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Towards a Perron-Frobenius Theorem for Strongly Aperiodic Stochastic Chains

Perron-Frobenius theorem is a fundamental tool in matrix analysis that has applications to complex networks, population dynamics, social learning, and numerous physical, engineering, and economic phenomena. However, since this theorem and many of its extensions can be applied only to a single matrix at a time, their applications in networked dynamical systems are limited to static networks. To extend the applicability of these results to time-varying networks, we generalize two assertions of the Perron-Frobenius theorem to sequences as well as continua of row-stochastic matrices. Our results have potential applications in areas such as distributed averaging, optimization, and estimation.


Introduction
Perron-Frobenius theorem is a foundational tool in linear algebra that is central to theory of Markov chain, and has many applications in database systems, complex networks, population dynamics, opinion dynamics, social learning, economic growth and income inequalities, and many other physical, social, and economic phenomena [1][2][3][4][5][6][7][8][9][10][11][12][13]. Its strength lies in connecting the limiting behavior of A k as k → ∞ with the structural (graph-theoretic) pattern of a fixed non-negative matrix A. For example, in the case of Google's PageRank algorithm, A denotes the transition matrix of a Markov chain modelling a web-surfer, and the theory relates the ergodic (long term) behavior of this Markov chain to the centrality of webpages on World Wide Web (WWW).
However, since this theorem and many of its extensions (e.g. [14][15][16][17]) apply only to fixed matrices, their applications in understanding networked dynamical systems are limited to static networks. By contrast, most real-world networks are time-varying due to changing connections and communication patterns, temporary link failures, etc. [18][19][20][21]. Dynamical processes over such networks are related to products of time-varying matrices that capture the network structure. Such products are natural generalizations of powers of time-invariant matrices. This motivates us to study the generalization of the Perron-Frobenius theorem to products of time-varying matrices, as such a generalization would enhance many of the applications of the original result to time-varying settings.
In view of the above, we extend two assertions of the Perron-Frobenius theorem to time-varying networks. Specifically, the original theorem implies that a weighted random walk over a static strongly connected network has a stationary distribution that is (a) unique and (b) positive. Analogously, we establish the generalization of strong connectivity for time-varying networks and show that for lazy random walks over such networks, the time-varying extension of stationary distribution (Kolmogorov's absolute probability sequence [22]) is (a) unique and (b) uniformly positive if and only if the network is strongly connected over time in the generalized sense. We believe that this fundamental study will help advance the state-of-the-art understanding of dynamical processes over real-world networked systems.
The paper is organized as follows. We first present a brief overview of all our main results in Section 2. Next, we formulate the problem of interest in Section 3 and discuss our main results in Section 4. We then provide a few applications of our main results in Section 5 and end with a few concluding remarks in Section 6. The proofs of all the results are provided in the Appendix.
Notation: Let N := {1, 2, . . .} denote the set of natural numbers and let N 0 := N {0}. Let R denote the set of real numbers, let R n denote the set of n-dimensional real-valued column vectors, and let R n×n denote the set of square matrices with real entries. For a given matrix A ∈ R n×n , we let a ij = (A) ij denote the entry in the i-th row and the j-th column of A.
Let I denote the identity matrix (of the known dimension), let O denote the all-zeros matrix (of the known dimension), and let 0 and 1 denote the vector (of the known dimension) with all entries equal to zero and the vector with all entries equal to one, respectively.
A vector v ∈ R n is said to be stochastic if v is non-negative and v T 1 = 1. A matrix A 0 ∈ R n×n is called row-stochastic or simply stochastic if A 0 is non-negative and if each row of A 0 sums up to 1, i.e., if A 0 ≥ O and A 0 1 = 1. Note that all matrix and vector inequalities are assumed to hold entry-wise. Let {A(t)} ∞ t=0 be a stochastic chain (a sequence of row-stochastic matrices in R n×n ). Then, for any two times t 1 , t 2 ∈ N 0 with t 1 < t 2 , we use the following notation to denote the backwards matrix product of {A(t)} ∞ t=0 corresponding to the time interval [t 1 , t 2 ].
A(t 2 : t 1 ) := A(t 2 − 1)A(t 2 − 2) · · · A(t 1 ) with the convention that A(t : t) := I for all t ∈ N 0 . For a matrix A ∈ R n×n and a subset S ⊂ [n], let A S be the principal sub-matrix of A corresponding to the rows and columns indexed by S. LetS := [n]\S, and let A SS denote the sub-matrix of A corresponding to the rows indexed by S and the columns indexed byS. For a sequence of matrices {A(t)} ∞ t=0 ⊂ R n×n and times t 0 , t 1 ∈ N 0 satisfying t 0 ≤ t 1 , let A S (t 1 : t 0 ) := (A(t 1 : t 0 )) S and A SS (t 1 : t 0 ) := (A(t 1 : t 0 )) SS .
An unweighted undirected graph with vertex set [n] and edge set E is denoted by G = ([n], E). On the other hand, a weighted time-varying directed graph with vertex set [n], edge set E(t) ⊂ [n] × [n], and edge weights {w ij (t) : Recall that G(t) is said to be strongly connected if, for any two nodes i, j ∈ [n], there exists a directed path from i to j in G(t).
For a weighted time-varying directed graph G(t) = ([n], E(t), W (t)), we let L(t) = (ℓ ij (t)) denote the weighted Laplacian matrix of G(t), defined by ℓ ij (t) = −w ij (t) for all i = j and ℓ ii (t) = j =i w ij (t) for all i ∈ [n]. In addition, for a given matrix A 0 , we let G(A 0 ) = ([n], E(A 0 ), A 0 ) denote the weighted directed graph whose weighted adjacency matrix is A 0 , i.e., we let E(A 0 ) = {(i, j) ∈ [n] × [n] : (A 0 ) ij > 0} and we construct G(A 0 ) in such a way that every (i, j) ∈ E(A 0 ) is a directed edge in G(A 0 ) with weight (A 0 ) ij .

Overview of the Main Results
To highlight the main results of this work and before discussing the their connection to the classical Perron-Frobenius Theorem in details, we state our main results here. These results are based on the three definitions provided below. We shall reproduce these definitions in Section and extensively discuss them with regards to the classical Perron-Frobenius Theorem in Section 3.
The first definition is an extension of strong aperiodicity [50] to time-varying stochastic chains.
Definition (Strong Aperiodicity). A stochastic chain {A(t)} ∞ t=0 is said to be strongly aperiodic if there exists a constant γ > 0 such that a ii (t) ≥ γ for all i ∈ [n] and all t ∈ N 0 .
The second object, an approximate reciprocal chain, is a central object of this work and we will see shortly its connection to the classical Perron-Frobenius Theorem.
t=0 is said to be approximately reciprocal if there exist constants p 0 , β ∈ (0, ∞) such that the following inequality holds for all S ⊂ [n] and all times t 0 , t 1 ∈ N 0 that satisfy t 0 < t 1 .
Finally, let us introduce a proper object extending the concept of Perron eigenvector for a stochastic matrix to time-varying chains and a proper extension of positive eigenvector to timevarying setting.
Definition (Uniformly Positive Absolute Probability Sequence). Let {A(t)} ∞ t=0 be a stochastic chain. A sequence of stochastic vectors {π(t)} ∞ t=0 is said to be a uniformly positive absolute probability sequence for and if there exists a constant p * > 0 such that π(t) ≥ p * 1 for all t ∈ N 0 .
With the introduction of these concepts, we are ready to assert the main results of this work.
Theorem 1 (An Analog of the Positive Eigenvector Assertion of the Perron-Frobenius Theorem). Suppose {A(t)} ∞ t=0 ⊂ R n×n is a strongly aperiodic stochastic chain. Then {A(t)} ∞ t=0 has a uniformly positive absolute probability sequence if and only if it is approximately reciprocal.
As we shall show, this theorem generalizes the following Perron-Frobenius assertion to timevarying matrices: an irreducible non-negative matrix has a positive principal left eigenvector.
Theorem 2 (An Analog of the Uniqueness Assertion of the Perron-Frobenius Theorem). Let {A(t)} ∞ t=0 ⊂ R n×n be a strongly aperiodic stochastic chain that is also approximately reciprocal. Then, {A(t)} ∞ t=0 admits a unique absolute probability sequence if and only if its infinite flow graph is connected.
We shall see that this result extends the following statement to the case of time-varying matrices: an irreducible non-negative matrix has a principal left eigenvector that is unique up to scaling.
We now state the continuous-time analogs of the above results, which are based on the following continuous-time analogs of the above definitions.
Definition (Uniformly Positive Absolute Probability Sequence in Continuous Time). Let {A(t)} t≥0 be a continuous-time stochastic chain and let Φ(·, ·) denote the state transition matrix for the dynamicsẋ(t) = Ax(t). Then, a continuum of stochastic vectors {π(t)} t≥0 is said to be a uniformly positive absolute probability sequence for holds for all t ≥ τ ≥ 0, and if there exists a constant p * > 0 such that π(t) ≥ p * 1 for all t ≥ 0.
The following is the natural continuous counterpart of approximate reciprocity.
Definition (Approximate Reciprocity in Continuous Time). A continuous-time stochastic chain {A(t)} t≥0 is said to be approximately reciprocal if there exist p 0 , β ∈ (0, ∞) such that   The rest of the paper is on the detailed discussion on these results, their connection to the classical Perron-Frobenius theorem, and their implications.
3 Classical Perron-Frobenius Theorem, Approximate Reciprocity, and Absolute Probability We first review the eigenvector assertions of the original Perron-Frobenius theorem. For this purpose, we need to recall the following property of irreducible matrices, which is often stated as a definition of irreducibility.
Lemma 1 (Section 8.3, [51]). A non-negative matrix A 0 ∈ R n×n is irreducible if and only if the associated digraph G(A 0 ) is a strongly connected directed graph.
In addition, we need the concepts of reciprocity and infinite flow graphs, which we reproduce from [52] below.
Definition 1 (Reciprocity/Cut-balance). A stochastic chain {A(t)} ∞ t=0 is said to be cut-balanced or reciprocal if there exists a constant α ∈ (0, 1) such that Intuitively, a stochastic chain is said to be reciprocal if the sequence of associated directed graphs is such that the total influence of any subset S of individuals on the complementary subsetS is comparable to the total reverse influence ofS on S, i.e., the ratio of the forward and the backward influences does not vanish in time.
Definition 2 (Infinite Flow Graph [52]). For a stochastic chain {A(t)} ∞ t=0 , we define its infinite flow graph to be the unweighted undirected graph G ∞ = ([n], E ∞ ) with vertex set [n] and edge set E ∞ , where Intuitively, there exists a link from a node j ∈ [n] to another node i ∈ [n] \ {j} in the infinite flow graph G ∞ if and only if node j exerts a long-term influence on node i in the time-varying directed graph G(A(t)).
We now observe that for a static chain A(t) = A 0 , the irreducibility of A 0 can be expressed in terms of the connectivity of the infinite flow graph G ∞ and the reciprocity of t=0 is reciprocal and its infinite flow graph G ∞ is connected.
We are now ready to recall two eigenvector assertions of the Perron-Frobenius theorem in the context of row-stochastic matrices. Lemma 2 enables us to state these assertions as follows.

Proposition 1 (Eigenvector Assertions of the Perron-Frobenius Theorem for Stochastic Ma
is reciprocal and if its infinite flow graph is connected, then A 0 has a stochastic principal left eigenvector π 0 ∈ R n that is (a) entry-wise positive, Note that the original theorem applies to left eigenvectors as well as to right eigenvectors. However, we choose to focus on the former because it is clear that all row-stochastic matrices have 1 as their principal right eigenvector. Moreover, our main results are centered on a concept that generalizes the notion of principal left eigenvectors to the case of time-varying row-stochastic matrices. This concept is defined below.
Definition 3 (Kolmogorov Absolute Probability Sequence [22]). Let {A(t)} ∞ t=0 be a stochastic chain. A sequence of stochastic vectors {π(t)} ∞ t=0 is said to be an absolute probability sequence for Note that every stochastic chain admits an absolute probability sequence [22]. Moreover, if {A(t)} ∞ t=0 is a static chain with A(t) = A 0 ∈ R n×n for all t ∈ N 0 , then the static sequence π(t) = π 0 , where π 0 ∈ R n is a stochastic vector satisfying π T 0 A 0 = π T 0 , is an absolute probability sequence for {A(t)} ∞ t=0 . Hence, absolute probability sequences are a time-varying analog of stochastic principal left eigenvectors.
This discussion naturally leads to the following question: can we generalize Proposition 1 to the class of non-static stochastic chains (or any sub-class thereof) using the notion of absolute probability sequences? The next section answers this question.

Results
We first extend Proposition 1 to discrete-time stochastic chains of the form {A(t) : t ∈ N 0 } and then to continuous-time stochastic chains of the form {A(t) : t ≥ 0}.

Discrete Time
To extend the first assertion of Proposition 1 to time-varying matrices (non-static chains), we need to extend the notion of positive principal left eigenvectors to the time-varying case. The following concept, first introduced in [52], offers such an extension.
Definition 4 (Class P * [52]). We let (Class-)P * be the set of all stochastic chains that admit uniformly positive absolute probability sequences, i.e., a sequence of stochastic vectors {π(t)} ∞ t=0 for which there exists a p * > 0 such that π(t) ≥ p * 1 for all t ∈ N 0 . (Note that the absolute probability sequence and the value of p * may vary from chain to chain).
It is worth noting that in the context of social learning, if {π(t)} ∞ t=0 is an absolute probability sequence for {A(t)} ∞ t=0 , then π i (t) denotes the Kolmogorov centrality or social power of agent i at time t, which quantifies how influential the i-th agent is relative to other agents at time t [52,53]. In view of Definition 4, this means that a stochastic chain belonging to Class P * describes a sequence of influence graphs in which the social power of every agent exceeds a fixed positive threshold p * at all times.
Since the definition of Class P * eludes simple interpretation, we would like to derive necessary and sufficient conditions for a given stochastic chain to belong to Class P * . To this end, we introduce the idea of approximate reciprocity, which is a weaker notion of reciprocity (Definition 1).

Definition 5 (Approximate Reciprocity). A stochastic chain {A(t)} ∞
k=0 is said to be approximately reciprocal if there exist constants p 0 , β ∈ (0, ∞) such that the following inequality holds for all S ⊂ [n] and all times t 0 , t 1 ∈ N 0 that satisfy t 0 < t 1 .
As it turns out, approximate reciprocity is a necessary condition for a given stochastic chain to belong to Class P * .

Proposition 2 (Necessary Conditions for Class
is approximately reciprocal. Interestingly, approximate reciprocity is also a sufficient condition for certain stochastic chains called strongly aperiodic chains to belong to Class P * . A given stochastic chain is called strongly aperiodic if all the diagonal entries of its matrices are uniformly bounded away from zero.
To connect approximate reciprocity, a property expressed in terms of sums of matrix entries, to Class P * , a concept associated with products of matrices, we need the following lemmas that help relate matrix sums to matrix products.
Lemma 4. Let n, σ ∈ N and i, j ∈ [n] be given. Let {B(t)} σ−1 t=0 ⊂ R n×n be a sequence of substochastic matrices and let t L := max{t ∈ {0, 1, . . . , σ − 1} : B ji (t) > 0}. Suppose there exist positive constants η i and η j such that In addition to the above lemmas, we need the notion of approximately stochastic chains, which we define below.
The constant ∆ will be referred to as the deviation from stochasticity of the sequence We are now well-equipped to establish approximate reciprocity as a sufficient condition for strongly aperiodic chains to lie in P * . To do so, we use inductive arguments involving approximately stochastic chains to prove a slightly more general result that asserts that the backward matrix products of the concerned chains can be uniformly lower-bounded by a multiple of the identity matrix. This general result is stated below and proved in the appendix.
⊂ R n×n is a substochastic chain with the following properties.
We now obtain the desired sufficient conditions as a straightforward consequence of the above proposition.
Corollary 1 (Sufficient Conditions for Class P * ). Suppose {A(t)} ∞ t=0 ⊂ R n×n is a strongly aperiodic stochastic chain, i.e., suppose there exists a γ > 0 such that As a direct consequence of Corollary 1 and Proposition 2, we obtain the following necessary and sufficient conditions for Class P * : a strongly aperiodic stochastic chain belongs to P * iff it is approximately reciprocal. Since a stochastic chain belongs to Class P * iff it has a uniformly positive absolute probability sequence, we obtain Theorem 1, whose statement is reproduced below. Observe how Theorem 1 parallels the first assertion of Proposition 1: the original theorem asserts that for a static network that is reciprocal and whose infinite flow graph is connected (i.e., a network defined by an irreducible matrix), there exists a positive principal left eigenvector. Analogously, Theorem 1 asserts that for a dynamic network that is approximately reciprocal, there exists a uniformly positive absolute probability sequence.
Next, we extend the second assertion of Proposition 1 (the unique eigenvector assertion of Perron-Frobenius theorem). For this purpose, we will need the following lemmas and definitions.

Definition 8 (Ergodicity for Stochastic Chains
To interpret the above definition, we first observe that in the distributed averaging dynamics x(t + 1) = A(t)x(t) with a starting time t 0 ∈ N 0 and an initial condition x(t 0 ) ∈ R n , we have For an ergodic chain, this means that lim t→∞ x(t) = π T (t 0 )x(t 0 )1, which is a consensus vector (i.e., all its entries are equal). Therefore, a stochastic chain being ergodic means that it always enables consensus among the nodes of the network, regardless of the starting time t 0 and the starting point x(t 0 ).
converges to a limit for all starting times t 0 ∈ N 0 and all initial conditions Put simply, a stochastic chain is infinite flow stable if (a) the states of all the nodes of the corresponding time-varying network converge to a limit asymptotically in time, and (b) if a consensus is necessarily reached among nodes that exert a long-term influence on each other.
We are now ready to extend the second assertion of Proposition 1 to dynamic stochastic chains. We thus reproduce the statement of Theorem 2 below.
Let {A(t)} ∞ t=0 ⊂ R n×n be a strongly aperiodic stochastic chain that is also approximately reciprocal. Then, {A(t)} ∞ t=0 admits a unique absolute probability sequence if and only if its infinite flow graph is connected.
Like Theorem 1, Theorem 2 closely parallels an assertion of the Perron-Frobenius theorem. In view of Proposition 1, the original theorem asserts that, if a matrix describes a static network that is reciprocal and whose infinite flow graph is connected, then its principal left eigenvector is unique. Analogously, Theorem 2 asserts that, if a stochastic chain describes a time-varying network that is approximately reciprocal and whose infinite flow graph is connected, then its absolute probability sequence is unique.
Besides, it is worth noting that stochastic chains that are approximately reciprocal and whose infinite flow graphs are connected are a time-varying analog of irreducible matrices (this follows immediately from Lemma 2). Therefore, we shall henceforth use the term irreducible chains to refer to such chains.

Continuous Time
We now extend our discrete-time results (Theorems 1 and 2) to continua of row-stochastic matrices, henceforth called continuous-time stochastic chains. Consider the following continuous-time analog of the discrete-time dynamics where A(t) = −L(t) is the negative of the Laplacian matrix of a given graph G(t). Throughout this section, we assume It is well-known [55,56] that under Assumption (4), the solution to (3) is unique and can be expressed as where the state-transition matrix Φ is the unique solution to the equation continuum It is also known that Φ(t, τ ) is row-stochastic for all t ≥ τ ≥ 0, Φ(τ, τ ) = I for all τ ≥ 0, and Therefore, for any sequence of increasing times {t k } ∞ k=1 ⊂ R ≥0 , if we let B(k) := Φ(t k+1 , t k ) for all k ∈ N 0 , then we have B(m : ℓ) = Φ(t m : t ℓ ) for all ℓ, m ∈ N with ℓ ≤ m. As a result, an application of Proposition 3 to the stochastic chain {B(k)} ∞ k=0 yields the following result.
It is clear from Lemma 6 above and from Lemma 8 of [57] that the discrete-time chain {Φ(t k+1 , t k )} ∞ k=0 lies in Class P * if the approximate reciprocity condition (7) and the strong aperiodicity condition Φ(t k+1 , t k ) ≥ γI are satisfied. As we shall see shortly, the following assumptions ensure that both these conditions are met.
Assumption 1 (Uniform Bound on Integral Weights [55]). There exists an M < ∞ and an increasing sequence Assumption 1 is sufficient to guarantee the strong aperiodicity condition Φ(t k+1 , t k ) ≥ γI for some γ > 0 and all k ∈ N 0 . This is evident from the proof of Lemma 8 in [55].
holds for all sets S ⊂ [n] and for all ℓ, m ∈ N 0 with ℓ ≤ m.
We now use Lemma 7 to show that approximate reciprocity in continuous time is equivalent to {A(t)} ∞ t=0 belonging to Class P * . To begin, we first define the continuous-time analogs of absolute probability sequences and Class P * .
Definition 10 (Continuous-time Absolute Probability Sequence [57]). A continuum of stochastic vectors {π(t)} t≥0 is said to be an absolute probability sequence for a continuous-time stochastic holds for all t ≥ τ ≥ 0, where Φ(·, ·) denotes the state transition matrix for the dynamics (3).
Definition 11 (Continuous-time Class P * [57]). We let (continuous-time Class-)P * be the set of all continuous-time stochastic chains that admit uniformly positive absolute probability sequences, i.e., a continuum of stochastic vectors {π(t)} t≥0 such that π(t) ≥ p * 1 for some scalar p * > 0 and all t ∈ N 0 . (Note that the absolute probability sequence and the value of p * may vary from chain to chain).
We are now ready to state the first main result of this section. It is Theorem 3, whose statement is reproduced below.
We now reproduce the statement of the desired theorem (Theorem 4).

Let {A(t)} t≥0 be a continuous-time stochastic chain that satisfies Assumptions 1 and 2. Then {A(t)} t≥0 admits a unique absolute probability sequence if and only if its infinite flow graph is connected.
To interpret Theorems 1 -4, we provide below a series of remarks in which we start from some existing interpretations of Proposition 1, which is based on the classical theorem, and we extend these interpretations to the case of time-varying networks.
Remark 1 (Implications for Markov Chains). The eigenvector assertions of the Perron-Frobenius theorem can be interpreted as follows: for a time-homogeneous Markov chain with transition probabilities given by an aperiodic irreducible matrix, the probability of visiting any given state converges asymptotically in time to a unique positive value, regardless of the initial probability distribution. Analogously, Theorems 1 and 2 can be interpreted as follows: given a starting time, for a backwardpropagating time-non-homogeneous Markov chain with transition probabilities given by strongly aperiodic irreducible chain, the probability of visiting any given state converges asymptotically in time to a unique positive value, regardless of the initial probability distribution. Although this limiting probability is a function of the starting time, it is bounded away from zero by a fixed threshold that does not depend on the starting time.
Remark 2 (Opinion Dynamics-Based Interpretation). In the context of opinion dynamics, the matrix A(t) can be interpreted as the influence matrix at time t, i.e., a ij (t) quantifies the extent to which agent i values agent j's opinion at time t (or equivalently, the extent to which agent j influences agent i at time t). Therefore, an irreducible chain (and hence also an irreducible matrix) describes a network in which every subset of agents influences the complementary subset persistently over the entire course of opinion evolution, which means that there exists no group of elite agents that dominate others forever. Additionally, as mentioned before, absolute probability sequences can be interpreted as quantifying the agents' social powers.
Therefore, an interpretation of the eigenvector assertions of the original theorem is as follows: in a static social network, the social power of every agent (given by the eigenvector centrality of the corresponding network node) is unique and positive if no subset of agents dominates other agents forever. Analogously, Theorems 1-2 can be interpreted as follows: in a time-varying social network, the time-varying social power of every agent (given by the Kolmogorov centrality of the corresponding network node) is unique and uniformly positive (lower-bounded by a constant positive threshold) if no subset of agents dominates other agents forever.

Remark 3 (Implications for Economic Growth).
Here, we follow the approach taken in [58]. Consider an economy with n sectors of activity, and let x i denote the activity level in the i-th sector. Then the evolution of activity levels may be expressed as x(t + 1) = A(t)x(t), wherein a ij (t) quantifies the number of activity units in Sector i that are required in the next economic cycle as a result of the completion of each activity unit in Sector j. In addition, if A(t) is a static matrix, then its principal left eigenvector gives the long-term economic value of the activity carried out in Sector i during the zeroth time period (t = 0) relative to the long-term values of the activities carried out in other sectors during the zeroth time period.
Therefore, the original theorem now has the following interpretation: if the activity evolution matrix is static and exhibits the properties of reciprocity and connectivity (via irreducibility), then the long-term economic value of the initial activity in any given sector relative to initial activities in other sectors will be positive and can be determined uniquely. Analogously, our main results imply that, if the activity evolution matrix is time-varying and exhibits the properties of approximate reciprocity and connectivity of the infinite flow graph, then, for any starting time period t = t 0 (as opposed to t = 0), the long-term economic value of initial activity in any given sector relative to initial activities in other sectors will be uniformly positive and can be uniquely determined.

Remark 4 (Implications for Population Dynamics).
Analogous to the economic growth model, we may consider a population comprised of n age groups instead of an economy comprised of n sectors. We may let x i denote the population size of the i-th age group and further let a ij (t) quantify the fraction of individuals that transition from the j-th age group to the i-th age group in the t-th time period for i = j + 1. In addition, we may let a 1j (t) quantify the number of births per parent in the j-th age group. In this setting, the entry π i (t 0 ) of the absolute probability sequence denotes the sensitivity of the long-term total population size to the size that the i-th age group has at time t 0 . One may then interpret the original theorem and our results in this context by drawing analogies with the economic growth context discussed above.

Applications
We now derive a few corollaries of our main results. Since some of these corollaries apply to random stochastic chains, we first define the relevant terms.

Infinite Flow Stability of Independent Random Chains
The concept of independent random chains is a straightforward extension of the concept of deterministic chains, as per the definition below. We now extend the notion of Class P * to independent random chains.
Definition 14 (Class P * for Independent Random Chains [52]). An independent random chain {A(t)} ∞ t=0 is said to belong to Class P * if the expected chain {E[A(t)]} ∞ t=0 belongs to Class P * .
We will also need the notion of feedback property, a weak notion of strong aperiodicity for independent random chains.
Definition 15 (Feedback Property for Independent Random Chains [52]). Let {A(t)} ∞ t=0 be an independent random chain. We say that {A(t)} ∞ t=0 has the feedback property if there exists a γ > 0 for all t ∈ N 0 and all distinct i, j ∈ [n], in which case γ is called the feedback coefficient.
In addition, we will use the term mutual ergodicity, which we define below.
Based on these concepts, we have the following result.

Rate of Convergence to Steady State
We now provide a result on the rate of convergence for the dynamics x(t + 1) = A(t)x(t) in terms of the quadratic comparison function where u is an arbitrary stochastic vector in R n .
Corollary 3. Let {A(t)} ∞ t=0 be an independent random chain with feedback property and feedback coefficient γ > 0, and suppose the expected chain {Ā(t)} ∞ t=0 is approximately reciprocal. In addition, let t q = 0 for q = 0 and let t q = argmin Then, for all q ≥ 1 and all stochastic vectors u ∈ R n ,

Class Ergodicity
To characterize the ergodicity of approximately reciprocal chains, we borrow the following definitions from [59].
Definition 17 (Jet). For a given set S ⊂ [n], a jet J in S is a sequence {J(t)} ∞ t=0 of subsets of S. A jet J in S is called proper if ∅ J(t) S for all t ∈ N 0 . In addition, for a jet J, the jet limit J * denotes lim t→∞ J(t) if it exists, in the sense that the sequence becomes constant after a finite time.
The following definition quantifies the interaction between two jets.
Definition 18. For a given stochastic chain {A(t)} ∞ t=0 and any two disjoint jets J u and J v in S, the total interaction between the two jets over the time interval [0, ∞), denoted by U A (J u , J v ), is defined by The next definition captures the idea of long-term interaction between two jets.
Finally, we reproduce the definition of a weak notion of ergodicity.

Some Other Applications
We now briefly discuss a few other applications of our main results.

Multiple Consensus:
We say that multiple consensus [61] occurs whenever lim t→∞ x(t) exists but is not necessarily a multiple of the consensus vector 1, meaning that different entries of x(t) may or may not converge to different limits. An immediate consequence of Theorem 3 above and Theorem 2 of [61] is that multiple consensus always occurs in the continuous-time is an approximately reciprocal chain that satisfies Assumption 1.

2.Éminence Grise Coalitions:
In essence, anéminence grise coalition (EGC, [57]) is a subset of the total agent population that has the ability to steer the opinions of all the individuals in the network to a desired consensus asymptotically in time. A direct consequence of Theorem 3 above and Corollary 3 of [57] is as follows: if {A(t)} t≥0 is an approximately reciprocal chain satisfying Assumption 1, then the size of a minimal EGC of a network with dynamicsẋ(t) = A(t)x(t) is the number of connected components in the infinite flow graph of {A(t)} t≥0 .

Distributed Optimization:
A typical distributed optimization framework consists of a network of n interacting agents with the common objective of minimizing the sum of n convex subject to the constraint that for each i ∈ [n], the function f i is known only to agent i. Notably, [62] provides a continuous-time algorithm for distributed optimization without requiring the associated stochastic chain {A(t)} t≥0 to be cutbalanced [52]. However, the results therein are based on an assumption involving an abstract concept called Class P * flows, the interpretation of which is aided significantly by results such as Theorem 3.

Distributed Learning/Hypothesis Testing:
In a typical distributed learning scenario, there is a set of possible states of the world, of which a subset of states are true. In addition, there is a network of interacting agents whose common objective is to learn the identity of the true state through mutual interaction as well as by performing private measurements on the state of the world. We note that [24] generalizes certain known results on distributed learning to networks described by random, independently distributed time-varying directed graphs. Importantly, the sequence of weighted adjacency matrices of all the networks considered therein are assumed to belong to Class P * . Hence, along with Definitions 13 and 14, Theorem 1 significantly facilitates our interpretation of the main results of [24].

Conclusion
We extended two eigenvector assertions of the classical Perron-Frobenius theorem to sequences as well as continua of row-stochastic matrices that satisfy the mild assumption of strong aperiodicity.
In the process, we established approximate reciprocity as an equivalent characterization of Class P * , a special but broad class of stochastic chains that subsumes a few important sub-classes such as cut-balanced (reciprocal) chains, doubly stochastic chains, and uniformly strongly connected chains [52]. We then discussed a few applications of our main results to problems in distributed learning, averaging with strongly quasi-non-expansive maps, etc. In future, we would like to examine whether it is possible to weaken the assumption of strong aperiodicity while retaining the essence of our main results. We would also like to extend our results to dependent random chains, as such chains are pivotal to the study of real-worlds networks subjected to random phenomena such as communication link failures. As yet another future direction, it would be interesting to attempt an extension of our results to sequences of non-negative matrices that are not necessarily row-stochastic. Such an extension would result in a complete generalization of the eigenvector assertions of the Perron-Frobenius theorem to the case of time-varying matrices.
Nevertheless, we believe that our main results, even in their present form, have the potential to find a significant number of applications other than those discussed above. This belief is rooted in the already wide applicability of the classical theorem.

Proof of Proposition 2
Proof. Consider any set S ⊂ [n], and letS := [n] \ S. Then, there exists a permutation matrix Q such that for all t ∈ N 0 . Let {π(t)} ∞ t=0 denote an absolute probability sequence for {A(t)} ∞ t=0 . Then one may verify that the corresponding absolute probability sequence for {Q T A(t)Q} ∞ t=0 is given by for all t ∈ N 0 . As a result, the following holds for all t ∈ N 0 . π T S (t + 1) π T S (t + 1) The above equation is essentially a pair of two vector equations, the second of which is π T S (t + 1)A SS (t) + π T S (t + 1)AS(t) = π T S (t).
Multiplying each side of this equation by the all-ones vector yields π T S (t + 1)A SS (t)1 + π T S (t + 1)AS(t)1 = π T S (t)1.
On the other hand, the row-stochasticity of A(t) implies that Combining (8) and (9) gives us On transposition, we obtain Since {A(t)} ∞ t=0 ∈ P * , there exists a p * > 0 such that π S (t + 1) ≥ p * 1. Therefore, Now, let k 0 , k 1 ∈ N be any two numbers such that k 0 < k 1 . Then, summing both the sides of (11) over the range t ∈ {k 0 , k 0 + 1, . . . , k 1 − 1} yields where we have used a telescoping sum on the right hand side.
which is the same as (2).
In order to prove this claim, suppose it is false for some s ∈ {0, . . . , n − 1}. Then, there exists a partition {T,T } of [n] such that there is no directed link from any node in T to any node inT in G (s) . This implies that which contradicts the definitions of the times τ 0 , . . . , τ n−1 , thereby proving the claim. Since the

Proof of Theorem 2
Proof. From Theorem 1, we know that {A(t)} ∞ t=0 admits a uniformly positive absolute probability sequence, i.e., {A(t)} ∞ t=0 ∈ P * . As a result, Theorem 4.4 of [52] implies that {A(t)} ∞ t=0 is infinite flow stable. Now, suppose that the infinite flow graph of {A(t)} ∞ t=0 is connected. Since the chain is also infinite flow stable, we know from Lemma 5 that the chain is ergodic. It now follows from Theorem 1 of [63] that {A(t)} ∞ t=0 admits a unique absolute probability sequence. On the other hand, suppose that the infinite flow graph of {A(t)} ∞ t=0 is not connected. Then, by Lemma 3.6 of [52], either there exists an initial condition (t 0 , x(t 0 )) with t 0 ∈ N and x(t 0 ) ∈ R n such that x(t + 1) = A(t)x(t) does not converge to a steady state (Case 1: lim t→∞ x(t) does not exist), or there exist indices i and j such that (i, j) ∈ [n] × [n] and lim sup t→∞ |x i (t) − x j (t)| > 0 (Case 2).
In the first case, we know that lim t→∞ A(t : t 0 ) does not exist (for otherwise, lim t→∞ x(t 0 ) = lim t→∞ A(t : t 0 )x(t 0 ) would exist). Hence, {A(t)} ∞ t=0 is not ergodic. Consider now the second case and suppose that {A(t)} ∞ t=0 is ergodic. Then, for every initial condition (t 0 , x(t 0 )), there exists a π(t 0 ) ∈ R n such that lim t→∞ x(t) = lim t→∞ A(t : t 0 )x(t 0 ) = π T (t 0 )x(t 0 )1, which implies that lim t→∞ x l (t) = lim t→∞ x m (t) for all l, m ∈ [n]. However, this contradicts the hypothesis of Case 2. Hence, {A(t)} ∞ t=0 cannot be ergodic. We have thus shown that if the infinite flow graph of {A(t)} ∞ t=0 is not connected, it is not ergodic. It now follows from Theorem 1 in [63] that if the infinite flow graph of {A(t)} ∞ t=0 is not connected, then the chain does not admit a unique absolute probability sequence.

Proof of Theorem 4
Proof. Observe that by repeating some of the arguments used to prove Theorem 3, we can show that Assumptions 1 and 2 continue to hold (if only with different constants) even if we augment the sequence {t k } ∞ k=0 by inserting into it an arbitrary constant τ ≥ 0. By Lemma 8 of [55], this further implies that the discrete-time chain {Φ(t ′ k+1 : t ′ k )} ∞ k=0 (where {t ′ k } ∞ k=0 denotes the augmented sequence t 1 , t 2 , . . . , τ, . . . ,) is strongly aperiodic. In addition, since {A(t)} t≥0 satisfies the uniform bound assumption (Assumption 1) in addition to the condition of approximate reciprocity, we know from Theorem 3 that {A(t)} t≥0 ∈ P * . By Definitions 10 and 11, this implies that {Φ(t ′ k+1 : t ′ k )} ∞ k=0 ∈ P * in discrete time. Hence, by Theorem 1, {Φ(t ′ k+1 : t ′ k )} ∞ k=0 is approximately reciprocal. Now, the infinite flow graph of {A(t)} t≥0 being connected is equivalent to

Proof of Corollary 3
Proof. We can repeat the arguments used in the proof of Corollary 2 to show that {E[A(t)]} ∞ t=0 ∈ P * . Therefore, this corollary is a straightforward consequence of Theorem 1 above, Lemma 4.2 of [52], and Theorem 5.2 of [52].

Proof of Corollary 4
Proof. The first assertion is a direct consequence of Theorem 1 above and Lemma 2 of [59]. The rest of the assertions follow from Theorem 1 above, from Corollary 3 and Theorem 4 of [59], and from the fact that strong aperiodicity implies weak aperiodicity.