Gabriel Vasile
email: gabriel.vasile@gipsa-lab.grenoble-inp.fr

Anne Guy D'urso
email: guy.durso@edf.fr

Eric Goarant

De Oliveira

Elena Lungu
email: elungu@sigintec.fr

Potential of Active Ultrasound Monitoring Systems for Jellyfish Detection

This paper investigates the potential of active ultrasound jellyfish monitoring systems to continuously detect, quantify and identify different species in turbulent underwater environments. The experimentations carried out in December 2015 at the Aquarium de Paris, France, are used to propose an unified acoustic signal processing framework for visualizing jellyfish passages.

I. INTRODUCTION

A fast increase in jellyfish population can be one important indicator of the recent climate changes in seawater. This may be due to the effect of anthropic activity, such as over fishing, global warming, pollution, or coastal development [START_REF] Brotz | Increasing jellyfish populations: trends in large marine ecosystems[END_REF].

Such clogging organism, like jellyfish, can pose a significant threat to power plant cooling water intake structures in the context of nuclear power production. In sufficient quantities, these organism can block intake screening equipment (for example, bar racks and traveling water screens), which can lead to reduced cooling water flow or, in extreme cases, structural failure of the screening equipment. Furthermore, the passage into the circulating water system can result in plugging of the condenser tubes. Cooling water blockage is a concern because it negatively affects facility reliability and results in a loss of revenue. One can cite the cluster of jellyfish that forced the Oskarshamn plant from Sweden, the site of one of the world's largest nuclear reactors, to shut down in 2013 by clogging the pipes conducting cool water to the turbines.

A way to reduce the impact of the clogging, is to detect the clogging organism to prevent the power plant. Two main methods have been developed and used for monitoring jellyfish: the towing and scooping techniques for samples collection [START_REF] Bernard | Long term (1981-2008) monitoring of the jellyfish Pelagia noctiluca (Cnidaria, Scyphozoa) on Mediterranean Coasts (Principality of Monaco and French Riviera)[END_REF]. All these methods require manual and heavy processes (jellyfish manipulation) or they depend on light and water condition.

In this context, an active ultrasound monitoring system has the potential to efficiently contribute to detect, quantify and identify in the water column the jellyfish species continuously. This paper investigates the first results obtained using the active ultrasound jellyfish monitoring system experimented in December 2015 at the Aquarium de Paris, France. The two types of jellyfish that were inspected are illustrated in Fig. 1.

The remainder of the paper is structured as follows. Section II shows the details of the experiment carried out in controlled environment, as well as the setup of the proposed active ultrasound monitoring system. In Section III, we illustrate the obtained results, whereas Section IV concludes the paper.

II. ACTIVE ULTRASOUND MONITORING SYSTEM

The current prototype system was developed in collaboration between the GIPSA-lab, the EDF R&D and the SIG-INTEC, and it is composed of a multi-static acoustic wall operating at 1 MHz central frequency. As illustrated in Fig. 2, it is composed of several transmitters (Ultran Group WS-100-1), receivers (Ultran Group KS-100-1) and a reflecting metallic mirror. The US transducers have been excited using Apex Microtechnology PA107 power amplifiers and the Keysight 33522B arbitrary signal generator. The received signals have been conditioned using Texas Instruments VCA 5807 low noise amplifiers and recorded using four synchronous TiePie HS4 oscilloscopes.

In the first step, the proposed ultrasound monitoring system has been calibrated in order to make the derived measurements independent on the transfer function of the acoustic transducers.

Calibration by substitution requires a reference standard, to which the output is reported. For our system of sensors, the calibration process determines a relationship between the input voltage controlling the EM US transducer and the output characteristic. Illustrated in Fig. 3, the calibration procedure has been realized using a Precision Acoustics ML4X50 hydrophone (operating frequency range 100 kHz to 1 MHz). Finally, the exact geometry of the acoustic system can be used to simulate and subtract this deterministic acoustic intensity component using, for example, the algorithm proposed in [START_REF] Julea | Simulation of SAR images and radar coding of georeferenced information for temperate glacier monitoring[END_REF], [START_REF] Landes | Simulation of SAR images and radar coding of georeferenced information for temperate glacier monitoring[END_REF] for radar signals.

In the current setup, each target (jellyfish) passage is obstructing the acoustic propagation channel between the emittermirror-receiver, which makes possible jellyfish monitoring. It acts as a conventional immersed 200 KHz bandwidth ultrasound system in multi-static configuration: the emitted waveform is a linear Hamming windowed chirp. The received signal is illustrated in Fig. 4.

For each ultrasound burst, the match filtering is performed at the reception in order to obtain its corresponding acoustic range profile as shown in Fig. 5. In order to avoid any eventual mismatches between the emitted and received acoustic wave (wave deformation induced by either the propagation channel or the conditioning system), the reference selection procedure proposed in [START_REF] Vasile | Optimal parameter estimation in heterogeneous clutter for high-resolution polarimetric SAR data[END_REF], [START_REF] Anghel | Modelbased parameter estimation of non-stationary signals using time warping and a measure of spectral concentration[END_REF], [START_REF] Vasile | Reference selection for an active ultrasound wild salmon monitoring system[END_REF] has been employed in the final processing algorithm.

III. RESULTS AND DISCUSSION

Two configurations have been tested in controlled water tank facilities:

• in a small test tank (10 m radius × 50 cm depth, used for temporary transportation of annimals) with clear water and no turbulence, • in a large tank (10 m × 20 m × 5 m, used in the standard aquarium show) with turbid water and in the presence of tides and air bubbles. One can notice the time axis is relatively short with respect to a slow jellyfish passage. This is because the acquisition mode is not continuous and the time stamp of each burst was not taken into account for these preliminary results.

The jellyfish detection profiles can be observed on Fig. 7 for the large tank experimental setup, also. This time, the passage detection profile is more noisy, especially for species 1.

IV. CONCLUSION

The active ultrasound monitoring system presented in this paper offers an alternative option to monitor jellyfish passages in turbulent underwater environments. Based on the coupling of conventional active signal processing algorithms with novel acoustic channel equalization techniques, the main advantages of the proposed system are its improved autonomy and the fact that it requires significantly less human intervention for monitoring and counting of jellyfish species. Future studies include the full performance analysis in real underwater conditions.

Fig. 1 .

 1 Fig. 1. Jellyfish used in the experiment at the Aquarium de Paris: (a) species 1 and (b) species 2.

Fig. 2 .

 2 Fig. 2. (a) setup of the ultrasound transducers on the mobile alluminium frame (0.8 × 1.2 meters) and (b) general view of the experimentation.

Fig. 3 .

 3 Fig. 3. (a) setup of the calibration procedure and (b) estimated transfer function for one KS-100-1 transducer.

Figs. 6 -

 6 Fig. 4. (a) received acoustic signal (time domain) and (b) spectrum of the received signal.

Fig. 5 .

 5 Fig. 5. Range profile after match filtering.

Fig. 6 .

 6 Fig. 6. Small test tank experimental setup (small circular test tank with 10 m radius × 50 cm depth with clear seawater): (a) species 1, (b) species 2. The maximum received acoustic signals time series after match filtering: (b) species 1, (d) species 2.

Fig. 7 .

 7 Fig. 7. Large test tank experimental setup (rectangular test tank with 10 m × 20 m × 5 m, with turbulent seawater): (a) species 1, (b) species 2. The maximum received acoustic signals time series after match filtering: (b) species 1, (d) species 2.

ACKNOWLEDGMENT

This work was supported by the FEDER project ACOU-EAU. The authors would like to thank the Aquarium de Paris team, France, for both allowing access to our research team and for helping us with the jellyfish manipulation for the experimentation campaign.


Michel Grabisch
email: michel.grabisch@univ-paris1.fr

Andagnieszka Rusinowska
email: agnieszka.rusinowska@univ-paris1.fr

Determining models of influence 1

Keywords: JEL Classification: C7, D7, D85 social network, opinion formation, aggregation function, influential coalition, algorithm

We consider a model of opinion formation based on aggregation functions. Each player modifies his opinion by arbitrarily aggregating the current opinion of all players. A player is influential on another player if the opinion of the first one matters to the latter. A generalization of an influential player to a coalition whose opinion matters to a player is called an influential coalition. Influential players (coalitions) can be graphically represented by the graph (hypergraph) of influence, and convergence analysis is based on properties of the hypergraphs of influence. In the paper, we focus on the practical issues of applicability of the model w.r.t. a standard framework for opinion formation driven by Markov chain theory. For a qualitative analysis of convergence, knowing the aggregation functions of the players is not required, one only needs to know the set of influential coalitions for each player. We propose simple algorithms that permit us to fully determine the influential coalitions. We distinguish three cases: the symmetric decomposable model, the anonymous model, and the general model.

Introduction -dynamic models of opinion formation

Models of opinion formation are widely studied in psychology, sociology, economics, mathematics, computer sciences, among others; for overviews, see, e.g., [START_REF] Jackson | Social and economic networks[END_REF], [START_REF] Acemoglu | Opinion dynamics and learning in social networks[END_REF]. A seminal model of opinion formation and imitation was introduced in [START_REF] Degroot | Reaching a consensus[END_REF]. In that model, individuals in a society start with initial opinions on a subject. The interaction patterns are described by a stochastic matrix, whose entry in row j and column k represents the weight "that player j places on the current belief of player k in forming j 's belief for the next period". These beliefs are updated over time.

While [START_REF] Degroot | Reaching a consensus[END_REF] assumes that players update their opinion by taking weighted averages of the opinions of all players, [START_REF] Grabisch | A model of influence based on aggregation functions[END_REF] investigate a model of opinion formation in which players update their beliefs according to arbitrary aggregation functions. [START_REF] Foerster | Anonymous social influence[END_REF] study a model of opinion formation in which ordered weighted averages are used in the process of updating information. In this paper, we consider the model of influence based on aggregation functions introduced in [START_REF] Grabisch | A model of influence based on aggregation functions[END_REF] and discuss practical issues of applying this model w.r.t. the "standard" framework for opinion formation driven by Markov chain theory. For a full qualitative analysis of the convergence of opinions, i.e., determining all the terminal classes (without their probabilities), it is sufficient to identify influential coalitions, which can be easily obtained by interviewing the agents. The aim of this paper is to show that a full qualitative analysis of convergence is feasible in practical situations. We introduce simple algorithms that permit us to fully determine the influential coalitions in three cases: the symmetric decomposable model (influential coalitions reduce to individuals), the anonymous model (only the number of agents matters, not their identity), and the general model. We show how clues on convergence can be obtained in a simple way, even without determining the reduced transition matrix.

There exists a vast literature that presents other variations and extensions of the DeGroot model. We briefly recall some of them. In particular, [START_REF] Jackson | Social and economic networks[END_REF] and [START_REF] Golub | Naïve learning in social networks and the wisdom of crowds[END_REF] investigate a model, in which players update their beliefs by repeatedly taking weighted averages of their neighbors' opinions. According to these authors, one of the issues regarding the DeGroot framework concerns necessary and sufficient conditions for convergence of the social influence matrix and reaching a consensus; see additionally [START_REF] Berger | A necessary and sufficient condition for reaching a consensus using DeGroot's method[END_REF]. [START_REF] Jackson | Social and economic networks[END_REF] also examines the speed of convergence of beliefs, and [START_REF] Golub | Naïve learning in social networks and the wisdom of crowds[END_REF] analyze, in the context of the DeGroot model, whether consensus beliefs are "correct", i.e., whether beliefs converge to the right probability, expectation, etc. The authors consider a sequence of societies, where each society is strongly connected and convergent in opinions, and described by its updating matrix. In each social network of the sequence, the belief of each player converges to the consensus limit belief. There is a true state of nature, and the sequence of networks is wise if the consensus limit belief converges in probability to the true state as the number of societies grows.

Several other generalizations of the DeGroot model can be found in the literature, e.g., models in which the updating of beliefs can vary in time and circumstances; see e.g., [START_REF] Demarzo | Persuasion bias, social influence, and unidimensional opinions[END_REF], [35], [START_REF] Lorenz | A stabilization theorem for dynamics of continuous opinions[END_REF], [START_REF] Friedkin | Social influence and opinions[END_REF], [START_REF] Friedkin | Social positions in influence networks[END_REF]. In the model described by [START_REF] Demarzo | Persuasion bias, social influence, and unidimensional opinions[END_REF], players in a network try to estimate some unknown parameter. This model allows updating to vary over time, i.e., a player may place more or less weight on his own belief over time. The authors study the case of multidimensional opinions, in which each player has a vector of beliefs. They show that, in fact, individuals' opinions can often be well approximated by a one-dimensional line, where a player's position on the line determines his position on all issues. [START_REF] Friedkin | Social influence and opinions[END_REF] and [START_REF] Friedkin | Social positions in influence networks[END_REF] study a similar framework, in which social attitudes depend on the attitudes of neighbors and evolve over time. In their model, players start with initial attitudes and then mix in some of their neighbors' recent attitudes with their starting attitudes. Also, other works in sociology related to influence are worth mentioning, e.g., the eigenvector-like notions of centrality and prestige ( [START_REF] Katz | A new status index derived from sociometric analysis[END_REF], [START_REF] Bonacich | Power and centrality: a family of measures[END_REF], [START_REF] Bonacich | Eigenvector-like measures of centrality for asymmetric relations[END_REF]), and models of social influence and persuasion by [START_REF] French | A formal theory of social power[END_REF] and [START_REF] Harary | Status and contrastatus[END_REF]; see also [START_REF] Wasserman | Social network analysis: Methods and applications[END_REF]. A sociological model of interactions on networks is also presented in [START_REF] Conlisk | Interactive Markov chains[END_REF]; see also [START_REF] Conlisk | A stability theorem for an interactive Markov chain[END_REF], [START_REF] Conlisk | Stability and monotonicity for interactive Markov chains[END_REF] and [START_REF] Lehoczky | Approximations for interactive Markov chains in discrete and continuous time[END_REF]. Conlisk introduces interactive Markov chains, in which every entry in a state vector at each time represents the fraction of the population with some attribute. The matrix depends on the current state vector, i.e., the current social structure is taken into account to model how sociological dynamics evolve. In [START_REF] Granovetter | Threshold models of collective behavior[END_REF] threshold models of collective behavior are discussed. These are models in which agents have two alternatives and the costs and benefits of each depend on how many other agents choose which alternative. The author focuses on the effect of individual thresholds (i.e., the proportion or number of others that make their decision before a given agent) on collective behavior, discusses an equilibrium in a process occurring over time and the stability of equilibrium outcomes. Another model of influence is studied in [START_REF] Asavathiratham | Influence model: a tractable representation of networked Markov chains[END_REF] and [START_REF] Asavathiratham | The influence model[END_REF]. This model consists of a network of nodes, each with a status evolving over time. The evolution of status acts according to an internal Markov chain, but the transition probabilities depend not only on the current status of the node, but also on the statuses of the neighboring nodes.

More research on interaction is presented in [START_REF] Hu | On authority distributions in organizations: controls[END_REF] and [START_REF] Hu | On authority distributions in organizations: equilibrium[END_REF], where the authors apply the command structure of [START_REF] Shapley | A Boolean model of organization authority based on the theory of simple games[END_REF] to model players' interactions using simple games. For each player, boss sets and approval sets are introduced, and based on these sets, a simple game called the command game for a player is built. In [START_REF] Hu | On authority distributions in organizations: equilibrium[END_REF], the authors introduce an authority distribution over an organization and the (stochastic) power transition matrix, in which the entry in row j and column k is interpreted as agent j 's "power" transfered to k . The authority equilibrium equation is defined. In [START_REF] Hu | On authority distributions in organizations: equilibrium[END_REF], multi-step commands are considered, where commands can be implemented through command channels.

There is also a vast literature on learning in the context of social networks; see e.g. [START_REF] Banerjee | A simple model of herd behavior[END_REF], [START_REF] Ellison | Learning, local interaction, and coordination[END_REF], [START_REF] Ellison | Rules of thumb for social learning[END_REF], [START_REF] Ellison | Word-of-mouth communication and social learning[END_REF], [START_REF] Bala | Learning from neighbours[END_REF], [START_REF] Bala | Conformism and diversity under social learning[END_REF], [START_REF] Gale | Bayesian learning in social networks[END_REF], [START_REF] Celen | Distinguishing informational cascades from herd behavior in the laboratory[END_REF], [START_REF] Banerjee | Word-of-mouth learning[END_REF]. In general, in models of social learning agents observe choices over time and update their beliefs accordingly, which is different from models where choices depend on the influence of others.

The paper is organized as follows. Section 2 presents fundamental material on models of influence based on aggregation functions, as well as establishing notation and terminology, and recalls an essential result, which is the basis for determining the qualitative part of the model of influence. Section 3 addresses the problem of determining a model of influence in practice and focuses on determining its qualitative part, which is sufficient for a qualitative analysis of convergence. Section 4 gives some concluding remarks.

A model of influence based on aggregation functions

In this section, we recapitulate a model of influence based on aggregation functions ( [START_REF] Grabisch | A model of influence based on aggregation functions[END_REF]). Consider a set } , {1, := n N



of players that have to make a yes-no decision on a certain issue. Each player has an initial opinion, which may change due to mutual interaction (influence) between players. By T S b , we denote the probability that the set S of 'yes'-voters becomes T after one step of influence. We assume that the process of influence may iterate, and therefore obtain a stochastic process of influence, depicting the evolution of the coalition of `yes'-players in time. We assume that the process is Markovian ( T S b , depends on S and T , but not on the whole history) and stationary ( T S b , is constant over time). The states of this finite Markovian process are all subsets N S  representing the set of 'yes'-players, and we also have the transition matrix

N T S T S b  , , ] [ := B , which is a n n 2 2  row-stochastic matrix.
For a qualitative description of the convergence of the process, it is sufficient to know the reduced matrix B ~ given by

   . otherwise 0, 0 > if 1, = ~, , T S T S b b
and equivalently represented by the transition graph

) , (2 = E N 
, where E is the set of arcs, its vertices are all possible coalitions, and the arc ) , ( T S from state S to state T exists if and only if

1 = ~,T S b . Definition 1. An n -place aggregation function is a mapping [0,1] [0,1] :  n A satisfying (i) 0 = ,0) (0,  A , 1 = ,1) (1, A (boundary conditions) (ii) If ' x x  then ) ' ( ) ( x x A A  (nondecreasingness).
To each player N i  we associate an aggregation function i A , which specifies the way player i modifies his opinion based on the opinions of the other players. Let ) , , ( :=

1 n A A  A denote the vector of aggregation functions. We compute )) (1 , ), (1 ( = ) (1 1 S n S S A A  A
, where S 1 is the characteristic vector of S , and

) (1 S i A
indicates the probability of player i saying 'yes' at the next step when the set of agents presently saying 'yes' is S . We assume that these probabilities are independent over the set of agents. Hence the probability of transition from the yes-coalition S to the yes-coalition T is given by

)). (1 (1 ) (1 = , S i T i S i T i T S A A b      (1)
A detailed study of convergence under this model is provided in [START_REF] Grabisch | A model of influence based on aggregation functions[END_REF]. It is shown, in particular, that three types of terminal class 2 can exist: singletons, cycles, and regular terminal classes. The first case occurs when a class is reduced to a single coalition (called the terminal state). The second one is the case where no convergence occurs because the process endlessly cycles over a sequence of coalitions, and the last case occurs when the class is a Boolean lattice of the form

} | 2 { L S K S N   
for some sets L K, . In any case, N and  are terminal states (called trivial terminal states).

We emphasize two particular aggregation functions. The first one is the well-known weighted arithmetic mean (WAM), defined by

i i n i n w x w x x  1 = 1 = ) , , ( WAM  , where ) , , ( = 1 n w w w  is a weight vector, i.e., n w [0,1]  with the property 1 = 1 = i n i w  .
Weighted arithmetic means are used in most models of opinion formation, e.g., the DeGroot model. Another noteable aggregation function is the ordered weighted arithmetic mean (OWA) 2 A class is a maximal collection of coalitions such that for any two distinct coalitions T S, in the class, there exists a sequence of transitions inside the class leading from S to T . A class is terminal if no transition to a coalition outside that class is possible.

( [START_REF] Yager | An ordered weighted averaging aggregation operators in multicriteria decision making[END_REF]), defined by

) ( 1 = 1 = ) , , ( OWA i i n i n w x w x x   ,
where w is a weight vector, and the inputs have been arranged in decreasing order:

) ( (1) n x x   
. Note that, unlike in the case of WAM, these weights do not act on inputs, but on the rank of the inputs, so that the minimum and the maximum are particular cases, by taking ,0,1) (0, =  w and ,0) (1,0, =  w , respectively. Applied to our context of influence where the input vectors are binary, if each agent aggregates his opinions according to an OWA, we obtain a model of anonymous influence, because each agent updates his opinion according to the number of agents saying 'yes', not to which agents say 'yes'. Models of anonymous influence have been studied in detail in [START_REF] Foerster | Anonymous social influence[END_REF].

Definition 2. Let i A be the aggregation function of agent i . A nonempty coalition N S  is yes-influential for i if (i) 0 > ) (1 S i A (ii) For all S S   , 0 = ) (1 S i A  . Similarly, a coalition S is no-influential for i if (i) 1 < ) (1 \S N i A (ii) For all S S   , 1 = ) (1 \S N i A  .
We denote by yes i

C

and no i C the collections of yes-and no-influential coalitions for i . Coalition S is yes-influential for player i if, when the players in S say 'yes' and every other player says 'no', i has a positive probability of saying `yes' (and similarly for no-influential coalitions). Moreover, S has no superfluous player. If an influential coalition is formed by only one player, then we call it an influential player. Note that these collections are never empty, since if no proper subcoalition of N were yes-or no-influential, then N would be both yes- and no-influential by Definition 2. More importantly, each such collection is an antichain in N 2 , that is, for any two distinct members of the collection

S S  , , S S    and S S    .
Influential players can easily be represented in a directed graph. Define yes A G , the graph of yes-influence, as follows: the set of nodes is the set of agents N , and there is an arc

) , ( i j from j to i if j is yes-influential on i .

The graph of no-influence no

A

G is defined similarly. The representation of influential coalitions requires the more complex notion of a hypergraph. Definition 3. We define the following concepts: [START_REF] Berge | Graphs and hypergraphs[END_REF] H is a pair

(i) A hypergraph

6

(iii) A directed hyperpath from i to j is a sequence

q q q q i S S i i S S i S S i ) , ( ) , ( ) , ( 1 2 2 2 1 1 1 0           
, where ; and

q q i j i i i i =: , , , , := 1 1 0   are nodes, ) , ( , ), , ( 1 1 q q S S S S        are hyperarcs such that 1    k k i S and k k i S    for all q k , 1, =  .
(ii) For each

T i   , there exists a nonempty i S   such that     = S S i and i S   is no-influential on i , i.e., no i i S C    .
In particular,

0 = ~,T b  for all   T , 1 = ,  b
, and

0 = ~,T N b for all N T  , 1 = ,N N b .
Recall that (1) is valid only if the probabilities of saying 'yes' are independent over the set of agents. Therefore, non-independence in this sense makes the determination of the transition matrix difficult. However, B ~ is insensitive to possible correlation between agents, because

1 = ~,T S b if and only if 0 > ) (1 S i A for every T i  and 1 < ) (1 S i A
for every T i   , regardless of the correlation between agents.

Determination of the model

An important issue concerns the determination of a model of influence of the above type in a practical situation. This implies that we are making essentially two assumptions:

1. Each agent aggregates the opinion of all the other agents to form his opinion in the next step.

2. The aggregation function is monotonically increasing. The latter assumption implies that anti-conformist behaviors (i.e., the more individuals say 'yes', the more I am inclined to say 'no') cannot be modeled in this framework.

General considerations

Complete determination of the model amounts to identifying either the transition matrix B or all the aggregation functions

n A A , , 1 
(assuming the absence of correlation).

Considering the size of the matrix B ( n n 2 2  ), statistical determination of B seems to be nearly impossible, unless a huge number of observations are made. As for the determination of the aggregation functions, the situation is even worse, since questioning agents about their aggregation functions (type, parameters) appears to be quite unrealistic. We know from Section 2 that knowledge of the reduced matrix B ~ is enough to obtain a qualitative description of the convergence of the model, which is insensitive to possible correlations between agents. Moreover, knowledge of

B ~ (size n 2
2 ) is equivalent by Theorem 1 to knowledge of the collections of all yes-and no-influential coalitions (size at most

          2 2 n n n
), which is, in turn, equivalent to knowledge of the hypergraphs of yes-and no-influence. In some favorable cases (e.g., the WAM model), the hypergraphs reduce to ordinary graphs. This immediately indicates two ways of identifying the (qualitative part of the) model: either by observation of the transitions, i.e., the evolution of the coalition of the 'yes' agents, or by interviewing the agents.

In the first case, observing a transition from S to

T yields 1 = ~,T S b
. In the second case, interviews permit us to determine influential coalitions or graphs of influence.

In the rest of this section, we mainly focus on the second approach. Concerning the first one, we only mention an important fact. The underlying assumptions of the model mean that the reduced matrix B ~ is not arbitrary and has specific properties. Recall that

1 = ~,T S b if and only if for all T i  , 0 > ) (1 S i A
and for all

T i   , 1 < ) (1 S i A
. This implies the following fact:

Fact 1 For a given N S  , N S ,  
, the candidates' transitions are all sets of the form

L K T  =
, where

1} = ) (1 | { = S i A N i K  1}. < ) (1 < 0 | { S i A N i L  
In other words, T  , the intersection of all possible transitions from S yields the set

1} = ) (1 | { = S i A N i K  , while T N  \ yields 0} = ) (1 | { = S i A N i K  
. When S increases, K increases, while K decreases. This fact permits us to detect, when B ~ is constructed from observations, possible deviations from the model (e.g., presence of anti-conformists).

Determination of influential coalitions

We may distinguish three cases, according to the type of underlying model: (i) WAM model (symmetric decomposable model): all aggregation functions are weighted arithmetic means;

(ii) OWA model (anonymous model): all aggregation functions are ordered weighted averages;

(iii) general model (no special assumptions).

The symmetric decomposable model. The case of the WAM model is particularly simple and has been studied in depth in [START_REF] Grabisch | A model of influence based on aggregation functions[END_REF]. It has been proved to be equivalent to a symmetric decomposable model. An aggregation model is decomposable if for every agent N i  , every yes-and no-influential coalition for agent i is a singleton. Now, an aggregation model is symmetric if a yes-influential coalition for i is also no-influential for i and vice versa, for every N i  . Note that the first property implies that the hypergraphs of yes-and no-influence reduce to ordinary graphs, while the second property implies that the two graphs are identical, and therefore the whole (qualitative) model is represented by a single graph representing influence. This makes interviewing agents particularly simple: it suffices to ask to every agent whom he asks for advice. Then, i asks j for advice is translated into the graph representing influence by an arc from j to i . In [START_REF] Grabisch | A model of influence based on aggregation functions[END_REF], we applied this technique to a real case, namely the manager network of Krackhardt ([34]). The agents are the 21 managers of a small manufacturing firm in the USA, and the network is obtained as follows: each agent k is asked if he/she thinks that agent i asks agent j for advice. An arc from j to i is placed in the graph representing influence if a majority of agents think that i asks j for advice. From the graph, and to the properties of symmetric decomposable models, many conclusions can be easily drawn on the convergence of the model. In particular, it is possible to detect the presence of regular terminal classes (Theorem 8 in [START_REF] Grabisch | A model of influence based on aggregation functions[END_REF]). There is also a simple criterion to determine if there is no regular terminal class: it suffices that for each agent i , there is an arc in the influence graph from ) ( cl i to every agent outside ) ( cl i , where ) ( cl i , the closure of i , is the set of agents who can reach i by a path in the influence graph.

The anonymous model.

According to the OWA model, agents do not change their opinion due to particular individuals, but due to the number of individuals saying 'yes'. Therefore, in general these are not decomposable models, and one needs to determine influential coalitions as in the general case. However, because according to these models the players are anonymous, a collection

yes i C
or no i C is composed of all sets of a given size s ,

n s   1
, and this is characteristic of an anonymous model. Therefore, under the assumption of anonymity, it suffices to ask to every agent i the following questions: Q1: Suppose that your opinion on some question is 'yes'. What is the minimal number of agents saying 'no' that may make you change your opinion? Q2: Suppose that your opinion on some question is 'no'. What is the minimal number of agents saying 'yes' that may make you change your opinion?

Assuming that the answers are respectively s and s , it follows that

}. |= | | 2 { = }, |= | | 2 { = yes no s S S s S S N i N i    C C
Now, it is easy to see that given s s  , for agent i , one can get the form of the weight vector w in the aggregation function

w OWA of agent i (Proposition 2 in [21]): ), 0 0 0 , , 0 0 0 ( = zeros 1 zeros 1                  s s w
where `` '' indicates any nonzero weight. In particular, all agents are yes-influential (respectively, no-influential) on i if and only if

0 > 1 w (respectively, 0 > n w
). As for under this model, it is shown in [START_REF] Foerster | Anonymous social influence[END_REF] (Proposition 3) that no cycle can occur, but the two other types of terminal classes may occur. Terminal states are easily detected as follows: S of size s is a terminal state if and only if for every S i  , the size of a no-influential coalition is at least 1   s n , and for every S i   , the size of a yes-influential coalition is at least 1  s . The absence of regular terminal classes can also be characterized only through influential coalitions, but this condition is more complex (see Corollary 3 in the aforementioned paper).

The general model.

We now address the general case, where no special assumption is made on the model, except the following: we assume that each agent is yes-and no-influential on himself, which means that 0 > ) (1

i i A , 1 < ) (1 \i N i A
(in other words, the agent trusts his opinion to a nonnull extent). This induces some simplification in the algorithm, but it would not be difficult to generalize it, in order to this limitation.

Interview for Agent

i 0. Set }} {{ = yes i i C , }} {{ = no i i C , i N i i \ no yes 2 = = N N . % no yes , i i N N
are the sets of candidate coalitions. 1. For each agent N j  , i j  , ask 3 : Q: Suppose that your opinion on some question is 'yes'. Would you be inclined to change your opinion if Agent j says 'no' ? If the answer is positive:

• add } { j to no i C , and discard } { j and all sets containing

j from no i N . • If  = no i N
, STOP (GO TO STEP 3).

Otherwise, discard

N • If  = no i N or if           2 |= | no n n i C
, STOP (GO TO STEP 3).

Set

S N N \ no no i i  .
3. Exactly as in Steps 1 and 2 for yes i C , Question 1 becomes: Suppose that your opinion on some question is 'no'. Would you be inclined to change your opinion if Agent j says 'yes'?, etc. 3 As in Step 2, it is possible to gather all these questions into a single one: Suppose that your opinion on some question is 'yes'. Would

C

We now give another example to illustrate how the reduced transition matrix B ~ can be obtained from the influential coalitions using Theorem 1. To this end, we suppose that the above algorithm has been applied to each agent, in order to obtain all influential coalitions. The condition that every agent is self-influential permits us to simplify the application of the theorem to determine each term (ii)

If S T  , condition (i) in Theorem 1 is always sufficient to check that 1 = ~,T S b ; moreover, if 0 = ~,T S b , then 0 = ~' ,T S b for every T T   ; similarly, if 1 = ~,T S b , then 1 = ~' ,T S b for every S T T    ; (iii) If S T  , condition (ii) in Theorem 1 is always sufficient to check that 1 = ~,T S b ; moreover, if 0 = ~,T S b , then 0 = ~' ,T S b for every T T   ; similarly, if 1 = ~,T S b , then 1 = ~' ,T S b for every T T S    .
Example 2. Consider a society {1,2,3,4} = N of 4 agents. Suppose that the following collections have been obtained (braces are omitted when denoting coalitions): , only condition (i) has to be checked. Lastly, observe that all the remaining sets contain 4. Thus condition (i) of Theorem 1 is never satisfied, since there is no

{1,234} = {1,2,34}, = 1 no 1 C C {2,134} = {2,34}, = yes 2 no 2 C C {3,12} = {2,3}, = yes 3 no 3 C C {4} = {12,
yes 4 C   S
which is included in 12. One can check that Fact 1 is satisfied. Observe that this approach is very useful to identify quickly all the possible T 's: it suffices to find the minimal one ( K ) and the maximal one ( L K  ).

The corresponding transition graph  is given in Figure 1 below. We now show that it is possible to get conclusions on convergence without computing B ~, by examining the hypergraphs, thanks to results presented in [START_REF] Grabisch | A model of influence based on aggregation functions[END_REF]. To this end, we need the notion of an ingoing hyperarc. We say that a coalition S has an ingoing hyperarc ). The advantage of Fact 3 is that it is not necessary to find all T S, such that 1 = ~,T S b (i.e., it is not necessary to know the transition graph) to check whether a given coalition is a terminal state (or to find all of them).

Concluding remarks

We have shown how, in a practical situation, one can determine a model of influence based on aggregation functions. Exact determination of such model, yielding the type and parameters of the aggregation function of each agent, appears to be out of reach without using complex procedures. What we show is that, on the contrary, it is easy to obtain the "qualitative part" of the model, which permits a full qualitative analysis of the convergence of opinions, that is, to determine all terminal classes. This is sufficient to predict whether a consensus will occur or, on the contrary, society will become polarized, or a cycle will appear, etc. Simple criteria are available to detect terminal states or the presence of regular terminal classes, without even determining the reduced transition matrix. We believe that this study will make the use of models of influence based on aggregation functions more familiar and easier to use.

) , ( T T    is normal if      = T T
. Note that due to our assumption that every player is self-influential, all hyperarcs are normal.

Fact 2

 2 Indeed, the following facts are easy to show. Suppose

Figure 1 :

 1 Figure 1: Transition graph (loops are omitted). The terminal states are denoted in red and larger font

  We define the hypergraphs

		H	yes A	no , A H	of yes-influence and no-influence as follows:
	for	yes H , the set of nodes is N , and there is a hyperarc	}) ( i { , C	for each	C C 	yes	(similarly
		A						i
	for	no H ).
		A
		Grabisch and Rusinowska [27] prove that the hypergraphs	H	yes	,	H	no	(equivalently, the
								A	A
	collections yes i C therefore contain the entire qualitative description of convergence. and no i N i  ) are equivalent to the reduced matrix Theorem 1. Consider an influence process B based on the aggregation functions A . Then B ~, and B ~ can be reconstructed from the hypergraphs yes A H and no A H as follows: for any N T S 2 ,  , 1 = ~,T S b if and only if (i) For each T i  , there exists a nonempty S S i   such that i S is yes-influential on i , C for all i.e., yes i i S C  

  indicates that the hyperarcs have been inverted, and H ˆ indicates that only normal hyperarcs are considered4 .This result can be translated in terms of influential collections as follows:Applying this fact to Example 1, we indeed find that the only terminal states are 23, 24 and 123. For example, 23 is a terminal state, because none of 1, 234 are subsets of 23 (condition

	Fact 3 A nonempty	S 	N	is a terminal state if and only if
	(i) For every	S i   , there is no	i yes T C 	such that	S T  ;
	)   hyperarc). , ( T T  in hypergraph Now, Theorem 3 in the aforementioned paper establishes that a nonempty H if S N T \   and S T    (and versa for an outgoing N S  is a terminal state if and only if S has no ingoing arc in the hypergraph no * yes ) ( A A H H  , where * () (ii) For every S i  , there is no no i  such that   = S T . (i) for 1 = i ), 4 is not a subset of 12 (condition (i) for 4 = i ), none of 2, 34 are disjoint from 23 T C (condition (ii) for 2 = i ), and none of 2, 3 are disjoint from 23 (condition (ii) for 3 = i

The authors acknowledge the support by the National Agency for Research (Agence Nationale de la Recherche), Project ANR-14-CE24-0007-01 ``CoCoRICo-CoDec''.

We give some examples. Example 1. (braces are omitted when denoting coalitions) Consider } {1,2,3,4,5 = N . We detail the process of interviewing Agent 1.

C

. We take agent 2. Suppose that your opinion on some question is 'yes'. Would Suppose that your opinion on some question is 'no'. Would you be inclined to change


Gabriel Vasile
email: gabriel.vasile@gipsa-lab.grenoble-inp.fr

Leandro Pralon

BLIND SOURCE SEPARATION IN POLARIMETRIC SAR INTERFEROMETRY

Keywords: Polarimetry, SAR, Interferometry, ICA

 in the context of ICA are proposed.

INTRODUCTION

Polarimetric target decomposition is one of the most powerful and widespread tools for POLSAR image interpretation. The analysis of the interaction between the illuminated area and the transmitted waveform, to each polarimetric state of the latter, allows for a better prediction of the basic scattering mechanisms present on the scene, and to more efficiently propose classification, detection and geophysical parameter inversion algorithms.

Many methods have been proposed in the literature to both decompose an image pixel into basic target vectors as well as to correctly retrieve quantitative information from them (parametrization). Concerning the latter, Cloude and Pottiers parameters (entropy, alpha and anisotropy) [START_REF] Cloude | An entropy based classification scheme for land applications of polarimetric SAR[END_REF] and Touzi Target Scattering Vector Model [START_REF] Touzi | Target scattering decomposition in terms of rollinvariant target parameters[END_REF] are the most employed ones, whose usefulness have already been demon-Thanks to the French Aerospace Lab (ONERA) for providing the Brétigny POLinSAR data. strated by several authors. Regarding the decomposition, the algorithms are mainly classified in either coherent, if they are based on the scattering matrix analysis, or incoherent, if their interest lies in the Hermitian, semidefinite positive coherence or covariance matrix. Incoherent target decomposition (ICTD) theory assumes that the scattering process in most natural media is a combination of coherent speckle noise and random vector scattering effects. Therefore, not only a statistical analysis is often required, but also it is a common practice to associate to the imaging cell the concept of average or dominant scattering mechanisms [START_REF] Cloude | A review of target decomposition theorems in radar polarimetry[END_REF]. The Eigenvector based ICTD manages to decompose an image pixel into the three most dominant scatters from the averaged coherence matrix. Furthermore, it has an intrinsic property that the derived scatters are orthogonal and uncorrelated, which for Gaussian clutters also means independence. The drawback of this kind of method emerge when the clutter is not Gaussian [START_REF] Vasile | Normalized coherency matrix estimation under the SIRV model. Alpine glacier POLSAR data analysis[END_REF][START_REF] Vasile | Optimal parameter estimation in heterogeneous clutter for high resolution polarimetric SAR data[END_REF] or not composed by orthogonal mechanisms, situations where the performance of the algorithm could be compromised.

BLIND SOURCE SEPARATION STRATEGY IN POLSAR

In [START_REF] Besic | Polarimetric incoherent target decomposition by means of independent component analysis[END_REF], a new Blind Source Separation strategy to polarimetric target decomposition was presented by incorporating the Independent Component Analysis (ICA). The ICA is a blind source separation technique based on higher order statistical moments and cumulants whose utility has already been explored in many different research areas, such as wireless communications, feature extraction and brain imaging applications [START_REF] Hyvarinen | Independent Component Analysis[END_REF]. The results presented in [START_REF] Besic | Polarimetric incoherent target decomposition by means of independent component analysis[END_REF] proved it to be a very promising area in polarimetry, mainly when non-Gaussian heterogeneous clutters (inherent to high resolution SAR systems) are under study. The theoretical potential in estimating similar entropy and first component, when compared to traditional eigenvector decomposition, but rather a second most dominant component independent with respect to the first one and unconstrained by the orthogonality introduces an alternative way of physically interpreting a polarimetric SAR image. The referred method is briefly summarized in three main steps: data selection, based on the statistical classification of the POLSAR image; estimation of independent components and parametrization of the derived target vectors.

As stated in [START_REF] Besic | Polarimetric incoherent target decomposition by means of independent component analysis[END_REF], the major drawback of the proposed method is the size of the observation dataset, which has to be somewhat larger than the size of the sliding window used in the well established methods. This constraint led the authors in [START_REF] Besic | Polarimetric incoherent target decomposition by means of independent component analysis[END_REF] to use an unsupervised classification algorithm rather than relying on a very large sliding window, jeopardizing the effectiveness of the method.

The use of a classification algorithm limits the performance of the method in the sense that the image is segmented in a priori defined number of classes with variable sizes, what can compromise the estimation of the target vectors parameters and, as a consequence, the correct interpretation of the scatters present in the area under study. One of the implications of the employment of a classification algorithm is that a class can contain more samples than it needs for a correct estimation of targets parameters, meaning that spatial resolution, highly degraded with the use of this approach, is worse than it could be. On the other hand, if a class do not contain samples enough, the parameters estimated can be biased, meaning that the values derived do not comply with ground truth.

Within this context, this paper considers a Monte Carlo simulation approach to evaluate the optimal size of a sliding window for various medias, simple ones composed by basic scatters such as helix, dipole, dihedral and trihedral and more complex ones like Surface, Double Bounce and Volume returns. The simulation procedure is similar to the one presented in [START_REF] Lee | Evaluation and bias removal of multilook effect on Entropy/Alpha/Anisotropy in polarimtric SAR decomposition[END_REF] to evaluate the bias of multilook effect on Cloude and Pottier [START_REF] Cloude | An entropy based classification scheme for land applications of polarimetric SAR[END_REF] parameters in Eigenvector based polarimetric SAR decomposition. An unsupervised classification algorithm is employed to identify within a RAMSES Xband image, sets of samples characterizing Surface, Double Bounce and Volume type of average scattering mechanisms. The mixing matrix and the covariance matrix for each of the aforementioned complex type of scatters are estimated using proper algorithms (Non-Circular Complex Fast-ICA algorithm and Fixed-point algorithm, respectively) and used to bootstrap random samples for the Monte Carlo simulation approach. It is important to highlight that, as reported in [START_REF] Besic | Polarimetric incoherent target decomposition by means of independent component analysis[END_REF], the mixing matrix and the covariance matrix are potentially different, thus the set of random samples used in the ICA and Eigenvector decomposition analysis are not the same.

BLIND SOURCE SEPARATION STRATEGY IN POLINSAR

In case of POLinSAR data, the same ICA algorithm as proposed in [START_REF] Besic | Polarimetric incoherent target decomposition by means of independent component analysis[END_REF] is emplyed, except that the dimension of the BSS problem is increased by two. The ICA techniques use a set of observation vectors (x) to retrieve the sources vector (s) and the mixing matrix (A), which gives the share of the sources in the observed process:

        x M 1 x M 2 x M 3 x S 1 x S 2 x S 3         =         A11 A12 A13 A14 A15 A16 A21 A22 A23 A24 A25 A26 A31 A32 A33 A34 A35 A36 A41 A42 A43 A44 A45 A46 A51 A52 A53 A54 A55 A56 A61 A62 A63 A64 A65 A66         •         s M 1 s M 2 s M 3 s S 1 s S 2 s S 3         . (1)
If we introduce the spatially averaged coherency matrix of the POLinSAR observation vector as T 6 , the mixing matrix A can be represented as the factorization of the covariance matrix:

T 6 = AA † = T 11 Ω 12 Ω * T 12 T 22 , (2)
with T 11 being the master POLSAR coherency matrix, T 22 the slave POLSAR coherency matrix and Ω 12 the interferometric coherency matrix.

After computing these three matrices from the previously derived mixing matrix, it is now possible to directly apply the POLinSAR coherence optimization proposed in [START_REF] Cloude | Polarimetric SAR interferometry[END_REF].

POLSAR EXPERIMENTAL RESULTS

An unsupervised classification algorithm developed for highly textured POLSAR data [START_REF] Formont | Statistical classification for heterogeneous polarimetric SAR images[END_REF] is employed to identify within a RAMSES X-band image acquired over Brétigny, France, sets of samples characterizing Surface, Double Bounce and Volume type of average scattering mechanisms. Fig. 1 presents the referred area in Red (HH+VV), Green (HV) and Blue (HH-VV) and the classification algorithm output.

Analyzing the H/α feature space, it is possible to concluded that Class 1 is mainly composed by Volume scatters, Class 5 is mainly composed by Surface and Class 6 is mainly composed by Double-Bounce scatters. Therefore samples, corresponding to each class were extracted from the referred set and the mixing matrix and covariance matrix were estimated for each of the described classes (c = 1, 2, 3) of mechanisms.

The parameters of the distribution are then extracted and they are used to generate simulated texture vectors to bootstrap random samples for the Monte Carlo simulation approach. The remaining steps of the simulated data generation are the same as previously described. It is important to highlight that, as reported in [START_REF] Besic | Polarimetric incoherent target decomposition by means of independent component analysis[END_REF], the mixing matrix and the covariance matrix are potentially different, thus the set of random samples used in the ICA and Eigenvector decomposition analysis are not the same. Fig. 2 presents the results of the polarimetric decomposition using both ICA and Eigenvector decomposition (PCA) as a function of the number of looks.

POLINSAR EXPERIMENTAL RESULTS

Figs. 3 and4 illustrate the coherence optimization results obtained using the proposed BSS technique -local sliding neighborhood approach, both in terms of optimized coherence and the associated interferometric phases. On can notice the unwrapped phase is converted to DEM using the ambiguity height (around 200 m for this POLinSAR data set). The obtained results are in good agreement with the available ground truth (top of the building at about 10 m).

For example, the hight difference between the 2nd and the 1st component shows that the two scattering mechanisms (building edge and ground) are well separated in Fig. 4-(b). Similar behavior has been reported in [START_REF] Colin | An interferometric coherence optimization method in radar polarimetry for high-resolution imagery[END_REF].

For quantitative performance assessment, the normalized log-ratio [START_REF] Touzi | A statistical and geometrical edge detector for SAR images[END_REF] between the ICA and PCA derived optimized

The ICA revels an improvement in R of exactly 27.05 dB computed over the entire test POLinSAR image (500 × 500 pixels).

CONCLUSION

This paper presented a new framework for applying BSS techniques with POLinSAR data for polarimetric coherence optimization and associated interferometric phases estimation. After analyzing the POLSAR estimation bias function of the number of looks, the POLinSAR mixing matrix is derived using the Non-Circular Complex Fast-ICA algorithm and Fixed-point algorithm.

The obtained results show improvements in terms of the derived optimized coherences and, in the same time, they remain consistent with the actual ground truth.

Fig. 1 .

 1 Fig. 1. Brétigny area under study: (a) RGB image, Red (HH+VV), Green (HV), Blue (HH-VV); (b) Statistical classification algorithm output.

Fig. 2 .

 2 Fig. 2. Entropy and Touzi TSVM parameters derived with ICA and Eigenvector polarimetric target decomposition (PCA) for a complex clutter types: Surface, Double-Bounce and Volume.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Brétigny area under study: ICA optimized coherence -1st component.


Michel Grabisch
email: michel.grabisch@univ-paris1.fr

Agnieszka Rusinowska
email: agnieszka.rusinowska@univ-paris1.fr

grabisch-rusinowska-igtr-revised International Game Theory Review

Keywords: Influence function, follower function, distributive lattice, aggregation function, convergence, terminal class. Subject Classification: C7, D7. 1

We present an application of lattice theory to the framework of influence in social networks. The contribution of the paper is not to derive new results, but to synthesize our existing results on lattices and influence. We consider a two-action model of influence in a social network in which agents have to make their yes-no decision on a certain issue. Every agent is preliminarily inclined to say either 'yes' or 'no', but due to influence by others, the agent's decision may be different from his original inclination. We discuss the relation between two central concepts of this model: influence function and follower function. The structure of the set of all influence functions that lead to a given follower function appears to be a distributive lattice. We also consider a dynamic model of influence based on aggregation functions and present a general analysis of convergence in the model. Possible terminal classes to which the process of influence may converge are terminal states (the consensus states and non trivial states), cyclic terminal classes and unions of Boolean lattices.

Introduction

Influence phenomena attract research interest in several scientific areas, e.g., in economics, sociology and social psychology, where both experiments on influence and theoretical models are proposed, in political science, in management and organizational studies. One of the particularly rich and elegant approaches to influence is based on using social networks. Economists and sociologists are interested in the study of how the structure of networks affects opinion and consen-sus formation, which the key players in a network are, or how important, central or influential an agent's neighbors are. Among numerous works along these lines one should mention measures of prestige [START_REF] Katz | A new status index derived from sociometric analysis[END_REF]), (eigenvector) centrality [START_REF] Bonacich | Power and centrality: a family of measures[END_REF]; [START_REF] Bonacich | Eigenvector-like measures of centrality for asymmetric relations[END_REF]; [START_REF] Borgatti | Centrality and network flow[END_REF]; [START_REF] Borgatti | Identifying sets of key players in a social network[END_REF]; [START_REF] Borgatti | A graph-theoretic perspective on centrality[END_REF]), and intercentrality by [START_REF] Ballester | Who's who in networks. Wanted: The key player[END_REF]. The seminal network interaction model of information transmission and opinion formation is presented in DeGroot [1974]. Several variations and generalizations of his model have been proposed for the last fifteen years, see, e.g., [START_REF] Demarzo | Persuasion bias, social influence, and unidimensional opinions[END_REF]; [START_REF] Golub | Naïve learning in social networks and the wisdom of crowds[END_REF]; [START_REF] Jackson | Social and economic networks[END_REF]; Krause [2000]; [START_REF] Lorenz | [END_REF]. In the sociological literature, an interpersonal influence model is investigated in [START_REF] Friedkin | Theoretical foundations for centrality measures[END_REF]; [START_REF] Friedkin | Social influence and opinions[END_REF]; [START_REF] Friedkin | Social positions in influence networks[END_REF]. Surveys of different approaches to influence are given, e.g., in Grabisch and Rusinowska [2010c] and [START_REF] Rusinowska | Different approaches to influence in social networks[END_REF]. For broad overviews of dynamic models of social influence we also refer to, e.g., [START_REF] Jackson | Social and economic networks[END_REF], [START_REF] Golub | Naïve learning in social networks and the wisdom of crowds[END_REF].

The present paper deals with a particular framework of influence introduced in Hoede and Bakker [1982] and studied, e.g., in [START_REF] Grabisch | Measuring influence in command games[END_REF]; Grabisch and Rusinowska [2010a]; Grabisch and Rusinowska [2011a]; Grabisch and Rusinowska [2010b]. We consider a social network in which agents (players) may influence each others when making decisions. Each agent has an inclination to choose one of the actions, but due to influence by others, the decision of the agent may be different from his original inclination. Such a transformation from the agents' inclinations to their decisions is represented by an influence function. An important concept in the influence model is that of follower of a coalition, which is defined as a player who decides according to the inclination of the coalition in question whenever all members of the coalition are inclined to choose the same action. The original model of influence was a two-action model in which agents say either 'yes' or 'no'. In Grabisch and Rusinowska [2010a] we investigate some tools to analyze such framework. In particular, we study a (weighted) influence index of a coalition upon an individual and different influence functions. Also some generalizations of the two-action model are proposed. In Grabisch and Rusinowska [2010b] we generalize the yes-no model of influence to a multi-choice framework in which agents have an ordered set of possible actions. The influence model in which every player has a continuum of actions is investigated in Grabisch and Rusinowska [2011b]. In [START_REF] Grabisch | Measuring influence in command games[END_REF] we study the relation between the yesno influence model and command games proposed by Hu and Shapley [2003b]; Hu and Shapley [2003a] who use the command structure to model players' interaction relations by simple games. In particular, we show that our framework of influence is more general than the framework of the command games.

In two of our works on influence we show that some structures in the framework of influence are lattices. In Grabisch and Rusinowska [2011a] we consider, among other issues, the relation between two central concepts of the influence model: influence function and follower function. We deliver sufficient and necessary conditions for a function to be a follower function and show that the structure of the set of all influence functions that lead to a given follower function is a distributive lattice. In Grabisch and Rusinowska [forthcoming] we analyze the decision process in which the mutual influence does not stop after one step but iterates. In particular, we propose a model based on aggregation functions, where the inclination of each agent is obtained as an aggregation of the inclinations of the others. We provide a general analysis of convergence in the aggregation model and find all terminal classes and states. We show that possible terminal classes to which the process of influence may converge are terminal states (the consensus states and non trivial states), cyclic terminal classes, and unions of Boolean lattices (called regular terminal classes).

The idea of applying lattice theory to social phenomena, in particular, to the issues of influence, is not really new and has been already considered in the previous literature. [START_REF] Monjardet | The presence of lattice theory in discrete problems of mathematical social sciences[END_REF] lists numerous domains of mathematical social sciences where lattice theory plays a significant role, e.g., cluster analysis, data analysis, game theory, assessment theory, social choice and consensus theory. He provides numerous references of the related literature. Also [START_REF] Caspard | Finite ordered sets -concepts, results and uses[END_REF] show that lattices are present in many fields of mathematical social sciences. The importance and usefulness of lattices for the analysis of influence is stressed, e.g., in [START_REF] Martin | Some algebraic structures for diffusion in social networks[END_REF]; [START_REF] Martin | Jointness and duality in algebraic approaches to dichotomous data[END_REF], and in several other works that this author refers to. He considers algebraic structures for diffusion (of information, control or opinion) in social networks. As discussed by the author, it is possible to recreate an influence network (not observed directly) by examining a certain structure of a lattice of distribution of information. Even in case of incomplete data, if gaps are moderate, then there are some techniques to construct a lattice of information. [START_REF] Martin | Some algebraic structures for diffusion in social networks[END_REF] argues that some diffusion processes give structures that are closed under union but not intersection, and are therefore different from another prominent technique for algebraic analysis of network data, based on Galois lattices (structures that are closed under intersection but not union). [START_REF] Martin | Jointness and duality in algebraic approaches to dichotomous data[END_REF] considers different algebraic techniques for the analysis of social data. Algebraic approaches to dichotomous data can be used as models of processes, as many social structures are algebraic and data produced by such structures have algebraic properties, or as data reduction and interpretation that do not correspond to particular social processes.

The contribution of the present paper is the synthesis of our existing results on lattices and influence (Grabisch and Rusinowska [2011a]; Grabisch and Rusinowska [forthcoming]), and not the derivation of new theorems. Following Martin and some others who confirm the importance of discussing models of influence and lattices, we would like to stress the interest of synthesizing our published results. While in the two related papers (Grabisch and Rusinowska [2011a]; Grabisch and Rusinowska [forthcoming]) we present and prove some results that link lattices with influence concepts, in none of these works we discuss the usefulness of the results to agents in choosing their influence techniques. The reasons why lattices present in our influence frameworks are useful for the influence analysis are different from the reasons discussed by other authors of the existing works on lattices and influence. It is partially due to the fact that we consider two specific November 11, 2013 0:17 WSPC/INSTRUCTION FILE grabisch-rusinowska-igtr-revised 4 Michel Grabisch, Agnieszka Rusinowska models of influence, and lattices help to answer specific 'influence questions' related to these models. We point out two particular issues. One of them concerns the choice of influence functions that lead to desirable follower functions, i.e., desirable set of followers. The second issue answers the question of how to aggregate the others' opinions in order to avoid some bad effects of the dynamic influence process. We elaborate more on these issues in the concluding section.

The remaining part of the paper is the following. In Section 2.1 we recapitulate basic concepts of posets and lattices. Some basic notions of directed graphs and aggregation functions that we need in the last part of the paper are recapitulated in Section 2.2. The one-step model of influence is presented in Section 3. Section 4 concerns applications of lattice theory to the one-step model of influence. In Section 5 we present an application of lattices to the dynamic model of influence based on aggregation functions. In Section 6 we present a short conclusion and a discussion of the usefulness of lattice theory in our works on influence that are systematized in the present paper.

Basic Notions

Lattices and posets

We recall here the necessary material on finite posets and lattices for the sequel (see, e.g., [START_REF] Davey | Introduction to lattices and orders[END_REF]).

Let N be a finite set. We write for two functions F, G :

2 N → 2 N that F ≤ G if F (S) ⊆ G(S) for all S ⊆ N . As usual, F = G means F (S) = G(S) for all S ⊆ N . F : 2 N → 2 N is isotone or monotone nondecreasing if S ⊆ T ⊆ N implies F (S) ⊆ F (T ).
If the first inclusion is reversed, then F is said to be antitone or monotone nonincreasing. A function is monotone if it is either isotone or antitone.

A partially ordered set (poset) (P, ≤) is a set P endowed with a partial order ≤, i.e., a binary relation being reflexive, antisymmetric and transitive. We recall that in any poset (P, ≤), Q ⊆ P is a downset of P if x ∈ Q and y ≤ x imply y ∈ Q. The set of all downsets of P is denoted by O(P ). For any two elements x, y ∈ L, x is covered by y or y covers x (denoted by x ≺ y or y ≻ x) if x < y and there is no z such that x < z < y.

A lattice L is a poset such that for any x, y ∈ L their least upper bound x ∨ y and greatest lower bound x ∧ y always exist. In a finite lattice (L, ≤), we denote by ⊤, ⊥ the top and the bottom elements of L. A lattice is said to be distributive if ∨, ∧ obey distributivity. A lattice is autodual if reversing the order relation, the same lattice is obtained (up to an isomorphism). A lattice is Boolean if it is isomorphic to some lattice of subsets (2 N , ⊆). Every Boolean lattice is distributive and autodual.

An element j ∈ L is join-irreducible if it is not the bottom element and it cannot be expressed as a supremum of other elements, or equivalently, if it covers only one element. The set of all join-irreducible elements is denoted by J (L). When the lattice is distributive, any element x ∈ L can be expressed in a unique way as an irredundant supremum of join-irreducible elements. is distributive, it suffices to know only J (L) to reconstruct all the lattice. More precisely, L is isomorphic to O(J (L)), the set of all downsets of J (L) (Birkhoff's theorem).

The height function h on L gives the length of a longest chain from ⊥ to any element in L. The height of the lattice is h

(⊤). A lattice is ranked if x ≻ y implies h(x) = h(y) + 1. If a lattice is distributive, then it is ranked and the length of any maximal chain from ⊥ to ⊤ is |J (L)|. A function τ : (L, ≤) → (L, ≤) is a dual closure operator if x ≥ τ (x) for all x ∈ L, τ is isotone, and τ (τ (x)) = τ (x).

Basic notions of digraphs and aggregation functions

For Section 5, we need some basic concepts on directed graphs and aggregation functions. For more details, see, e.g., [START_REF] Seneta | Non-negative matrices[END_REF], [START_REF] Horn | Matrix analysis[END_REF], [START_REF] Meyer | Matrix analysis and applied linear algebra[END_REF], [START_REF] Jackson | Social and economic networks[END_REF].

Let X = {x 1 , . . . , x m } be a finite set of states. We consider the m × m transition matrix P = [p ij ] i,j=1,...,m , where p ij is the probability that the next state s(t + 1) is

x j knowing that the current state s(t) is x i , i.e., p ij = Prob(s(t+1) = x j | s(t) = x i ).
This matrix is row-stochastic.

We associate to P its transition directed graph (digraph) Γ, whose set of nodes is the set of states X, and there is a directed edge from x i to x j if and only if p ij > 0.

In such a case we say that x j is a successor of x i .

A path in Γ is a sequence of nodes x j1 , . . . , x j k such that x j l+1 is a successor of x j l for l = 1, . . . , k -1. A cycle is a path for which the first and last nodes coincide, and its length is the number of edges.

A (strongly) connected component (called also a class) is any subset S of nodes such that there is a path from any node to any other node of S, and which is maximal for this property. Obviously a single node can be a class.

If the graph has only one class (which is then X), then the matrix is said to be irreducible. A class is transient if there is an edge going outside it, otherwise the class is said to be recurrent or terminal. The set of all classes C 1 , . . . , C k forms a partition of X.

A terminal class C is said to be periodic of period p if the greatest common divisor of the length of all cycles in C is p. If p = 1, then we say that the class is aperiodic.

When the matrix is irreducible, if its (unique) class is aperiodic, we say that the matrix is aperiodic too. Usually, aperiodic (and therefore irreducible) matrices are called ergodic.

An n-place aggregation function is any mapping

A : [0, 1] n → [0, 1] satisfying (1) A(0, . . . , 0) = 0, A(1, . . . , 1) = 1 (boundary conditions) (2) If x ≤ x ′ then A(x) ≤ A(x ′ ) (nondecreasingness).
Aggregation functions are well-studied and there exist many families of them, November 11, 2013 0:17 WSPC/INSTRUCTION FILE grabisch-rusinowska-igtr-revised 6 Michel Grabisch, Agnieszka Rusinowska e.g., all kinds of means (geometric, harmonic, quasi-arithmetic) and their weighted version, weighted ordered averages, any combination of minimum and maximum (lattice polynomials or Sugeno integrals), Choquet integrals, triangular norms, copulas; see Grabisch et al. [2009].

In the rest of the paper, by 1 S = ((1 S ) 1 , . . . , (1 S ) n ) with S ⊆ {1, . . . , n} we mean the characteristic vector of S, i.e., (1 S ) i = 1 if and only if i ∈ S and 0 otherwise.

The One-step Model of Influence

Let us recapitulate the model of influence originally introduced in Hoede and Bakker [1982], and next investigated e.g. in [START_REF] Grabisch | Measuring influence in command games[END_REF]; Grabisch and Rusinowska [2010a]; Grabisch and Rusinowska [2011a].

We consider a social network with the set N := {1, ..., n} of agents (players). The model is general and there is no restriction on the social network. Each agent k ∈ N is supposed to make a certain yes-no decision and has an inclination i k either to say 'yes' (denoted by 1) or 'no' (denoted by 0). Let i = (i 1 , ..., i n ) denote an inclination vector and I := {0, 1} n the set of all inclination vectors. Let for any S ⊆ N , |S| ≥ 1,

I S := {i ∈ I | ∀k, j ∈ S, i k = i j }, (1)
i.e., I S is the set of all inclination vectors under which all members of coalition S have the same inclination. By i S we denote the value i k for some k ∈ S, i ∈ I S . Agents may influence each other in the network, and due to these influences, the decision of an agent may be different from his (original) inclination. Formally, each inclination vector i ∈ I is transformed into a decision vector B(i) = ((B(i)) 1 , ..., (B(i)) n ), where B : I → I, i → B(i) is the influence function. The set of all influence functions is denoted by B.

In this paper, we will use the set notation, i.e., S will denote the set of yesinclined agents and it corresponds to the inclination vector (1 S , 0 N \S ), i.e., to the n-vector i

= (i 1 , ..., i n ) such that i k = 1 if k ∈ S and i k = 0 if k ∈ N \ S.
Consequently, S := N \ S is the set of players that are inclined to say 'no'. If S corresponds to i, then B(i) will be denoted by B(S), where B(S) ⊆ N is the set of agents whose decision is 'yes', and B(S) := N \ B(S) is the set of agents whose decision is 'no'. Consequently, an influence function B : I → I can also be seen as a mapping from 2 N to 2 N .

Next, we define the concept of a follower which is one of the key concepts of the influence model. A follower of a given coalition is an agent who decides according to the inclination of the coalition in question in every situation in which all members of the coalition have the same inclination. Let B ∈ B. The follower function under the influence function B is a mapping F B : 2 N → 2 N given by all players that always decide according to the inclination of this (unanimous) coalition. The set of all follower functions is denoted by F. In Grabisch and Rusinowska [2010a] we show that F B is isotone and F B (S) ∩ F B (T ) = ∅ whenever S ∩ T = ∅.

F B (S) := {k ∈ N | ∀i ∈ I S , (B(i)) k = i S }, ∀S ⊆ N, S = ∅, (2)
One can check that, using the set notation, the follower function is defined as

F B (S) = S ′ ⊇S B(S ′ ) ∩ S ′ ⊆N \S B(S ′ ), ∀S ⊆ N, S = ∅, (3)
and F B (∅) := ∅. Assume F B is not identically the empty set. The kernel K(B) of influence function B is the set of 'truly' influential coalitions, i.e., the set of minimal coalitions having followers:

K(B) := {S ∈ 2 N | F B (S) = ∅, and S ′ ⊂ S ⇒ F B (S ′ ) = ∅}. ( 4
)
The kernel is well defined due to isotonicity of F B .

Applying Lattice Theory to the Influence Concepts

In Grabisch and Rusinowska [2011a] we study the exact relation between two concepts: influence function and follower function. While both an influence function B and a follower function F B are mappings from 2 N to 2 N , F B should satisfy some conditions (like isotonicity) which need not be satisfied by B.

We define the mapping Φ : B → (2 N ) (2 N ) by B → Φ(B) := F B , i.e., Φ assigns to every influence function B, the follower function F B under B. We have Φ(B) =: F. This mapping is obviously neither a surjection nor an injection. Note that:

Remark 1. The function Φ is not monotone (neither isotone nor antitone), so it fails to be a dual closure operator. Table 1 shows this fact. Clearly, B ′ ≥ B, but Φ(B) and Φ(B ′ ) are incomparable. We deliver the sufficient and necessary conditions that a function from 2 N to 2 N is the follower function of a certain influence function. Moreover, if a function F : 2 N → 2 N is a follower function, we find examples of B's such that F B = F . We have the following result:

Theorem 1 (Grabisch and Rusinowska [2011a]). A function F : 2 N → 2 N is a follower function of some B ∈ B (i.e., F B = F , or Φ(B) = F ) if and only if it satisfies the following three conditions: (1)

F (∅) = ∅; (2) F is isotone; (3) If S ∩ T = ∅, then F (S) ∩ F (T ) = ∅.
Moreover, the smallest and greatest influence functions belonging to Φ -1 (F ) are respectively the influence functions B F and B F , defined by, in set notation:

B F (S) = F (S), B F (S) = F (S), ∀S ⊆ N.
(5)

We call these influence functions the lower and upper inverses of F .

We are interested in Φ -1 (F ) which, for a given follower function F , is the set of all possible influence functions that lead to that F . We determine the algebraic structure of Φ -1 (F ). We already know that all elements of the inverse of F are between B F and B F , with the usual order ≤ on functions. This means that (Φ -1 (F ), ≤) is a poset, which is a subset of ([B F , B F ], ≤). Let us write for simplicity:

D S := B F (S) \ B F (S), S ⊆ N. (6)
An element of [B F , B F ] can be denoted by the 2 n -dim vector (T ∅ , . . . , T N ), where T S ⊆ D S for each S ⊆ N . Using this notation, B F and B F are denoted by (∅, . . . , ∅) and (D ∅ , . . . , D N ), respectively. We have the following:

Theorem 2 (Grabisch and Rusinowska [2011a]). Let B := (T ∅ , . . . , T N ) = B F be an element of Φ -1 (F ). Then for any S ⊆ N such that D S \ T S = ∅, and any k ∈ D S \ T S , B ′ := (T ∅ , . . . , T S ∪ {k}, . . . , T N ) is an element of Φ -1 (F ) if and only if one of the following conditions is NOT satisfied:

(1) For any S ′ ⊃ S, k ∈ B(S ′ );

(2) For any S ′ ⊆ N \ S, k ∈ B(S ′ ).

The main result, fully describing the structure of Φ -1 (F ), is the following.

Theorem 3 (Grabisch and Rusinowska [2011a]). For any F ∈ F, the set Φ -1 (F ), endowed with the usual ordering of functions, has the following properties:

(1) The greatest and least elements are B F and B F = F .

(2) It is a lattice, with supremum and infimum given by, for any S ∈ 2 N : (5) The lattice is distributive and its height is h = S⊆N |D S |.

(B ∨ B ′ )(S) := B(S) ∪ B ′ (S) (B ∧ B ′ )(S) := B(S) ∩ B ′ (S) (3) Φ -1 (F ) is autodual, i.
We would like to comment on the results (4) of Theorem 3. From lattice theory (see Section 2.1), we know that when the lattice is distributive, then it is enough to know its join-irreducible elements. Any element of the lattice can be written in a unique way as an (irredundant) supremum of join-irreducible elements. Hence, knowing only these elements permits to reconstruct the entire lattice. Moreover, we can generate all elements of the lattice in the following way. Let J denote the set of its join-irreducible elements, with the same order relation ≤. We consider an antichain A of J , i.e., a subset of J such that any two elements B, B ′ of A are incomparable, which means that neither B ≤ B ′ nor B ′ ≤ B hold. Then B := B ′ ∈A B ′ is an element of the lattice Φ -1 (F ), and generating the entire lattice amounts to generating all antichains of J . We can check whether (k S ∅) is an element of Φ -1 (F ) by using Theorem 2 with (T ∅ , . . . , T N ) = (∅, . . . , ∅), i.e., we only need to check if one of the two following conditions fails:

(1)

∀S ′ ⊃ S, k ∈ F (S ′ ); (2) ∀S ′ ⊆ N \ S, k ∈ F (S ′ ). Note that if (k S ∅) is not an element of Φ -1 (F ), then necessarily (k S ∅) is.
These results are illustrated in the following example, borrowed from Grabisch and Rusinowska [2011a].

Example 1. Let n = 3 and the function F be defined as in Table 2. The upper and lower inverses are given in Table 3. We compute the join-irreducible elements of Φ -1 (F ). The sets D S are given in Table 4.

The set J of join-irreducible elements contains 8 elements, given by: • For S = 3, k = 1: (1 3 ∅) belongs to Φ -1 (F ), so it is a join-irreducible element.

• For S = 1, k = 3: (3 1 ∅) belongs to Φ -1 (F ), so it is a join-irreducible element. • For S = 2, k = 1: (1 2 ∅) belongs to Φ -1 (F ),
• For S = 3, k = 3: (3 3 ∅) belongs to Φ -1 (F ), so it is a join-irreducible element.

• For S = 12, k = 1: (1 12 ∅) does not belong to Φ -1 (F ), so (

1 12 1 3 ∅) is a join- irreducible element. • For S = 12, k = 3: (3 12 ∅) does not belong to Φ -1 (F ), so (3 12 3 3 ∅) is a join- irreducible element. • For S = 13, k = 1: (1 13 ∅) does not belong to Φ -1 (F ), so (1 13 1 2 ∅) is a join- irreducible element. • For S = 23, k = 3: (3 23 ∅) does not belong to Φ -1 (F ), so (3 23 3 1 ∅) is a join- irreducible element.
From this we can generate all elements of Φ -1 (F ). For example, {(1 12 1 3 ∅), (3 1 ∅), (1 2 ∅)} is an antichain of J , and therefore (3

1 1 2 1 3 1 12 ) = (∅, 3, 1, 1, 1, ∅, ∅, ∅) is an element of Φ -1 (F ).
Let us finish this section with a short remark concerning atomicity of the lattice Φ -1 (B). As shown by Example 1, not all join-irreducible elements are atoms. More precisely, all elements of the form (k S ∅) are atoms, but elements of the form (k S k S ∅) are not atoms, and these elements appear as soon as (k S ∅) does not belong to Φ -1 (B). It seems difficult then to investigate in the general case when this happens.

Lattices in the Iterated Model of Influence

Next, we generalize the yes-no model of influence in a social network with a single step of mutual influence to a framework with iterated influence.

Given the current set S of 'yes'-voters, there is a certain probability b S,T that the set of 'yes'-voters after one step of influence is T . If we assume that the process of influence may iterate (several rounds in the discussion), we obtain a stochastic process which depicts the evolution of the coalition of 'yes'-agents along time. We assume that:

(1) The process is Markovian, i.e., the probability b S,T depends on S (the present situation) and T (the future situation), and not on the whole history.

(2) The process is stationary, i.e., b S,T is constant over time.

States of this finite Markovian process are therefore all subsets S ⊆ N , representing the set of 'yes'-agents; its transition matrix B := [b S,T ] S,T ⊆N is a 2 n × 2 n rowstochastic matrix. In Grabisch and Rusinowska [forthcoming] we propose a model of influence based on aggregation functions. In other words, to each agent k ∈ N we associate an aggregation function A k which specifies the way agent k modifies his opinion from the opinion of the other agents and himself. More precisely, the output A k (1 S ) ∈ [0, 1] of agent k's aggregation function is the probability that k says 'yes' after one step of influence, when the current situation is 1 S (i.e., the agents in S say 'yes' and the others say 'no'). We denote by A := (A 1 , . . . , A n ) the vector of aggregation functions. Specifically, supposing that S is the set of agents saying 'yes', we compute A(1 S ) = (A 1 (1 S ), . . . , A n (1 S )), that is, vector A(1 S ) indicates the probability of each agent to say 'yes' after influence.

By using the model based on aggregation functions, we find the Markovian influence model. In particular, if we assume that the probabilities to say 'yes' are independent among agents, then the probability of transition from the yes-coalition

S to the yes-coalition T is b S,T = k∈T A k (1 S ) k ∈T (1 -A k (1 S )) (7)
which determines B. To calculate the probability b S,T , we then need to multiply the probabilities of every agent in T to say 'yes' and of every agent outside T to say 'no' after one step of influence, assuming that all agents in S and only them say 'yes' in the current situation. It follows that deterministic models correspond to aggregation functions satisfying A k (1 S ) ∈ {0, 1} for all k ∈ N , hence they reduce to Boolean functions B k : 2 N → {0, 1} which are nondecreasing and nonconstant. Conversely, any deterministic influence model B : 2 N → 2 N satisfying B(∅) = ∅, B(N ) = N and being nondecreasing is a particular aggregation model, with nondecreasing nonconstant Boolean functions B 1 , . . . , B n defined by B k (1 S ) = 1 if B(S) ∋ k, and 0 otherwise.

We provide a general analysis of convergence of influence functions based on aggregation functions. We have the following result:

Theorem 4 (Grabisch and Rusinowska [forthcoming]). Suppose B is obtained from an aggregation model, with aggregation functions A 1 , . . . , A n . Then terminal classes are:

(1) either singletons {S}, S ∈ 2 N , (2) or cycles of nonempty sets {S 1 , . . . , S k } of any length 2 ≤ k ≤ n ⌊n/2⌋ (and therefore they are periodic of period k) with the condition that all sets are pairwise incomparable (by inclusion), (3) or collections C of nonempty sets with the property that C = C 1 ∪ • • • ∪ C p , where each subcollection C j is a Boolean lattice [S j , S j ∪ K j ], S j = ∅, S j ∪ K j = N , and at least one K j is nonempty.

We The terminal states can represent the two consensus situations ({N } where everybody in the society says 'yes', and {∅} where everybody says 'no'), and situations of the polarized society (where the individuals within the class say 'yes' while the others say 'no'). These terminal states which are not the consensus states are called nontrivial terminal states. In the second type of terminal classes, states form a cycle of nonempty sets, and the period of the class determines after how many steps a state is reached again. A Boolean terminal class [S, S ∪ K] can be interpreted as a situation where agents in S finally decided to say 'yes' and agents outside S ∪ K finally decided to say 'no', while the agents in K change their opinion nondeterministically forever. For a general case of regular terminal classes, i.e., with more than one interval, the interpretation is more complex and depends on the transitions between the intervals.

Examples 2, 3, and 4 show the existence of (nontrivial) terminal states, cyclic classes, and regular terminal classes, respectively.

Example 2. Consider N = {1, 2, 3} and the following aggregation functions:

A(1 0 0) = (1 x 0) A(0 1 0) = (x 1 0) A(0 0 1) = (0 0 1) A(1 1 0) = (1 1 0) A(1 0 1) = (x 1 1) A(0 1 1) = (1 x 1) with arbitrary 0 < x < 1. The digraph for the Markov chain is presented in Fig. 1. Apart from the consensus states {123} and {∅}, we have also nontrivial terminal states {12} and {3}. A(1 0 0) = (0 1 0) A(0 1 0) = (0 0 1) A(0 0 1) = (1 0 0) A(1 1 0) = (0 1 1) A(1 0 1) = (1 1 0) A(0 1 1) = (1 0 1).

The digraph for the Markov chain is given in Fig. 2. Then {1, 2, 3} and {12, 23, 13} are cyclic classes.

A 1 (1 0 0) = 1 A 2 (1 0 0) = 0.5 A 3 (1 0 0) = 0 A 1 (0 1 0) = 0 A 2 (0 1 0) = 0.5 A 3 (0 1 0) = 0 A 1 (0 0 1) = 0 A 2 (0 0 1) = 0.5 A 3 (0 0 1) = 0.5 A 1 (1 1 0) = 1 A 2 (1 1 0) = 0.5 A 3 (1 1 0) = 0 A 1 (1 0 1) = 1 A 2 (1 0 1) = 0.5 A 3 (1 0 1) = 0.5 A 1 (0 1 1) = 1 A 2 (0 1 1) = 0.5 A 3 (0 1 1) = 1.
This gives the digraph for the Markov chain as presented in Fig. 3. Clearly, {1, 12} is a regular terminal class.

Regular terminal classes can be periodic, as Example 5 shows.

Example 5. Consider N = {1, 2, 3} and the following aggregation functions: A(1 0 0) = A(1 1 0) = (0 0 1)

A(1 0 0) = A(1 1 0) = (0 x 1) A(0 0 1) = A(0 1 1) = (1 x 0) A(0 1 0) = A(0 0 0) = (0 0 0) A(1 0 1) = A(1 1 1) = (1 1 1)
A(0 0 1) = (1 x 0) A(0 1 0) = A(0 0 0) = (0 0 0) A(1 0 1) = A(0 1 1) = A(1 1 1) = (1 1 1)
with arbitrary 0 < x < 1. Then {1, 3, 12} is a periodic class of period 2 with 3 sets (see Fig. 4, right).

Concluding Remarks

In this paper, some selected results on the model of influence that are related to lattices have been presented. First of all, we should clarify that the relation between the two works (Grabisch and Rusinowska [2011a]; Grabisch and Rusinowska [forthcoming]) lies in the framework of influence that we consider and in the importance of lattices for the influence analysis, rather than in the type of lattice structures that we find. The first work deals with the one-step model of influence, and the second one concerns the dynamic and extended version of the previous work. It appears that the structure of the set of all influence functions that lead to a given follower function is a distributive lattice. Moreover, in the iterated model of influence, for influence functions based on aggregation, some terminal classes are unions of Boolean lattices. Hence, in both models, lattices are "present" and appear to be useful as discussed below.

As introduced in Section 3, the follower function assigns to every coalition the set of followers of that coalition. Some individuals might be interested in having some other concrete agents as their followers or as followers of specific coalitions that these individuals belong to. In other words, such individuals might be interested in specific follower functions, and therefore they should learn which 'influence techniques' (i.e., which influence functions) lead to the specific follower functions. It would be therefore helpful to them to know the set of all influence functions that lead to the particular follower function. This set is Φ -1 (F ) and since it appears to be a distributive lattice, it is enough to know its join-irreducible elements. The essential help for this problem comes from the recapitulated Theorem 3 proved in Grabisch and Rusinowska [2011a].

In numerous dynamic models of influence, possible terminal classes in which the process of influence might end up are reduced only to the consensus states. In Grabisch and Rusinowska [forthcoming] we show that if the individuals are allowed to aggregate opinions arbitrarily, not only by using the weighted arithmetic means, then other terminal classes may appear. As a result, the society may be polarized into 'yes'-and 'no'-individuals, or there may be a cycle of nonempty sets, or the society may be divided into three types of individuals: 'yes'-individuals, 'no'-individuals, and the ones that change their opinion non-deterministically forever. This is known due to the recapitulated Theorem 4 proved in Grabisch and Rusinowska [forthcoming]. One can imagine that some individuals, or even the society as a whole, would like to avoid some of these terminal classes. For instance, the lattice structure is not a particularly desirable terminal class, as it represents situations in which there exist some individuals that permanently change their opinions. Under such 'lattice November 11, 2013 0:17 WSPC/INSTRUCTION FILE grabisch-rusinowska-igtr-revised 16 Michel Grabisch, Agnieszka Rusinowska structure situations', neither reaching the consensus nor having two stable groups of individuals with the opposite opinions is possible. Consequently, the decision process, for instance, in committees and all kinds of decision-making bodies, seems to be more difficult under the lattice structure. Our results show that it is possible to 'manipulate the future' by not using some ways to aggregate opinions. For (sufficient and necessary) conditions to avoid the particular types of terminal classes in the dynamic model of influence, we refer to Grabisch and Rusinowska [forthcoming].

Obviously, there are several possibilities to continue this line of research, both by considering other concepts in the framework of influence as well as by analyzing social phenomena different from the social influence.

  and F B (∅) := ∅. F B (S) is the set of followers of S under B. In other words, the follower function assigns to every coalition the set of followers of that coalition, i.e., -igtr-revised Lattices in Social Networks with Influence 7

  e., (Φ -1 (F ), ≤) and (Φ -1 (F ), ≥) are isomorphic. The duality is expressed as follows: to each element B := (T ∅ , . . . , T S , . . . , T N ) ofΦ -1 (F ) corresponds the element B ′ := (D N \ T N , . . . , D S \ T S , . . . , D ∅ \ T ∅ ).(4) There are S⊆N |D S | join-irreducible elements, one for each k ∈ D S , S ⊆ N , either of the form (k S ∅) if this element belongs to Φ -1 (F ), otherwise of the form (k S k S ∅), where the notation (k S ∅) is a shorthand for (∅, . . . , ∅, k, ∅, . . . , ∅), where k is at position S, and similarly for (k S k S ∅). -igtr-revised Lattices in Social Networks with Influence 9

  -igtr-revised Lattices in Social Networks with Influence 11

  Fig. 1. Example of (nontrivial) terminal states

Fig. 2 .

 2 Fig. 2. Example of cyclic classes

Fig. 3 .

 3 Fig. 3. Example of a regular terminal class

Fig. 4 .

 4 Fig. 4. Examples of a periodic terminal class

  This means that if L

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	Lattices in Social Networks with Influence 5

Table 1 .

 1 Remark 1 -Φ is not monotone

Table 4 .

 4 so it is a join-irreducible element. Example 1 -The sets D S

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	S	∅	1	2	3	12	13	23	123
	D S	∅	3	1	13	13	1	3	∅

  call terminal states those terminal classes of the first type, cyclic terminal classes those of the second type, and regular terminal classes those terminal classes of the

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	third type. Regular terminal classes formed by a single Boolean lattice [S, S ∪ K] are
	called Boolean terminal classes.

Acknowledgments

This research project is supported by the National Agency for Research (Agence Nationale de la Recherche), Reference: ANR-09-BLAN-0321-01. We thank two anonymous referees for helpful comments.