Sustained oscillations in multi-topic belief dynamics over signed networks Anastasia Bizyaeva Alessio Franci and Naomi Ehrich Leonard Abstract We study the dynamics of belief formation on mul-

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Sustained oscillations in multi-topic belief dynamics over signed networks

Anastasia Bizyaeva, Alessio Franci, and Naomi Ehrich Leonard

Abstract— We study the dynamics of belief formation on mul-

tiple interconnected topics in networks of agents with a shared

belief system. We establish sufﬁcient conditions and necessary

conditions under which sustained oscillations of beliefs arise

on the network in a Hopf bifurcation and characterize the

role of the communication graph and the belief system graph

in shaping the relative phase and amplitude patterns of the

oscillations. Additionally, we distinguish broad classes of graphs

that exhibit such oscillations from those that do not.

I. INTRODUCTION

Having the means to evaluate what can happen when a

group of social agents forms beliefs on a set of related topics

is key to understanding belief propagation in human social

networks and to enabling decentralized decision-making in

teams of robots and other distributed technological systems.

Dynamic models of social belief formation provide a tool for

systematic investigation of belief processes and for principled

design of distributed algorithms for decision-making.

In this paper we investigate conditions under which oscil-

lations emerge in the beliefs of agents in social networks.

Temporal oscillations in attitudes and beliefs may be an

important feature of individual cognition [1]. Oscillations in

beliefs are common in social systems; e.g., periodic swings in

public opinion between more conservative and more liberal

attitudes are characteristic of the American electorate [2].

In a multi-robot problem such as task allocation, it may be

important to reliably promote or avoid oscillations. Design-

ing oscillations will also be necessary for building electronic

circuits with complicated, but well controlled, oscillation

patterns as those needed for neuromorphic applications [3].

However, sustained oscillations are rarely observed in

popular models of belief formation. Classically, formation

of beliefs or opinions on a single topic is modeled as a

discrete-time or continuous-time linear weighted averaging

process on a network [4], [5]. For the multi-topic scenario,

multi-dimensional averaging models have been investigated,

e.g., see [6]–[12]. According to linear models, the beliefs

of agents in a static social network typically converge to an

equilibrium. The study of these models is thus concerned

with characterizing the agents’ beliefs at steady state.

Recently, an alternative modeling paradigm for social

opinion formation was proposed that assumes the belief

Supported by ONR grant N00014-19-1-2556, ARO grant W911NF-18-

1-0325, DGAPA-UNAM PAPIIT grant IN102420, and Conacyt grant A1-

S-10610, and by NSF Graduate Research Fellowship DGE-2039656.

A. Bizyaeva and N.E. Leonard are with the Dept. of Mechanical and

Aerospace Engineering, Princeton University, Princeton, NJ, 08544 USA;

bizyaeva@princeton.edu, naomi@princeton.edu.

A. Franci is with the Dept. of Mathematics, National Autonomous Uni-

versity of Mexico, 04510 Mexico City, Mexico; afranci@ciencias.unam.mx

or opinion update rules of agents to be nonlinear [13],

[14]. The nonlinearity is deceptively simple: each agent

saturates information it accumulates from its social network.

The imposition of a saturating function is a well-motivated

and mild extension of classic averaging models [13], [14].

Despite the simplicity, networked beliefs that follow this non-

linear update rule can have dramatically different properties

from those predicted by classic averaging models, including

sustained oscillations. These nonlinear dynamics are also

general. Beyond opinion formation, they are closely related

to recurrent neural network and neuromorphic electronic

circuit models. To date, analysis of these nonlinear dynamics

focused on characterizing multi-stable equilibria [13]–[15].

In this paper we add to this body of work and present

novel analysis that characterizes the emergence of belief

oscillations and their properties as a function of design

parameters including mixed-sign network structure.

Our main contributions are as follows. 1) We establish

sufﬁcient conditions and necessary conditions for the onset

of stable sustained oscillations in belief dynamics. 2) We

characterize the relative phase and amplitude patterns of the

oscillations in terms of the parameters of the model.

Section II reviews mathematical preliminaries. Section III

introduces the belief dynamics model. Section IV presents

the main results. Classes of graphs that can lead to oscilla-

tions are distinguished from those that cannot in Section V.

Section VI presents numerical examples.

II. MATHEMATICAL PRELIMINARIES

A. Notation

For x=a+ib =reiφ ∈C,x=a−ib =re−iφ is

its complex conjugate, |x|=√xx =rits modulus, and

arg(x)its argument φ. The inner product of vectors v,wis

hw,vi=wTv.0∈RNis the zero vector and diag(v)is

the diagonal matrix with diagonal entries the elements of v.

The spectrum of A∈Rn×nis σ(A) = {λ1, . . . , λn}

and its spectral radius ρ(A) = max{|λi|, λi∈σ(A)}.

The kernel of Ais N(A) = {v∈Rns.t. Av=0}.

An eigenvalue λ∈σ(A)is a leading eigenvalue of Aif

Re(λ)≥Re(µ)for all µ∈σ(A). A leading eigenvalue λ

of Ais a dominant eigenvalue if λ=ρ(A). Given vectors

v,wor matrices M, N, we say vwif vi> wifor all i

and MNif Mij > Nij for all i, j. For matrices M, N ∈

Rm×n, the element-wise Hadamard product MN∈Rm×n

is deﬁned as (MN)ij =Mij Nij . For matrices A=

(aij )∈Rm×n, B = (bij )∈Rl×kthe Kronecker product

arXiv:2210.00353v2 [math.OC] 22 Mar 2023

A⊗B∈Rml×nk is deﬁned as

A⊗B=





a11B . . . a1nB

.....

am1B . . . amnB





.

A real square matrix Ahas the strong Perron-Frobenius

property if it has a unique dominant eigenvalue λ=ρ(A)

satisfying λ≥ |λi|for all λi6=λin σ(A)and its

corresponding eigenvector satisﬁes v0.Ais irreducible

if it cannot be transformed into an upper triangular matrix

through similarity transformations. Ais eventually positive

(eventually nonnegative) if there exists a positive integer k0

such that Ak0N×N(Ak0N×N) for all integers k > k0.

Proposition II.1. [16, Theorem 2.2] The following state-

ments are equivalent for a real square matrix A: (1) A

and AThave the strong Perron-Frobenius property; (2) Ais

eventually positive; (3) ATis eventually positive.

B. Signed graphs

A graph G= (V,E)is a set of nodes V={1, . . . , N}and

a set of edges E. We assume the graph is simple, i.e., there

is at most one edge between any two nodes. The adjacency

matrix A= (aik)of Gsatisﬁes aik = 0 if eik 6∈ E and

aik 6= 0 otherwise. Ais weighted if nonzero entries aik ∈R.

Gis unweighted if aik ∈ {0,1}or signed unweighted if

aik ∈ {0,1,−1}for all i, k ∈ V.Gis undirected whenever

aik =aki for all i, k ∈ V, and directed otherwise.

The in-degree of node ion Gis Pkaik. A path on Gis

a ﬁnite or inﬁnite sequence of edges that joins a sequence

of nodes. Gis strongly connected if there exists a path from

any node to any other node. Gis strongly connected if and

only if Ais an irreducible matrix. A switching matrix Mis

a diagonal matrix with diagonal entries that are either 1or

−1. Two graphs G1,G2with adjacency matrices A1, A2are

switching equivalent whenever A1=MA2M.

C. Hopf bifurcation

Assume without loss of generality that (x, p)=(0,0) is

an equilibrium of a system ˙

x=f(x, p), where xis the state

and pa parameter. Then (0,0) is a Hopf bifurcation point

if it satisﬁes the following: i) The Jacobian Df(0,0) has a

complex conjugate pair of eigenvalues ±iω(0);ii) No other

eigenvalues of Df(0,0) lie on the imaginary axis; iii) Let

λ(p) = r(p)+iω(p),λ(p) = r(p)−iω(p)be the eigenvalues

of Df(x, p)that are smoothly parametrized by pand for

which r(0) = 0; then ∂r

∂p (0,0) 6= 0. We use Lyapunov-

Schmidt reduction methods [17, Chapter VIII] to study the

limit cycles that emerge through a Hopf bifurcation.

III. BELIEF FORMATION MODEL

We study a nonlinear model of Nahomogeneous agents

forming beliefs about Notopics, adapted from [13], [14].

zij ∈Ris the belief of agent iabout topic j. Whenever

zij >0(<0), agent iis in favor of (in opposition to) topic

j, and when zij = 0 it has a neutral belief on the topic. The

magnitude |zij |signiﬁes the strength of commitment to the

belief on topic j. The total belief state of agent iis the vector

Zi= (zi1, . . . , ziNo)∈RNo, and the total network belief

state is Z= (Z1,...,ZNa)∈RNaNo. We say agents iand

kagree (disagree) on topic jif they both form a non-neutral

belief on the topic and sign(zij ) = sign(zkj )(6= sign(zkj )).

Agent iupdates its belief on topic jin continuous time as

˙zij =−d zij +uS1

αzij +γPNa

k=1

k6=i

(Aa)ikzkj 

+PNo

l6=j

l=1

β(Ao)jlzil +δ(Ao)jl PNa

k=1

k6=i

(Aa)ikzkl (1)

where S1, S2:R→Rare bounded saturation functions

satisfying Sr(0) = 0,S0

r(0) = 1,S00

r(0) = 0,S000

r(0) 6= 0,

with an odd symmetry Sr(−y) = −Sr(y)where r∈

{1,2}.S1saturates same-topic information and S2saturates

inter-topic information. These saturations reﬂect that social

network inﬂuence on each topic is bounded, and that an agent

is maximally affected by small changes in its neighbors’

beliefs on a topic when their weighted average is close to

zero. Parameter d > 0represents the agents’ resistance to

forming strong beliefs. Parameter u≥0regulates the amount

of attention agents allocate towards their social interactions,

or their susceptibility to social inﬂuence.

There are two signed directed graphs underlying the belief

formation process. One is the communication graph Ga=

(Va,Ea, sa), with signed adjacency matrix Aa∈RNa×Na.

(Aa)ik = 1 means agent iis cooperative towards agent k,

and (Aa)ik =−1means it is antagonistic towards agent

k. The other is the belief system graph Go= (Vo,Eo, so),

with signed adjacency matrix Ao∈RNo×No. The graph

Goencodes the logical interdependence between different

topics in the set Vo. Whenever (Ao)jl = 1(−1), topic j

is positively (negatively) aligned with topic laccording to

the belief system, and whenever (Ao)jl = 0, topic jis

independent of topic l. In the model (1) we assume that all

agents form beliefs following a single shared belief system.

The gains α, γ, β, δ ≥0regulate the relative strengths

of inﬂuence on beliefs in (1). αis the strength of agent

self-reinforcement of already-held beliefs; βis the strength

of agent internal adherence to the belief system Go.γis

the strength of agent social imitation, i.e. to mimic the

beliefs of neighbors towards whom the agent is cooperative

and to oppose the beliefs of those towards whom it is

antagonistic. δis agent ideological commitment; when δ

is large, agents evaluate their neighbors’ inﬂuence more

holistically according to the belief system Gorather than

through pure imitation along each topic. An illustration of

these four effects and their respective cumulative weights in

the model (1) is shown in Fig. 1.

IV. INDECISION-BREAKING AND OSCILLATIONS

We establish sufﬁcient conditions for the onset of small-

amplitude periodic oscillations in the dynamics of beliefs (1).

The indecision state Z=0in which all agents have neutral

beliefs on all topics is an equilibrium of (1) for all parameter

values. To establish the onset of oscillations we ﬁrst study

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Sustainedoscillationsinmulti-topicbeliefdynamicsoversignednetworksAnastasiaBizyaeva,AlessioFranci,andNaomiEhrichLeonardAbstractWestudythedynamicsofbeliefformationonmul-tipleinterconnectedtopicsinnetworksofagentswithasharedbeliefsystem.Weestablishsufcientconditionsandnecessaryconditionsunderwhichsu...

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Sustained oscillations in multi-topic belief dynamics over signed networks Anastasia Bizyaeva Alessio Franci and Naomi Ehrich Leonard Abstract We study the dynamics of belief formation on mul-

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