Delft University of Technology
A tutorial on modeling and analysis of dynamic social networks. Part II
Proskurnikov, Anton V.; Tempo, Roberto
DOI
10.1016/j.arcontrol.2018.03.005
Publication date
2018
Document Version
Accepted author manuscript
Published in
Annual Reviews in Control
Citation (APA)
Proskurnikov, A. V., & Tempo, R. (2018). A tutorial on modeling and analysis of dynamic social networks.
Part II.
Annual Reviews in Control
,
45
, 166-190. https://doi.org/10.1016/j.arcontrol.2018.03.005
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A Tutorial on Modeling and Analysis of Dynamic Social Networks. Part II.
✩
Anton V. Proskurnikov
a,b,c,∗
, Roberto Tempo
d
a
Delft Center for Systems and Control, Delft University of Technology, Delft, The Netherlands
b
Institute for Problems of Mechanical Engineering of the Russian Academy of Sciences, St. Petersburg, Russia
c
ITMO University, St. Petersburg, Russia
d
CNR-IEIIT, Politecnico di Torino, Torino, Italy
Abstract
Recent years have witnessed a significant trend towards filling the gap between Social Network Analysis (SNA)
and control theory. This trend was enabled by the introduction of new mathematical models describing
dynamics of social groups, the development of algorithms and software for data analysis and the tremendous
progress in understanding complex networks and multi-agent systems (MAS) dynamics. The aim of this
tutorial is to highlight a novel chapter of control theory, dealing with dynamic mod els of social networks and
processes over them, to the attention of the broad research community. In its first part [1], we have considered
the most classical models of social dynamics, which have anticipated and to a great extent inspired the recent
extensive studies on MAS and complex networks. This paper is the second part of the tutorial, and it is
focused on more recent mod els of social processes that have been developed concurrently with MAS theory.
Future perspectives of control in social and techno-social systems are also discussed.
Keywords: Social network, opinion dynamics, multi-agent systems, distributed algorithms.
1. Introduction
Originating from the early studies on sociome-
try [2, 3], Social Network Analysis (SNA) has quick ly
grown into an interdisciplinary science [4–7] that
has found applications in political sciences [8, 9],
medicine [10], economics [11, 12], crime prevention
and security [13, 14]. The recent breakthroughs in
algorithms and s oftware for big data analysis have
made SNA an efficient tool to study online s ocial
networks and media [15, 16] with millions of users.
The development of SNA has inspired many impor-
tant concepts of mo dern network science [17–20] such
as cliques and communities, centrality measures, re-
silience, graph’s density and clustering coefficient.
Employing many mathematical and algorithmic
tools, SNA has however benefited little from the re-
cent progress in systems and control [21–23]. The
realm of s ocial sciences has remained almost un-
✩
The paper is supported by Russian Science Foundation
(RSF) grant 14-29-00142, hosted by IPME RAS.
∗
Corresponding author
Email address: anton.p.1982@ieee.org (Anton V.
Proskurnikov)
touched by control theory, despite the long-term stud-
ies on social group dynamics [24–26] and “sociocyber-
netics” [27–30]. This gap between SNA and control
can be explained, to a great extent, by the lack of
dynamic models of social processes and mathemati-
cal armamentarium for their analysis. Focusing on
topological properties of networks, SNA and network
science have paid much less attention to dynamics
over th em, except f or some special processes such
as e.g. r an dom walks, branching and queueing pro-
cesses, percolation and contagion dynamics [19, 20].
The recent years have witnessed an important ten-
dency towards fi lling th e gap between SNA and con-
trol theory, enabled by the rapid progress in multi-
agent s y stems and dynamic network s. The emerging
branch of control th eory, stu dying s ocial processes,
is very young and even has no name yet. However,
the interest of sociologists to this new area and un-
derstanding that “coordination and control of social
systems is the foundational problem of sociology” [31]
leaves no doubt that it should become a key instru-
ment to examine social networks and dynamics over
them. Without aiming to prov ide a exhaustive survey
of “social control theory” at its dawn, this tutorial fo-
Preprint submitted to Annual Reviews in Control March 31, 2018
© 2018 Manuscript version made available under CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/
cuses on the most “mature” results, primarily dealing
with mechanisms of opinion formation [31–36].
In the first part of this tutorial [1], the most classi-
cal models of opinion formation have been discussed
that have anticipated and inspired the “boom” in
multi-agent and networked control, witnessed by the
past decades. This paper is the second part of the
tutorial and deals with more recent d y namic mod-
els, taking into account effects of time-varying graphs,
homophily, negative influence, asynchronous interac-
tions and q uantization. The theory of such models
and multi-agent contr ol have been developed concur-
rently, inspiring and reinforcing each other.
Whereas analysis of the classical models addressed
in [1] is mainly based on lin ear algebra and matrix
analysis, th e models discussed in this part of the tu-
torial require more sophisticated and diverse mathe-
matical tools. The page limit makes it impossible to
include th e detailed proofs of all results discussed in
this part of the tutorial; for many of them, we have to
omit the proofs or provide only their brief sketches.
The paper is organized as f ollows. Section 2 intro-
duces preliminary concepts and some notation used
throughout the paper. Section 3 considers basic re-
sults, concerned with properties of the non-stationary
French-DeGroot and Abelson models. In Section 4 we
consider bounded confidence models, wher e the inter-
action graph is opinion-dependent. Section 5 is de-
voted to dynamic models based on asynchronous gos-
siping interactions. Section 6 introduces some mod-
els, exploiting the idea of negative influence. Sec-
tion 7 concludes the tutorial.
2. Preliminaries and notation
In this section we introduce some notation; basic
concepts regarding opinion formation mod eling are
also recollected for the reader’s convenience.
2.1. Notation
We use m : n to den ote the set {m, m + 1, . . . , n}
(here m, n are integer and m ≤ n). Given a vector
x ∈ R
n
, |x| stands for its Euclidean norm |x| =
√
x
⊤
x.
Each non-negative matrix A = (a
ij
)
i,j∈V
corre-
sponds to the weighted graph G[A] = (V, E[A], A),
whose arcs repr esent positive entries of A: a
ij
> 0 if
and only if (j, i) ∈ E(A). Being untypical f or graph
theory (where the entry a
ij
> 0 is encoded by the arc
(i, j)), this notation is convenient in social dynamics
modeling [1] and multi-agent control [37, 38].
Dealing with algorithms’ complexity, we use stan-
dard L an dau-Knuth notation [39]. Given two posi-
tive fun ctions f(n), g(n) of the natural argument n,
g(n) = O(f(n)) stands for the estimate |g(n)| ≤
C|f(n)|, where C is some constant, and f (n ) =
Ω(g(n)) means that lim
n→∞
f(n)/g(n) > 0 (i.e. f (n
k
) ≥
c
0
g(n
k
) for a constant c
0
> 0 and a sequence n
k
→ ∞.
2.2. Agent-based modeling of opinion evolution
From th e sociological viewpoint [31], an in divid-
ual’s op inion stands for his/her cognitive orienta-
tion towards some object (e.g. issue, event, ac-
tion or another individual). Mathematically, opin-
ions are scalar or vector quantities of interest, asso-
ciated with social actors. Depending on the specific
model, opinions may represent signed attitudes [40–
42], subj ective certainties of belief [43, 44] or proba-
bilities [45, 46]. In this tutorial, we deal with models
where opinions can attain a continuum of values and
are represented by real numbers or vectors. Dynam-
ics of real-valued opinions obey ordinary differential
or recurrent equation and are much better studied by
the systems and control community than the evolu-
tion of discrete (finite-valued) opinions. For this rea-
son, many important models w ith finite-valued opin-
ions [47–55] are beyond the scope of this tutorial.
Historically, the first approach to social dynamics
modeling originates from mathematical biology [56–
59], portraying social behaviors as interactions of
multiple “species” or compartments [60]. Dealing
with a social group, a compartment is a subgroup
whose members are featured by some behavior or
hold the s ame position on some issue. Interacting
as indecomposable entities, compartments can grow
or decline. The models describing these processes are
called compartmental and broadly used in mathemat-
ical biology and evolutionary game theory [60–62],
as exemplified by th e SIR/SIS models of epidemic
spread and the Lotka-Volterra predator-prey dynam-
ics. Compartmental models describe how the dis-
tribution of individuals between the compartments
evolves, paying no attention to behaviors of specific
social actors. T his statistical approach is typical for
sociodynamics [63–65], representing the state of a so-
ciety by a point in some configuration space and has
lead to statistical model of opinion formation, de-
scribing how the distribution of opinions evolves over
time. Similar in sp irit to models arising in continuum
mechanics, such models are often referred to as Eu le-
rian [66–68] or continuum [69–71] opinion dynamics.
2
In this tutorial, we focus on agent-based models
of opinion formation, describing how the opinion of
each individual social actor, or agent, evolves under
the influence of the remaining individuals. The col-
lective behavior of a social group is constituted by
the numerous individual behaviors. Such “bottom-
up” models of social dynamics, called also aggrega-
tive [72], are similar in spirit to agent-based models
of self-organization in complex physical and biological
systems [73–75]. Unlike statistical models, adequate
for very large social groups, agent-based models can
describe both small-size and large-scale communities.
Throughout this paper, we deal with a closed com-
munity of n ≥ 2 agents, indexed 1 through n.
2.3. M odels of consensus and Abelson’s puzzle
As have been discussed in the first part of this tuto-
rial [1], the first agent-based model of opinion forma-
tion was introduced by French [76] and later studied
and extended by Harari [77, 78] and DeGroot [45].
The French-DeGroot model describes the discrete-
time evolution of the agents opinions x
1
, . . . , x
n
∈ R,
whose stack vector x(k) = (x
1
(k), . . . , x
n
(k))
⊤
∈ R
n
at step k = 0, 1, . . . obeys the averaging dynamics
x(k + 1) = W x(k), k = 0, 1, . . . (1)
where W = (w
ij
) is a stochastic matrix. The
continuous-time counterpart of (1), proposed by
Abelson [40], is the Laplacian flow dynamics [79]
˙x(t) = −L[A]x(t), t ≥ 0, (2)
where A = (a
ij
) is a non-negative matrix of “contact
rates” and L[A] stands for the Laplacian matrix of the
correspondin g weighted graph [1, 79 ]. The asymp-
totic consensus of opinions appears to be the most
typical behavior of the systems (1) and (2), the rele-
vant criteria are considered in [1]. At the same time,
real social groups often fail to reach consensus and
exhibit clustering of opinions and other “irregular”
behaviors. This has lead Abelson to the fundamental
problem, called the community cleavage prob lem [31]
or Abelson’s diversity puzzle [80]: to find mathemati-
cal models, able to explain these disagreement effects.
The original formulation of Abelson [40] was as fol-
lows: “we are naturally lead to inquire what on earth
one must assume in order to generate the b im odal
outcome of community cleavage studies”.
One reason for community cleavage is the absence
of connectivity: consensus of opinions in the mod-
els (1) and (2) cannot be established when the cor-
respond ing interaction graph G[W ] or G[A] has no
directed spanning tree. Although social networks
are usually dens ely connected [81], they may contain
some “radical” groups [82], closed to social infl uence.
For instance, consensus is not possible in presence
for s everal stubborn individuals (or zealots) [53, 83 ],
whose opinion remains unchanged x
i
≡ x
i
(0). Fur-
ther development of this idea naturally leads [1] to
the Friedkin-Johnsen theory of social influence net-
works [84, 85] with “partially stubborn ” agents.
Stubborness is however not the only factor lead-
ing to the community cleavage; in this part of the
tutorial we consider other models of opinion forma-
tion where op inions can both converge to consensus or
split into several clusters. Many of these models are
based on the ideas, proposed in the seminal Abelson’s
works [40, 72] and extend the classical models (1),(2).
3. T he models by French-DeGroot and Abel-
son wit h time-varying interaction graphs
Non-stationary counterparts of the models (1)
and (2) h ave been thorough ly studied in regard to
consensus and synchronization in multi-agent net-
works. In this tutorial, only some r esults are consid-
ered that directly related to social dynamics; detailed
overview of consensus algorithms can be found e.g. in
the recent mon ographs and surveys [37, 38, 86–90].
3.1. The time-varying French-DeGroot model
We start with a time-varying counterp art of (1),
where W is replaced by a sequence (W (k))
k≥0
x(k + 1) = W (k)x(k), k = 0, 1, . . . (3)
Obviously, all solutions to (3) are bounded and the
sequences min
i
x
i
(k) and max
i
x
i
(k), k = 0, 1, . . ., are
respectively non-decreasing and non-increasing.
As discussed in [1], even for the static case W (k) ≡
W the opinions do not always converge. For instance,
when the graph G[W ] is periodic, the system (1) has a
periodic solution. The convergence problem for time-
varying system (3) still remains a challenge, and u p to
now only sufficient convergence conditions have been
obtained. One of them is given by the following im-
portant result, proved independently in [91–93].
Lemma 1. Let δ > 0 exist such that the sequence of
n × n stochastic matrices (W (k))
k≥0
satisfies at any
time k ≥ 0 the following three conditions:
(a) (non-vanishing couplings) w
ij
(k) ∈ {0} ∪ [δ, 1];
3
(b) (self-confidence) w
ii
(k) ≥ δ ∀i ∈ 1 : n;
(c) (type-symmetry) w
ij
(k) > 0 ⇐⇒ w
ji
(k) > 0.
Then the limit ¯x = lim
k→∞
x(k) exists for any x(0) ∈
R
n
, being an equilibrium point: W (k)¯x = ¯x for suffi-
ciently large k. If agents i and j interact persistently
∞
X
k=0
w
ij
(k) = ∞,
then their limit opinions coincide ¯x
i
= ¯x
j
.
Introducing the undirected graph of persistent in-
teractions G
∞
= (V, E
∞
), whose nodes stand for the
agents and arcs (i, j) connect pairs of persistently in-
teracting agents, the last statement of Lemma 1 can
be formulated as follows: in each connected compo-
nent of G
∞
, the opinions reach consensus.
We give a sketch of the proof of Lemma 1, follow-
ing the ideas f rom [91] and proposed in [94] for more
general systems of recurrent inequalities. It s uffices
to consider the case of connected graph G
∞
. Indeed,
if (i, j) 6∈ E
∞
, then w
ij
(k) > 0 only for finite number
of instants k thanks to condition (a). In other words,
k
0
≥ 0 exists such that w
ij
(k) = 0 for k > k
0
unless i
and j persistently interact. R enumbering the agents,
for k > k
0
the matrix W (k) is block diagonal
W (k) =
W
11
(k) . . . 0
.
.
.
.
.
.
.
.
.
0 . . . W
rr
(k)
,
where the stochastic submatrices W
ii
(k) correspond
to connected components of G
∞
. Hence (3) for k > k
0
is decoupled into r independent systems.
Let G
∞
be connected and j
1
(k), . . . , j
n
(k) be
the permutation of indices, sorting the op inions
x
1
(k), . . . , x
n
(k) in the ascending order, that is,
y
i
(k) = x
j
i
(k)
(k) satisfy the following inequalities
min
i
x
i
(k) = y
1
(k) ≤ y
2
(k) ≤ . . . ≤ y
n
(k) = max
i
x
i
(k).
We are going to prove the following statement: for
any k ≥ 0 and i = 1, . . . , n − 1, there exists k
′
> k
(depending on both k,i), satisfying the inequality
y
i+1
(k
′
) ≤ δy
i
(k) + (1 − δ)y
n
(k), (4)
where δ > 0 is the constant from condition (a).
To prove this, divide the agents into two sets I =
{j
1
(k), . . . , j
i
(k)} and J = {j
i+1
(k), . . . , j
n
(k)}. Since
G
∞
is connected, an arc between I and J should exist,
and hence there exist K > k, such that w
qp
(K) ≥ δ
for some p ∈ I, q ∈ J. Let k
0
stand for the mini-
mal such K. Since x
s
(k) ≤ y
i
(k) for any s ∈ I and
the agents from I and J do not interact at times
k, k + 1, . . . , k
0
− 1, it can be shown that x
r
(k
0
) ≤
y
i
(k) ∀r ∈ I. Also x
r
(k
0
) ≤ y
n
(k
0
) ≤ y
n
(k) ∀r ∈ J
since y
n
(k) is non-increasing in k. For r ∈ I, one has
x
r
(k
0
+ 1) ≤ x
r
(k
0
) +
1−w
rr
(k
0
)≤1−δ
z
}| {
X
l6=r
w
rl
(k
0
) [ x
l
(k
0
)
|
{z}
≤y
n
(k
0
)
−x
r
(k
0
)] ≤
≤ x
r
(k
0
) + (1 − δ)[y
n
(k
0
) − x
r
(k
0
)] ≤
≤ δx
r
(k
0
) + (1 − δ)y
n
(k
0
) ≤ δy
i
(k) + (1 − δ)y
n
(k).
Recalling that w
qp
(k
0
) ≥ δ an d p ∈ I, similarly to the
previous inequality one obtains
x
q
(k
0
+ 1) ≤ w
qp
(k
0
)x
p
(k
0
) + (1 − w
qp
(k
0
))y
n
(k
0
) =
= y
n
(k
0
) − w
qp
(k
0
)[y
n
(k
0
) − x
p
(k
0
)] ≤
≤ δx
p
(k
0
) + (1 − δ)y
n
(k
0
) ≤ δy
i
(k) + (1 − δ)y
n
(k).
Denoting k
′
= k
0
+ 1, for any index ρ ∈ I
′
= I ∪ {q}
one has x
ρ
(k
′
) ≤ δy
i
(k) + (1 − δ)y
n
(k). Since I
′
con-
tains i + 1 different indices, one arrives at (4). Since
y
n
(k) is bounded from below and non -in cr easing, it
converges to a limit y
n
(k) → M
∗
as k → ∞. Pass-
ing to the limit as k → ∞ in (4), the corresponding
sequence k
′
= k
′
(i, k) also tends to ∞ and thus
lim
k→∞
y
i+1
(k) ≤ δ lim
k→∞
y
i+1
(k) + (1 − δ)M
∗
.
Applying this to i = n − 1, one has
M
∗
≤ lim
k→∞
y
n−1
(k) ≤
lim
k→∞
y
n−1
(k) ≤
lim
k→∞
y
n
(k) = M
∗
, and therefore y
n−1
(k) −−−→
k→∞
M
∗
. Iterating this procedure for i = n −2, . . . , 1, one
proves th at y
i
(k) → M
∗
, i.e. consensus of opinions is
established. Obviously, any consensus vector c
1
n
is
an equilibrium point, which finishes the proof.
The convergence of opinions in (3) can be reformu-
lated in terms of matrix products convergence [93].
Corollary 2. Under the assumptions of Lemma 1,
the limit of the matrix products exist
¯
W = lim
k→∞
W (k) . . . W (1)W (0). (5)
Renumbering of the agents,
¯
W is block-diagonal
¯
W =
¯
W
11
. . . 0
.
.
.
.
.
.
.
.
.
0 . . .
¯
W
rr
,
4
