I Introduction
There has been great interest over the past few years in agentbased network models of opinion dynamics that describe how individuals’ opinions on a topic evolve over time as they interact [1, 2]. The seminal discretetime French–Harary–DeGroot model [3, 4, 5] (or DeGroot model for short) assumes that each individual’s opinion at the next time step is a convex combination of his/her current opinion and the current opinions of his/her neighbours. This weighted averaging aims to capture social influence, where individuals exert a conforming influence on each other so that over time, opinions become more similar (and thus giving rise to the term “influence network”). For networks satisfying mild connectivity conditions, the opinions reach a consensus, i.e. the opinion values are equal for all individuals.
Since then, and to reflect realworld networks, much focus has been placed on developing models of increasing sophistication to capture different sociopsychological features that may be involved when individuals interact. The Hegselmann–Krause model [6, 7, 8, 9] introduced the concept of bounded confidence, which is used to capture homophily, i.e. the phenomenon whereby individuals only interact with those other individuals whose opinion values are similar to their own. The Altafini model [10, 11, 12, 13] introduced negative edge weights to model antagonistic or competitive interactions between individuals (perhaps arising from mistrust). An individual’s propensity to assimilate information in a biased manner, by more heavily weighting opinions close to his or her own, is studied in [14]. The Friedkin–Johnsen model generalised the DeGroot model by introducing the idea of “stubbornness”, where an individual remains (at least partially) attached to his or her initial opinion [15, 16]. Of particular note is that the DeGroot and Friedkin–Johnsen models have been empirically examined [17, 18, 16]. For more detailed discussions on opinion dynamics modelling, we refer the reader to [1, 2, 19].
Recently in [20], a multidimensional extension to the Friedkin–Johnsen was proposed to describe a network of individuals who simultaneously discuss a set of logically interdependent topics. That is, an individual’s position on Topic may influence his/her position on Topic due to his/her view of constraints or relations between the two topics. Such interdependencies are captured in the model by a “logic matrix”. This interdependence can greatly shift the final opinion values on the set of topics since now the interdependencies and the social influence from other individuals both affect opinion values. The model is used in [21] to explain that the shift in the US public’s opinions on the topic of whether the 2003 Invasion of Iraq was justified was due to shifting opinions on the logically interdependent topic of whether Iraq had weapons of mass destruction. The set of topics, the interdependent functionalities between the topics, and the mechanism by which an individual processes such interdependencies forms a “belief system” as termed by Converse in his now classical paper [22]. For networks where all individuals have the same logic matrix, a complete stability result is given using algebraic conditions in [20] and using graphtheoretic conditions in [23]. Of course, the assumption that all individuals have the same logic matrix is restrictive. Heterogeneous logic matrices were considered in [21], but at least one individual is required to exhibit stubbornness in order to obtain a stability result.
This paper will also consider a generalisation of the multidimensional model proposed in [20] for the evolution of opinions in belief systems, going beyond [20, 21] by analysing the effects of the logic matrix, including especially heterogeneity of the logic matrices among the individuals, on the limiting opinion distribution. We first establish a general convergence result for the model with heterogeneous logic matrices on strongly connected networks. Then, we provide a set of results which enables the systematic determination of whether for a given topic, the opinions of the individuals will reach a consensus, or will reach a state of persistent disagreement.
We find that the nature of the heterogeneity of the logic structure among the individuals, viz. the logical interdependencies between topics, and the structure itself, plays a major role in determining whether opinions on a given topic reach a consensus or fail to do so. If the logical interdependencies do not have a cascade structure, then consensus is always secured. When the logical interdependencies have a cascade structure, and by considering topics at the top of a cascade structure to be axiom(s) that an individual’s belief system is built upon, we establish that discussion of the axiomatic topics will lead to a consensus. In contrast, we discover that persistent disagreement can arise in the topics at the bottom of the cascade when certain types of heterogeneity exist in the logic matrices. A preliminary work [24] considers the special case of lower triangular logic matrices, but we go well beyond that in this paper by considering general logic matrix structures and providing a comprehensive account of the results.
We discover that if there is a failure to reach a consensus, then it is typically not minor; in general a strong diversity of opinions will eventually emerge. In more detail, a network is said to exhibit weak diversity [25] if opinions eventually converge into clusters where there is no difference between opinions in the same cluster (consensus is the special case of one single cluster). Strong diversity occurs when the opinions converge to a configuration of persistent disagreement, with a diverse range of values (there may be clusters of opinions with similar, but not equal, values within a cluster). Weak diversity is a common outcome in the Hegselmann–Krause model, with the network becoming disconnected into subgroups associated with the clusters. In strongly connected networks, weak diversity also emerges in the Altafini model (specifically polarisation of two opinion clusters) when the network is “structurally balanced”. However, sign reversal of some selected edges may destroy the structural balance of the network, causing the opinions to converge to a consensus at an opinion value of zero, indicating that the polarisation phenomenon is not robust to changes in the network structure.
There has been a growing interest to study models which are able to capture the more realistic outcome of strong diversity in networks which remain connected [26, 25]. The DeGroot model shows that social influence in a connected network acts to bring opinions closer together until a consensus is achieved, meaning some other sociopsychological process must be at work to generate strong diversity. The Friedkin–Johnsen model attributes strong diversity to an individual’s stubborn attachment to his/her initial opinion [15]. In contrast, [27] considers a model where an individual’s susceptibility to interpersonal influence is dependent on the individual’s current opinion; strong diversity is verified as a special case. The papers [26, 25] consider two features that might give rise to strong diversity, the first being “social distancing”, and the second being an individual’s “desire to be unique”. Experimental studies are inconclusive with regards to the existence of ubiquitous and persistent antagonistic interpersonal interactions (there might be limited occurrences in the network over short time spans) [20], while it is unlikely that an individual has the same level of stubborn attachment to his or her initial opinion value for months or years.
In contrast to these works, we identify for the first time in the literature that strong diversity can arise because of the structure of individuals’ belief systems, and show that heterogeneity among belief systems plays a crucial role. In the model, each individual is concurrently undergoing two driver processes; individuallevel belief system dynamics to secure logical consistency of opinions across a set of topics, and interpersonal influence to reach a consensus. Our findings explain that when the two drivers do not interfere with each other, a consensus is reached, whereas conflict between the two drivers leads to persistent disagreement even though all individuals are trying to reach a consensus. This gives a new and illuminating perspective as to why strong diversity can last for extended periods of time in connected networks.
The rest of the paper is structured as follows. In Section II, we provide notations, an introduction to graph theory and the opinion dynamics model. At the same time, a formal problem statement is given. The main results are presented in Section III, with simulations given in Section IV for illustration, and conclusions in Section V.
Ii Background and Formal Problem Statement
We begin by introducing some mathematical notations used in the paper. The entry of a matrix is denoted . A matrix is said to be nonnegative (respectively positive) if all are nonnegative (respectively positive). We denote as being nonnegative and positive by and , respectively. A matrix is said to be rowstochastic (respectively, rowsubstochastic) if there holds (respectively, if there holds and ). The transpose of a matrix is denoted by . Let and denote, respectively, the
column vectors of all ones and all zeros. The
identity matrix is given by . Two matrices and are said to be of the same type, denoted by , if and only if . The Kronecker product is denoted by . The infinity norm and spectral radius of a square matrix is and , respectively. A square matrix is primitive if [28, Definition 1.12].Iia Graph Theory
The interaction between individuals in a social network, and the logical interdependence between topics can modelled using a weighted directed graph. To that end, we introduce some notations and concepts for graphs. A directed graph is a triple where node is in the finite, nonempty set of nodes . The set of ordered edges is . We denote an ordered edge as , and because the graph is directed, in general the existence of does not imply existence of . An edge is said to be outgoing with respect to and incoming with respect to . Selfloops are allowed, i.e. may be in . The matrix associated with captures the edge weights. More specifically, if and only if . If is nonnegative, then all edges have positive weights, while a generic may be associated with a signed graph , having signed edge weights.
A directed path is a sequence of edges of the form where are unique, and . Node is reachable from node if there exists a directed path from to . A graph is said to be strongly connected if every node is reachable from every other node. A square matrix is irreducible if and only if the associated graph is strongly connected. A directed cycle is a directed path that starts and ends at the same vertex, and contains no repeated vertex except the initial (which is also the final) vertex. The length of a directed cycle is the number of edges in the directed cyclic path. A directed graph is aperiodic if there exists no integer that divides the length of every directed cycle of the graph [28]. Note that any graph with a selfloop is aperiodic.
A signed graph is said to be structurally balanced (respectively structurally unbalanced) if the nodes can be partitioned (respectively cannot be partitioned) into two disjoint sets such that each edge between two nodes in the same set has a positive weight, and each edge between nodes in different sets has a negative weight [29].
The following is a useful result employed in the paper.
Lemma 1 ([28, Proposition 1.35]).
The graph , with , is strongly connected and aperiodic if and only if is primitive.
Note that the irreducibility of (implied by the strong connectivity property of ) implies that if a exists such that , then for all .
IiB The MultiDimensional DeGroot Model
In this paper, we investigate a recently proposed multidimensional extension to the DeGroot and FriedkinJohnsen models [20, 21], which considers the simultaneous discussion of logically interdependent topics.
Formally, consider a population of individuals discussing simultaneously their opinions on topics, with individual and topic index set and , respectively. Individual ’s opinions on the topics at time , are denoted by . In this paper, we adopt a standard definition of an opinion [21]. In particular, is individual ’s attitude towards topic , which takes the form of a statement, with representing ’s support for statement , representing rejection of statement , and representing a neutral stance. The magnitude of denotes the strength of conviction, with being maximal support/rejection. Mild assumptions are placed on the network and individual parameters in the sequel to ensure that for all , and thus the opinion values are always well defined.
In the multidimensional DeGroot model, evolves according to
(1) 
where the nonnegative scalar represents the influence weight individual accords to the vector of opinions of individual . Thus, the influence matrix , with entry , can be used to define the graph that describes the interpersonal influences of the individuals. We assume that for all and for all , which implies that is rowstochastic. The matrix , with entry , is termed the logic matrix. In [20, 21], the authors elucidate that represents the logical interdependence between the topics as seen by individual . We note that the are assumed to be heterogeneous (i.e. ). Indeed, a critical aspect of this paper is to study how the structure of the s, especially heterogeneity, can determine whether certain topics have opinions that reach a consensus or a persistent disagreement.
We now illustrate with a simple example how is used by individual to obtain a set of opinions consistent with any logical interdependencies between each topic, and in doing so, motivate that certain constraints must be imposed on due to the problem context (these constraints are implicitly imposed in [20, 21], but without motivation).
Suppose that there are two topics. Topic 1: The exploration of Space is important to mankind’s future. Topic 2: The exploration of Space should be privatised. Using Topic 1 as an example, and according to the definition of an opinion given above Eq. (1), represents individual ’s maximal support of the importance of Space exploration, while represents maximal rejection that Space exploration is important. Now, suppose that individual has , i.e. individual initially believes with maximal conviction that Space exploration is important and initially believes with some (but not absolute) conviction that Space exploration should not be privatised^{1}^{1}1Note that we do not require to be rowstochastic and nonnegative, though the of this example is.. Let
(2) 
This tells us that individual ’s opinion on the importance of Space exploration is unaffected by his or her own opinion on whether Space exploration should be privatised. On the other hand, individual ’s opinion on Topic depends positively on his or her own opinion on Topic , perhaps because individual believes privatised companies are more effective. In the absence of opinions from other individuals, individual ’s opinions evolves as
(3) 
which yields , i.e. individual eventually believes that Space exploration should be privatised. Thus, moves from , where individual ’s opinions are inconsistent with the logical interdependence as captured by , to the final state , which is consistent with the logical interdependence. Eq. (3), with opinion vector and the logical interdependencies captured by , models individual ’s belief system. (We explained qualitatively what a belief system was in the Introduction, and have now given the mathematical formulation.)
In general, one might expect, as do we in this paper, that an individual’s belief system without interpersonal influence from neighbours will eventually become consistent. For a topic which is independent of all other topics, one also expects that for all . To ensure the belief system is consistent, we impose the following assumption.
Assumption 1.
The matrix , for all
, is such that each eigenvalue of
is either 1 or has modulus less than 1. If an eigenvalue of is 1, then it is semisimple^{2}^{2}2By semisimple, we mean that the geometric and algebraic multiplicities are the same. Equivalently, all Jordan blocks of the eigenvalue are 1 by 1.. For all and , there holds , and the diagonal entries satisfy .The assumptions on the eigenvalues of ensure that Eq. (3) converges to a limit, and are necessary and sufficient for individual ’s belief system to eventually become consistent. The other assumptions lead to desirable properties for the system Eq. (1). Specifically, the reasonable assumption that means topic is positively correlated with itself. The constraint for all and ensures that implies for all (see [20]), and also implies that if topic is independent of all other topics, i.e. for all , then . The wellstudied special case where topics are totally independent is . We are now in a position to formally define this paper’s objective.
IiC Objective Statement
This paper is focused on establishing the effects of the set of logic matrices on the evolution of opinions, and in particular the limiting opinion configuration. First, we record two assumptions on the logic matrix and the network topology, which will hold throughout this paper.
Assumption 2.
For every , there holds .
Assumption 3.
The influence network is strongly connected, is rowstochastic, and .
Assumption 2 implies that, for every , the graphs and have the same structure (but possible with different edge weights, including weights of opposing signs). This means that all individuals have the same view on which topics have dependent relationships with which other topics, but the assigned weights (and signs) may be different. This assumption ensures that the scope of this paper is reasonable, because otherwise the assumption that does not hold would introduce too many different scenarios to analyse.
Objective 1.
Let a set of logic matrices and an influence network be given, satisfying Assumptions 1, 2 and 3. Suppose that each individual ’s opinion vector evolves according to Eq. (1). Then, for each and generic initial conditions , this paper will investigate a method to systematically determine when there exists, and when there does not exist, an such that
(4) 
We will show that of a certain structure always guarantees consensus, and conversely, that of a certain other structure will lead to disagreement in certain identifiable topics.
Next, we provide further discussion to motivate Objective 1, including our interest in heterogeneous . The dynamics of the form Eq. (1) is a variation on the model studied in [20, 21], and we explain our interest in this particular variation by explaining in detail the differences between Eq. (1) and work in [20, 21].
For convenience, denote the vector of opinions for the entire influence network as . Supposing that the logic matrices were indeed homogeneous, i.e. for all , we can verify that much of the analysis becomes rather easy. For then one could write the influence network dynamics as
(5) 
and limiting behaviour is characterised by the following result.
Theorem 1 ([20, Theorem 3]).
For completeness and to aid discussion, we also record the Friedkin–Johnsen variant to Eq. (1), which is given as
(6) 
Here, the parameter represents individual ’s susceptibility to interpersonal influence, while represents the level of stubborn attachment by individual to his/her initial opinion . This paper studies the special case where there are no stubborn individuals, i.e. for all , and thus Eq. (6) is equivalent to Eq. (1). The paper [20] mainly focuses on the considerable challenge of obtaining complete convergence results for the model in Eq. (6) but with a homogeneous , and aside from some short remarks, does not investigate the effect of on the final opinion distribution (assuming the opinions do in fact converge to a steady state). The paper [21] secures a convergence result for heterogeneous but makes an assumption that there is at least one somewhat stubborn individual. Unlike [20] and [21], the key focus of this paper is to investigate the effect of the structure of , including heterogeneity, on the final opinion distribution.
We explain this further. If and for all , then existing results establish that under Assumption 3, a strong diversity of opinions emerges [20], with obviously no effects arising from the matrix. On the other hand, consider the case of homogeneous logic matrices and no stubbornness. For any satisfying Assumption 3, it is known that where
is a left eigenvector of
associated with the simple eigenvalue at , having entries , and normalised to satisfy [28]. Combining with Theorem 1, we can conclude that under Assumption 3 and if and for all , the opinions of all individuals on any given topic reach a consensus. That is, for all , there holds .In contrast, this paper assumes heterogeneous and no stubbornness among individuals. If we can show that opinions on a given topic fail to reach a consensus in the general case of , and instead strong diversity emerges, then this failure must be attributed to the structure, and the heterogeneity, of the among individuals. This would constitute a novel insight into the emergence of strong diversity in strongly connected networks, linking it for the first time to differences in individuals’ belief systems as opposed to stubbornness [15], a desire to be unique [25, 26], or social distancing [25].
To conclude this subsection, we now provide the definition of “competing logical interdependencies” which will be important in some scenarios for characterising the final opinions.
Definition 1 (Competing Logical Interdependence).
An influence network is said to contain individuals with competing logical interdependencies on topic if there exist individuals such that for some , and have nonzero entries and that are of opposite signs.
In other words, individuals with competing logical interdependencies are those who, when having the same opinion on topic , move in opposite directions on the opinion spectrum for topic . Such occurrences can be prevalent in society. Using the example in Section IIB, one might have an individual with
(7) 
because considers that private companies are profitdriven, and therefore cannot be ethically trusted with the exploration of Space. Then, from Eq. (3), one has that , i.e. individual eventually firmly believes Space exploration should not be privatised. In particular, .
In light of Assumption 2, if two individuals have competing interdependencies on topic , then for every individual , there is necessarily some individual with whom individual has competing logical interdependence on topic : the nonzero entries and are of opposite signs for some .
Remark 1.
Recall that is individual ’s set of constraints/functional dependencies between topics in ’s belief system. Thus, heterogeneity of may arise for many different reasons, such as education, background, or expertise in the topic. For example, if the set of topics is related to sports, a professional athlete may have very different weights in compared to someone that does not pursue an active lifestyle. Competing interdependencies may also arise for contentious issues, such as gun control discussions in the USA. Interestingly, [30] showed that when presented with the same published statement on an issue, different people could take opposite positions on the issue.
In the next section, we provide the set of main theoretical results of this paper to address Objective 1.
Iii Main Results
The main results are presented in two parts. First, we establish a general convergence result for the networked system. Then, we analyse the limiting opinion distribution and the role of the set of logic matrices in determining whether opinions for a given topic reach consensus or fail to do so. In order to place the focus on the theoretical results and interpretations as social phenomena, all proofs are presented to the Appendix.
Iiia Convergence
The network dynamics of Eq. (1) are given by
(8) 
and we define the system matrix above as . To begin, we rewrite the network dynamics Eq. (8) into a different form to aid analysis by introducing a coordinate transform (actually a reordering). In particular, define , for as the vector of all individuals’ opinions on the topic. Then, captures all of the individuals’ opinions on the topics. One obtains that
(9) 
where is a diagonal matrix with the diagonal element of being , the entry of . It follows that
(10) 
We denote the matrix in Eq. (10) as , with block matrix elements . We now show how the system Eq. (10) can be considered as a consensus process on a multiplex (or multilayered) signed graph.
Consider the matrix in Eq. (10), with the associated graph , and the matrix in Eq. (8), with associated graph . Clearly, the two graphs are the same up to a reordering of the nodes. In , with node set , one can consider the node subset , as a layer of the multilayer graph with vertices associated with the opinions of individuals on topic . In , with node set , one can consider the node subset , as a layer of a multilayer graph with vertices associated with the opinions of individual on topics . This is illustrated in Fig. 1, where each layer is identified by a dotted green ellipse border. A key motivation to study and the dynamical system Eq. (10) is that all the block diagonal entries of are nonnegative and irreducible because Assumption 1 indicates that has positive diagonal entries. This means that the edges between nodes in the subset , have positive weights, and this property greatly aids in the checking of the structural balance or unbalance of the network given and .
Verify from the rowstochastic property of and the rowsum property of in Assumption 1 that the entries of satisfy for all . We therefore conclude that Eq. (10) has the same dynamics as the discretetime Altafini model (see e.g. [10, 11]).
Remark 2.
Although Eq. (10) has the same dynamics as the discretetime Altafini model, a number of important differences exist. First, the context of negative edge weights is entirely different: in the Altafini model, implies individual mistrusts individual [10]. In contrast, Eq. (10) assumes nonnegative influence , and the negative edge weights arise from negative logical interdependencies in . Moreover the network structure of is affected by both the influence network and the logic matrix graphs .
The main convergence result is given as follows.
Theorem 2.
Suppose that for a population of individuals, the vector of the individuals’ opinions evolves according to Eq. (10), with interpersonal influences captured by . Suppose further that Assumptions 1, 2, and 3 hold. Then, for any initial condition , there exists some such that there holds exponentially fast. If for all and , then for all and and .
Having established that the opinion dynamical system always converges, we now address Objective 1 by studying the influence of in determining the limiting opinion vector .
IiiB Consensus and Disagreement of Each Topic
We now explain how to use the logic matrices to systematically determine whether opinions on a given topic will reach a consensus or not. We defer discussion of the social interpretation of the theoretical results until Section IIIC, and illustrate some of the conclusions drawn in this section with selected simulations in Section IV.
Consider the graph associated with for some , which is a signed graph if there are negative offdiagonal entries in . It turns out (see Theorem 3 immediately below) that if for all are irreducible, then all topics will reach a consensus (although the consensus value for two different topics and may be different). We remark that irreducible logic matrices correspond to which are strongly connected, and thus for any two topics , there is a (possibly signed) directed path from to . In other words, all topics are directly or indirectly dependent on all other topics.
Theorem 3.
Let the hypotheses in Theorem 2 hold. Suppose that (i) for all and , and (ii) that Assumptions 1, 2, and 3 hold. Suppose further that are irreducible^{4}^{4}4Under Assumption 2, irreducibility of one implies the same for all.. Then, for all , exponentially fast, where . Moreover,

If there are no competing logical interdependencies, as given in Definition 1, and are structurally balanced^{5}^{5}5Under Assumption 2 and in the absence of competing logical interdependencies, the presence or absence of structural balance for one implies the same for all., then for almost all initial conditions, .

If there are no competing logical interdependencies, and are structurally unbalanced, then .

If there are competing logical interdependencies, then .
Further to the conclusions of Theorem 3, one can obtain the following result for the case where consensus to a nonzero opinion value is achieved.
Corollary 1.
Let the hypotheses in Theorem 3 hold. Suppose that there are no competing logical interdependencies, and are structurally balanced. For with node set , define two disjoint subsets of nodes and so that each edge between two nodes in or two nodes in has a positive weight, and each edge between two nodes in and has a negative weight. Then, for any , there holds

if or .

if and .
Consider now the more general case where for all are reducible. Thus, is no longer strongly connected. The logic matrices of all individuals can be expressed in a lower block triangular form through an inessential reordering of the topic set. From Assumption 2, we further conclude that there exists a common permutation matrix such that, for all , is lower block triangular. Without loss of generality, we therefore assume that the topics are ordered such that, for each ,
(11) 
where is irreducible for any and are positive integers such that . Decompose the opinion set into disjoint subsets for where
(12) 
with . Though reducible may seem to be restrictive, they are in fact common given the problem context since they imply a cascade logical interdependence structure among the topics. This may be representative of an individual who obtains by sequentially building upon an axiom or axioms (the first block matrices). The two topics of the Space exploration example given in Eq. (2) constitute one such example of a belief system driven by an axiom (Topic 1).
From the perspective of the graph , the expression in Eq. (11) enables to be divided into strongly connected components which are “closed” or “open”. (This is related to a concept called the condensation of a graph, see [28]). Formally, we say that a subgraph is a strongly connected component of if is strongly connected and any other subgraph of strictly containing is not strongly connected. A strongly connected component of a graph is said to be closed if there are no incoming edges to from a node outside of , and is said to be open otherwise. The simplest possible strongly connected component is a single node, and it would be closed if there were no incoming edges to it. Figure 2 shows an example of a graph divided into strongly connected components (identified by the dotted line encircling a set of nodes), with the blue and purple components being closed, and the green and orange components being open. Following the notation in Eq. (11) and Eq. (12), we have for the example in Fig. 2, , , and , , , .
If the topic set corresponds to a closed strongly connected component of , then clearly in Eq. (11), for all . One can then use Theorem 3 and Corollary 1 to establish that for every , there holds exponentially fast, with . That is, all opinions in topic reach a consensus. If, on the other hand, the topic set corresponds to an open strongly connected component of , then the results presented below can be employed sequentially in order to establish whether opinions on a given topic have reached a consensus. By “sequentially”, we mean that we analyse the topic sets with in the order . Under Assumption 2, define for each topic , the set
(13) 
where is the entry of . In other words, identifies all topics that topic is logically dependent upon. Because of Assumption 2, the set is the same for all individuals . In Fig. 2, for example is .
We present necessary and sufficient conditions that ensure every topic in the subset reaches a consensus of opinions in two theorems, the first for the case when the subset is a singleton (e.g. in Fig. 2), and the second for when has at least two elements (e.g. in Fig. 2).
Theorem 4.
Let the hypotheses in Theorem 2 hold. Assume that (i) for all and , and (ii) that is decomposed as in Eq. (11). Suppose that , as defined in Eq. (12), is a singleton, and let as defined in Eq. (13) be nonempty. Suppose further that all topics satisfy . Then, for some if and only if there exists a such that
(14) 
If such a exists, then .
The key necessary and sufficient condition involves Eq. (14), which is somewhat complex and nonintuitive. We now provide a corollary which studies the condition in Eq. (14) for some situations which are important or of interest in the social context. Discussion and interpretation of these formal results are provided in the following Section IIIC.
Corollary 2.
Adopting the hypotheses in Theorem 4, the following hold:

If for all , then satisfies Eq. (14).

Suppose that , . If for all and , then there exists a satisfying Eq. (14).

Suppose that , . Suppose further that for all . Then, there exists a satisfying Eq. (14) if either (i) the sign of and are equal for all and or (ii) the sign of and are opposite for all and . In the case of (i), , and in the case of (ii), .
When is not a singleton, the analysis becomes significantly more involved. To that end, we first introduce some additional notation. Define
(15) 
as the set of topics not in that the topics in depend upon. For example, in Fig. 2, , , , and . Note that if is a singleton, we have . Perhaps unsurprisingly, Theorem 4 requires that consensus must first occur for topics in , on which the topics in depend. The following theorem also has the requirement that consensus occur for all topics in .
Theorem 5.
Let the hypotheses in Theorem 2 hold. Assume that (i) for all and , and that (ii) is decomposed as in Eq. (11). Suppose that , as defined in Eq. (12), has at least two elements. Let , as defined in Eq. (15), be nonempty and suppose further that all topics satisfy . Then, for all if and only if, for every , there exists a such that
(16) 
If such a set of exist, then for all .
Similar to above, we now present a corollary which gives sufficient conditions for Eq. (5) in two scenarios.
Corollary 3.
IiiC Discussion and Social Interpretations
We conclude this section by providing some discussion and comments on the main results, focusing in particular on the theorems and corollaries in Section IIIB. Overall, the outcomes we have established depend on the graphical structures on the one hand, and on the numerical values (including their signs) of the entries on the other. This dependence sometimes flows simply from the signs (the presence or absence of competing logical interdependencies), and sometimes from the precise values of the . Further, when consensus on a topic occurs, it is evident that sometimes a value 0 is always the outcome, and sometimes a nonzero value dependent on the initial opinions of those topics in the closed and strongly connected components of .
It is clear from Theorem 3 that for any topic set corresponding to a closed and strongly connected component of , every topic will reach a consensus. One interpretation is that a closed and strongly connected component corresponds to having a topic(s) that is an axiom (or axioms) upon which an individual builds his or her belief system (see below Eq. (12)). Our results show that discussion of axiomatic topics will always lead a consensus under the model Eq. (1) (a consensus might not be reached if, as in Eq. (6), there is stubbornness present).
Theorem 3 and Corollaries 2 and 3 also illustrate that competing logical interdependencies, if present, can play a major role in determining the final opinion values. For instance, see Theorem 3 Part 3, where given a topic set corresponding to a closed and strongly connected component of , all opinion values for all topics in converge to the neutral value at 0 whenever competing interdependencies are present in the topics in . Also, the presence of any competing logical interdependencies in topic is enough to prevent the sufficient conditions detailed in Corollary 2 Item 1), 3), and 4) and Corollary 3 Item 2) from being satisfied. Of particular note is Corollary 2 Item 1). When is a singleton, heterogeneity in the entries of is not enough to prevent a consensus of opinions on topic ; competing logical interdependences are required. This last finding is a surprising, and nonintuitive result.
The sufficient condition in Corollary 2 Item 2) requires for all . This is not as restrictive as it first seems: one possible scenario is that all elements of belong to topics from the same closed and strongly connected component in , with the component being structurally unbalanced, or having competing logical interdependencies. The same can be said for Corollary 3 Item 1). Part of the sufficient condition for Corollary 2 Item 4) is that for all , . This will always hold if are topics that are part of the same closed and strongly connected component in .
From numerous simulations, we frequently observed that minor heterogenieties in the entries of among the individuals (e.g. if the
were all selected from a uniform distribution) were often sufficient to create disagreement among the opinions on topic
. We observed this in many different examples, except in the case of Corollary 2 Item 1), where and are both singletons, since existence of competing logical interdependencies was proven to be a necessary and sufficient condition for disagreement.It is also clear from Theorems 3, 4 and 5 that disagreement is possible only in topic sets associated with an open strongly connected component of . Put another way, belief systems with a cascade logical structure, viz. reducible in the form of Eq. (11), including heterogeneity among individuals’ belief systems, play a significant role in generating disagreement when social networks discuss multiple logically interdependent topics. Looking at Eq. (1), one can see two separate processes occurring: the DeGroot component describes interpersonal influence between individuals in an effort to reach a consensus, while the logic matrix by itself (as in Eq. (3)) captures an intrapersonal effort to secure logical consistency of opinions across several topics. These two drivers may or may not end up in conflict, and the presence of conflict or lack thereof determines whether opinions of a certain topic reach a consensus or fail to do so. Our results in Theorems 4 and 5 identify when such conflict can occur.
Remark 3.
Theorems 4 and 5 establish necessary and sufficient conditions for topic to reach a consensus under a particular hypothesis. Specifically, it is assumed that for the set under consideration, there holds
(17) 
That is, all other topics that one or more topics depend upon are assumed to have reached a consensus. Based on numerous simulations, we believe the requirement that Eq. (17) holds is also a necessary condition for to reach a consensus. In other words, if any topic fails to reach a consensus, we conjecture that all will also fail to reach a consensus. Confirming this would provide yet another indication that networks with belief systems having a cascade logic structure more readily result in disagreement. We leave this to future investigations.
Iv Simulations
We now provide simulations to illustrate some of the results in Section III using a network of individuals, with
(18) 
Note that satisfies Assumption 3. Initial conditions are generated by selecting each from a uniform distribution in , and the same initial conditon vector is used for all simulations. We consider 5 topics, i.e. .
In the first simulation, we use two logic matrices:
(19a)  
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