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In this article, we investigate limitations of importing methods based on algorithmic information theory from monoplex networks into multidimensional networks (such as multilayer networks) that have a large number of extra dimensions (i.e., aspects). In the worst-case scenario, it has been previously shown that node-aligned multidimensional networks with non-uniform multidimensional spaces can display exponentially larger algorithmic information (or lossless compressibility) distortions with respect to their isomorphic monoplex networks, so that these distortions grow at least linearly with the number of extra dimensions. In the present article, we demonstrate that node-unaligned multidimensional networks, either with uniform or non-uniform multidimensional spaces, can also display exponentially larger algorithmic information distortions with respect to their isomorphic monoplex networks. However, unlike the node-aligned non-uniform case studied in previous work, these distortions in the node-unaligned case grow at least exponentially with the number of extra dimensions. On the other hand, for node-aligned multidimensional networks with uniform multidimensional spaces, we demonstrate that any distortion can only grow up to a logarithmic order of the number of extra dimensions. Thus, these results establish that isomorphisms between finite multidimensional networks and finite monoplex networks do not preserve algorithmic information in general and highlight that the algorithmic information of the multidimensional space itself needs to be taken into account in multidimensional network complexity analysis.

We present the algebraic representation and basic algorithms for MultiAspect Graphs (MAGs). A MAG is a structure capable of representing multilayer and time-varying networks, as well as higher-order networks, while also having the property of being isomorphic to a directed graph. In particular, we show that, as a consequence of the properties associated with the MAG structure, a MAG can be represented in matrix form. Moreover, we also show that any possible MAG function (algorithm) can be obtained from this matrix-based representation. This is an important theoretical result since it paves the way for adapting well-known graph algorithms for application in MAGs. We present a set of basic MAG algorithms, constructed from well-known graph algorithms, such as degree computing, Breadth First Search (BFS), and Depth First Search (DFS). These algorithms adapted to the MAG context can be used as primitives for building other more sophisticated MAG algorithms. Therefore, such examples can be seen as guidelines on how to properly derive MAG algorithms from basic algorithms on directed graphs. We also make available Python implementations of all the algorithms presented in this paper.

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 investigates 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. Our theoretical investigation first explores fundamental and basic conditions for connecting the sequential growth of information with sequential interdimensional structures such as time in dynamic networks, and secondly it presents open problems demanding future investigation. Although there may be a dissonance between sequential information dynamics and sequential topology in the general case, we demonstrate that incompressibility dissolves it, preventing both the algorithmic dynamics and the interdimensional structure of multidimensional networks from displaying a snapshot-like behavior (as characterized by any arbitrary mathematical theory). Thus, beyond methods based on statistics or probability as traditionally seen in random graphs and complex networks models, representational incompressibility implies a necessary underlying constraint in the multidimensional network topology. We argue that the study of how isomorphic transformations and their respective algorithmic information distortions can characterize sequential interdimensional structures in (multidimensional) networks helps the analysis of network topological properties while being agnostic to the chosen theory, algorithm, computation model, and programming language.

We study emergent information in populations of randomly generated networked computable systems that follow a Susceptible-Infected-Susceptible contagion (or infection) model of imitation of the fittest neighbor. These networks have a scale-free degree distribution in the form of a power-law following the Barabási-Albert model. We show that there is a lower bound for the stationary prevalence (or average density of infected nodes) that triggers an unlimited increase of the expected emergent algorithmic complexity (or information) of a node as the population size grows.

This article presents a theoretical investigation of generalized encoded forms of networks in a uniform multidimensional space. First, we study encoded networks with (finite) arbitrary node dimensions (or aspects), such as time instants or layers. In particular, we study these networks that are formalized in the form of multiaspect graphs. In the context of node-aligned non-uniform (or node-unaligned non-uniform and uniform) multidimensional spaces, previous results has shown that, unlike classical graphs, the algorithmic information of a multidimensional network is not in general dominated by the algorithmic information of the binary sequence that determines the presence or absence of edges. In the present work, first we demonstrate the existence of such algorithmic information distortions for node-aligned uniform multidimensional networks. Secondly, we show that there are particular cases of infinite nesting families of finite uniform multidimensional networks such that each member of these families is incompressible. From these results, we also recover the network topological properties and equivalences in irreducible information content of multidimensional networks in comparison to their isomorphic classical graph counterpart in the previous literature. These results together establish a universal algorithmic approach and set limitations and conditions for irreducible information content analysis in comparing arbitrary networks with a large number of dimensions, such as multilayer networks.

Different graph generalizations have been recently used in an ad-hoc manner to represent multilayer networks, i.e. systems formed by distinct layers where each layer can be seen as a network. Similar constructions have also been used to represent time-varying networks. We introduce the concept of MultiAspect Graph (MAG) as a graph generalization that we prove to be isomorphic to a directed graph, and also capable of representing all previous generalizations. In our proposal, the set of vertices, layers, time instants, or any other independent features are considered as an aspect of the MAG. For instance, a MAG is able to represent multilayer or time-varying networks, while both concepts can also be combined to represent a multilayer time-varying network and even other higher-order networks. Since the MAG structure admits an arbitrary (finite) number of aspects, it hence introduces a powerful modelling abstraction for networked complex systems. This paper formalizes the concept of MAG and derives theoretical results useful in the analysis of complex networked systems modelled using the proposed MAG abstraction. We also present an overview of the MAG applicability.

As one of the main subjects of investigation in data science, network science has been demonstrated a wide range of applications to real-world networks analysis and modeling. For example, the pervasive presence of structural or topological characteristics, such as the small-world phenomenon, small-diameter, scale-free properties, or fat-tailed degree distribution were one of the underlying pillars fostering the study of complex networks. Relating these phenomena with other emergent properties in complex systems became a subject of central importance. By introducing new implications on the interface between data science and complex systems science with the purpose of tackling some of these issues, in this article we present a model for a network game played by complex networks in which nodes are computable systems. In particular, we present and discuss how some network topological properties and simple local communication rules are able to generate a phase transition with respect to the emergence of incompressible data.

Challenging the standard notion of totality in computable functions, one has that, given any sufficiently expressive formal axiomatic system, there are total functions that, although computable and \"intuitively\" understood as being total, cannot be proved to be total. In this article we show that this implies the existence of an infinite hierarchy of time complexity classes whose representative members are hidden from (or unknown by) the respective formal axiomatic systems. Although these classes contain total computable functions, there are some of these functions for which the formal axiomatic system cannot recognize as belonging to a time complexity class. This leads to incompleteness results regarding formalizations of computational complexity.

When mining large datasets in order to predict new data, limitations of the principles behind statistical machine learning pose a serious challenge not only to the Big Data deluge, but also to the traditional assumptions that data generating processes are biased toward low algorithmic complexity. Even when one assumes an underlying algorithmic-informational bias toward simplicity in finite dataset generators, we show that current approaches to machine learning (including deep learning, or any formal-theoretic hybrid mix of top-down AI and statistical machine learning approaches), can always be deceived, naturally or artificially, by sufficiently large datasets. In particular, we demonstrate that, for every learning algorithm (with or without access to a formal theory), there is a sufficiently large dataset size above which the algorithmic probability of an unpredictable deceiver is an upper bound (up to a multiplicative constant that only depends on the learning algorithm) for the algorithmic probability of any other larger dataset. In other words, very large and complex datasets can deceive learning algorithms into a ``simplicity bubble'' as likely as any other particular non-deceiving dataset. These deceiving datasets guarantee that any prediction effected by the learning algorithm will unpredictably diverge from the high-algorithmic-complexity globally optimal solution while converging toward the low-algorithmic-complexity locally optimal solution, although the latter is deemed a global one by the learning algorithm. We discuss the framework and additional empirical conditions to be met in order to circumvent this deceptive phenomenon, moving away from statistical machine learning towards a stronger type of machine learning based on, and motivated by, the intrinsic power of algorithmic information theory and computability theory.

This work presents some outcomes of a theoretical investigation of incompressible high-order networks defined by a generalized graph representation. We study some of their network topological properties and how these may be related to real-world complex networks. We show that these networks have very short diameter, high k-connectivity, degrees of the order of half of the network size within a strong-asymptotically dominated standard deviation, and rigidity with respect to automorphisms. In addition, we demonstrate that incompressible dynamic (or dynamic multilayered) networks have transtemporal (or crosslayer) edges and, thus, a snapshot-like representation of dynamic networks is inaccurate for capturing the presence of such edges that compose underlying structures of some real-world networks.