ‡Oxford Immune Algorithmics – RG1 3EU – Reading – U.K. Algorithmic Dynamics Lab, Unit of Computational Medicine, Department of Medicine Solna, Center for Molecular Medicine, Karolinska Institute – SE-171 77 – Stockholm – Sweden. Algorithmic Nature Group, Laboratoire de Recherche Scientifique (LABORES) for the Natural and Digital Sciences – 75005 – Paris – France. Emails: [email protected], [email protected], [email protected], [email protected] 2 algorithmic complexity whose invariance theorem guarantees that different representation will have convergent values [15,28,34] (up to a constant that only depends on the choice of the universal programming languages). Of course, the complication is how to achieve the estimation of those universal measures which by virtue of being universal are also semi-computable and their application require, therefore, a much higher degree of methodological care compared to those measures that are simply computable such as those based on traditional statistical approaches, e.g., entropy, or graph-theoretic approaches, e.g., node degree. In this article, we apply the results on computable labeling and algorithmic randomness introduced in [14,29,52]. In particular, we extend these ones for string-based representations of classical graphs to string-based representations of multiaspect graphs (MAGs), as shown in [1] . MAGs are formal representations of dyadic (or 2-place) relations between two arbitrary n-ary tuples [45,46] and have shown fruitful representational properties to network modelling [2,3,45,49] and analysis of multidimensional networks [19, 44, 47, 48], such as dynamic networks and multilayer networks. First, we compare the algorithmic complexity and incompressibility of snapshot-dynamic networks and multiplex networks with more general forms of dynamic networks and multilayer networks, respectively. In turn, dynamic networks and multilayer networks are considered as distinct types of multidimensional (or high-order) networks. Secondly, we demonstrate some multidimensional topological properties of incompressible general multidimensional networks. To tackle these problems in the present paper, we apply a theoretical approach by putting forward definitions, lemmas, theorems, and corollaries. In Sections II and III, we present some background results upon which we build the contributions of this paper. In Section IV, we investigate the algorithmic randomness of snapshot-dynamic networks and multiplex networks through the calculation of the worst-case lossless compression of the characteristic string of the network. This way, one can compare these two kinds of networks with more general forms of dynamic networks and multilayer networks, respectively. Further, in Section V, we investigate some multidimensional network topological properties of incompressible multidimen- sional networks. Such topological properties include, more specifically, degree distribution of composite vertices, composite diameter, and composite k-connectivity. As shown in [1], if the randomness deficiency is asymptotically bounded above by a logarithmic term of the network size, these findings follow from the fact that incompressible multidimensional networks inherit the topological properties from their respective isomorphic graphs [45]. Hence, we also demonstrate the presence of a new multidimensional network topological property that may not correspond to underlying structural constraints in real-world networks, e.g., of snapshot-dynamic networks. In particular, we show the presence of transtemporal or crosslayer edges (i.e., edges linking vertices at non-sequential time instants or layers). II. PRELIMINARY DEFINITIONS AND NOTATION A. Multiaspect graphs We directly base our definitions and notation regarding classical graphs on [12, 13, 27]; and regarding generalized graph representation of dyadic relations between n-tuples (i.e. multiaspect graphs) on [45, 46]. Definition II.1. Let G = (A ; E ) be a multiaspect graph