On incompressible multidimensional networks
Felipe S. Abrahão, Klaus Wehmuth, Hector Zenil, Artur Ziviani
11 On incompressible multidimensional networks ∗ Felipe S. Abrah˜ao † , 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-caseirreducible information content analysis, the demand for new graph abstractions increases. This article presents an investigation ofincompressible multidimensional networks defined by generalized graph representations. In particular, we mathematically study thelossless incompressibility of snapshot-dynamic networks and multiplex networks in comparison to the lossless incompressibility ofmore general forms of dynamic networks and multilayer networks, from which snapshot-dynamic networks or multiplex networksare particular cases. We show that incompressible snapshot-dynamic (or multiplex) networks carry an amount of algorithmicinformation that is linearly dominated by the size of the set of time instants (or layers). This contrasts with the algorithmicinformation carried by incompressible general dynamic (or multilayer) networks that is of the quadratic order of the size ofthe set of time instants (or layers). Furthermore, we prove that incompressible general multidimensional networks have edgeslinking vertices at non-sequential time instants (layers or, in general, elements of a node dimension). Thus, representationalincompressibility implies a necessary underlying constraint in the multidimensional network topology.
Index Terms
Algorithmic information, Complex networks, Lossless compression, Multidimensional systems, Network topology
I. I
NTRODUCTION
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 graphtheory and network science, proper representations of multidimensional networks into new extensions of graph-theoreticalabstractions has become an important topic of investigation [30, 31, 36]. In a general sense, a multidimensional network (alsoknown as high-order network) is any network that has additional representational structures. For example, this is the caseof 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 networksas well as network topological properties in such generalized graph representations. Unlike traditional methods from randomgraphs theory, such as the probabilistic method, and from data compression algorithm analysis, which is dependent on the chosenprogramming language, we present a theoretical investigation of algorithmic complexity and algorithmic randomness. First, thisenables 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 ingeneralized 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 studyingdata compression of individual objects [6, 26, 34, 43, 51]. On the one hand, first note that the source coding theorem [20] inclassical information theory ensures that, as ∣ V ( G )∣ = n → ∞ , every recursively labeled representation of a random graph G on n vertices and edge probability p = / in the classical Erd˝os–R´enyi model G( n, p ) —from an independent and identicallydistributed stochastic process—is expected to be losslessly incompressible with probability arbitrarily close to one [32, 34]. Inthis sense, as shown in [25, 33, 38, 52], approaches to network complexity based on classical (or statistical) information theoryhave presented useful tools to find, estimate, or measure underlying graph-theoretic topological properties in random graphsor in complex networks.On the other hand, for some graphs G (with n vertices) displaying maximal degree-sequence entropy [53] or exhibitinga Borel-normal distribution of presence or absence of edges [7–9], the edge set E ( G ) is computable and, therefore, isalgorithmically compressible on a logarithmic order [15, 28, 34]. That is, even though their inner structures might seem tobe 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 estimationsare not invariant to language description in general [52]. This comes from the reliance on probability distributions that requiremaking a choice of feature of interest relevant to the measure of interest. For example, the node degrees for the study of theirdistribution in a network disregards other representations of the same object (network) and its possible underlying generatingmechanism. While some of these statistical approaches may be useful when a feature of interest is selected they can onlycapture 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] a r X i v : . [ c s . I T ] A p r 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 ishow to achieve the estimation of those universal measures which by virtue of being universal are also semi-computable andtheir application require, therefore, a much higher degree of methodological care compared to those measures that are simplycomputable such as those based on traditional statistical approaches, e.g., entropy, or graph-theoretic approaches, e.g., nodedegree.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 -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 multidimensionalnetworks [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 networkswith more general forms of dynamic networks and multilayer networks, respectively. In turn, dynamic networks and multilayernetworks are considered as distinct types of multidimensional (or high-order) networks. Secondly, we demonstrate somemultidimensional topological properties of incompressible general multidimensional networks. To tackle these problems in thepresent 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 ofthe worst-case lossless compression of the characteristic string of the network. This way, one can compare these two kinds ofnetworks 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, compositediameter, and composite k-connectivity. As shown in [1], if the randomness deficiency is asymptotically bounded above bya logarithmic term of the network size, these findings follow from the fact that incompressible multidimensional networksinherit the topological properties from their respective isomorphic graphs [45]. Hence, we also demonstrate the presence of anew multidimensional network topological property that may not correspond to underlying structural constraints in real-worldnetworks, 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. P RELIMINARY DEFINITIONS AND NOTATION
A. Multiaspect graphs
We directly base our definitions and notation regarding classical graphs on [12, 13, 27]; and regarding generalized graphrepresentation of dyadic relations between n -tuples (i.e. multiaspect graphs) on [45, 46]. Definition II.1.
Let G = ( A , E ) be a multiaspect graph (MAG), where E is the set of existing composite edges of the MAGand A is a class of sets, each of which is an aspect . Each aspect σ ∈ A is a finite set and the number of aspects p = ∣ A ∣ iscalled the order of G . By an immediate convention, we call a MAG with only one aspect as a first order MAG, a MAG withtwo aspects as a second order
MAG and so on. Each composite edge (or arrow) e ∈ E may be denoted by an ordered p -tuple ( a , . . . , a p , b , . . . , b p ) , where a i , b i are elements of the i -th aspect with ≤ i ≤ p = ∣ A ∣ .Note that A ( G ) denotes the class of aspects of G and E ( G ) denotes the composite edge set of G . We denote the i -thaspect of G as A ( G )[ i ] . So, ∣ A ( G )[ i ]∣ denotes the number of elements in A ( G )[ i ] . In order to match the classical graphcase, we adopt the convention of calling the elements of the first aspect of a MAG as vertices . Therefore, we also denote theset A ( G )[ ] of elements of the first aspect of a MAG G as V ( G ) . Thus, a vertex should not be confused with a compositevertex. The set of all composite vertices v of G is defined by V ( G ) = p ⨉ i = A ( G )[ i ] and the set of all composite edges e of G is defined by E ( G ) = p ⨉ n = A ( G )[( n − ) ( mod p ) + )] ,so that, for every ordered pair ( u , v ) with u , v ∈ V ( G ) , we have ( u , v ) = e ∈ E ( G ) . Also, for every e ∈ E ( G ) , we have ( u , v ) = e such that u , v ∈ V ( G ) . Thus, E ( G ) ⊆ E ( G ) Definition II.2.
We say a directed
MAG (or graph) without self-loops is a traditional directed MAG (or graph), denoted as G d = ( A , E ) . Definition II.3.
We say an undirected
MAG (or graph) without self-loops is a simple
MAG (or graph), denoted as G c = ( A , E ) ,so that the set of all possible composite edges E is subjected to a restriction in the form E ( G c ) ⊆ E c ( G c ) ∶= {{ u , v } ∣ u , v ∈ V ( G c )} ,where there is Y ⊆ E ( G c ) such that { u , v } ∈ E ( G c ) ⇐⇒ ( u , v ) ∈ Y ∧ ( v , u ) ∈ Y ∧ u ≠ v We have directly from this Definition II.3 that ∣ E c ( G c )∣ = ∣ V ( G c )∣ − ∣ V ( G c )∣ .Concerning the presence or absence of composite edges in E ( G c ) , we defined the characteristic string [1] of a simple MAGby previously fixing an arbitrary ordering of all possible composite edges. However, this condition becomes unnecessary forrecursively labeled families, since this ordering is already embedded in Definition II.13. See also Lemma III.4. Definition II.4.
Let ( e , . . . , e ∣ E c ( G c )∣ ) be any arbitrarily fixed ordering of all possible composite edges of a simple MAG G c .We say that a string x ∈ { , } ∗ with l ( x ) = ∣ E c ( G c )∣ is a characteristic string of a simple MAG G c iff , for every e j ∈ E c ( G c ) , e j ∈ E ( G c ) ⇐⇒ the j -th digit in x is ,where ≤ j ≤ l ( x ) .We define the composite diameter of G in an analogous way to diameter in classical graphs: Definition II.5.
The composite diameter D E ( G ) is the maximum value in the set of the minimum number of steps (throughcomposite edges) in E ( G ) necessary to reach a composite vertex v from a composite vertex u , for any u , v ∈ V ( G ) .See also [45] for paths and distances in MAGs. Moreover, as in [45]: Definition II.6.
We say a traditional MAG G d is isomorphic to a traditional directed graph G when there is a bijective function f ∶ V ( G d ) → V ( G ) such that an edge e ∈ E ( G d ) if, and only if, the edge ( f ( π o ( e )) , f ( π d ( e ))) ∈ E ( G ) , where π o is a functionthat returns the origin vertex of an edge and the function π d is a function that returns the destination vertex of an edge.Thus, the reader may notice that the aspects in A determine how the set E will be defined and, therefore, they determinethe type of network that the MAG is univocally representing: for example, a time-varying graph (TVG) as in [19, 49] or amultilayer graph as in [44]. In the particular case of dynamic networks, as defined in [19, 49], we have that: Definition II.7.
Let G t = ( V , E , T ) be a second order MAG representing a time-varying graph (TVG), where V is the setof vertices, T is the set of time instants, and E ⊆ V × T × V × T is the set of (composite) edges. We denote the set of timeinstants of the graph G t = ( V , E , T ) by T ( G t ) = { t , t , . . . , t ∣ T ( G t )∣− } . Also, let V ( G t ) denote the set of vertices of G t and ∣ V ( G t )∣ denote the cardinality of the set of vertices in G t .Immediately, one indeed has from Definition II.1 [45, 46] that classical graphs are particular cases of MAGs, which haveonly one aspect: Definition II.8.
A labeled undirected graph G = ( V, E ) without self-loops is a labeled graph with a restriction E c in the edgeset E such that each edge is an unordered pair with E ⊆ E c ( G ) ∶= {{ x, y } ∣ x, y ∈ V } where there is Y ⊆ V × V such that { x, y } ∈ E ⊆ E c ( G ) ⇐⇒ ( x, y ) ∈ Y ∧ ( y, x ) ∈ Y ∧ x ≠ y We also refer to these graphs as classical (or simple labeled ) graphs.The terms vertex and node may be employed interchangeably in this article. However, note that we rather choose to use theterm node preferentially within the context of networks, where nodes may realize operations, computations or would have somekind of agency, like in real-world networks. Thus, we rather choose to use the term vertex preferentially in the mathematicalcontext of graph theory. That is, the adjacency matrix of this graph is symmetric and the main diagonal is null.
B. Algorithmic information theory
From [15,17,28,34], we will briefly recover in this section some of the main definitions, concepts, and notation in algorithmicinformation theory (aka Kolmogorov complexity theory or Solomonoff-Kolmogorov-Chaitin complexity theory). Let { , } ∗ be the set of all finite binary strings. Let l ( x ) denote the length of a finite string x ∈ { , } ∗ . In addition, let ∣ X ∣ denote thenumber of elements (i.e., the cardinality) in a set, if X is a finite set. Let ( x ) denote the string which is a binary represenationof the number x . In addition, let ( x ) L denote the representation of the number x ∈ N in language L . Let L U denote a binaryuniversal programming language for a universal Turing machine U . Let L ′ U denote a binary prefix-free (or self-delimiting )universal programming language for a prefix universal Turing machine U . Let ⟨ ⋅ , ⋅ ⟩ denote an arbitrary recursive bijectivepairing function. This notation can be recursively extended to ⟨ ⋅ , ⟨ ⋅ , ⋅ ⟩⟩ and, then, to an ordered tuple ⟨ ⋅ , ⋅ , ⋅ ⟩ . This iterationcan be recursively applied with the purpose of defining finite ordered tuples ⟨⋅ , . . . , ⋅⟩ . Definition II.9.
The (unconditional) plain algorithmic complexity (also known as C-complexity, plain Kolmogorov complexity,plain program-size complexity or plain Solomonoff-Komogorov-Chaitin complexity) of a finite binary string w , denoted by C ( w ) , is the length of the shortest program w ∗ ∈ L U such that U ( w ∗ ) = w . The conditional plain algorithmic complexityof a binary finite string y given a binary finite string x , denoted by C ( y ∣ x ) , is the length of the shortest program w ∈ L U such that U (⟨ x, w ⟩) = y . Note that C ( y ) = C ( y ∣ (cid:15) ) , where (cid:15) is the empty string. We also have the joint plain algorithmiccomplexity of strings x and y denoted by C ( x, y ) ∶= C (⟨ x, y ⟩) and the C-complexity of information in x about y denoted by I C ( x ∶ y ) ∶= C ( y ) − C ( y ∣ x ) . Notation II.10.
Let ( e , . . . , e ∣ E ( G )∣ ) be a previously fixed ordering (or indexing) of the set E ( G ) . For an (composite) edgeset E ( G ) , let C ( E ( G )) ∶= C (⟨ E ( G )⟩) , where ⟨ E ( G )⟩ denotes the (composite) edge set string ⟨⟨ e , z ⟩ , . . . , ⟨ e n , z n ⟩⟩ such that z i = ⇐⇒ e i ∈ E ( G ) ,where z i ∈ { , } with ≤ i ≤ n = ∣ E ( G )∣ . Thus, in the simple MAG case (as in Definition II.3) with the ordering fixed inDefinition II.4, we will have that ⟨ E ( G c )⟩ denotes the (composite) edge set string ⟨⟨ e , z ⟩ , . . . , ⟨ e n , z n ⟩⟩ such that z i = ⇐⇒ e i ∈ E ( G c ) ,where z i ∈ { , } with ≤ i ≤ n = ∣ E c ( G c )∣ . The same applies analogously to the conditional, joint, and C-complexity ofinformation cases from Definition II.9. Definition II.11.
The (unconditional) prefix algorithmic complexity (also known as K-complexity, prefix Kolmogorov com-plexity, prefix program-size complexity or prefix Solomonoff-Komogorov-Chaitin complexity) of a finite binary string w ,denoted by K ( w ) , is the length of the shortest program w ∗ ∈ L ′ U such that U ( w ∗ ) = w . The conditional prefix algorithmiccomplexity of a binary finite string y given a binary finite string x , denoted by K ( y ∣ x ) , is the length of the shortest program w ∈ L ′ U such that U (⟨ x, w ⟩) = y . Note that K ( y ) = K ( y ∣ (cid:15) ) , where (cid:15) is the empty string. Similarly, we have the joint prefixalgorithmic complexity of strings x and y denoted by K ( x, y ) ∶= K (⟨ x, y ⟩) , the K-complexity of information in x about y denoted by I K ( x ∶ y ) ∶= K ( y ) − K ( y ∣ x ) , and the mutual algorithmic information of the two strings x and y denoted by I A ( x ; y ) ∶= K ( y ) − K ( y ∣ x ∗ ) . Notation II.12.
Analogously to Notation II.10, for an (composite) edge set E ( G ) , let K ( E ( G )) ∶= K (⟨ E ( G )⟩) denote K (⟨⟨ e , z ⟩ , . . . , ⟨ e n , z n ⟩⟩) such that z i = ⇐⇒ e i ∈ E ( G ) ,where z i ∈ { , } with ≤ i ≤ n = ∣ E ( G )∣ . The same applies analogously to the simple MAG case (as in Notation II.10) andthe conditional, joint, K-complexity of information, and mutual cases from Definition II.11. Note that w ∗ denotes the lexicographically first p ∈ L U such that l ( p ) is minimum and U ( p ) = w . C. Algorithmically random multiaspect graphs
In order to study C-randomness (i.e., plain algorithmic randomness) of simple MAGs analogously to classical graphs, firstone needs to extend the concept of labeling in classical graphs to families of simple MAGs [1]:
Definition II.13.
A family F G c of simple MAGs G c (as in Definition II.3) is recursively labeled iff there are programs p ′ , p ′ ∈ { , } ∗ such that, for every G c ∈ F G c and for every a i , b i , j ∈ N with ≤ i ≤ p ∈ N , the following hold at the sametime:(I) if ( a , . . . , a p ) , ( b , . . . , b p ) ∈ V ( G c ) , then U (⟨⟨ a , . . . , a p ⟩ , ⟨ b , . . . , b p ⟩ , p ′ ⟩) = ( j ) (II) if ( a , . . . , a p ) or ( b , . . . , b p ) does not belong to any V ( G c ) with G c ∈ F G c , then U (⟨⟨ a , . . . , a p ⟩ , ⟨ b , . . . , b p ⟩ , p ′ ⟩) = (III) if ≤ j ≤ ∣ E c ( G c )∣ = ∣ V ( G c )∣ − ∣ V ( G c )∣ ,then U (⟨ j, p ′ ⟩) = ⟨⟨ a , . . . , a p ⟩ , ⟨ b , . . . , b p ⟩⟩ = ( e j ) (IV) if ≤ j ≤ ∣ E c ( G c )∣ = ∣ V ( G c )∣ − ∣ V ( G c )∣ does not hold for any V ( G c ) with G c ∈ F G c , then U (⟨ j, p ′ ⟩) = ⟨⟨ a , . . . , a p ⟩ , ⟨ b , . . . , b p ⟩⟩ = ⟨ ⟩ Thus, in Definition II.13, note that p ′ and p ′ define a family of finite MAGs (with arbitrarily fixed order p ) for whichthere is a unique ordering for all possible composite edges in the sets E c ( G c ) ’s and this ordering does not depend on thechoice of the MAG G c in the family F G c . Indeed, we will see in Lemma III.4 that, given an arbitrarily fixed order p , there isa recursively labeled infinite family that contains every possible MAG of order p , so that such a unique ordering of compositeedges is shared by each one the member of the family.Second, we extend the definition of plain algorithmically random classical graphs in [14, 34] to simple MAGs. First, in theclassical graph case, we have in [14, 34]: Definition II.14.
A classical graph G with ∣ V ( G )∣ = n is δ ( n ) -C-random if, and only if, it satisfies C ( E ( G ) ∣ n ) ≥ ( n ) − δ ( n ) ,where δ ∶ N → N n ↦ δ ( n ) is the randomness deficiency function.Then, in the simple MAG case, an analogous definition holds as introduced in [1]: Definition II.15.
We say a simple MAG G c is δ (∣ V ( G c )∣) -C-random iff it satisfies C ( E ( G c ) ∣ ∣ V ( G c )∣) ≥ (∣ V ( G c )∣ ) − δ (∣ V ( G c )∣) ,where δ ∶ N → N n ↦ δ ( n ) is the randomness deficiency function.With respect to weak asymptotic dominance of function f by a function g , we employ the usual O ( g ( x )) for the big O notation when f is asymptotically upper bounded by g ; and with respect to strong asymptotic dominance by a function g , weemploy the usual o ( g ( x )) when g dominates f .As also introduced in [1], one can also apply this same concept of algorithmic randomness to the prefix-free (or self-delimited)version, i.e., K-randomness: Definition II.16.
We say a simple MAG G c is O ( ) -K-random iff it satisfies K ( E ( G c )) ≥ (∣ V ( G c )∣ ) − O ( ) III. B
ACKGROUND RESULTS
In this section, we briefly recover some previous results.
A. Algorithmic information theory
First of all, it is important to remember some basic and important relations in algorithmic information theory [16, 17, 28, 34].
Lemma III.1.
For every x, y ∈ { , } ∗ and n ∈ N , C ( x ) ≤ l ( x ) + O ( ) (1) K ( x ) ≤ l ( x ) + O ( lg ( l ( x ))) (2) C ( y ∣ x ) ≤ C ( y ) + O ( ) (3) K ( y ∣ x ) ≤ K ( y ) + O ( ) (4) C ( y ∣ x ) ≤ K ( y ∣ x ) + O ( ) ≤ C ( y ∣ x ) + O ( lg ( C ( y ∣ x ))) (5) C ( x ) ≤ C ( x, y ) + O ( ) ≤ C ( y ) + C ( x ∣ y ) + O ( lg ( C ( x, y ))) (6) K ( x ) ≤ K ( x, y ) + O ( ) ≤ K ( y ) + K ( x ∣ y ) + O ( ) (7) C ( x ) ≤ K ( x ) + O ( ) (8) K ( n ) = O ( lg ( n )) (9) K ( x ) ≤ C ( x ) + K ( C ( x )) + O ( ) (10) I A ( x ; y ) = I A ( y ; x ) ± O ( ) (11)Note that the inverse relation K ( x, y ) + O ( ) ≥ K ( y ) + K ( x ∣ y ) + O ( ) does not hold in general in Equation (7). In fact,one can show that K ( x, y ) = K ( y ) + K ( x ∣ ⟨ y, K ( y )⟩) ± O ( ) , which is the key step to prove Equation (11). In this way, wehave that the notion of network topological (algorithmic) information of a simple MAG G c , i.e., the computably irreducibleinformation necessary to determine/compute ⟨ E ( G c )⟩ , is formally captured by I A (⟨ E ( G c )⟩ ; ⟨ E ( G c )⟩) = K (⟨ E ( G c )⟩) ± O ( ) .In the present article, wherever the concept of information is mentioned, we are in fact referring to algorithmic information;and wherever the term topological information appears, it is referring to the computably irreducible information necessary todetermine/compute the composite edge set E ( G c ) , which is structured and represented by string ⟨ E ( G c )⟩ . B. Multiaspect graphs and graphs isomorphism
In order to represent multidimensional networks, we are basing our work on a generalized graph representation of dyadicrelations between n -tuples [45, 46] called multiaspect graphs (MAGs). Directly from [45], one has that a simple MAG isbasically equivalent to a classical (i.e., simple labeled) graph: Corollary III.2.
For every simple MAG G c of order p > , where all aspects are non-empty sets, there is a unique (up to agraph isomorphism) classical graph G G c = ( V, E ) with ∣ V ( G )∣ = p ∏ n = ∣ A ( G c )[ n ]∣ that is isomorphic (as in Definition II.6) to G c . We also have that the concepts of walk , trail , and path become well-defined for MAGs analogously to within the contextof graphs. See [45, Section 3.5]. C. Algorithmically random multiaspect graphs
In this section, we briefly recover some previous results on algorithmically random graphs in [1]. These are obtained bybasing our previous work on algorithmically random classical (i.e., labeled simple) graphs in [14, 29, 52].In order to study plain algorithmic randomness (i.e., C-randomness) of simple MAGs, we extended in [1] the conceptof labeling in classical graphs to simple MAGs as restated in Definiton II.13. Then, we directly extended the definition ofalgorithmically random classical graphs to simple MAGs as restated in Definition II.16. Thus, by combining Corollary III.2with well-known inequalities in algorithmic information theory [17, 28, 34], we showed in [1] that:
Theorem III.3.
Let F G c ≠ ∅ be an arbitrary recursively labeled family of simple MAGs G c . Then, for every G c ∈ F G c , G c is ( δ (∣ V ( G c )∣) + O ( log (∣ V ( G c )∣))) -C-randomiff G is ( δ (∣ V ( G )∣) + O ( log (∣ V ( G )∣))) -C-random,where G is isomorphic (as in Definition II.6) to G c . Thus, Theorem III.3 establishes an algorithmic-informational cost of performing a transformation of a MAG into itsisomorphically correspondent graph, and vice-versa. Although such an isomorphism (as in Definition II.6) is an abstract mathematical equivalence, when dealing with structured data representations of objects (e.g., as the strings ⟨ E ( G c )⟩ or ⟨ E ( G )⟩ ),this equivalence must always be performed by computable procedures. Therefore, there is always an associated necessaryalgorithmic information, which corresponds to the length of the minimum program that performs these transformations.Theorem III.3 gives a worst-case algorithmic-informational cost in a logarithmic order of the network size for plain algorithmiccomplexity (i.e., for non-prefix-free universal programming languages). However, even in the prefix case (as in Definition II.11),we also showed in [1] that there is a algorithmic-informational cost upper bounded by a positive constant.We also showed in [1] that, given an arbitrarily fixed order p , there is a recursively labeled (see Definition II.13) infinitefamily that contains every possible MAG of order p : Lemma III.4.
There is a recursively labeled infinite family F G c of simple MAGs G c with arbitrary symmetric adjacency matrixsuch that every one of them has the same order p . Thus, Lemma III.4 ensures that one can follow the same ordering (or indexing) for the set E c ( G c ) , where this ordering doesnot depend on the choice of G c , so that every possible MAG of order p with vertex labels in N belongs to such an infinitefamily satisfying Lemma III.4.In addition, the basic algorithmic-informational properties of the binary string that determines the presence or absence of acomposite edge in Definition II.4 (i.e., the characteristic string) can be properly defined for recursively labeled families: Corollary III.5.
Let F G c be a recursively labeled family of simple MAGs G c . Then, for every G c ∈ F G c and x ∈ { , } ∗ , where x is the characteristic string of G c , the following relations hold: C ( E ( G c ) ∣ x ) ≤ K ( E ( G c ) ∣ x ) + O ( ) = O ( ) (12) C ( x ∣ E ( G c )) ≤ K ( x ∣ E ( G c )) + O ( ) = O ( ) (13) K ( x ) = K ( E ( G c )) ± O ( ) (14) I A ( x ; E ( G c )) = I A ( E ( G c ) ; x ) ± O ( ) = K ( x ) − O ( ) = K ( E ( G c )) ± O ( ) (15)In summary, Theorem III.3 shows that the plain algorithmic complexity of simple MAGs and of its respective isomorphicclassical graphs are roughly the same, except for the amount of algorithmic information necessary to encode the length ofthe program that performs this isomorphism on an arbitrary universal Turing machine. In fact, regarding the connectionsthrough composite edges, this shows that not only “most” of the network topological properties of such graph are inheritedby the MAG (and vice-versa), but also “most” of those that derives from the graph’s topological incompressibility. Thatis, more formally, every network topological property regarding the connections through composite edges that derives fromthe MAG G c being ( δ (∣ V ( G c )∣) + O ( log (∣ V ( G c )∣))) -C-random is inherited by G c from its isomorphic graph G , if G is ( δ (∣ V ( G )∣) + O ( log (∣ V ( G )∣))) -C-random and ∣ V ( G )∣ is large enough. And the inverse inheritance also holds. For example,we extended in [1] some results from [14, 34] on plain algorithmically random classical graphs to simple MAGs: Corollary III.6.
Let F G c be an arbitrary recursively labeled infinite family of simple MAGs G c . Then, the following hold forlarge enough G c ∈ F G c :1) The degree d ( v ) of a composite vertex v ∈ V ( G c ) in a δ (∣ V ( G c )∣) -C-random MAG G c ∈ F G c satisfies ∣ d ( v ) − ( ∣ V ( G c )∣ − )∣ = O (√∣ V ( G c )∣ ( δ (∣ V ( G c )∣) + O ( log (∣ V ( G c )∣)))) o (∣ V ( G c )∣) -C-random MAGs G c ∈ F G c have ∣ V ( G c )∣ ± o (∣ V ( G c )∣) disjoint paths of length 2 between each pair of composite vertices u , v ∈ V ( G c ) . In particular, o (∣ V ( G c )∣) -C-randomMAGs G c ∈ F G c have composite diameter . Thus, an incompressible simple MAG under randomness deficiency δ (∣ V ( G c )∣) = o (∣ V ( G c )∣) tends to be an expected “almostregular” graph in the limit when the network size increases indefinitely; for sufficiently large set of composite vertices, theseMAGs also cross a phase transition in which the diameter between composite vertices becomes ; with respect to k-connectivity,as defined in [14], they are ∣ V ( G c )∣ ± o (∣ V ( G c )∣) -connected.Furthermore, from Definition II.16, one can build an infinite family of simple MAGs in which every member is O ( ) -K-random and, in turn, also retrieve their plain algorithmically randomness [1]: Lemma III.7.
There is a recursively labeled infinite family F G c of simple MAGs G c that are O ( ) -K-random. Theorem III.8.
Let F G c be a recursively labeled infinite family of simple MAGs G c such that, for every G c ∈ F G c and n ∈ N ,if x ↾ n is its characteristic string and n = ∣ E c ( G c )∣ , then x ∈ [ , ] ⊂ R is O ( ) -K-random. Thus, every MAG G c ∈ F G c is O ( log (∣ V ( G c )∣)) -C-random and O ( ) -K-random. In addition, there is such a family F G c with Ω = x ∈ [ , ] ⊂ R , where Ω is the halting probability [15, 17] . IV. S
NAPSHOT - LIKE MULTIDIMENSIONAL NETWORKS
This section presents a theoretical investigation of the consequences of the results in Section III-C to some of the commonrepresentations of dynamic networks and multilayer networks. As we will explain and formalize in Section IV-B, we choosea differential approach to the dynamic and multilayer case, so that both become particular cases of general multidimensionalnetworks while keeping their own distinct physical interpretation of what each ‘dimension’ (or aspect) [45, 46, 48] represents.In this way, we first present the investigation of the algorithmic complexity of snapshot-based representations of dynamicnetworks. Then, in Section IV-B, we introduce the same kind of investigation for multiplex networks.
A. Snapshot-dynamic networks
In the context of real-world complex networks analysis, one may highlight some important representation models of dynamicnetworks, such as, time-varying graphs (TVGs) [19, 45, 46, 49], temporal networks (TNs) [39, 42], temporal graphs (TGs) [35],and snapshot networks (SNs) [42, 49]. In this direction, we follow the same unifying and universal approach in [49] withthe purpose of showing that a particular class of dynamic networks (in the case, snapshot-dynamic networks) displays lessirreducible information content than a more general representation of dynamic networks such as TVGs. However, studying theadvantages and disadvantages of each representation model in terms of network analysis is not in our current scope. Thus, wefocus on studying a general snapshot-based representation of dynamic networks and its algorithmic randomness in relation totime-varying graphs (TVGs) and its algorithmic randomness. Note that TVGs are second order multiaspect graphs (MAGs)[19, 47, 49].The main idea is to: first, briefly discuss equivalences of some of the main representations of snapshot-like dynamic networks;secondly, study the algorithmic randomness of snapshot-dynamic networks, which can be represented by a particular class ofTVGs; and, then, compare with the algorithmic randomness of general undirected dynamic networks, which are arbitrary simpleTVGs, i.e., second order simple MAGs.Except for the cases in which the pertinence of the vertices in each time instant (and not only its connectivity in eachtime instant) do matter in the network analysis—see node-alignment below—, one can easily show that a snapshot-basedrepresentation as in [42] is equivalent to the snapshot-based representation in [49]. To this end, note that, in [42], a snapshotnetwork is defined as a sequence of graphs G i = ( V i , E i ) in the form ( G , . . . , G t max ) . On the other hand, in [49], a snapshotnetwork is a TVG composed of only spatial edges, i.e., edges that connect two vertices at the same time instant only. In otherwords, a snapshot-like dynamic network in [49] is a special case of dynamic network that can be solely represented by, forexample, the main diagonal blocks of the adjacency matrix of the isomorphic graph to the TVG in Figure 1. Fig. 1. The adjancency matrix of the isomorphic graph to the TVG, which represents a sequentially coupled node-aligned dynamic network.
For the sake of simplicity, we call an arbitrary TVG G t = ( V , E , T ) composed of only spatial edges as a spatial TVG.Therefore, the sequence of vertex sets ( V , . . . , V t max ) in G i = ( V i , E i ) may be mapped onto a larger vertex set V = ⋃ V i ∈( V ,...,V tmax ) V i such that G t = ( V , E , T ) , ∣{ G i ∣ G i ∈ ( G , . . . , G t max )}∣ = ∣ T ( G t )∣ , and ( u, v ) ∈ E i ( G i ) ⇐⇒ ( u, t i , v, t i ) ∈ E ( G t ) , (16)where ≤ i ≤ t max . Inversely, a spatial TVG can be univocally represented by a sequence of graphs ( G , . . . , G t max ) as in[42] by simply assuming V i = V j for every ≤ i ≤ t max and ≤ j ≤ t max , so that the equivalence in Equation (16) also holds. In most cases, a node-alignment [21, 30] hypothesis (i.e., V i = V j for every ≤ i ≤ t max and ≤ j ≤ t max ) is assumed, sothat no additional information would be needed to determine which vertices do not belong to a specific time instant. Thus, forthe present purposes of this article, we assume that the snapshot network is node-aligned.Nevertheless, an interesting future research would be to investigate the worst-case scenarios in which additional informationis needed to recover the original snapshot network that is not node-aligned from the spatial TVG. Indeed, an irreducibleinformation dependency on the sequence ( V ∖ V , . . . , V ∖ V t max ) of excluded vertices may take place in a similar manner tothe one on the companion tuple in [1].In any event, note that retrieving the correspondent spatial TVG [45, 49] from the snapshot network [42] is alwaysstraightforward by the addition of empty nodes [30]] (or, in MAG terminology, unconnected composite vertices [46]), sinceevery TVG is defined on the set of composite vertices [45] and, therefore, is always node-aligned by definition.Another formalization of a snapshot-like dynamic network may be through restricting the set E ( G t ) of all possible compositeedges of a TVG G t into another set E ′ ( G t ) such that, for every e ∈ E ( G t ) and i, j ∈ N , e = ( u, t i , v, t j ) ∈ E ′ ( G t ) ⇐⇒ j = f ( i ) ,where f ∶ { , f ( ) , f ( f ( )) , . . . , f − (∣ T ( G t )∣ − )} → { f ( ) , f ( f ( )) , . . . , ∣ T ( G t )∣ − } is a strictly increasing bijectivefunction defined on any recursive iteration. Thus, these restricted TVGs G t = ( V , E ′ , T ) , where E ′ ( G t ) ⊆ E ′ ( G t ) , maypresent advantages when considering a more realistic scenario in which relations or communications between nodes are notinstantaneous or demand non-equal time intervals over time. In addition, if one allows temporal edges (i.e., edges connectingthe same node at two distinct time instants [49]), which are directly analogous to coupling edges [30] in the multilayer case,some assumptions like the one that guarantees the transitivity on a node in the case it is disconnected within one or moresnapshots become unnecessary.In fact, unlike the multilayer case in which there may not be a physical interpretation of the necessity of preserving thepairwise ordering of layers—as we will also discuss in Section IV-B—, this assumption of transitivity is represented by arestriction on the set of temporal (or coupling) edges: we say a TVG is sequentially coupled if all the temporal (or coupling)edges are connecting the same node u from the time instant t i to the time instant t i + only and, for every ≤ i ≤ t max , everynode u has a temporal (or coupling) edge to itself from the time instant t i to the time instant t i + . The reader is invited tonote that sequentially coupling is an even stronger restriction of the set of coupling edges than saying that the couplings are diagonal and/or categorical as in [30]—see also Section IV-B. That is, if the network is sequentially coupled, there is no othertemporal (or coupling) edge connecting a node u at time instant t i to the same node u at time instant t j than the case inwhich j = i + . An example of sequentially coupled networks are the temporal networks as defined in [11], should they bealso node-aligned.Thus, one can see that a snapshot-like dynamic network in the form ( V , E ′ , T ) enables one to better represent the cases inwhich the sequential coupling does not hold in general; or, in other words, in which some nodes may not relay information forfuture communication in forthcoming time instants. Related to this issue, it is a consequence of the result we will demonstratein Section V that arbitrary incompressible simple TVGs are not sequentially coupled. See Corollary V.2.In any event, we have that this representation of snapshot-like dynamic networks in the form ( V , E ′ , T ) can also be reducedto spatial TVGs ( V , E , T ′′ ) —i.e., without the restrictions in the set of composite edges—by injectively mapping the set of timeintervals onto another set T ′′ ( G t ) of time instants while preserving the previous ordering: for example, one runs a recursivebijective procedure that makes t ′′ i ≡ ( t i , t f ( i ) ) , where ∣ T ′′ ( G t )∣ ≤ ∣ T ( G t )∣ − , and e = ( u, t i , v, t f ( i ) ) ∈ E ′ ( G t ) ⇐⇒ ( u, t ′′ i , v, t ′′ i ) ∈ E ( G t ) .Again, as also may occur with snapshot networks ( G , . . . , G t max ) that are not node-aligned, retrieving the spatial TVGfrom a snapshot-like dynamic network ( V , E ′ , T ) is straightforward, whereas the inverse conversion may require additionalinformation—in this latter particular case to determine which is the time interval ( t i , t f ( i ) ) that each t ′′ i is representing (inthe case these intervals are not uniformly equal). Thus, a future investigation of the worst-case information dependency ofa non-uniform function f ( i ) will be necessary; and the irreducible topological information (i.e., the irreducible informationnecessary to determine/compute ⟨ E ′ ⟩ ) carried by an arbitrary ( V , E ′ , T ) may be less compressible than that of spatial TVGs.In this way, for the purposes of this article, we assume hereafter the representation of a snapshot-like dynamic network asa spatial TVG; and we call them simply as snapshot-dynamic network . Note that underlying properties in snapshot-dynamicnetworks, like the sequential coupling, are fixed. Therefore, the algorithmic information of a spatial TVG and the algorithmicinformation of a spatial TVG with the addition of sequential couplings can only differ by a constant (that only depends on thechosen language L U ) and, thus, is negligible in our forthcoming results.It is straightforward to calculate the maximum number of spatial directed edges e ∈ E ( G t ) in a traditional TVG G dt =( V , E , T ) . Note that a traditional MAG is a (directed or undirected) MAG without (composite) self-loops [1, 45]. (SeeDefinition II.2). Moreover, from [19, 49], we have that a TVG is a second order MAG. (See Definition II.7). Therefore,we define a traditional directed
TVG G dt = ( V , E , T ) as a TVG without self-loops. This way, we will have that there are ∣ T ( G dt )∣ (∣ V ( G dt )∣ − ∣ V ( G dt )∣) possible spatial directed edges.From a simple graph (i.e., an undirected graph without self-loops) perspective, we may consider spatial TVGs as sequencesof simple graphs. We call a simple TVG G ut = ( V , E , T ) as a particular case of a simple second order MAG, where a simpleMAG (see Definition II.3)) is defined in [1] as a traditional undirected MAG. This way, we will have that there are ∣ T ( G ut )∣ ( ∣ V ( G ut )∣ − ∣ V ( G ut )∣ ) possible spatial undirected edges in simple spatial TVGs.Now, let G ′ t denote an arbitrary simple spatial TVG G ut = ( V , E , T ) that belongs to a recursively labeled infinite family F G ′ t of all possible simple TVGs with vertex labels in N . The existence of such a family is guaranteed by Lemma III.4. Directlyfrom Definition II.4, we have the binary string that univocally represents the presence or absence of an edge in E ( G ′ t ) ; and wecall it the characteristic string of G ′ t [1]. In this sense, from Corollary III.5, one can see that a characteristic string promptlycontains all the information that is necessary to computably retrieve the entire G ′ t , except for the information required to applya previously known recursive way to label the composite vertices and order the composite edges.It is also important to note that the information encoded in the characteristic string may be displaying a decompressed formof its algorithmic information content. To tackle this issue, we define the algorithmic-informational version of the characteristicstring, and thus formalizing such a notion of topological (algorithmic) information—see Theorem IV.1, Corollary III.5, andSection III-A—that is potentially agnostic with respect to node labeling or indexing: Definition IV.1.
Let F G c be a recursively labeled family of simple MAGs G c . Let p ′′ , p ′′ ∈ L U be fixed and only depend onthe choice of the family F G c . We say a binary string y ∈ { , } ∗ is an algorithmically characteristic string of G c with respectto F G c iff U (⟨ y, p ′′ ⟩) = ⟨ E ( G c )⟩ and U (⟨⟨ E ( G c )⟩ , p ′′ ⟩) = y .Thus, if y is such an algorithmically characteristic string, it is immediate to show that C ( E ( G c )∣ y ) ≤ K ( E ( G c )∣ y ) + O ( ) = O ( ) and C ( y ∣ E ( G c )) ≤ K ( y ∣ E ( G c )) + O ( ) = O ( ) hold independently of the choice of G c in F G c .On the other hand, it may not be the case that the opposite implication ( i.e., K ( E ( G c )∣ y ) + O ( ) = O ( ) and K ( y ∣ E ( G c )) + O ( ) = O ( ) implying the existence of constants p ′′ , p ′′ ∈ L U such that U (⟨ y, p ′′ ⟩) = ⟨ E ( G c )⟩ and U (⟨⟨ E ( G c )⟩ , p ′′ ⟩) = y ) does hold ingeneral—and this should be an interesting future research. For example, a possible question in this direction might be whetherit is possible or not to construct a recursively labeled infinite family of MAGs in which there is an infinite subfamily of MAGsthat are K-trivial [28], but not computable, with respect to a string y .In any event, from the proof of Corollary III.5 presented in [1], we have that Definiton IV.1 is always satisfiable by takingthe algorithmically characteristic string y as, for instance, the very characteristic string. This holds because of the recursivelylabeling of the entire family of MAGs, as in Definition II.13. However, since one surely knows there are ( (∣ V ( G ′ t )∣ ∣ T ( G ′ t )∣) − ∣ V ( G ′ t )∣ ∣ T ( G ′ t )∣ ) − ∣ T ( G ′ t )∣ ( ∣ V ( G ′ t )∣ − ∣ V ( G ′ t )∣ ) non-existent non-spatial undirected edges (including non-spatial undirected edges that are sequential couplings) and G ′ t is asecond order simple MAG G c , one can compress the characteristic string of G ′ t in such a way that the resulting algorithmicallycharacteristic string retains the algorithmic information carried, or conveyed, by the usual characteristic string: Theorem IV.1.
Let G ′ t = ( V , E , T ) be a simple spatial TVG that belongs to a recursively labeled infinite family F G ′ t of simpleTVGs. Then, there is a binary string y ∈ { , } ∗ that is an algorithmically characteristic string of G ′ t such that K ( y ) ≤ l ( y ) + O ( ) ≤ ∣ T ( G ′ t )∣ ( ∣ V ( G ′ t )∣ − ∣ V ( G ′ t )∣ ) + O ( log (∣ V ( G ′ t )∣ ∣ T ( G ′ t )∣)) , K ( x ) ≤ K ( y ) + O ( ) , K ( x ∣ y ∗ ) ≤ K ( x ∣ y ) + O ( ) ≤ O ( ) , K ( y ) ≤ K ( x ) + O ( ) , and K ( y ∣ x ∗ ) ≤ K ( y ∣ x ) + O ( ) ≤ O ( ) hold, where x is the characteristic string of G ′ t .Proof. The main idea of the proof is based on showing a Turing equivalence between the string y and its respective characteristicstring x . Then, as in the proof of Corollary III.5 presented in [1], we will recover the Turing equivalence between thecharacteristic string x and the string ⟨ E ( G ′ t )⟩ . So, first let y ′ ∈ { , } ∗ be any arbitrary binary string with l ( y ′ ) = ∣ T ( G ′ t )∣ ( ∣ V ( G ′ t )∣ − ∣ V ( G ′ t )∣ ) Let p ′ ∈ { , } ∗ be a binary string that represents an algorithm running on a prefix universal Turing machine U that takes j ∈ N as input and returns the j -th edge in E c ( G ′ t ) . The existence of such p ′ is guaranteed by the definition of recursivelylabeling in [1] (see Definition II.13). Moreover, p ′ is independent of the choice of G ′ t in the family F G ′ t . Let s ∈ { , } ∗ bea binary string that represents an algorithm running on a prefix universal Turing machine U that:1) takes p ′ , y ′ and j as inputs;2) calculates U (⟨ j ′ , p ′ ⟩) for every j ′ ≤ j ;3) enumerates all the spatial edges e j ′ = U (⟨ j ′ , p ′ ⟩) as a subsequence of the sequence of possible undirected edges in ( e , . . . , e j ) ;4) and returns: a) , if U (⟨ j, p ′ ⟩) is not a spatial edge;b) , if U (⟨ j, p ′ ⟩) is the i -th spatial edge and the i -th digit in y ′ is ;c) , if U (⟨ j, p ′ ⟩) is the i -th spatial edge and the i -th digit in y ′ is .Note that deciding whether an edge e is spatial or not follows directly from deciding whether t u = t v or not in e = ( u, t u , v, t v ) ,which is a decidable (and computationally cheap) procedure. Let s ∈ { , } ∗ be a binary string that represents an algorithmrunning on a prefix universal Turing machine U that:1) takes s , p ′ , y ′ and ( (∣ V ( G ′ t )∣ ∣ T ( G ′ t )∣) − ∣ V ( G ′ t )∣ ∣ T ( G ′ t )∣ ) as inputs;2) calculates U (⟨ j, y ′ , p ′ , s ⟩) for every j with ≤ j ≤ ( (∣ V ( G ′ t )∣ ∣ T ( G ′ t )∣) − ∣ V ( G ′ t )∣ ∣ T ( G ′ t )∣ ) ;3) and returns the binary string x = z . . . z n such that z i = ⇐⇒ U (⟨ i, y ′ , p ′ , s ⟩) = ,where z i ∈ { , } with ≤ i ≤ n = ( (∣ V ( G ′ t )∣ ∣ T ( G ′ t )∣) −∣ V ( G ′ t )∣ ∣ T ( G ′ t )∣ ) .Now, let y = ⟨ k, y ′ , p ′ , s , s ⟩ , where k is the self-delimiting binary representation of ( (∣ V ( G ′ t )∣ ∣ T ( G ′ t )∣) − ∣ V ( G ′ t )∣ ∣ T ( G ′ t )∣ ) ∈ N .We know one can encode k in O ( log ((∣ V ( G ′ t )∣ ∣ T ( G ′ t )∣) )) bits. Therefore, since p ′ , s , and s are fixed and independentof the choice of G ′ t , we will have that, by the minimality of the prefix algorithmic complexity, K ( x ) ≤ K ( y ) + O ( ) ≤ l (⟨ k, y ′ , p ′ , s , s ⟩) + O ( ) ≤≤ l ( y ′ ) + O ( log ((∣ V ( G ′ t )∣ ∣ T ( G ′ t )∣) )) and K ( x ∣ y ∗ ) ≤ K ( x ∣ y ) + O ( ) ≤ O ( ) .On the other hand, in order to show that K ( y ) ≤ K ( x ) + O ( ) , let s ∈ { , } ∗ be a binary string that represents an algorithmrunning on a prefix universal Turing machine U that: Alternatively, in the case the sequential couplings were not excluded in the representation of the snapshot-dynamic network, one can add here a clausealso returning if U (⟨ j, p ′ ⟩) is a sequential coupling. The reader is invited to note that the theorem holds anyway.
1) takes p ′ , s , s , and x as inputs;2) enumerates all the spatial edges using program p ′ ;3) and builds the binary string y ′ = y ′ . . . y ′ k ′ such that:a) y ′ i = , if j corresponds to the i -th spatial edge and the j -th digit of x is ;b) y ′ i = , if j corresponds to the i -th spatial edge and the j -th digit of x is ;4) finally, s returns the binary string ⟨ k, y ′ , p ′ , s , s ⟩ = y .Note that x was already given as input and l ( x ) = ( (∣ V ( G ′ t )∣ ∣ T ( G ′ t )∣) − ∣ V ( G ′ t )∣ ∣ T ( G ′ t )∣ ) .Therefore, since p ′ , s , s , and s are fixed and independent of the choice of G ′ t , we will have that, by the minimality ofthe prefix algorithmic complexity, K ( y ) = K (⟨ k, y ′ , p ′ , s , s ⟩) ≤ K (⟨ x, p ′ , s , s , s ⟩) + O ( ) ≤ K ( x ) + O ( ) and K ( y ∣ x ∗ ) ≤ K ( y ∣ x ) + O ( ) ≤ O ( ) Now, let p be a binary string that represents the algorithm running on a universal Turing machine that:1) receives the string x as its input;2) for ≤ j ≤ l ( x ) , reads each j -th bit of x ;3) calculates U (⟨ j, p ′ ⟩) ;4) and, from the outputs e j of these programs ⟨ j, p ′ ⟩ , returns the string ⟨⟨ e , z ⟩ , . . . , ⟨ e n , z n ⟩⟩ , where: z j = , if the j -thbit of x is ; and z j = , if the j -th bit of x is .Thus, since p ′ is fixed, we will have that there is a self-delimiting binary encoding of p such that U (⟨ x, p ⟩) = ⟨⟨ e , z ⟩ , . . . , ⟨ e n , z n ⟩⟩ holds. Then, by the minimality of K (⋅) , we will have that K ( E ( G c ) ∣ x ) ≤ l ( p ) ≤ O ( ) Analogously to program p , using program p ′ instead of p ′ in order to build the string x from ⟨ E ( G c )⟩ , we will have anotherprogram q such that there is a self-delimiting binary encoding of q such that U (⟨⟨⟨ e , z ⟩ , . . . , ⟨ e n , z n ⟩⟩ , q ⟩) = x holds and, by the minimality of K (⋅) , we have that K ( x ∣ E ( G c )) ≤ l ( q ) ≤ O ( ) Note that p, q, p ′ , p ′ , s , s , s are fixed. Therefore, in order to finish the proof, just let p ′′ be the binary string that representsthe algorithm running on a universal Turing machine U that receives y as input and returns the value of U (⟨ U ( y ) , p ⟩) = ⟨⟨ e , z ⟩ , . . . , ⟨ e n , z n ⟩⟩ = ⟨ E ( G c )⟩ .Similarly, let p ′′ be the binary string that represents the algorithm running on a universal Turing machine U that receives ⟨ E ( G c )⟩ as input and returns the value of U (⟨ U (⟨⟨ E ( G c )⟩ , q ⟩) , p ′ , s , s , s ⟩) = y .Thus, for every simple spatial TVG G ′ t = ( V , E , T ) that belongs to a recursively labeled infinite family F G ′ t of simple TVGs,we will have that, from Lemma III.1, Theorem IV.1, and Corollary III.5, C ( E ( G ′ t ) ∣ (∣ V ( G ′ t ))∣ ∣ T ( G ′ t ))∣)) ≤ K ( E ( G ′ t )) + O ( ) ≤≤ ∣ T ( G ′ t )∣ ( ∣ V ( G ′ t )∣ − ∣ V ( G ′ t )∣ )+ O ( log (∣ V ( G ′ t )∣ ∣ T ( G ′ t )∣)) holds. On the other hand, as shown in [1], we directly have from Lemma III.7, Corollary III.5, and Theorem III.8 that, forarbitrary O ( ) -K-random simple TVGs G t , K ( E ( G t )) ≥ ( (∣ V ( G t )∣ ∣ T ( G t )∣) − ∣ V ( G t )∣ ∣ T ( G t )∣ ) − O ( ) and C ( E ( G t ) ∣ (∣ V ( G t ))∣ ∣ T ( G t ))∣)) ≥ ( (∣ V ( G t )∣ ∣ T ( G t )∣) − ∣ V ( G t )∣ ∣ T ( G t )∣ )− O ( log (∣ V ( G t )∣ ∣ T ( G t )∣)) .Therefore, for large enough sets of time instants, O ( ) -K-random simple TVGs carry at least ( (∣ V ( G t )∣ ∣ T ( G t )∣) − ∣ V ( G t )∣ ∣ T ( G t )∣ ) − ∣ T ( G t )∣ ( ∣ V ( G t )∣ − ∣ V ( G t )∣ )− O ( log (∣ V ( G t )∣ ∣ T ( G t )∣)) more topological (or network) irreducible information than any simple spatial TVG could carry. B. Multiplex networks
Other network models of increasing importance in network science are those in which the nodes and/or connections betweennodes may have distinct features [11, 18, 23, 30, 31, 45] other than the time dependency. For example, one may differentiate inthe case of: ● social networks, – between connection features, such as friendship, family, professional colleagues, face-to-face interaction, email interac-tion, etc [18, 21, 30]; – or between node features, such as gender, online platform, school, city, company, etc [18, 30]; ● biological networks, – between connection features, such as distinct nature of protein-protein reactions [23], neuronal interactions (in particular,either through chemical or ionic channels) [11], etc; – or between node features, such as distinct species, gene pool, community, location [40], etc; ● multimodal transportation networks, – between connection features, such as bus network, the subway network, the air transportation network [31, 44], etc; – or between node features, such as airline companies [44], etc.Usually in the modeling of complex networked systems, each of these features has been called a layer [11, 30, 44, 46].Particularly, in this section, we focus on connection features, analyzing the theoretical characteristics of the so called multiplex networks [21,30]. A multiplex network is a type of multilayer network. We first briefly discuss some representation equivalences.Then, we apply an analogous investigation to that of Section IV-A.Before presenting multiplex networks, it is important to recover some definitions and nomenclatures from the previousliterature on the subject, clarifying conditions or assumptions behind the mathematical concepts, so as to enable unambiguousapplications of our theoretical results in future studies of real-world networks. In doing so, we are not only grounding ournomenclature on, but also following the same purpose of terminology unification in [30]. In this way, we choose an approachin order that one can distinguish between multilayer networks and dynamic networks, even though both can be viewed as justdistinct physical interpretations of the same kind of network’s aspect (or dimension).To this end, we compare a well-known definition of multilayer network as in [30] with the multidimensional (i.e., high-order)approach formalized in [45, 46]. In [30] (or, equivalently, in [11]), a multilayer network M = ( V M , E M , V, L ) is understood asan interconnected and/or intraconnected labeled collection of graphs. Each of these graphs represents a ‘layer’ whose (distinctor not) vertex sets can be either linked within this ‘layer’ or linked to a vertex in another ‘layer’. Formally, a multilayer network M = ( V M , E M , V, L ) [30] is defined by: ● V denotes the set of all possible vertices v ; ● L = { L a } da = denotes a collection of d ∈ N sets L a composed of elementary layers α ∈ L a ; ● V M ⊆ V × L × ⋯ × L d denotes the subset of all possible vertices that belong to a ‘layer’ L × ⋯ × L d ; ● E M ⊆ V M × V M denotes the set of interlayer and/or intralayer edges connecting two node-layer tuples ( v, α , . . . , α d ) ∈ V M . In fact, one can even improve this inequality to show that C ( E ( G ′ t ) ∣ (∣ V ( G ′ t ))∣ ∣ T ( G ′ t ))∣)) ≤ O (∣ T ( G ′ t )∣ ( ∣ V ( G ′ t )∣ −∣ V ( G ′ t )∣ )) . An interlayer edge is defined as (( u, α , . . . , α d ) , ( v, β , . . . , β d )) ∈ E M such that ( α , . . . , α d ) ≠ ( β , . . . , β d ) and an intralayeris defined as (( u, α , . . . , α d ) , ( v, β , . . . , β d )) ∈ E M such that ( α , . . . , α d ) = ( β , . . . , β d ) . In addition, one defines a coupling edge (( u, α , . . . , α d ) , ( v, β , . . . , β d )) ∈ E M as an interlayer edge with u = v .Actually, there are in fact some equivalences between a MAG [45] (see Definition II.1) and a multilayer M = ( V M , E M , V, L ) : V is the usual set of vertices V ( G ) ≡ A ( G )[ ] ; each set L a is the a -th aspect A ( G )[ a ] of a MAG G ; V M is a subset of the setof all composite vertices V ( G ) , where a node-layer tuple ( v, α , . . . , α d ) ∈ V M is a composite vertex v ∈ V ( G ) ; E M ≡ E ( G ) is a subset of the set of all composite edges E ( G ) . The only distinctive characteristic of M = ( V M , E M , V, L ) and G = ( A , E ) is the possibility that one or more vertices do not belong to one or more aspects of a MAG. Therefore, if a multilayer network M is node-aligned [21, 30], i.e., V M = V × L × ⋯ × L d , this multilayer network M is equivalent to a ( d + ) -order MAG.Thus, a node-aligned multilayer network with d layers is a network that can be mathematically represented by a ( d + ) -orderMAG.In most cases, as mentioned in Section IV-A, a node-alignment hypothesis can be assumed [30], except for the cases inwhich the pertinence of the vertices in each layer (and not only its connectivity) do matter in the network analysis. In thisregard, we also leave as future research the investigation of the worst cases for the information needed to retrieve V M from V × L × ⋯ × L d . Due to this possibility, the algorithmic information carried by M may be larger than that of the correspondentMAG G with d + aspects and E M = E ( G ) . Anyway, for the purposes of the present article (as we did in the dynamic case inSection IV-A), we assume hereafter that the multilayer networks M are node-aligned, so that ⟨ E M ⟩ denotes the string ⟨ E ( G )⟩ with respect to such correspondent MAG G , where E M ∶= E ( G ) .Besides the above formalities, we choose a distinguishing interpretation of the two currently studied multidimensionalstructures in complex networks: dynamism and “multilayerism”. In accordance with [44–46], and unlike [11, 22, 30] (wheredynamic networks are considered as a particular type of multilayer networks), we are considering a time-varying graph topology,like in dynamic networks in Section IV-A, and a multilayer topology, like the ones of the above described M , as two distinctmultidimensional structures.At a first glance, both the mathematical representations of a finite time progression and of a finite number of different layerscan be performed by an ordered set of labels (or indexes). However, besides distinct physical properties, a multilayer networkmay not need to obey a sequential indexing of layers that corresponds to a meaningful ordering of the physical counterparts ofeach layer. This promptly differs from the sequential coupling introduced in Section IV-A. For example, node features (e.g.,species, gender, or company) or connection features (e.g., bus network or email interaction) in the multilayer case do not havean intrinsic underlying structure that indicates the direction that the information is ‘flowing through’ an edge, whereas theopposite holds in principle for time instants in dynamic networks. Moreover, as we will show in Section V, some topologicalproperties, such as the presence of crosslayer edges, may happen to make sense in multilayer networks, whereas, in dynamicnetworks—specially, in snapshot-dynamic networks—the presence of transtemporal edges may not.In addition, we are assuming a general meaning of the nomenclature ‘multidimensional’ so as to encompass both eachindividual node feature and each type of the individual node features. We say that each type of individual feature, suchas being an arbitrary set of time instants or being an arbitrary set of layers, is a node dimension . Thus, a node dimensioncorresponds to an aspect of a MAG, as in [44–46]. This is in consonance with a common understanding of a dimension asbeing an aspect or property in which a particular object can assume different values, names, etc.However, note that this differs from some usages of the term in the literature of network science, where for example one maysay that a node u linked to a node v through a family relation lies on a different ‘dimension’ than that of a node u ′ linked toa node v ′ through a professional relation. In this sense, the particular object that can assume different values is the connectionitself and the different values are the nodes. This way, any element of an aspect of a MAG other than the set of verticesrepresents a ‘dimension’. In fact, such usage of the term may be also found in an overlap with that of multiplex networks[10, 11]. Thus, in order to avoid ambiguities, we call this kind of dimension as connection dimension . As a consequence, itderives directly from our chosen nomenclature that every connection dimension belongs to a node dimension.In this way, our approach ensures that in any case one employs multidimensional networks, it can either mean networkswith more than one node dimension, e.g., a dynamic network, or networks with more than one connection dimension, e.g.,a social network with only two types of interactions, like family and professional. Thus, unlike for example in [10, 11],in which multidimensional networks refers basically to multiplex networks, we adopt the convention of defining a generalmultidimensional network as a network that has more than one node dimension and more than one connection dimension.As a consequence, both multilayer networks and dynamic networks become particular cases of general multidimensionalnetworks. Also note that a network with only one node dimension and connection dimension is a monoplex (i.e., single-layeror monolayer) network [21, 24, 30] and, consequentially, is totally equivalent to a graph.In addition, we define a high-order network as a network that can be mathematically represented by a high-order MAG,i.e., a MAG with two or more aspects, each containing two or more elements [1, 44, 48]. Therefore, every node-aligned generalmultidimensional network is a high-order network. And this distinction will be reflected in our nomenclature and in our notation. See the first paragraph of this section for more examples of these individual features. In fact, it is shown in [46, Section 3.3] that one can, for instance, construct a main-component graph m ( G ) , which is definedas the MAG G ) with the unconnected composite vertices excluded. In this sense, if one allows the a priori exclusion of arbitrarycomposite vertices, it is possible to define MAGs that are not node-aligned, which would establish a complete equivalencebetween general multidimensional networks and high-order networks according to our nomenclature. However, in consonancewith Section II-A and [1], we choose to stick with the notion of a MAG defined on a full set of possible composite vertices inthis article. Hereafter, unless specified differently, multidimensional networks refers to node-aligned general multidimensional(i.e., high-order) networks and dimension refers to a node dimension. Whereas we are focusing on the multiplex case in thissection, we will return to higher-order network properties in Section V.As in [21, 30], and similarly in [24], a multiplex network M is a particular case of multilayer networks M = ( V M , E M , V, L ) that are diagonally coupled, categorical, and potentially layer-connected, where L = { L a } a = = { L } and ∣ L ∣ ≥ . From [30],we have that: a network M is diagonally coupled if and only if, for every interlayer edge (( u, α ) , ( v, β )) ∈ E M , where α ≠ β , one has that u = v ; a network M is categorically coupled if and only if, for every ( u, α ) , ( u, β ) ∈ V M , one has that (( u, α ) , ( u, β )) ∈ E M . Also in consonance with [30], we define here a condition for multiplex networks in order to ensure thatthe categorical couplings always apply to each pair of layers for at least one vertex u ∈ V : a network M is potentially layer-connected if and only if one has that V α ∩ V β ≠ ∅ , where V γ ∶= { v ∣ ( v, γ ) ∈ V M } and α, β, γ ∈ L are arbitrary. This property isparticularly important if M is not node-aligned. In general, most multiplex networks are considered to be node-aligned [30].In fact, for example in [11, 41], multiplex networks are defined already assuming a node-alignment hypothesis. Moreover, aswe already saw for node-aligned multilayer networks, we will have that any node-aligned multiplex network M is equivalentto a particular type of second order MAG. Hereafter, unless specified differently, we will consider only node-aligned multiplexnetworks.Let G M be the second order simple MAG that is equivalent to an undirected node-aligned multiplex network M . As inSection IV-A, note that diagonal coupling and categorical coupling are fixed. Let G ′M = ( V , E , L ) denote the second ordersimple MAG G M with all the interlayer edges excluded. In addition, G ′M belongs to a recursively labeled infinite family F G ′M of arbitrary simple second order MAGs. Then, analogously to Section IV-A, the algorithmic information of G M and thealgorithmic information of G ′M can only differ by a constant (that only depends on the chosen language L U ) and, thus, willbe negligible in our forthcoming results.With these definitions and nomenclature clarified in this section—as the reader may notice—, G ′M is in fact an equivalentrepresentation of a spatial simple TVG, except for a change in notation and nomenclature. Therefore, we can now directlytranslate Theorem IV.1 and all the other results from Section IV-A: Corollary IV.2.
Let G ′M = ( V , E , L ) belong to a recursively labeled infinite family F G ′M of simple second order MAGs.Then, there is a binary string y ∈ { , } ∗ that is an algorithmically characteristic string of G ′M such that K ( y ) ≤ l ( y ) + O ( ) ≤∣ L ( G ′M )∣ ( ∣ V ( G ′M )∣ − ∣ V ( G ′M )∣ ) + O ( log (∣ V ( G ′M )∣ ∣ L ( G ′M )∣)) ,where K ( x ) ≤ K ( y ) + O ( ) , K ( x ∣ y ∗ ) ≤ K ( x ∣ y ) + O ( ) ≤ O ( ) , K ( y ) ≤ K ( x ) + O ( ) ,and K ( y ∣ x ∗ ) ≤ K ( y ∣ x ) + O ( ) ≤ O ( ) hold and x is the characteristic string of G ′M . Therefore, for large enough sets L , O ( ) -K-random undirected node-aligned multilayer networks with L = { L } (i.e.,simple second order MAGs) carry at least ( (∣ V ( G ′M )∣ ∣ L ( G ′M )∣) − ∣ V ( G ′M )∣ ∣ L ( G ′M )∣ )− ∣ L ( G ′M )∣ ( ∣ V ( G ′M )∣ − ∣ V ( G ′M )∣ )− O ( log (∣ V ( G ′M )∣ ∣ L ( G ′M )∣)) more topological (or network) irreducible information than any node-aligned undirected multiplex network could carry. See the construction of program s in the proof of Theorem IV.1. Moreover, the reader is invited to note that (if both the set of vertices and the set of layers are large enough) Theorem V.3will imply that incompressible node-aligned general multilayer networks cannot be diagonally coupled and, therefore, cannotbe multiplex networks. To this end, just replace an aspect corresponding to a set of layers with the first aspect (which is theset of vertices) and vice-versa.V. M
ULTIDIMENSIONAL DEGREE , CONNECTIVITY , DIAMETER , AND NON - SEQUENTIAL INTERDIMENSIONAL EDGES
We saw in Section IV-A that snapshot-dynamic networks inevitably can carry only a number of bits of computably irreducibleinformation upper bounded by O (∣ T ∣ ( ∣ V ∣ −∣ V ∣ )) . In Section IV-B, we showed that the same also occurs for multiplex networks.Thus, one can define a non-empty class of (undirected) networks whose topological information characterizes a snapshot-likestructure, in particular containing snapshot-dynamic networks or multiplex networks: Definition V.1.
Let G ′ c be a simple second order MAG that belongs to a recursively labeled infinite family F G ′ c of simplesecond order MAGs. We say that the network mathematically represented by the MAG G ′ c is an (undirected) algorithmicallysnapshot-like multidimensional network with respect to dimension (i.e., aspect) A ( G )[ i ] if and only if there is an algorithmicallycharacteristic string y = ⟨ y ′ , w ⟩ ∈ { , } ∗ of G ′ c such that w ∈ {∅} ∪ { , } ∗ is independent of the choice of G ′ c ∈ F G ′ c and y ′ = ⟨ x , . . . , x ∣ A ( G )[ i ]∣ ⟩ ∈ { , } ∗ , where x α = ⟨ z , . . . , z k j ⟩ ∈ { , } ∗ , ≤ α ≤ ∣ A ( G )[ i ]∣ , k j = ( ∣ A ( G )[ j ]∣ ) , j ≠ i , z h ∈ { , } , ≤ h ≤ k j , and, for every h -th element e ′ of { e ∣( u, α, v, α ) ∈ E c ( G ′ c )} , we have z h = ⇐⇒ e ′ ∈ E ( G ′ c ) .In other words, an algorithmically snapshot-like (undirected) multidimensional network is a network that can be totallyrepresented by a second order simple MAG whose composite edge set can be algorithmically determined by only informinga sequence of presences or absences of composite edges connecting two nodes within the same node dimension. Note thatthe proof of Theorem IV.1 and Corollary IV.2 guarantees that Definition V.1 is satisfiable, for example, by snapshot-dynamicnetworks or multiplex networks that are node-aligned. The reader is also invited to note that Definition V.1 can be generalizedfor simple MAGs with more than two aspects.As we will show in this section, although an incompressible general multidimensional network can carry much moreinformation in its topology than any algorithmically snapshot-like multidimensional network—see Sections IV-A and IV-B—,it displays some properties that may not be seen in real-world multidimensional networks, e.g., in those that can be univocallyrepresented by algorithmically snapshot-like multidimensional networks.First, since a TVG is just a second order MAG, it is immediate to show in Corollary V.1 that the previously studiedCorollary III.6, which holds for arbitrary order p [1], also applies to simple TVGs. Thus, for the sake of exemplification, westart with the dynamic case. Then, we generalize to the multidimensional case. Corollary V.1.
Let F G t be a recursively labeled infinite family F G t of simple TVGs G t that are O ( log (∣ V ( G t )∣)) -C-random.Then, the following hold for large enough G t ∈ F G t , where V ( G t ) = V ( G t ) × T ( G t ) :1) The degree d ( v ) of a composite vertex v ∈ V ( G t ) in a MAG G t ∈ F G t satisfies ∣ d ( v ) − ( ∣ V ( G t )∣ − )∣ = O (√∣ V ( G t )∣ ( O ( log (∣ V ( G t )∣)))) .2) G t has ∣ V ( G t )∣ ± o (∣ V ( G t )∣) disjoint paths of length 2 between each pair of composite vertices u , v ∈ V ( G t ) .3) G t has (composite) diameter . From Lemma III.7, it is also immediate that there is an incompressible TVG that satisfies the conditions of Corollary III.6. Infact, this follows from Theorem III.8 by assuming a family of initial segments of a K-random (i.e., prefix algorithmically random)real number. For example, one may take initial segments of the halting probability (or Chaitin’s constant) [1]. Thus, fromLemma III.7, Corollary III.5, and Theorem III.8, we have that Corollary III.6 is satisfiable with δ (∣ V ( G c )∣) = O ( log (∣ V ( G t )∣)) and, therefore, resulting in the satisfiability of Corollary V.1 .Now, let a transtemporal edge be a composite edge e = ( u, t i , v, t j ) ∈ E ( G t ) with j ≠ i ± and j ≠ i . Thus, the short compositediameter and high k -connectivity of a O ( log (∣ V ( G t )∣)) -C-random simple TVG ensures the existence of transtemporal edgesin G t : Corollary V.2.
Let G t be any large enough simple TVG satisfying Corollary V.1 with ( − o (∣ V ( G t ) × T ( G t )∣) ∣ V ( G t ) × T ( G t )∣ ) − = o (∣ T ( G t )∣) . Then, between every pair of vertices u, v ∈ V ( G t ) and time instants t i , t j ∈ T ( G t ) with j > i + , there is at least onetranstemporal edge e ∈ E ( G t ) .Proof. It is a particular case of Theorem V.3.In fact, Corollary V.2 holds as a particular case of undirected high-order networks (i.e., undirected node-aligned generalmultidimensional networks), as we will demonstrate below. The multilayered case with just one additional aspect besides theset of vertices is totally analogous to Corollary V.2. For the multidimensional case, we will have that the first aspect still isthe set of vertices. The second aspect in turn may be the set T ( G c ) = A ( G c )[ ] of time instants or it may be the first layertype L ( G c ) = A ( G c )[ ] ). The further aspects are any other layer type L k ( G c ) = A ( G c )[ k + ] , where k ≤ ∣ A ( G c )∣ − , orany other node dimension. Note that, unlike in [30], we refer to each element of L k ( G c ) as a layer and to each ( α , . . . , α k ) ∈ L ( G c ) × ⋯ × L k ( G c ) as a layer tuple (or composite layer), instead of, respectively, a elementary layer and a layer. Moreover,we refer to each set L k ( G c ) as a layer type and to each arbitrary set L i ( G c ) × ⋯ × L j ( G c ) as a multilayer type .By generalizing the temporal case, let a crosslayer edge be a composite edge e = ( u, . . . , x ki , . . . , x ps , v, . . . , x kj . . . , x ps ′ ) ∈ E ( G c ) with j ≠ i ± and j ≠ i . In fact, in accordance with Section IV-B, both transtemporal and crosslayer edges are particularcases of what we call by a non-sequential interdimensional edge , should the aspect A ( G c )[ k ] corresponds to an arbitrarynode dimension of the general multidimensional network. This way, by noting that Corollary III.6 applies to simple MAGswith order p ≥ (as proved in [1]), Corollary V.2 becomes indeed a particular case of: Theorem V.3.
Let G c be any large enough O ( log (∣ V ( G c )∣)) -C-random simple MAG with order p ≥ that satisfies Corol-lary III.6 with δ (∣ V ( G c )∣) = O ( log (∣ V ( G c )∣)) such that ( − o (∣ V ( G c )∣) ∣ V ( G c )∣ ) − = o ⎛⎜⎜⎝ ∣ V ( G c )∣∣ V ( G c )∣ ⨉ ≤ h ≤ p,h ≠ k ≤ p ∣ A ( G c )[ h ]∣ ⎞⎟⎟⎠ ,where ≤ k ≤ p . Then, between every pair of composite vertices ( u, . . . , x ki , . . . , x ps ) and ( v, . . . , x kj . . . , x ps ′ ) in V ( G c ) with j > i + , there is: at least one crosslayer edge e ∈ E ( G c ) , if L k − ( G c ) = A ( G c )[ k ] ; at least one transtemporaledge e ∈ E ( G c ) , if T ( G c ) = A ( G c )[ k ] ; or at least one non-sequential interdimensional edge e ∈ E ( G c ) , if A ( G c )[ k ] corresponds to any arbitrary node dimension.Proof. Given a proper interpretation of the k -th aspect, both transtemporal edges and crosslayer edges are particular cases ofnon-sequential interdimensional edges. Thus, we will prove only the general case. First, if ( u, . . . , x ki , . . . , x ps , v, . . . , x kj . . . , x ps ′ ) ∈ E ( G c ) ,then it immediately satisfies the definition of non-sequential interdimensional edge. On the other hand, since G c is large enough,satisfying Corollary III.6 with δ (∣ V ( G c )∣) = O ( log (∣ V ( G c )∣)) ≤ o (∣ V ( G c )∣) ,then the composite diameter becomes . Therefore, it suffices to investigate the cases in which there is a ( v ′ , . . . , x kz , . . . , x ps ′′ ) ∈ V ( G c ) with x kz ∈ A ( G c )[ k ] such that ( u, . . . , x ki , . . . , x ps , v ′ , . . . , x kz , . . . , x ps ′′ ) ∈ E ( G c ) and ( v ′ , . . . , x kz , . . . , x ps ′′ , v, . . . , x kj . . . , x ps ′ ) ∈ E ( G c ) From Corollary III.6, we have that, for every pair of composite vertices ( u, . . . , x ki , . . . , x ps ) and ( v, . . . , x kj . . . , x ps ′ ) , there are ∣ V ( G c )∣ ± o (∣ V ( G c )∣) disjoint paths of length 2 between ( u, . . . , x ki , . . . , x ps ) and ( v, . . . , x kj . . . , x ps ′ ) . Now, note that ∣ A ( G c )[ k ]∣ = ∣ V ( G c )∣∣ V ( G c )∣ ⨉ ≤ h ≤ p,h ≠ k ≤ p ∣ A ( G c )[ h ]∣ .But, since G c can have arbitrarily large sets V ( G c ) and ( − o (∣ V ( G c )∣) ∣ V ( G c )∣ ) − = o ⎛⎜⎜⎝ ∣ V ( G c )∣∣ V ( G c )∣ ⨉ ≤ h ≤ p,h ≠ k ≤ p ∣ A ( G c )[ h ]∣ ⎞⎟⎟⎠ , then the number of possible distinct composite vertices ( v ′ , . . . , x kz , . . . , x ps ′′ ) with z = i or z = j will be always smaller thanor equal to lim ∣ V ( G c )∣→∞ ⎛⎝∣ V ( G c )∣ ⨉ ≤ h ≤ p,h ≠ k ≤ p ∣ A ( G c )[ h ]∣⎞⎠ == lim ∣ V ( G c )∣→∞ ∣ V ( G c )∣ ⎛⎜⎜⎝ ∣ V ( G c )∣∣ V ( G c )∣ ⨉ ≤ h ≤ p,h ≠ k ≤ p ∣ A ( G c )[ h ]∣ ⎞⎟⎟⎠ − << lim ∣ V ( G c )∣→∞ ∣ V ( G c )∣ ( − o (∣ V ( G c )∣) ∣ V ( G c )∣ ) == lim ∣ V ( G c )∣→∞ ∣ V ( G c )∣ − o (∣ V ( G c )∣) Thus, in the limit, it will eventually be strictly smaller than the number of distinct composite vertices connecting ( u, . . . , x ki , . . . , x ps ) and ( v, . . . , x kj . . . , x ps ′ ) . Therefore, connecting these two composite vertices, there will be at least one composite vertex ( v ′ , . . . , x kz , . . . , x ps ′′ ) with i + < z , z + < j , z < i , or j < z .VI. C ONCLUSIONS
In this work, we have studied the plain and prefix algorithmic randomness of multidimensional networks that can be formallyrepresented by multiaspect graphs (MAGs). We have dealt with the lossless incompressibility of multidimensional networks,especially node-aligned undirected multilayer networks or dynamic networks.First, we have compared time-varying graphs with other snapshot-like representations of dynamic networks. We have shownthat incompressible snapshot-dynamic networks carry an amount of topological algorithmic information (i.e., the irreducibleinformation necessary to computationally determine the graph-theoretic representation of the respective network) that is linearlydominated by the size of the set of time instants. To this end, we have applied a study of a worst-case lossless compressionof the algorithmically characteristic string of the network using the theoretical tools from algorithmic information theory.Then, after a careful analysis of previous nomenclature and assumptions in the literature, we have shown that the sameresults can be applied to multiplex networks, where the set of layers plays the role of the set of time instants instead. Inthis regard, we have shown that both snapshot-dynamic networks and multiplex networks are two particular and distinct casesof (algorithmically) snapshot-like multidimensional networks, but that are equivalent (except by a constant that only dependson the chosen universal programming language) in terms of algorithmic information. In addition, we have shown that themaximum amount of topological information of an incompressible snapshot-like multidimensional network is much smallerthan the amount of topological information of an incompressible multidimensional network.Secondly, we have investigated some topological properties of incompressible multidimensional networks. To this end, wehave applied previous results for incompressible MAGs. We have shown that these networks have very short diameter, high k-connectivity, and degrees on the order of half of the network size within a strong-asymptotically dominated standard deviation.Therefore, these theoretical findings relate lossless compression of multidimensional networks with their network topologicalproperties. For example, these properties are expected to happen in both artificial or real-world multidimensional networkswith a network topology that carries a maximal and irreducible information content. In this way, such theoretical results maygive rise to future tools that could be applied to, for instance, network summarization algorithms and the reducibility problem(i.e., the problem of finding the aggregate graph that represents the original multidimensional network and preserves its coreproperties during network analysis).Furthermore, we have also shown the presence of transtemporal or crosslayer edges (i.e., edges linking vertices at non-sequential time instants or layers) in those incompressible multidimensional networks. Although representations of these generalforms of multidimensional networks may carry much more topological information than that of snapshot-like multidimensionalnetworks, this presence of transtemporal or crosslayer edges may not correspond to some underlying structures of real-worldnetworks. Specifically, this is the case of snapshot-dynamic networks, where transtemporal edges would not have any physicalcorrespondence to connections that ‘jump across time instants’. On the other hand, in the crosslayer case, it can make sensefor some multilayer networks. Thus, with the purpose of bringing algorithmic randomness to the context of multidimensionalnetworks, our theoretical results suggest that estimating or analyzing both the incompressibility and the network topologicalproperties of real-world networks cannot be taken into a universal approach without previously taking into account underlyingtopological constraints. Similarly to what we have shown for snapshot-based representations of multidimensional networks, thealgorithmic randomness of certain networks may be strongly dependent on the underlying constraints in the structure of thenetwork; and, in turn, as we have shown, the incompressibility (i.e., algorithmic randomness) of multidimensional networksimplies underlying constraints in the structure of the network.This study and approach is also key to move forward from statistical approaches to non-statistical challenges in the contextof network science, data science, and beyond, such as those related to model generation, feature selection, data dimensionality reduction, and summarization. For example, in dynamic multilayer networks, one important challenge is to disentangle thecause and effect between and inside different networks over time, a problem relevant to almost every area of science whereprocesses can be represented as networks and interactions as connections to other networks.VII. A CKNOWLEDGMENTS
Authors acknowledge the partial support from CNPq through their individual grants: F. S. Abrah˜ao (313.043/2016-7), K.Wehmuth (312599/2016-1), and A. Ziviani (308.729/2015-3). Authors acknowledge the INCT in Data Science – INCT-CiD(CNPq 465.560/2014-8). Authors also acknowledge the partial support from CAPES/STIC-AmSud (18-STIC-07), FAPESP(2015/24493-1), and FAPERJ (E-26/203.046/2017). H. Zenil acknowledges the Swedish Research Council (Vetenskapsr˚adet)for their support under grant No. 2015-05299. R
EFERENCES[1] Felipe S. Abrah˜ao, Klaus Wehmuth, Hector Zenil, and Artur Ziviani,
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