Table Of Content(cid:73)
A Tutorial on Modeling and Analysis of Dynamic Social Networks. Part I
Anton V. Proskurnikova,b,c,∗, Roberto Tempod
aDelft Center for Systems and Control, Delft University of Technology, Delft, The Netherlands
bInstitute of Problems of Mechanical Engineering of the Russian Academy of Sciences (IPME RAS), St. Petersburg, Russia
cITMO University, St. Petersburg, Russia
dCNR-IEIIT, Politecnico di Torino, Torino, Italy
Abstract
In recent years, we have observed a significant trend towards filling the gap between social network analysis
7
1and control. This trend was enabled by the introduction of new mathematical models describing dynamics of
0social groups, the advancement in complex networks theory and multi-agent systems, and the development of
2
modern computational tools for big data analysis. The aim of this tutorial is to highlight a novel chapter of
r
control theory, dealing with applications to social systems, to the attention of the broad research community.
a
MThis paper is the first part of the tutorial, and it is focused on the most classical models of social dynamics
and on their relations to the recent achievements in multi-agent systems.
6
Keywords: Social network, opinion dynamics, multi-agent systems, distributed algorithms.
]
Y
S
1. Introduction notes a structure, constituted by social actors (indi-
.
s
vidualsororganizations)andsocial ties amongthem.
c
The 20th century witnessed a crucial paradigm
[ Sociometry has given birth to the interdisciplinary
2 shift in social and behavioral sciences, which can science of Social Network Analysis (SNA) [4–7], ex-
be described as “moving from the description of so-
v tensively using mathematical methods and algorith-
7cial bodies to dynamic problems of changing group
mic tools to study structural properties of social net-
0life”[1]. Unlikeindividualisticapproaches,focusedon
3 works and social movements [8]. SNA is closely
individual choices and interests of social actors, the
6 related to economics [9, 10], political studies [11],
0emerging theories dealt with structural properties of
medicine and health care [12]. The development of
.
1socialgroups, organizationsandmovements, focusing
SNA has inspired many important concepts of mod-
0on social relations (or ties) among their members.
7 ern network theory [13–15] such as e.g. cliques and
A breakthrough in the analysis of social groups
1 communities, centrality measures, small-world net-
:was enabled by introducing a quantitative method
v work, graph’s density and clustering coefficient.
ifor describing social relations, later called sociome-
X On a parallel line of research, Norbert Wiener in-
try [2, 3]. The pioneering work [2] introduced an
r troduced the general science of Cybernetics [16, 17]
aimportant graphical tool of sociogram, that is, “a
with the objective to unify systems, control and in-
graph that visualizes the underlying structure of a
formation theory. Wiener believed that this new sci-
group and the position each individual has within
enceshouldbecomeapowerfultoolinstudyingsocial
it” [2]. The works [2, 3] also broadly used the term
processes, arguing that “society can only be under-
“network”, meaning a group of individuals that are
stood through a study of the messages and communi-
“bound together” by some long-term relationships.
cation facilities which belong to it” [17]. Confirming
Later, the term social network was coined, which de-
Wiener’s ideas, the development of social sciences in
the 20th century has given birth to a new chapter
(cid:73)The paper is supported by Russian Science Foundation of sociology, called “sociocybernetics” [18] and led to
(RSF) grant 14-29-00142 hosted by IPME RAS.
the increasing comprehension that “the foundational
∗Corresponding author
problem of sociology is the coordination and control
Email address: [email protected] (Anton V.
Proskurnikov) of social systems” [19]. However, the realm of social
Preprint submitted to Annual Reviews in Control March 8, 2017
systems has remained almost untouched by modern very young, and its contours are still blurred. With-
control theory in spite of the tremendous progress in out aiming to provide a complete and exhaustive sur-
control of complex large-scale systems [20–22]. vey of this novel area at its dawn, this tutorial fo-
Thegapbetweenthewell-developedtheoryofSNA cuses on the most “mature” dynamic models and on
and control can be explained by the lack of mathe- the most influential mathematical results, related to
maticalmodels, describingsocialdynamics, andtools them. Thesemodelsandresultsaremainlyconcerned
for quantitative analysis and numerical simulation of with opinion formation under social influence.
large-scale social groups. While many natural and This paper, being the first part of the tutorial, in-
engineered networks exhibit “spontaneous order” ef- troducespreliminarymathematicalconceptsandcon-
fects [23] (consensus, synchrony and other regular siders the four models of opinion evolution, intro-
collective behaviors), social communities are often duced in 1950-1990s (but rigorously examined only
featured by highly “irregular” and sophisticated dy- recently): the models by French-DeGroot, Abelson,
namics. Opinions of individuals and actions related Friedkin-Johnsen and Taylor. We also discuss the
to them often fail to reach consensus but rather ex- relations between these models and modern multi-
hibitpersistentdisagreement,e.g. clusteringorcleav- agent control, where some of them have been subse-
age[19]. Thisrequirestodevelopmathematicalmod- quently rediscovered. In the second part of the tuto-
els that are sufficiently “rich” to capture the behav- rial more advanced models of opinion evolution, the
ior of social actors but are also “simple” enough to current trends and novel challenges for systems and
be rigorously analyzed. Although various aspects of control in social sciences will be considered.
“social” and “group” dynamics have been studied in The paper is organized as follows. Section 2 in-
thesociologicalliterature[1,24],mathematicalmeth- troducessomepreliminaryconcepts, regardingmulti-
ods of SNA have focused on graph-theoretic proper- agentnetworks, graphsandmatrices. InSection3we
ties of social networks, paying much less attention to introduce the French-DeGroot model and discuss its
dynamics over them. The relevant models have been relation to multi-agent consensus. Section 4 intro-
mostly confined to very special processes, such as e.g. duces a continuous-time counterpart of the French-
random walks, contagion and percolation [14, 15]. DeGroot model, proposed by Abelson; in this section
The recent years have witnessed an important ten- the Abelson diversity problem is also discussed. Sec-
dency towards filling the gap between SNA and dy- tions 5 and 6 introduce, respectively, the Taylor and
namical systems, giving rise to new theories of Dy- Friedkin-Johnsen models, describing opinion forma-
namical Social Networks Analysis (DSNA) [25] and tion in presence of stubborn and prejudiced agents.
temporal or evolutionary networks [26, 27]. Advance-
ments in statistical physics have given rise to a new 2. Opinions, Agents, Graphs and Matrices
science of sociodynamics [28, 29], which stipulates
analogies between social communities and physical In this section, we discuss several important con-
systems. Besides theoretical methods for analysis of cepts, broadly used throughout the paper.
complex social processes, software tools for big data
analysis have been developed, which enable an inves- 2.1. Approaches to opinion dynamics modeling
tigation of Online Social Networks such as Facebook In this tutorial, we primarily deal with models of
and Twitter and dynamical processes over them [30]. opinion dynamics. As discussed in [19], individu-
Without any doubt, applications of multi-agent als’ opinions stand for their cognitive orientations to-
and networked control to social groups will become a wards some objects (e.g. particular issues, events
key component of the emerging science on dynamic or other individuals), for instance, displayed atti-
social networks. Although the models of social pro- tudes [34–36] or subjective certainties of belief [37].
cesses have been suggested in abundance [19, 29, 31– Mathematically, opinions are just scalar or vector
33], only a few of them have been rigorously ana- quantities associated with social actors.
lyzed from the system-theoretic viewpoint. Even less Upto now, system-theoretic studies onopiniondy-
attention has been paid to their experimental vali- namics have primarily focused on models with real-
dation, which requires to develop rigorous identifica- valued (“continuous”)opinions,whichcanattaincon-
tion methods. A branch of control theory, address- tinuumofvaluesandaretreatedassomequantitiesof
ing problems from social and behavioral sciences, is interest, e.g. subjective probabilities [38, 39]. These
2
modelsobeysystemsofordinarydifferentialordiffer- reader familiar with graph theory and matrix theory
ence equations and can be examined by conventional may skip reading the remainder of this section.
control-theoretictechniques. Adiscrete-valuedscalar Henceforththeterm“graph”strandsforadirected
opinion is often associated with some action or de- graph (digraph), formally defined as follows.
cision taken by a social actor, e.g. to support some
Definition 1. (Graph)AgraphisapairG = (V,E),
movementorabstainfromitandtovotefororagainst
whereV = {v ,...,v }andE ⊆ V×V arefinitesets.
a bill [29, 40–45]. A multidimensional discrete-valued 1 n
The elements v are called vertices or nodes of G and
opinion may be treated as a set of cultural traits [46]. i
the elements of E are referred to as its edges or arcs.
Analysis of discrete-valued opinion dynamics usually
require techniques from advanced probability theory Connections among the nodes are conveniently en-
that are mainly beyond the scope of this tutorial. coded by the graph’s adjacency matrix A = (a ). In
ij
Models of social dynamics can be divided into two graph theory, the arc (i,j) usually corresponds to the
major classes: macroscopic and microscopic models. positive entry a > 0. In multi-agent control [56, 57]
ij
Macroscopic models of opinion dynamics are similar and opinion formation modeling it is however conve-
in spirit to models of continuum mechanics, based on nient1 to identify the arc (i,j) with the entry a > 0.
ji
Euler’s formalism; this approach to opinion model-
ing is also called Eulerian [47, 48] or statistical [40]. Definition 2. (Adjacency matrix) Given a graph
Macroscopic models describe how the distribution of G = (V,E), a nonnegative matrix A = (a ) is
ij i,j∈V
opinions (e.g. the vote preferences on some election adapted to G or is a weighted adjacency matrix for G
or referendum) evolves over time. The statistical ap- if (i,j) ∈ E when a > 0 and (i,j) (cid:54)∈ E otherwise.
ji
proach is typically used in “sociodynamics” [28] and
Definition 3. (Weighted graph) A weighted graph
evolutionary game theory [9, 49] (where the “opin-
is a triple G = (V,E,A), where (V,E) is a graph and
ions” of players stand for their strategies); some of
A is a weighted adjacency matrix for it.
macroscopic models date back to 1930-40s [50, 51].
Microscopic, or agent-based, models of opinion for-
Any graph (V,E) can be considered as a weighted
mation describes how opinions of individual social
graph by assigning to it a binary adjacency matrix
actors, henceforth called agents, evolve. There is
an analogy between the microscopic approach, also (cid:40)
1, (j,i) ∈ E
calledaggregative [52], andtheLagrangianformalism A = (aij)i,j∈V, aij =
0, otherwise.
inmechanics[47]. Unlikestatisticalmodels,adequate
for very large groups (mathematically, the number of On the other hand, any nonnegative matrix A =
agents goes to infinity), agent-based models can de- (a ) is adapted to the unique graph G[A] =
ij i,j∈V
scribe both small-size and large-scale communities. (V,E[A],A). Typically, the nodes are in one-to-one
With the aim to provide a basic introduction to correspondence with the agents and V = {1,...,n}.
social dynamics modeling and analysis, this tutorial
is confined to agent-based models with real-valued Definition 4. (Subgraph) The graph G = (V,E)
scalar and vector opinions, whereas other models are contains the graph G(cid:48) = (V(cid:48),E(cid:48)), or G(cid:48) is a subgraph
either skipped or mentioned briefly. All the models, of G, if ∅ =(cid:54) V(cid:48) ⊆ V and E(cid:48) ⊆ (V(cid:48)×V(cid:48))∩E.
consideredinthispaper,dealwithanidealisticclosed
Simplyspeaking,thesubgraphisobtainedfromthe
community,whichisneitherleftbytheagentsnorcan
graph by removing some arcs and some nodes.
acquire new members. Hence the size of the group,
denoted by n ≥ 2, remains unchanged.
Definition 5. (Walk) A walk of length k connecting
nodeitonodei(cid:48) isasequenceofnodesi ,...,i ∈ V,
2.2. Basic notions from graph theory 0 k
where i = i, i = i(cid:48) and adjacent nodes are con-
0 k
Social interactions among the agents are described
nected by arcs: (i ,i ) ∈ E for any s = 1,...,k. A
s−1 s
by weighted (or valued) directed graphs. We intro-
duceonlybasicdefinitionsregardinggraphsandtheir
1This definition is motivated by consensus protocols and
properties; a more detailed exposition and examples
othermodelsofopiniondynamics,discussedinSections3-6. It
of specific graphs can be found in textbooks on graph
allows to identify the entries of an adjacency matrix with the
theory, networks or SNA, e.g. [4, 9, 10, 53–55]. The influence gains, employed by the opinion formation model.
3
3
1
4
2
(a) (b)
(a) (b)
Figure 1: Examples of graphs: (a) a directed tree with root 4;
(b) a cyclic graph of period 4
Figure3: Strongcomponentsofarootedgraph(a)andagraph
without roots (b)
Any node of a graph is contained in one and only
one strong component. This component may corre-
(a) (b) spond with the whole graph; this holds if and only
if the graph is strong. If the graph is not strongly
Figure 2: A quasi-strongly connected graph (a) and one of its
connected, then it contains two or more strong com-
directed spanning trees (b)
ponents, and at least one of them is closed. A graph
is quasi-strongly connected if and only if this closed
walk from a node to itself is a cycle. A trivial cycle strong component is unique; in this case, any node of
of length 1 is called a self-loop (v,v) ∈ E. A walk this strong component is a root node.
without self-intersections (ip (cid:54)= iq for p (cid:54)= q) is a path. Definition 7 is illustrated by Fig. 3, showing two
graphswiththesamestructureofstrongcomponents.
It can be shown that in a graph with n nodes the
The graph in Fig. 3a has the single root node 4, con-
shortest walk between two different nodes (if such a
stituting its own strong component, all other strong
walk exists) has the length ≤ n−1 and the shortest
components are not closed. The graph in Fig. 3b has
cycle from a node to itself has the length ≤ n.
two closed strong components {4} and {5,6,...,10}.
Definition 6. (Connectivity) A node connected by
walks to all other nodes in a graph is referred to as
2.3. Nonnegative matrices and their graphs
a root node. A graph is called strongly connected or
Inthissubsectionwediscusssomeresultsfromma-
strong if a walk between any two nodes exists (and
trix theory, regarding nonnegative matrices [59–62].
hence each node is a root). A graph is quasi-strongly
connected or rooted if at least one root exists.
Definition 8. (Irreducibility) A nonnegative matrix
A is irreducible if G[A] is strongly connected.
The“minimal”quasi-stronglyconnectedgraphisa
directed tree (Fig. 1a), that is, a graph with only one
Theorem 1. (Perron-Frobenius) The spectral radius
rootnode,fromwhereanyothernodeisreachablevia
ρ(A) ≥ 0 of a nonnegative matrix A is an eigenvalue
only one walk. A directed spanning tree in a graph G
of A, for which a real nonnegative eigenvector exists
is a directed tree, contained by the graph G and in-
cludingallofitsnodes(Fig.2b). Itcanbeshown[56] Av = ρ(A)v for some v = (v ,...,v )(cid:62) (cid:54)= 0, v ≥ 0.
1 n i
that a graph has at least one directed spanning tree
if and only if it is quasi-strongly connected. Nodes of If A is irreducible, then ρ(A) is a simple eigenvalue
a graph without directed spanning tree are covered and v is strictly positive v > 0∀i.
i
by several directed trees, or spanning forest [58].
Obviously, Theorem 1 is also applicable to the trans-
Definition 7. (Components) A strong subgraph G(cid:48) posed matrix A(cid:62), and thus A also has a left nonneg-
of the graph G is called a strongly connected (or ative eigenvector w(cid:62), such that w(cid:62)A = ρ(A)w(cid:62).
strong)component,ifitisnotcontainedbyanylarger Besides ρ(A), a nonnegative matrix can have other
strongsubgraph. Astrongcomponentthathasnoin- eigenvalues λ of maximal modulus |λ| = ρ(A). These
coming arcs from other components is called closed. eigenvalueshavethefollowingproperty[62, Ch.XIII].
4
Lemma 2. If A is a nonnegative matrix and λ ∈ C Definition 11. (Stochasticity and substochasticity)
is its eigenvalue with |λ| = ρ(A), then the algebraic A nonnegative matrix A (not necessarily square) is
(cid:80)
and geometric multiplicities of λ coincide (that is, all calledstochastic ifallitsrowssumto1(i.e. a =
j ij
Jordan blocks corresponding to λ are trivial). 1∀i) and substochastic if the sum of each row is no
(cid:80)
greater than 1 (i.e. a ≤ 1∀i).
j ij
For an irreducible matrix, the eigenvalues of max-
imal modulus are always simple and have the form The Gershgorin Disc Theorem [60, Ch. 6] implies
ρ(A)e2πmi/h, where h ≥ 1 is some integer and m = that ρ(A) ≤ 1 for any square substochastic matrix
0,1,...,h−1. This fundamental property is proved A. If A is stochastic then ρ(A) = 1 since A has an
e.g. in [60, Sections 8.4 and 8.5] and [61, Section 8.3]. eigenvector of ones 1 =∆ (1,...,1)(cid:62): A1 = 1. A sub-
stochastic matrix A, as shown in [63], either is Schur
Theorem 3. Letan irreduciblematrixAhaveh ≥ 1
different eigenvalues λ ,...,λ on the circle {λ ∈ C : stable or has a stochastic submatrix A(cid:48) = (aij)i,j∈V(cid:48),
1 h
where V(cid:48) ⊆ {1,...,n}; an irreducible substochastic
|λ| = ρ(A)}. Then, the following statements hold:
matrix is either stochastic or Schur stable [61, 63].
1. each λ has the algebraic multiplicity 1;
i
2. {λ ,...,λ } are roots of the equation λh = rh; 2.4. M-matrices and Laplacians of weighted graphs
1 h
3. if h = 1 then all entries of the matrix Ak are In this subsection we introduce the class of M-
strictly positive when k is sufficiently large; matrices2 [59, 61] that are closely related to nonneg-
4. if h > 1, the matrix Ak may have positive diag- ative matrices and have some important properties.
onal entries only when k is a multiple of h.
Definition 12. (M-matrix) A square matrix Z is an
M-matrix if it admits a decomposition Z = sI −A,
Definition 9. (Primitivity) An irreducible nonneg-
where s ≥ ρ(A) and the matrix A is nonnegative.
ative matrix A is primitive if h = 1, i.e. λ = ρ(A) is
the only eigenvalue of maximal modulus; otherwise,
For instance, if A is a substochastic matrix then
A is called imprimitive or cyclic.
Z = I −A is an M-matrix. Another important class
of M-matrices is given by the following lemma.
It can be shown via induction on k = 1,2,... that
if A is a nonnegative matrix and B = (bij) = Ak, Lemma 5. Let Z = (zij) satisfies the following two
then bij > 0 if and only if in G[A] there exists a walk conditions: 1) zij ≤ 0 when i (cid:54)= j; 2) zii ≥ (cid:80)j(cid:54)=i|zij|.
of length k from j to i. In particular, the diagonal Then, Z is an M-matrix; precisely, A = sI − Z is
entry (Ak)ii is positive if and only if a cycle of length nonnegative and ρ(A) ≤ s whenever s ≥ maxizii.
kfromnodeitoitselfexists. Hence,cyclicirreducible
matrices correspond to periodic strong graphs. Indeed,ifs ≥ maxizii thenA = sI−Z isnonnegative
(cid:80)
and ρ(A) ≤ max (s−z + |z |) ≤ s thanks to
i ii j(cid:54)=i ij
Definition 10. (Periodicity) A graph is periodic if the Gershgorin Disc Theorem [60].
it has at least one cycle and the length of any cycle is NoticingthattheeigenvaluesofZ andAareinone-
divided by some integer h > 1. The maximal h with to-onecorrespondenceλ (cid:55)→ s−λandusingTheorem1
such a property is said to be the period of the graph. and Lemma 2, one arrives at the following result.
Otherwise, a graph is called aperiodic.
Corollary 6. Any M-matrix Z = sI −A has a real
Thesimplestexampleofaperiodicgraphisacyclic eigenvalue λ = s − ρ(A) ≥ 0, whose algebraic and
0
graph (Fig. 1b). Any graph with self-loops is aperi- geometric multiplicities coincide. For this eigenvalue
odic. Theorem 3 implies the following corollary. there exist nonnegative right and left eigenvectors v
and p: Zv = λ v, p(cid:62)Z = λ p(cid:62). These vectors are
0 0
Corollary 4. An irreducible matrix A is primitive
positive if the graph G[−Z] is strongly connected. For
if and only if G[A] is aperiodic. Otherwise, G[A] is
any other eigenvalue λ one has Reλ > λ , and hence
0
periodic with period h, where h > 1 is the number of
Z is non-singular (detZ (cid:54)= 0) if and only if s > ρ(A).
eigenvalues of the maximal modulus ρ(A).
Many models of opinion dynamics employ stochas- 2The term “M-matrix” was suggested by A. Ostrowski in
tic and substochastic nonnegative matrices. honorofMinkowski,whostudiedsuchmatricesinearly1900s.
5
Non-singular M-matrices are featured by the fol- goal of French, however, was not to study consensus
lowing important property [59, 61]. and learning mechanisms but rather to find a mathe-
matical model for social power [68, 74, 75]. An indi-
Lemma 7. Let Z = sI − A be a non-singular M-
vidual’ssocialpowerinthegroupishis/herabilityto
matrix, i.e. s > ρ(A). Then Z−1 is nonnegative.
control the group’s behavior, indicating thus the cen-
Anexampleofasingular M-matrixistheLaplacian trality of the individual’s node in the social network.
(orKirchhoff)matrixofaweightedgraph[54,64,65]. French’s work has thus revealed a profound relation
between opinion formation and centrality measures.
Definition 13. (Laplacian) Given a weighted graph
G = (V,E,A), its Laplacian matrix is defined by 3.1. TheFrench-DeGrootmodelofopinionformation
The French-DeGroot model describes a discrete-
−aij, i (cid:54)= j time process of opinion formation in a group of n
L[A] = (lij)i,j∈V, where lij = (cid:80)a , i = j. (1) agents, whose opinions henceforth are denoted by
ij
j(cid:54)=i x ,...,x . First we consider the case of scalar opin-
1 n
ions x ∈ R. The key parameter of the model
The Laplacian is an M-matrix due to Lemma 5. i
is a stochastic n × n matrix of influence weights
Obviously, L[A] has the eigenvalue λ = 0 since
0
W = (w ). The influence weights w ≥ 0, where
L[A]1 = 0, where n is the dimension of A. The zero ij ij
n
j = 1,...,n may be considered as some finite re-
eigenvalue is simple if and only if the graph G[A] has
source, distributed by agent i to self and the other
a directed spanning tree (quasi-strongly connected).
agents. Given a positive influence weight w > 0,
ij
Lemma 8. For an arbitrary nonnegative square ma- agent j is able to influence the opinion of agent i at
trix A the following conditions are equivalent each step of the opinion iteration; the greater weight
is assigned to agent j, the stronger is its influence
1. 0 is an algebraically simple eigenvalue of L[A];
on agent i. Mathematically, the vector of opinions
2. if L[A]v = 0, v ∈ Rn then v = c1 for some c ∈
n x(k) = (x (k),...,x (k))(cid:62) obeys the equation
R (e.g. 0 is a geometrically simple eigenvalue); 1 n
3. the graph G[A] is quasi-strongly connected. x(k+1) = Wx(k), k = 0,1,.... (2)
which is equivalent to the system of equations
The equivalence of statements 1 and 2 follows from
n
Corollary 6. The equivalence of statements 2 and 3 (cid:88)
x (k+1) = w x (k), ∀i k = 0,1,.... (3)
was in fact proved in [34] and rediscovered in recent i ij j
j=1
papers [66, 67]. A more general relation between the
Hence w is the contribution of agent j’s opinion at
kernel’s dimension dimkerL[A] = n−rankL[A] and ij
thegraph’sstructurehasbeenestablished3 in[58,65]. each step of the opinion iteration to the opinion of
agent i at its next step. The self-influence weight
w ≥ 0 indicates the agent’s openness to the as-
ii
3. The French-DeGroot Opinion Pooling
similation of the others’ opinions: the agent with
One of the first agent-based models4 of opinion wii = 0 is open-minded and completely relies on the
formation was proposed by the social psychologist others’opinions, whereastheagentwithwii = 1(and
French in his influential paper [68], binding together wij = 0∀j (cid:54)= i) is a stubborn or zealot agent, “an-
SNA and systems theory. Along with its generaliza- chored” to its initial opinion xi(k) ≡ xi(0).
tion, suggested by DeGroot [38] and called “iterative More generally, agent’s opinions may be vectors
opinion pooling”, this model describes a simple pro- of dimension m, conveniently represented by rows
cedure, enabling several rational agents to reach con- xi = (xi1,...,xim). Stackingtheserowsontopofone
sensus [69–71]; it may also be considered as an algo- another, one obtains an opinion matrix X = (xil) ∈
rithm of non-Bayesian learning [72, 73]. The original Rn×m. The equation (2) should be replaced by
X(k+1) = WX(k), k = 0,1,.... (4)
3As discussed in [55, Section 6.6], the first studies on the Every column x (k) = (x (k),...,x (k))(cid:62) ∈ Rn
i 1i ni
Laplacian’srankdatebackto1970sandweremotivatedbythe
of X(k), obviously, evolves in accordance with (2).
dynamics of compartmental systems in mathematical biology.
4As was mentioned in Section 2, a few statistical models of Henceforth the model (4) with a general stochastic
social systems had appeared earlier, see in particular [50, 51]. matrixW isreferredtoastheFrench-DeGroot model.
6
3.2. History of the French-DeGroot model program [79]. Toobtainasimpleralgorithmofreach-
Aspecialcaseofthemodel(2)hasbeenintroduced ing consensus, a heuristical algorithm was suggested
by French in his seminal paper [68]. This paper first in [80], replacing the convex optimization by a very
introduces a graph G, whose nodes correspond to the simple procedure of weighted averaging, or opinion
agents; it is assumed that each node has a self-loop. pooling [81]. Developing this approach, the proce-
An arc (j,i) exists if agent j’s opinion is displayed to dure of iterative opinion pooling (4) was suggested
agent i, or j “has power over” i. At each stage of the in [38]. Unlike [77, 80], the DeGroot procedure was a
opinion iteration, an agent updates its opinion to the decentralized algorithm: each agent modifies its opin-
mean value of the opinions, displayed to it, e.g. the ion independently based on the opinions of several
weightedgraphinFig.4correspondstothedynamics “trusted” individuals, and there may be no agent
aware of the opinions of the whole group. Unlike the
x (k+1) 1/2 1/2 0 x (k)
1 1 French model [68], the matrix W can be an arbitrary
x2(k+1) = 1/3 1/3 1/3x2(k). (5) stochastic matrix and the opinions are vector-valued.
x (k+1) 0 1/2 1/2 x (k)
3 3
3.3. Algebraic convergence criteria
Obviously, the French’s model is a special case of
In this subsection, we discuss convergence proper-
equation (2), where the matrix W is adapted to the
ties of the French-DeGroot model (2); the properties
graph G and has positive diagonal entries; further-
for the multidimensional model (4) are the same.
more, in each row of W all non-zero entries are equal.
A straightforward computation shows that the dy-
Hence each agent uniformly distributes influence be-
namics (2) is “non-expansive” in the sense that
tween itself and the other nodes connected to it.
minx (0) ≤ ··· ≤ minx (k) ≤ minx (k+1),
i i i
i i i
maxx (0) ≥ ··· ≥ maxx (k) ≥ maxx (k+1)
i i i
i i i
for any k = 0,1,.... In particular, the system (2)
Figure 4: An example of the French model with n=3 agents is always Lyapunov stable5, but this stability is not
asymptotic since W always has eigenvalue at 1.
French formulated without proofs several condi- The first question, regarding the model (4), is
tions for reaching a consensus, i.e. the convergence whether the opinions converge or oscillate. A more
x (k) −−−→ x of all opinions to a common “unani- specific problem is convergence to a consensus [38].
i ∗
k→∞
mous opinion” [68] that were later corrected and rig- Definition 14. (Convergence)Themodel(2)iscon-
orously proved by Harary [53, 76]. His primary inter- vergent ifforanyinitialconditionx(0)thelimitexists
est was, however, to find a quantitative characteris-
x(∞) = lim x(k) = lim Wkx(0). (6)
tics of the agent’s social power, that is, its ability to
k→∞ k→∞
influencethegroup’scollectiveopinionx (theformal
∗ A convergent model reaches a consensus if x (∞) =
definition will be given in Subsect. 3.5). 1
... = x (∞) for any initial opinion vector x(0).
A general model (4), proposed by DeGroot [38], n
takes its origin in applied statistics and has been sug- The convergence and consensus in the model (2)
gested as a heuristic procedure to find “consensus of are equivalent, respectively, to regularity and full reg-
subjective probabilities” [77]. Each of n agents (“ex- ularity6 of the stochastic matrix W.
perts”) has a vector opinion, standing for an individ-
Definition 15. (Regularity) We call the matrix W
ual (“subjective”) probability distribution of m out-
regular if the limit W∞ = lim Wk exists and fully
comes in some random experiment; the experts’ goal k→∞
regular if, additionally, the rows of W∞ are identical
is to “form a distribution which represents, in some
(that is, W∞ = 1 p(cid:62) for some p ∈ Rn).
sense, a consensus of individual distributions” [77]. n ∞ ∞
This distribution was defined in [77] as the unique
Nash equilibrium [78] in a special non-cooperative 5This also follows from Lemma 2 since ρ(W)=1.
6Our terminology follows [62]. The term “regular matrix”
“Pari-Mutuel” game (betting on horse races), which
sometimes denotes a fully regular matrix [82] or a primitive
can be found by solving a special optimization prob-
matrix [83]. Fully regular matrices are also referred to as SIA
lem, referred now to as the Eisenberg-Gale convex (stochasticindecomposableaperiodic)matrices[56,57,84,85].
7
Lemma 2 entails the following convergence criterion. As shown in the next subsection, Theorem 12 can
be derived from the standard results on the Markov
Lemma 9. [62, Ch.XIII] The model (2) is conver-
chains convergence [82], using the duality between
gent(i.e. W isregular)ifandonlyifλ = 1istheonly
Markov chains and the French-DeGroot opinion dy-
eigenvalue of W on the unit circle {λ ∈ C : |λ| = 1}.
namics. Theorem12hasanimportantcorollary,used
The model (2) reaches consensus (i.e. W is fully reg-
in the literature on multi-agent consensus.
ular) if and only if this eigenvalue is simple, i.e. the
corresponding eigenspace is spanned by the vector 1. Corollary 13. Let the agents’ self-weights be posi-
tive w > 0∀i. Then, the model (2) is convergent.
Using Theorem 3, Lemma 9 implies the equivalence ii
It reaches a consensus if and only if G[W] is quasi-
of convergence and consensus when W is irreducible.
strongly connected (i.e. has a directed spanning tree).
Lemma 10. For an irreducible stochastic matrix W
the model (2) is convergent if and only if W is prim- It should be noted that the existence of a directed
itive, i.e. Wk is a positive matrix for large k. In this spanningtreeisingeneralnot sufficientforconsensus
case consensus is also reached. in the case where W has zero diagonal entries. The
second part of Corollary 13 was proved8 in [76] and
Since an imprimitive irreducible matrix W has
included, without proof, in [53, Chapter 4]. Numer-
eigenvalues {e2πki/h}h−1, where h > 1, for almost all7
k=0 ous extensions of this result to time-varying matrices
initial conditions the solution of (2) oscillates.
W(k) [56, 86–89] and more general nonlinear consen-
3.4. Graph-theoretic conditions for convergence sus algorithms [90, 91] have recently been obtained.
Some time-varying extensions of the French-DeGroot
For large-scale social networks, the criterion from
model, namely, bounded confidence opinion dynam-
Lemma 9 cannot be easily tested. In fact, conver-
ics [92] and dynamics of reflected appraisal [93] will
gence of the French-DeGroot model (2) does not de-
be discussed in Part II of this tutorial.
pendontheweightsw ,butonlyonthegraphG[W].
ij
In this subsection, we discuss graph-theoretic condi-
3.5. The dual Markov chain and social power
tions for convergence and consensus. Using Corol-
Notice that the matrix W may be considered as
lary 4, Lemma 10 may be reformulated as follows.
a matrix of transition probabilities of some Markov
Lemma 11. If the graph G = G[W] is strong, then
chainwithnstates. Denotingbyp (t)theprobability
i
the model (2) reaches a consensus if and only if G of being at state i at time t, the row vector p(cid:62)(t) =
is aperiodic. Otherwise, the model is not convergent
(p (t),...,p (t)) obeys the equation
1 n
and opinions oscillate for almost all x(0).
p(k+1)(cid:62) = p(k)(cid:62)W, t = 0,1,... (7)
Considering the general situation, where G[W] has
more than one strong component, one may easily no- The convergence of (2), that is, regularity of W im-
tice that the evolution of the opinions in any closed plies that the probability distribution converges to
strong component is independent from the remaining the limit p(∞)(cid:62) = lim p(k)(cid:62) = p(0)(cid:62)W∞. Con-
k→∞
network. Two different closed components obviously sensus in (2) implies that p(∞) = p , where p is
∞ ∞
cannotreachconsensusforageneralinitialcondition. the vector from Definition 15, i.e. the Markov chain
This implies that for convergence of the opinions “forgets” its history and convergence to the unique
it is necessary that all closed strong components are stationary distribution. Such a chain is called reg-
aperiodic. For reaching a consensus the graph G[W] ular or ergodic [62, 94]. The closed strong com-
shouldhavethe only closedstrongcomponent(i.e. be ponents in G[W] correspond to essential classes of
quasi-strongly connected), which is aperiodic. Both states, whereas the remaining nodes correspond to
conditions, in fact, appear to be sufficient. inessential (or non-recurrent) states [94]. The stan-
Theorem 12. [10, 86] The model (2) is convergent dard ergodicity condition is that the essential class is
if and only if all closed strong components in G[W] unique and aperiodic, which is in fact equivalent to
are aperiodic. The model (2) reaches a consensus if thesecondpartofTheorem12. ThefirstpartofThe-
and only if G[W] is quasi-strongly connected and the orem 12 states another known fact [94]: the Markov
only closed strong component is aperiodic.
8Formally,[53,76]addressonlytheFrenchmodel,however,
7“Almost all” means “all except for a set of zero measure”. the proof uses only the diagonal entries’ positivity w >0∀i.
ii
8
chain always converges to a stationary distribution if
and only if all essential classes are aperiodic.
Assuming that W is fully regular, one notices that
x(k+1) = Wkx(0) −−−→ (p(cid:62)∞x(0))1n. (8) Figure 5: The graph of the French-DeGroot model with two
k→∞
stubborn agents (source nodes) 1 and 3.
The element p can thus be treated as a measure
∞i
of social power of agent i, i.e. the weight of its initial
then the opinions reach a consensus (the source node
opinion x (0) in the final opinion of the group. The
i is the only closed strong component of the graph).
greater this weight is, the more influential is the ith
If more than one stubborn agent exist (i.e. G[W]
individual’s opinion. A more detailed discussion of
has several sources), then consensus among them is,
social power and social influence mechanism is pro-
obviously, impossible. Theorem 12 implies, however,
videdin[68,75]. Thesocialpowermaybeconsidered
that typically the opinions in such a group converge.
as a centrality measure, allowing to identify the most
“important” (influential) nodes of a social network. Corollary 14. Let the group have s ≥ 1 stubborn
This centrality measure is similar to the eigenvector agents, influencing all other individuals (i.e. the set
centrality [95], which is defined as the left eigenvec- of source nodes is connected by walks to all other
tor of the conventional binary adjacency matrix of nodes of G[W]). Then the model (2) is convergent.
a graph instead of the “normalized” stochastic adja-
cency matrix. Usually centrality measures are intro- Indeed, source nodes are the only closed strong com-
duced as functions of the graph topology [96] while ponents of G[W], which are obviously aperiodic.
theirrelationstodynamicalprocessesovergraphsare In Section 6 it will be shown that under the as-
not well studied. French’s model of social power in- sumptions of Corollary 14 the final opinion x(∞) is
troduces a dynamic mechanism of centrality measure fully determined by the stubborn agents’ opinions9.
and a decentralized algorithm (7) to compute it.
Example 2. Consider the French-DeGroot model,
Example 1. Consider the French model with n =
corresponding to the weighted graph in Fig. 5
3 agents (5), corresponding to the graph in Fig. 4.
Onecanexpectthatthe“central”node2corresponds x (k+1) 1 0 0x (k)
1 1
to the most influential agent in the group. This is x2(k+1) = 13 13 31x2(k).
confirmed by a straightforward computation: solving x (k+1) 0 0 1 x (k)
3 3
the system of equations p(cid:62) = p(cid:62)W and p(cid:62)1 = 1,
∞ ∞ ∞
oneobtainsthevectorofsocialpowersp(cid:62) = (2, 3, 2). Itcanbeshownthatthesteadyopinionvectorofthis
∞ 7 7 7
model is x(∞) = (x (0),x (0)/2+x (0)/2,x (0))(cid:62).
1 1 3 3
3.6. Stubborn agents in the French-DeGroot model
Although consensus is a typical behavior of the 4. Abelson’s Models and Diversity Puzzle
model (2), there are situations when the opinions do
not reach consensus but split into several clusters. In his influential work [34] Abelson proposed a
One of the reasons for that is the presence of stub- continuous-time counterpart of the French-DeGroot
born agents (called also radicals [97] or zealots [98]). model (2). Besides this model and its nonlinear ex-
tensions, he formulated a key problem in opinion for-
Definition 16. (Stubbornness) An agent is said to
mationmodeling, referredtoasthecommunity cleav-
be stubborn if its opinion remains unchanged inde-
age problem [19] or Abelson’s diversity puzzle [99].
pendent of the others’ opinions.
If the opinions obey the model (2) then agent i is 9ThisfactcanalsobederivedfromtheMarkovchaintheory.
stubborn x (k) ≡ x (0) if and only if w = 1. Such In the dual Markov chain (7), stubborn agents correspond to
i i ii
an agent corresponds to a source node in a graph absorbing states. TheconditionfromCorollary14impliesthat
all other states of the chain are non-recurrent, i.e. the Markov
G[W], i.e. a node having no incoming arcs but for
chain is absorbing [83] and thus arrives with probability 1 at
the self-loop (Fig. 5). Theorem 12 implies that if
one of the absorbing states. Thus the columns of the limit
G[W] has the only source, being also a root (Fig. 3a), matrix W∞, corresponding to non-stubborn agents, are zero.
9
4.1. Abelson’s models of opinion dynamics 4.2. Convergence and consensus conditions
To introduce Abelson’s model, we first consider Note that Corollary 6, applied to the M-matrix
an alternative interpretation of the French-DeGroot L[A] and λ = 0, implies that all Jordan blocks, cor-
0
(cid:80)
model(2). Recallingthat1−w = w , onehas responding to the eigenvalue λ = 0, are trivial and
ii j(cid:54)=i ij 0
for any other eigenvalue λ of the Laplacian L[A] one
(cid:88)
xi(k+1)−xi(k) = wij[xj(k)−xi(k)] ∀i. (9) has Reλ > 0. Thus, the model (12) is Lyapunov
(cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125)
j(cid:54)=i stable (yet not asymptotically stable) and, unlike the
∆xi(k) ∆(j)xi(k)
French-DeGroot model, is always convergent.
The experiments with dyadic interactions (n = 2)
Corollary 15. For any nonnegative matrix A the
show that “the attitude positions of two discussants
limit P∞ = lim e−L[A]t exists, and thus the vector of
... move toward each other” [52]. The equation (9)
t→∞
stipulates that this argument holds for simultaneous opinions in (12) converges x(t) −−−→ x∞ = P∞x(0).
t→∞
interactions of multiple agents: adjusting its opinion
The matrix P∞ is a projection operator onto the
x (k) by ∆ x (k), agent i shifts it towards x (k) as
i (j) i j
Laplacian’snullspacekerL[A] = {v : L[A]v = 0}and
x(cid:48) = x +∆ x =⇒ |x −x(cid:48)| = (1−w )|x −x |. is closely related to the graph’s structure [58, 102].
i i (j) i j i ij j i
Similar to the discrete-time model (2), the sys-
The increment in the ith agent’s opinion ∆x (k) is
i tem (12) reaches a consensus if the final opinions co-
the “resultant” of these simultaneous adjustments.
incide x∞ = ... = x∞ for any initial condition x(0).
Supposing that the time elapsed between two steps 1 n
Obviously, consensus means that kerL[A] is spanned
of the opinion iteration is very small, the model (9)
by the vector 1 , i.e. P∞ = 1 p(cid:62), where p ∈ Rn is
can be replaced by the continuous-time dynamics n n ∞
some vector. By noticing that x = 1 is an equilib-
n
(cid:88) rium point, one has P1 = 1 and thus p(cid:62)1 = 1.
x˙i(t) = aij(xj(t)−xi(t)), i = 1,...,n. (10) n n ∞ n
Since P commutes with L[A], it can be easily shown
j(cid:54)=i
that p(cid:62)L[A] = 0. Recalling that L[A] has a nonneg-
∞
Here A = (aij) is a non-negative (but not necessarily ative left eigenvector p such that p(cid:62)L[A] = 0 due to
stochastic) matrix of infinitesimal influence weights
Corollary 6 and dimkerL[A] = 1, one has p = cp,
∞
(or “contact rates” [34, 52]). The infinitesimal shift
where c > 0. Combining this with Lemma 8, one
oftheithopiniondx (t) = x˙ (t)dtisthesuperposition
i i obtains the following consensus criterion.
oftheinfinitesimal’sshiftsa (x (t)−x (t))dtofagent
ij j i
i’s towards the influencers. A more general nonlinear Theorem 16. The linear Abelson model (12)
mechanism of opinion evolution [34, 35, 52] is reaches consensus if and only if G[A] is quasi-
strongly connected (i.e. has a directed spanning tree).
(cid:88)
x˙ (t) = a g(x ,x )(x (t)−x (t)) ∀i. (11) In this case, the opinions converge to the limit
i ij i j j i
j(cid:54)=i
lim x (t) = ... = lim x (t) = p(cid:62)x(0),
1 n ∞
Hereg : R×R → (0;1]isacouplingfunction,describ- t→∞ t→∞
ingthecomplexmechanismofopinionassimilation10. where p ∈ Rn is the nonnegative vector, uniquely
∞
Inthissection,wemainlydealwiththelinear Abel- defined by the equations p(cid:62)L[A] = 0 and p(cid:62)1 = 1.
∞ ∞ n
son model (10), whose equivalent matrix form is
Similar to the French-DeGroot model, the vector p
∞
x˙(t) = −L[A]x(t), (12) may be treated as a vector of the agents’ social pow-
ers, or a centrality measure on the nodes of G[A].
where L[A] is the Laplacian matrix (1). Recently the
It is remarkable that a crucial part of Theorem 16
dynamics (12) has been rediscovered in multi-agent
wasprovedbyAbelson[34],whocalledquasi-strongly
controltheory[56,100,101]asacontinuous-timecon-
connected graphs “compact”. Abelson proved that
sensus algorithm. We discuss the convergence prop-
the null space kerL[A] consists of the vectors c1 if
n
erties of this model in the next subsection.
and only if the graph is “compact”, i.e. statements 2
and 3 in Lemma 8 are equivalent. He concluded that
10The reasons to consider nonlinear couplings between the
“compactness” is necessary and sufficient for consen-
individualsopinions(attitudes)andpossibletypesofsuchcou-
sus; the proof, however, was given only for diagonal-
plingsarediscussedin[35]. Manydynamicmodels,introduced
in [35], are still waiting for a rigorous mathematical analysis. izable Laplacian matrices. In general, the sufficiency
10
