||1 Scientific context
Social networks are a key feature of society. They play a major role in fields such as politics, economics and culture. Best
examples of how social networks are involved
in our daily life include information diffusion, decision making and online reputation systems. Over the past decades, key
features of the topology and dynamics of social
networks have been captured. Yet, little is known regarding the possibility of controlling such systems. The Ph.D. thesis aims at
bringing new theoretical results regarding
the control of these systems. This finds applications in opinion dynamics, participative governance or epidemiology.
A social network can be influenced either as a master regulator (designer of online social networks) or as a simple user. While the
regulator can modify the global structure
of the network, a simple user cannot act on the whole system and is instead limited to its local neighborhood in the network. We
will focus on the user’s point of view,
which presents the most challenging questions. We wonder to which extent and how a small group of users can consistently
modify the global behavior of the social
network. The answer will necessarily be of a decentralized nature. Formally, we will view a social network as a dynamical system.
A state is associated with each user of
the network . Only a part of the users may influence the system. The evolution of each user’s state is governed by its own
dynamics, function of the whole state and the
inputs of the system. In order to be in accordance with the social network context, we make the following assumptions :
• (local dynamics) each user’s dynamics satisfies the decentralized nature of networks : it depends only on the states and
eventually control inputs of his/her neighbors.
• (limited control) only users who may influence the system have an input control.
• (bounded influence) because of the limited capacity of users, the control set is bounded.
The control of this class of systems presents several challenges. It calls for local and bounded control laws. Also, social networks
are large scale systems . Thus, the
proposed methods should be computationally oriented and numerically tractable.
3 Potential methods
The control of multi-agent systems subject to constraints of locality and boundedness has little been explored in the past. One
possibility to design the bounded control
is to use a saturated feedback loop. Results regarding saturated feedback exist in the non-decentralized settings [7, 4], generally
involving Linear Matrix Inequalities
(LMI), but remain to be established in the present context. Moreover, the existing results provide controllability condition in the
form of algebraic equa- tions such as
rank conditions  or controllability grammian. While these conditions can be verified numerically, they are hard to interpret
using social network concepts. Therefore,
an important part of the project is to find ways to translate the obtained algebraic conditions into topological ones, e.g. linking
controllability of the system to the
centrality of the controlled individuals [5, 2]. In order to understand the system dynamics, the Ph.D. thesis will be focused on the
study of small size networks.
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