Multivaluedness in Networks: Theory

This brief note reports the fundamental phenomenon of implicit multivaluedness exhibited from one output to the other of two node-systems with a common input—referred to as counter-cascaded<xref ref-type="fn" rid="fn1"><sup>1</sup></xref> systems—under the appropriate conditions. The novel concepts of immanence and transcendence are introduced upon which the formulation and prove of a necessary and sufficient condition for multivaluedness are based; this is the main result of this note. Next, subsequent consequences of this result are presented. Among these is the fact that this result also holds for cascaded generalized systems. The novel application of structural complexity reduction in directed networks presented next, demonstrates the utility of multivaluedness and is itself a contribution to the theory of signals and systems. The significance of this brief presented here is that it contributes toward the theory of systems and networks as well as toward the arsenal of tools for studying networks.<fn id="fn1"><label><sup>1</sup></label><p>In <xref rid="fig1" ref-type="fig">Fig. 1</xref> the systems <inline-formula> <tex-math notation="LaTeX">${T}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">${M}$ </tex-math></inline-formula> are <italic>counter-cascaded systems</italic> with common input <inline-formula> <tex-math notation="LaTeX">${u}$ </tex-math></inline-formula>, considering outputs <inline-formula> <tex-math notation="LaTeX">${v}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">${w}$ </tex-math></inline-formula>; similarly are <inline-formula> <tex-math notation="LaTeX">${N}~\circ ~{T}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">${M}$ </tex-math></inline-formula>, considering outputs <inline-formula> <tex-math notation="LaTeX">${x}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">${w}$ </tex-math></inline-formula>.</p></fn>


I. INTRODUCTION
The area of Network Analysis and Complex Networks has rapidly expanded into a very active and vibrant field of research, with ever more fundamental theoretical results and novel applications being reported.In order to give a glimpse of the diverse nature of the objects of study, i.e. complex networks, note that, size-wise, real life networks range from a few nodes to billions of nodes and beyond.Structure-wise, they range from highly homogeneously structured networks through to amorphously unstructured and even randomly structured networks.Character-wise, they vary from uniformly cooperative or competitive to heterogeneously mixed cooperative and competitive factions contained within.Furthermore, the mathematical descriptions of nodes in a complex network range from uniform (identical) in some networks, to diverse (different) in others.For these reasons, graph-theoretical methods are indispensable for description and analysis of network problems.
In the literature, the meaning of the term "network analysis" is rather diverse.Of particular interest to us here, is the extended definition of Zaidi [1], namely that it encapsulates the Financial Support Acknowledgment: This work was supported in part by the Carl and Emily Fuchs Foundation's Chair in Systems and Control Engineering at the University of the Witwatersrand, Johannesburg, South Africa.study of theory, methods and algorithms applicable to graphbased models representing interconnected real-world systems.From this perspective, the collection of interconnected elements of a finite element analysis of a distributed structure or physical field and a complex interconnection of nonlinear dynamical systems are instances of complex network analyses [2][3], the former undirected and the latter directed.Both an excellent account of the theory and overview of current research directions in complex networks, can be found in [4].
Even though complex networks might not always have explicit inputs (causes) and outputs (effects), there are always internal (i.e.local) inputs and outputs of interest when considering a single node or a collection of nodes.A deeper understanding of the global behavior and dynamics of a complex network usually requires a deeper understanding of the mechanisms of behavior at a more detailed level in the network.For this reason, oftentimes it requires one to relate two effects brought about by the very same global or local cause, in order to gain deeper insight.In this paper we study this aspect in detail.
The outline of this paper is as follows: Section II presents a simple yet powerful theoretical result that gives a necessary and sufficient condition under which two different (sets of) effects v ∈ V and x ∈ X, produced by the common cause u ∈ U , are to be related by a well-defined mapping.Due to the minimal underlying assumptions and the simplicity of the set-theoretic argument used, the results derived is very general.Next, several consequences of the main result are addressed.By virtue of an example, Section III demonstrates the use of these results when applied to structural reductions in complex networks that we will call functional uniformization and nodal rationalization.The conclusion follows in Section IV.

II. THEORY: IMMANENCE VERSUS TRANSCENDENCE
In order to provide a definite and concrete context 2 for the presentation and discussion that follow, we consider complex networks consisting of complex configurations of nonlinear (dynamical) systems.In such networks, we will study occurrences of counter-cascaded systems, i.e. configurations of the kind shown in Fig. 1.Generally, U , V , W and X can be very general sets, with M : U → W , T : U → V and N : V → X mappings.For the selected context, unless stated otherwise, these mappings are nonlinear operators with domains and ranges being subsets of real vector spaces; typically T is a 2 Concreteness, here, does not restrict generality.nonlinear operator describing some nonlinear system, with M a nonlinear operator describing either yet another nonlinear system or an input ancillary system, and N is a nonlinear operator representing some output ancillary system.In some applications M might be the identity operator, as does N .For the purpose of our presentation here, N is redundant because it can be absorbed into T by replacing T with N •T .However, for applications of this work in other areas, it has a distinct and explicit purpose, as will be reported on in the future.Finally, S ⊂ W × X usually represents a multivalued function, strictly called a relation.
In order to simplify the notation used here, we will not distinguish a system from its mathematical representation and thus use the same symbol for both.Furthermore, in order to retain the generality of the results presented, we will talk of "mappings" rather than "operators" as required by the nonlinear systems context.
Definition II.1.(Immanence, Transcendence) In Fig. 1, the mapping T is called immanent with respect (or relative) to the ordered pair of mappings (M, N ) if, for every w ∈ M (U ) there exists an If T is not immanent with respect to (M, N ), then it is called transcendent with respect to (M, N ).

Notes.
a.For the sake of conciseness, we will sometimes use the statement "T is (M, N )-immanent" as an abbreviation of the statement "T is immanent with respect to (M, N )" and similarly for statements about transcendence.b.For a general complex network, in order for two nodes to be analyzed for immanence or transcendence, their inputs have to be connected together.c.Since collections of nodes can be clustered to form supernodes, which are themselves nodes, this definition and all subsequent results apply to supernodes without explicit further mention.
Assuming M and N generally to be many-to-one, we are now in a position to state and prove the main result: 3 If such x exists, then it is unique.To see this assume that at least two such elements x 1 and x 2 exist implying that Theorem II.2.(Well-Defined Mapping) The mapping T is immanent relative to (M, N ) if and only if N • T • M −1 is well-defined (i.e.single-valued). 4te.Before proceeding with the proof, first observe that for each element u ∈ U , there exist elements w u := M (u) and x u := N (T (u)).Next, we associate w u and x u by writing x u = S(w u ) for every u ∈ U .This can be compactly expressed as S := N • T • M −1 .Here, S defines a relation.If for every pair of distinct elements u 1 , u 2 ∈ U we have that w u1 = w u2 implies that x u1 = x u2 , then S is well-defined.
Proof.We first prove the "only if" part.Suppose that T is immanent with respect to (M, N ).Now, if S is not well-defined, then there exist at least two distinct elements u, u ∈ U such that M (u) = M (u ) but x := N (T (u)) = N (T (u )) =: x .This contradicts the consequence of immanence, namely that N (T (u)) = x for all u ∈ M −1 (w) and consequently S is well-defined.
Conversely, to prove the "if" part, suppose T is transcendent with respect to (M, N ).Then, for some w ∈ W , there are distinct elements u, u ∈ M −1 (w) for which x := N (T (u)) = N (T (u )) =: x , implying that S is not well-defined because S(w) = x and S(w) = x and yet x = x .This concludes the converse via the contrapositive and completes the proof.
An equivalent statement of this result follows: is not well-defined (i.e.multivalued).
To our knowledge this result, identifying all those situations when the outputs of two counter-cascaded subsystems are functionally related (as well as when not), is a novel result.Some immediate consequences of Theorem II.2 now follow.
Corollary II.4.(Existence of a Unique Faithful Model) For a given mapping T , a unique faithful model or modeling mapping S exists if and only if T is immanent with respect to (M, N ).
If T is immanent with respect to (M, N ), then there exists a unique mapping w → S(w) which yields a unique faithful model of T , as perceived through M and N , that is, S(w) = N •T •M −1 (w) for every w ∈ W .Even though these modeling problems are exactly solvable, in principle, it might happen that the prescribed class of models does not contain S, in which case the best that can be achieved is to choose the "best" approximating model S opt from the class prescribed, based on some optimization criterion.
On the other hand, if T is transcendent with respect to (M, N ), then there exists a (multivalued) relation S ≡ N • T • M −1 , which cannot be described by any mapping, whatsoever, and hence no faithful model exists.Therefore, the only alternative remaining is to find the "best" approximating mapping S opt to the relation S, based on some criterion.
Corollary II.5.Let M , T , N and S be as depicted in Fig. 1.
a.If M , N are given and T assumes the canonical form If M is many-to-one, T is not of canonical form and N is one-to-one, then T is (M, N )-transcendent.d.If M is one-to-one, N is many-to-one and T is (M, N )immanent, then the mapping S is many-to-one.e.Given a fixed mapping S, if there exists a T satisfying N • T = S • M , then T is (M, N )-immanent for every mapping N .

Notes.
a.If M is many-to-one and T is not of the canonical form T = F • M , then the (M, N )-immanence of T depends on the choice of N .b.In Corollary II.5(e), if N is invertible, then the expression T = N −1 • S • M gives an explicit formula for T .However, if N is not invertible then T satisfies the expression unless additional information about T is available, we can merely test candidate mappings T to determine if they satisfy this expression.....In the latter case, is T (M, N )-immanent or (M, N )-transcendent?
Now, a little thought reveals the following to be true for configurations similar to that shown in Fig. 1, but with additional exogenous inputs entering: Lemma II.6.(Resolution of Exogenous Inputs) Suppose that along some of the paths considered to yield the mappings M and T , there are additional causes entering.Then this configuration can be transformed to that in Fig. 1 by augmenting the input space U with a direct sum component for each additional exogenous input entering.After this transformation, proceed as before to define the mappings M and T .Note.Since, by assumption N does not take the input u ∈ U directly, this lemma does not apply to it.In more general situations where N shares an input with M , replacing T with I ⊕ T , the structure depicted in Fig. 1 applies once more.The symbol ⊕ represents the direct sum binary operation.
In the case of counter-cascaded systems, there are two possible directions to be considered for immanence or transcendence.The next definition expands on the previous definition to cover both possibilities.For this, N will effectively be removed by choosing it to be the identity mapping.
Definition II.7.(Bi-immanence, Bi-transcendence) In Fig. 1, if T is (M, I)-immanent and M is (T, I)-immanent, then T and M are called bilaterally immanent or bi-immanent.Similarly, if T and M are (M, I)and (T, I)-transcendent, respectively, then T and M are called bilaterally transcendent or bi-transcendent.
Note.Considering the two possible directions along two counter-cascaded systems, in principle, all of the following four cases are possible: immanent-immanent (I-I), immanent-transcendent (I-T), transcendent-immanent (T-I) and transcendent-transcendent (T-T).For the case I-I the mapping relating the outputs is a bijection while, for the case T-T, there is no mapping that relates the two outputs in either direction.The case I-T implies that such a mapping exists in one direction but not in the other; similarly for the case T-I.
Discussion.An important insight obtained from the above theoretical development is that, for cases with the mappings T and M given, but with T transcendent with respect to (M, I), the only way to resolve this situation, if at all possible, is to design an appropriate output ancillary mapping N .If this proves to be impossible, then the above theorems imply that more design freedom is required.For example, we can allow M to be engineered or redesigned in an attempt to find a pair (M, N ) rendering T to be immanent.If still not successful, M could be fixed and T be redesigned.If still not successful, then no choice remains other than a complete redesign.However, at any stage we could settle for approximation, then knowing that a solution does not exist which implies nonzero approximation error as a consequence of the prevailing transcendence.
In some applications, however, it happens that the mapping S is given instead, and the mapping T then follows as a consequence of S and the particular application's context and constraints.For such cases, it might be necessary to adapt either the context and/or the constraints, in order to obtain a T that is (M, N )-immanent with M and N implied by the application's context and constraints.

III. APPLICATION: NODAL RATIONALIZATION
We will now apply the results of the previous section to structural reduction in complex networks.We start by introducing the necessary terminology.The process of expressing node mappings in factored form, with the right-most factors chosen from as few as possible unique ones, will be referred to as functional uniformization.Furthermore, the process of minimizing the number of nodes in a functionally uniformized network, by merging as many nodes as is possible to share common right-most factors and (node) inputs, will be termed nodal rationalization.This form of structural transformation of a network results in a reduction in the number of nodes, with each of the newly formed nodes having either multiple inputs or multiple outputs or both.
Consider the simple yet general four-node network with immediate neighbor interaction, shown in Fig. 2. We will use the same symbol to present both the node and the mathematical mapping describing its behavior, i.e. its mathematical description.For example, A identifies the upper-most node in Fig. 2; it also represents the mathematical mapping A(•, •) that describes this node's behavior.
For clarity of presentation, we will relate back to earlier theoretical results by using the nonlinear systems representation employed in the previous section.Unless additional information is available, no structural reduction of this network is possible.So, suppose that C is (A, I)-immanent.Then, according to Corollary II.4,there exists a modelling mapping C, as indicated in Fig. 3. Following Lemma II.6, we can combine the inputs feeding into nodes A and C to obtain a common vector input feeding into both A and C, as depicted by the bold line in Fig. 4(a).Fig. 4(b) shows that the mapping C can be replaced by the composition C • A as follows to Corollary II. 4. This means that we can now replace node C of the network with a "node" C which has a single input, fed by the output of node A. The output of node C then replaces the output of node C, feeding into nodes B and D as shown in Fig. 5(a).To reduce this network to a three-node network requires us to merge A and C into a single node with mathematical description (I ⊕ C) • A resulting in it having a vector output which feeds into nodes B and D via the bold edges in the graph shown in Fig. 5(b).Now, if there were no further immanence present in the network, then Fig. 5(b) shows the simplest network to which the original network can be structurally reduced, using nodal rationalization.
Next, additionally, assume that D is (B, I)-immanent.Once again, according to Corollary II.4,there exists a modelling mapping D such that D = D • B. Following the same procedure as above, the network can now be reduced to the form shown in Fig. 6(a)-effectively a two-node network as shown in Fig. 6(b).To see this simply define the two nodes to have the mathematical descriptions 5 (I ⊕ C) • A and (I ⊕ D) • B, respectively, resulting in the interconnecting edges to become vector valued.This reduction is striking, considering the generality of the mathematical descriptions of the four nodes of the original network. 5For economy of presentation and for readability, we represent the two identity mappings I A and I B , operating on the ranges of A and B, respectively, using the same symbol, namely I.With the insight developed as our discussion unfolded, we look back and point out an important and surprising observation: if our example complex network was completely void of immanence, i.e. all counter-cascaded node pairs were transcendent, then by Theorem II.2, the original network would not in any way have been structurally reducible using nodal rationalization.In other words, except perhaps for cosmetic changes, Fig. 2 would then represent the simplest form possible for this network and imposing nodal rationalization in such a case, would yield unavoidable and unresolvable approximation or modeling errors.
A note about bi-immanence and bi-transcendence is in order: for our example here, if bi-immanence was present, then we would have had the option to interchange the roles of relevant systems, thus giving us more options while still producing the same results.
In conclusion we point out that, in this example application, we stretched the presence of immanence to the limit in order to demonstrate the compactness of representation produced by the nodal rationalization.However, in real life complex network investigations, nodal rationalization will usually only be applied selectively to expose important latent properties that would otherwise have gone unnoticed.

IV. CONCLUSION
We presented a theoretical result which states a necessary and sufficient condition for multivaluedness when attempting to relate two outputs (effects) resulting from the same local or global input (cause) in a complex network Subsequent corollaries provide further useful results for determining multivaluedness, given special conditions.
A simple application, namely nodal rationalization, was demonstrated, using a simple yet very general four-node complex network.It shows the presented results' potential to contribute toward the arsenal of tools for studying complex networks and systems in general.
Further work is in progress to specialize these theoretical results to applications in distributed measurement in networks, big data and neural networks, and in the analysis of signal processing algorithms.

Figure
Figure

Fig. 4 .Fig. 5 .
Fig. 5. Reduced network: (a) With node C represented by C •A thus sharing the existing node A. (b) With the outputs of the new node consolidated.

Fig. 6 .
Reduced network: (a) With nodes C and D represented by C • A and D • B, respectively.(b) With the outputs of the new nodes consolidated.