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Pretopological structuring of a finite set endowed with a family of binary relationships
Cynthia Basileu, Nadia Kabachi et Michel LamureAbstract
In this paper, we propose a mathematical model for analyzing the « topological » structure induced on a finite set by a family of binary relationships. Given a family of reflexive binary relationships, we are able to endow it with different pretopological structures, using pretopology which is an extension of topology.
A pretopological structure being defined, we can compute structure indicators generalizing usual indicators in graph theory as, for example, the degree of a node, the diameter of the graph, the centrality indicators, the closeness indicators…
In fact, advantage of our modeling is twice: on one hand, it allows to work on a family of graphs and not only on one graph. On the other hand, indicators we propose can be computed not only on nodes but also on any subsets of nodes.
So, in this paper, we first present basics of pretopology and then the different ways to derive a pretopological structure from a family of graphs.
In a second part, we propose different structure indicators with reference to those existing in graph theory.
The third part will focus on simulated cases to illustrate mathematical results developed in the two first parts.
In the last part, we shall develop interest of our approach in the field of multiagents systems in view to apply out modeling to social networks and, more generally, to complex systems.
Pretopology, graph theory, structural indicators, complex systems, social networks, multiagent systems
Table of contents
Full text
Introduction
When we analyze works focused on modeling complex systems, in particular modeling of social networks, we note that graph theory is widely used. This can be explained by the fact that, in the framework of complex systems, we must pay a great attention to relationships between components of these systems. Most of these works are based on papers from (Albert & Barabasi, 1999), (Barabasi, 2002), (Barrat, Barthelemy &Vespigniani, 2008), (Erdos & Renyi, 1959), (Newman, 2003), (Scott, 2000), (Strogatz, 2001), (Wasserman & Faust, 1994), (Watts, 1999) having played a major role.
These works share a common pattern: generally, they consider finite set of agents E, endowed with a binary relationship leading to an undirected graph. Then a “topological” analysis can be performed through indicators defined in the framework of the graph theory: diameter, degree of a node, centrality, assortativity, shortest path and so on.
However, we can wonder if considering one undirected relationship is sufficient to handle complexity of relationships linking agents in the real world. According to our opinion, the answer is no, for at least two reasons.
The first reason is that, in real world, relationships between agents evolve according to their environment. This question is handled by means of random graphs in the complex systems theory. A lot of applications use this model in the field of human and social sciences. Concerning health of population, one example is the modeling of epidemics spreading. In a same way, in the field of environment, the fighting against aerial pollution uses concept of random graphs (Basileu, 2011), (Ben Amor, Bui & Lamure, 2010), (Murat, Ben Amor, Bui, Lamure & Courel, 2009).
The second reason is that using only one undirected relationship or graph is unaware to complexity of reality in which multiple and directed relationships are linking agents.
This paper deals with this problem: using more than one relationship and considering directed relationships. For that, we consider a finite set E endowed with a finite family of binary relationships. We suppose they are reflexive ones (which does not reduce the generality of our modeling). The problem to be solved is endowing E with an alike topological structure, based on this family. For that, we use pretopology which allows associating a “pretopological” structure to such a family of relationships. In a second step, we are able to define structure indicators for analyzing this pretopological structure by generalizing usual indicators of graph theory.
So in paragraph 2, we present basic concepts of pretopology and formal relationships existing between pretopology and graph theory. The reader who wants to have more information on these topics can read (Belmandt, 1993), (Belmandt, 1994), (DaludVincent, Brissaud & Lamure, 2011), (DaludVincent, Brissaud & Lamure, 2001), (DaludVincent, Brissaud & Lamure, 2009), (Lamure, 1987).
In paragraph 3, we present pretopological indicators while illustrating them by very simple examples.
In paragraph 4, we give some considerations about pretopology and multiagent systems, before concluding in paragraph 5.
Graphs and Pretopology
Pretopological Structures
Let us consider a finite set E, a pretopological structure on E is defined by means of two functions a(.) and i(.), respectively named pseudoclosure and interior functions. They must satisfy the following properties
Definition 2.1.1
The pseudoclosure a(.) verifies:
The interior i(.) verifies:
Then, the 3uple (E,i(.),a(.)) is called pretopological space.
Most often, a(.) and i(.) are defined by duality using complementation operator, i.e., if A^{c} denotes the complementary set of A, subset of E, we put:
In this paper, we consider two particular types of pretopological spaces: the V type and the V_{D} type spaces.
Definition 2.1.2
Given a pretopological space (E,i(.),a(.)),
If the pseudoclosure verifies (i), (ii) and the first following property:
(E,i(.),a(.)) is said of V type.
If the pseudoclosure verifies (i), (ii) and the first following property:
(E,i(.),a(.)) is said of V_{D }type.
Obviously, we get:
Proposition 2.1.3
If a pretopological space is V_{D} type, it is of a V type.
Nota: The name V or V_{D} type is due to importance in that kind of space of neighborhood concept. The neighborhood is defined in the same way as in topology.
Definition 2.1.4
(E,i(.),a(.)) is a V type space, a subset V of E is a neighborhood of x element of E if and only if
If V(x) denotes the family of neighborhoods of x, i.e.
, we prove that (cf .[5a]):
(E,i(.),a(.)) being a V (or V_{D}) type space, we get:
This last result is a very important one as it gives a practical mean to endow E with a pretopological structure from a family of reflexive binary relationships.
In particular, we can define the basis of neighborhoods of x, for any x in E by putting:
Definition 2.1.6
(E,i(.),a(.)) is a V type space, x element of E, a family B(x) of subsets of E is said a basis of the family V(x) of neighborhoods of x if and only if
Pretopology induced from a family of binary relationships
From now, we consider a family of reflexive binary relationships. For each relationship R_{i}, we can define different pretopological structures by considering the two following subsets:
the set of successors of x in the relationship,
the set of predecessors of x in the relationship.
2.2.1. Pretopologies of successors
Let us put:
Proposition 2.2.1
a_{s}(.) and i_{s}(.) determine on E a pretopological structure and the space (E,i_{s}(.),a_{s}(.)) is of V type.
Proof: immediate, it is sufficient to use definitions.
Pretopology defined on E by a_{s}(.) and i_{s}(.) is called the strong pretopology induced by the family (R_{i}).
Let us also put:
Proposition 2.2.2
a_{w}(.) and i_{w}(.) determine on E a pretopological structure and the space (E,i_{w}(.),a_{w}(.)) is of V_{D} type.
Proof: immediate, it is sufficient to use definitions.
Pretopology defined on E by a_{w}(.) and i_{w}(.) is called the weak pretopology induced by the family (R_{i}).
Example
Let E={0,1,2,3,4,5,6,7}, we consider on E the three relationships R_{1}, R_{2} and R_{3} illustrated by the following figures (figures 1, 2 et 3)
Let A = {5}, then:
a_{s}(A)={2,4,5},
a_{w}(A)={2,3,4,5,7}.
Let A = {0,4,6}, then:
a_{s}(A)={0,1,3,4,5,6}, i_{s}(A)=Ø.
a_{w}(A)=E, i_{w}(A)=Ø.
As in topology, we can define concepts of open and closed subsets, of opening and closure. We also can define other functions for transforming subsets of E: the derivative, the coherency, the half inner frontier, the half outer frontier, the frontier and the exterior. In the following paragraph, we shall how interesting are some of these functions.
At least, an important advantage of pretopology, relatively to graph theory, is that different notions of connectivity can be defined in full accordance with connectivity concepts in graph theory. Thus, a « topological » analysis is possible of a graph or a structure induced by a family of graphs.
Other pretopological structures can be defined from a family of binary relationships. We briefly give some of them hereafter.
2.2.2. Pretopology of predecessors
The two mappings a(.) and i(.) are defined as follows:
In that case, the family B(x) is
2.2.3. Pretopology of predecessorssuccessors
The two mappings a(.) and i(.) are defined as follows:
In that case, the family B(x) is
2.2.4. Pretopology of collaterals
The two mappings a(.) and i(.) are defined as follows:
In that case, the family B(x) is
Structure Indicators
In this paragraph, we only consider the strong or the weak pretopology of successors induced by a family of reflexive binary relationships on E.
We shall focus to generalize different indicators of the graph theory to the framework of pretopology. In the following, we shall not precise strong or weak pretopology as proposed definitions hold whatever the type of pretopological structure.
Pseudoclosure and Interior ratio
Definition 3.1.1
Given a pretopological space (E,i(.),a(.)), for any non empty subset A of E, we call :
 Pseudoclosure ratio of A the parameter, noted R_{a}(A), defined by
,
 Interior ratio of A the parameter, noted R_{i}(A), defined by
These two ratios fulfill the following properties:
Proposition 3.1.2
Given a pretopological space (E,i(.),a(.)),
(i)
(ii) A closed subset of E is equivalent to R_{a}(A)=1
(iii)
(iv) A open subset of E is equivalent to R_{i}(A)=1
Proof:
By using definitions of a(.) and i(.), we directly obtain (i) and (iii). Concerning (ii), if A is a closed subset of E, a(A)=A, which leads to the result. It is the same thing for (iv) by noting that A open subset of E means A=i(A).
Note: These two ratios play in pretopological spaces the same role than in and out degrees in graph theory. They measure the capacity, for a subset A of E, for diffusing or for receiving influence from its complementary. We can note than these two indicators are defined for any subset A of E, not only for singletons of E. The following diagram (fig. 4) illustrates what can happen for a given subset A of E in the representing plane (R_{a}(.), R_{i}(.)).
Figure 4: representing space of pseudoclosure and interior ratios
The four corners, numbered from 1 to 4, correspond to very particular cases concerning the subset A of E, relatively to emission and reception of influence.
If we consider the subset A, A = {0,4,6}, from the previous example, we get:
a_{s}(A)={0,1,3,4,5,6}, i_{s}(A)=Ø.
a_{w}(A)=E, i_{w}(A)=Ø.
Furthermore:
.
Closeness Indicator
In graph theory, we can compute the closeness coefficient for any node x. This coefficient measures intensity of links between neighbors of x. The definition of the closeness coefficient is the following, in the case of an undirected graph (Newman, 2003).
Definition 3.2.1
For any node u of an undirected graph, the closeness coefficient, notes C(u), is defined as follows:
where nl(u) is the number of links between the neighbors of u and d(u) its degree.
Let us consider a subset A of E and a(A)A. This last subset is defined as the set of neighbors of elements of A, in the sense of neighbors in graph theory.
In pretopological spaces, the set a(A)A is defined as the half outer frontier (cf. previous paragraph) and is denoted o(A).
Let us compute R_{i}(o(A)) the interior ratio of o(A). If this ratio takes a value close to 1, that means that o(A) does not diffuse influence to its complementary. So, if the elements of o(A) diffuse influence, it is only inside o(A) and we have a high level of links density in o(A).
However, we have no information on that level of diffusion. So, let us consider:
c(.) is the coherency function. c(A) is the set of elements of A not isolated in A, in the sense where if x belongs to c(A), some neighbors of x are in A.
Let us now define the following coefficient ds(.):
If this coefficient ds(.) computed on c(o(A)), i.e. ds(c(o(A))), takes a value close to 0, that means there is a few isolated points in o(A).
Then, if simultaneously, ds(c(o(A))) is close to 0 and R_{i}(o(A)) is close to 1, we are faced to a case where o(A) diffuses a lot but internally only. That means a high relational density between neighbors of A.
By consequence, the couple (ds(A),R_{i}(A)) allows to characterize the « closeness » of elements of A, for any A subset of E.
Correlation Coefficient and assortativity function
Concerning undirected graphs, two other coefficients are widely used to analyze them. These two coefficients are the correlation degree of a node and the assortativity function.
The correlation degree of a node i is defined as follows.
where k_{i}denotes the degree of the nodei and V(i) denotes the set of neighbors of i, i.e. the set of nodes j such as {(i, j)} is an edge of the graph (for an undirected graph).
Assortativity function is then defined by:
where N_{k}denotes the number of nodes with degree k.
Let us put:
Definition 3.3.1
For any element x of a finite pretopological space (E,i(.),a(.)), the correlation coefficient of x, denoted dc(x), is defined as:
This coefficient is a direct generalization of the same coefficient defined in graph theory. It measures the same thing: the more the elements y of a({x}) have adherent points, the greater is dc(x) and conversely.
The assortativity function ca(.) is generalized as follows :
Definition 3.3.2
where N_{k} denotes the number of elements of E such as
.
Let us consider the following example where weak and strong pretopologies are induced by the three binary relationships described in the following table (cf. table 1).
node x 
_{1}(x) 
_{2}(x) 
_{3}(x) 
0 
0,3,6 
0,3,6,7 
0,4 
1 
0,1,7 
0,1,4 
1,4,7 
2 
1,2,5 
1,2,5 
2,4,5 
3 
3,4 
1,3,6 
0,3,5 
4 
3,4,5,7 
3,4,5,7 
1,4,5 
5 
0,1,2,5 
0,2,5 
4,5 
6 
4,6 
3,4,6 
1,3,6,7 
7 
2,5,7 
2,4,5,7 
0,4,7 
Table 1: data on the relationships defined on E.
The computation of dc(x) for any x element of E is listed in the following table (cf. table 2):
node x 
dc(x) – weak pretopology 
dc(x) – strong pretopology 
0 
3,00 
1,00 
1 
3,83 
1,00 
2 
3,33 
1,00 
3 
4,00 
1,00 
4 
3,88 
1,00 
5 
4,00 
0,67 
6 
3,00 
1,00 
7 
4,40 
1,00 
Table 2: coefficients dc
We can then derive the assortativity function, illustrated by the following diagram (cf. figure 5).
Figure 5: assortativity functions
Centrality Coefficient (eigenvalues)
In graph theory, the centrality coefficient is defined to measure influence of a node in the graph. This coefficient is defined from the adjacency matrix A of the graph as follows.
where denotes the greatest eigenvalue of the matrix A. In that case, ce(x) is given by the x^{th} component of the eigenvector associated to this eigenvalue.
Let us consider A defined by its general term
Let be the greatest eigenvalue of A. It is then possible to generalize the centrality coefficient by putting, in a similar way as in graph theory:
Definition 3.4.1
ce(x) is also given by the x^{th} component of the eigenvector associated to this eigenvalue.
Nota: in fact, this is equivalent defining a graph from the pretopology by putting:
and using the usual definition of graphs, in the case of directed graphs however.
Multiagents System based simulation
In order to operationalize this mathematical model and achieve its computer implementation for simulation purposes, we will use the multiagent approach which allows us to incorporate real data and social aspects, changes in the environment, and so on...
This simulation approach based on a MultiAgent System (MAS) (Hare & Deadman, 2004), (Weiss, 1999) allows distributed simulations as close to natural processes. It also ensures implementation of known phenomena closer to their image in the model and marry them to a qualitative approach from social sciences.
For modeling pandemic propagation, the MAS paradigm allows simulation at different level (individuals and global) and thus understanding this complex system over time.
The architecture of our MAS is composed of three hybrid agents’ types:  an individual (which complete a professional travel or other),  medical staff and  decision makers. Each of these agents will have to move from a geographical area to another by taking on public or private transportations that will have a direct impact on the evolution of the pandemic. It is an entity generating dynamics that take place in a territory. At each moment, the health of Agents will related to the epidemic; the change health state of an individual is deterministic: Susceptible (S), infected (E); (I) infectious or immune (R). Infected agent is an agent who contracted the virus but it is not contagious. The duration of these phases is more or less long because it depends on in inherent features to the individual (antibody...) and of the treatment (prevention, time of reaction…).
An agents’ environment defines the properties of the world in which an agent can and does act. In our MAS model, the environment describes only the topology of the territory or geographic area (towns, departments, region, etc.).
The system dynamics is driven by interactions between agents (people). In our MAS model, agents are interacting with each other within the framework of their professional activities, friendly, sporting activity...Thus; agents are connected by a set of relationships described before. The epidemic spread occurs through the social networks. We note that individual can only affect other individuals through interactions that occur when they are at the same location at the same time.
Experimental part:
This experiment deals with the problem of analyzing consequences of an epidemic outbreak. A pretopological model has been defined on a set of people living in a French area. French department is the geounit. Different kinds of relationships have been considered between people: friendship, at work, family and so on (five are considered in the experiment). A specific behavior has been defined for any kind of people, in particular concerning their way of transportation (public or private transportation). An epidemic outbreak was simulated and its spreading computed according to the pretopological structure induced by the different relationships. By means of the software platform REPAST, a prototype was developed, enabling the computation of some pretopological indicators.
To calculate these indicators, we performed a simulation of five days as a worked week. The agents are divided into four departments of the RhôneAlpes region in the following table:
S 
E 
I 
R 
Total 

H 
76 
0 
2 
0 
78 
P 
0 
0 
0 
4 
4 
E 
0 
0 
0 
8 
8 
Total 
76 
0 
2 
12 
90 
Table: Distribution of agents in four departments at t = 0
Nota: H, P, E denote the three categories of population we considered: H for “ordinary” inhabitants, P for Medical staff and E for workers to be protected.
At the beginning, the following window displays what happens: two people are supposed infected (red points).
We consider that, by assumption, the whole of the health workforces as well as people to be protected are vaccinated at the beginning of the epidemic. In addition, we suppose the infected people are in two different departments: one in the department of the Rhone and the other in that of Isere. The characteristics of these agents differ little. They are in the same age class, they have the same level of risk (equal to 0), they use both public transportation and they go and see the nearest doctor (relatively to their home). But, one (in the Rhone) lives alone whereas the other (in Isere) lives in couple.
From this distribution of the agents, and taking account of the five types of relations (professional, household, leisure...) integrated in the model, we obtain, at the fifth day, the following distribution of states of the agents (green=S, orange=E, red=I and blue=R):
In this experiment, taking into account the relations initially definite on the population, one notes that the agents are reached much in the department of the Rhone.
In the department of the Rhone, one inhabitant is deceased during the fourth day, information is integrated the fifth day in the morning and the inhabitant is eliminated from all the relations at this moment.
This simulation related to a unit E of 90 inhabitants. Each inhabitant of E is numbered from 1 to 90. Let us consider unit A defined by: A = {6, 8, 13, 15, 22}, a set of 5 inhabitants in the Rhone. We will illustrate on this unit A, calculations of the pretopologic parameters introduced into the preceding section. For that, we will calculate step by step, i.e. day after day, the adherence and the interior of A. The weak pretopology, of V_{D} type, has been used. In that case, pseudoclosure of any subset A of E is simply calculated by taking the union of the pseudoclosure of any of its singletons. That makes possible to very simply calculate a(A) and i(A).
Then, we get the following result for a(A), at the end of the first day:
a(J_{1}, A) = {4,5,6,8,9,12,13,15,16,19,22,23,27,65}.
We then compute the interior of A, at the end of the first day:
i(J_{1}, A) = {6}.
This leads us to the following result for the pseudoclosure ratio and interior ratio at the end of the first day:
We can note that R_{a(J1,A)}is close to 1 and that R_{i(J1,A) }is close to 0. For this scenario, that means a little people outside A are infected by people in A and that people in A who are infected are infected by people outside A.
According to the pretopologic definitions, the derivative of A, at the end of the first day is:
d(J_{1}, A) = {4,5,9,12,16,19,23,27,65}
From this derivative, we deduce:
c(J_{1}, A) = Ø
We can note that no element of A is included in the derivative of A. That means A is made up only of isolated points, in A, for this scenario.
The calculation of the pretopologic parameter ds(J_{1},A) = 5 = Card(A) enables us to confirm this result. In what follows, we will take again the whole of these calculations for days 2 to 5.
After computing pseudoclosure of days 2 to 5, we obtain for a(A):
a(J_{2}, A) = {6, 23, 8, 13, 9, 65, 27, 4, 12, 16, 15, 23, 22}
a(J_{3}, A) = {6, 9, 8, 65, 27, 13, 4, 12, 16, 15, 23, 22}
a(J_{4}, A) = {6, 8, 9, 5, 65, 12, 4, 13, 27, 16, 15, 22, 23, 18}
a(J_{5}, A) = {6, 16, 8, 9, 65, 27, 13, 4, 15, 22, 23}
Then, for i (A):
i(J_{2}, A) = {6}
i(J_{3}, A) = Ø
i(J_{4}, A) = {6}
i(J_{5}, A) = Ø
This carries out us to obtain the reports of adherence:
And for the reports of interior:
The following diagram locates A in the plane (R_{a}(.), R_{i}(.)), for any of the five days.
The calculation of the derivative of A gives:
d(J_{2}, A) = {4,8,9,12,13,16,23,27,65}
d(J_{3}, A) = {4,9,12,16,23,27,65}
d(J_{4}, A) = {4,5,9,12,16,18,23,27,65}
d(J_{5}, A) = {4,9,16,23,27,65}
From where one from of deduced his coherence:
c(J_{2}, A) = {8,13}
c(J_{3}, A) = Ø
c(J_{4}, A) = Ø
c(J_{5}, A) = Ø
What brings to the values below for the parameter ds (. , A):
ds(J_{2},A) = 3
ds(J_{3},A) = 5 = Card A.
ds(J_{4},A) = 5 = Card A.
ds(J_{5},A) = 5 = Card A.
These results allow to say that unit A is made up only of isolated points, in A, for days 3, 4 and 5.
The example illustrated in this experimentation shows the advantage that there is using pretopological concepts for analyzing how a social network evolves. Using the Anylogic Platform a more sophisticated tool has been developed. More complete experiments have been performed, on the whole territory of France, with a sample of French population at 1/1000, department by department. Epidemic outbreaks have ben simulated by using real data in the paste. Results of simulation gave a very good approximate of what happened. In addition, strategies for fighting have been simulated and their results on the level of sick people have been measured.
Conclusion
As we can see, with pretopology, it is possible to proceed a structural analysis of a finite set endowed by a family of binary relationships. In that framework, we can generalize some usual coefficients. In this paper, we have presented a generalized version of some of them. Actually, other indicators are studied for a better modelling of the structure induced on E by a family of relationships.
The modeling and simulation of pandemic propagation was based on the MAS paradigm. This type of simulation allows decision makers to understand those complex systems and their dynamic over time. In addition it facilitates the back and forth between the virtual experimentation on the simulator and the reality on the ground, and thereby the possibility to refine the results.
In this paper, we do not deal with stochastic aspects as in random graphs theory. However, this aspect has been handled in other works ((Basileu, 2011), (Ben Amor, Bui & Lamure, 2010). For that, we mixed two kinds of mathematical concepts: pretopological concepts and random sets concepts. This led us to stochastic pretopology. Stochastic pretopology was successfully applied to aerial pollution problems as well as to providing aid to decision making in case of an epidemic outbreak.
In parallel, computation tools have been designed through a library: Pretopolib. This library is currently evolving by addition of possibilities for computation of pretopological transforms.
A lot of problems remain to be solved. For example, with the different concepts of connectivity in pretopological spaces, how can we analyse setting up of communities in social networks? How can we detect “weak” nodes or subsets of nodes from the data issued from the relationships? And so on.
At last, an important point is to determine statistical concepts for dealing with ground data. Some preliminary results have been obtained under some conditions o the type of the random sets. From these results, we can expect to be able getting result about estimating and testing parameters.
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